Departamento de F´ısica Teo´rica e Instituto de F´ısica Teo´ricaUniversidad Auto´noma de MadridPROBING THE FOUNDATIONSOF THE STANDARDCOSMOLOGICAL MODELMemoria de Tesis Doctoral presentada porMiguel Zumalaca´rregui Pe´rezpara optar al t´ıtulo de Doctor en Ciencias F´ısicasPrograma de Doctorado de F´ısica Teo´ricaTrabajo supervisado por los DoctoresJuan Garc´ıa-Bellido Capdevila, Profesor Titulardel Departamento de F´ısica Teo´rica, Universidad Auto´noma de MadridPilar Ruiz-Lapuente, Investigadora Cient´ıficadel Instituto de F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficasyTomi S. Koivisto, Investigador Postdoctoraldel Instituto de Astrof´ısica Teo´rica, Universidad de Oslo.Octubre de 2012iiA mis padres.iiiivContentsStatement ixResumen y Conclusiones xiPreface xvAcknowledgements xixNotation and Acronyms xxiI Foundations 11 The Standard Cosmological Model 31.1 Ingredients and Preparation 41.1.1 General Relativity 41.1.2 Metric Ansatz: The Copernican Principle 51.1.3 Inflation: Symmetry and Initial Conditions 61.1.4 Standard and Dark Matter 71.1.5 The Cosmological Constant 71.2 The Need to Go Beyond 81.2.1 Theoretical Issues 81.2.2 Observational Reasons 101.3 Exotic Tastes and Alternative Recipes 121.3.1 Inhomogeneous Models 121.3.2 Dynamical Dark Energy 141.3.3 Modified Gravity 162 Observational Probes of Cosmology 232.1 Geometry and Dynamics 242.2 Primordial Nucleosynthesis 262.3 Cosmic Microwave Background Radiation 262.3.1 CMB Distance Priors 282.4 Large Scale Structure 292.4.1 The Baryon Acoustic Oscillation Scale 312.4.2 Other LSS Probes 342.5 Type Ia Supernovae 35vCONTENTS2.5.1 Local Expansion Rate 362.6 Local Gravity Tests 37II Results 393 Large Scale Homogeneity and Non-Copernican Void Models 413.1 Lemaˆıtre-Tolman-Bondi Models 433.1.1 The Adiabatic GBH Model 463.2 The Baryon Acoustic Scale in LTB Universes 473.2.1 Free-falling Scales in the LTB Metric 493.2.2 BAO Scale Evolution Beyond Zero Order 503.2.3 The Physical BAO Scale at Early Times and on the Lightcone 503.2.4 Comparison with the Observed BAO Scale 523.2.5 The Alcock-Paczynski Effect in LTB Models 533.3 Analysis and Results 543.3.1 Observational Data 543.3.2 MCMC Analysis 583.3.3 Homogeneous Models 593.3.4 Inhomogeneous Models 623.3.5 Model Comparison 663.4 Discussion 684 Phenomenological Modifications: Entropic Gravity 734.1 Gravity and Thermodynamics 744.2 Modifications from Surface Terms 754.2.1 Single Fluid 764.2.2 Adding a Cosmological Constant 774.3 Modifications from Quantum Corrections to the Entropy-area Law 794.4 Dark Energy from a Generalized Entropy-area Law 804.5 On the Evolution of Perturbations 844.6 Discussion 865 Standard Matter and Scalar Fields: Disformal Quintessence 895.1 Dark Energy from the Disformal Relation 905.1.1 Disformal Dark Energy 915.1.2 A Variation of the Model 925.2 Disformal Quintessence 935.2.1 Background Evolution 935.2.2 Cosmological Perturbations 975.2.3 Observational Constraints 985.3 Discussion 1016 General Relativity and Scalar Forces: Disformal Coupling 1036.1 A Test Particle in a Disformal Metric 1056.2 The Zoo of Disformally Related Theories 1066.2.1 Disformal Curvature: The Conformal Frame 108viCONTENTS6.2.2 Equations in the Einstein Frame 1106.3 Background Cosmology 1136.3.1 An Example Model: Disformally Coupled Dark Matter 1136.4 Cosmological Perturbations 1176.4.1 Small Scale Limit 1186.4.2 Structure Formation for Disformally Coupled Matter 1186.4.3 Viable Scenarios 1236.5 The Disformal Screening Mechanism 1246.5.1 Pressure Instability 1256.5.2 The Scalar Field in Dense, Non-relativistic Environments 1256.6 Discussion 1287 Conclusions and Outlook 131III Appendices 135A MCMC Analysis 137B CMBEasy 139C Equations for Generalized k-essence 141D Disformal Relations 143D.1 Disformal Geodesics 144D.2 General Disformal Coupling Perturbations 144D.3 Lagrangian Derivatives for Disformal Quintessence 145List of Tables 147List of Figures 149Bibliography 151viiCONTENTSviiiStatementThe results presented in this dissertation (Chapters 3-6) are based on original work done incollaboration with other researchers during the course of my PhD, from September 2008 toAugust 2012. The content of these Chapters is based on the released publications [1–5], aswell as another ongoing project which is at a mature stage [6]. These references are listedbelow for convenience. In particular, Chapter 3 is based on Reference [2], Chapter 4 onReferences [4, 5], Chapter 5 is based on Reference [3] and Chapter 6 is based on References[1, 6]. The introductory Chapters 1 and 2 are mainly a review. The exposition there hasbeen very influenced by References [7–10], as well the works cited in the text. Although theyrepresent my own vision about some of the topics in which the results are framed, no claimof originality is made about the introductory Chapters. Appendices A and B describe someof the tools necessary for the analysis. Appendix D contains some lengthy equations thatwere not necessary to include in the body of the discussion.Miguel Zumalaca´rregui Pe´rezMadrid, October 20121. “Screening Modifications of Gravity through Disformally Coupled Fields”T. S. Koivisto, D. F. Mota and M. Zumalacarregui. [arXiv:1205.3167].2. “Tension in the Void: Cosmic Rulers Strain Inhomogeneous Cosmologies”M. Zumalacarregui, J. Garcia-Bellido and P. Ruiz-Lapuente.JCAP 1210, 009 (2012), [arXiv:1201.2790].3. “Disformal Scalar Fields and the Dark Sector of the Universe”M. Zumalacarregui, T. S. Koivisto, D. F. Mota and P. Ruiz-Lapuente.JCAP 1005, 038 (2010), [arXiv:1004.2684].4. “Constraining Entropic Cosmology”T. S. Koivisto, D. F. Mota and M. Zumalacarregui.JCAP 1102, 027 (2011), [arXiv:1011.2226].5. “Modified Entropic Gravity and Cosmology”M. Zumalacarregui. AIP Conf. Proc. 1458, 539 (2011), [arXiv:1202.1281].6. “The Cosmology of Disformally Coupled Matter”T. S. Koivisto, D. F. Mota and M. Zumalacarregui. To appear.ixStatementxResumen y ConclusionesCuando una persona tiene la suerte de vivirdentro de una historia, de vivir un mundoimaginario, el dolor de este mundodesaparece. Mientras la historia siga, larealidad deja de existir.Paul Auster1La cosmolog´ıa moderna ha alcanzado un gran desarrollo como cienc´ıa emp´ırica y predic-tiva en las u´ltimas de´cadas. El incremento en cantidad y calidad de los datos observacionales,unido a una so´lida descripcio´n teo´rica, han permitido el establecimiento de un Modelo Cos-molo´gico Esta´ndar capaz de explicar una gran variedad de feno´menos. Numerosas observa-ciones cosmolo´gicas apuntan a que el universo esta´ experimentando una e´poca de expansio´nacelerada. Las supernovas lejanas de tipo Ia son menos luminosas de lo que cabr´ıa esperaren un universo que contiene u´nicamente materia. El estudio de las anisotrop´ıas del fondoco´smico de microondas combinado con la medida de la tasa de expansio´n local, implica queel universo tiene secciones espaciales planas y esta´ dominado por una densidad de energ´ıacon presio´n negativa. Los datos sobre formacio´n de estructuras a gran escala concuerdancon este escenario. La tasa acrecio´n de materia hacia las regiones ma´s densas indica que lacantidad total de materia esta´ muy por debajo de la que se necesita para que el universo seaespacialmente plano. La misma f´ısica responsable de las anisotrop´ıas del fondo co´smico demicroondas imprime una escala caracter´ıstica en la funcio´n de correlacio´n de materia, quepuede ser medida en catalogos de gala´xias en distintas e´pocas. Esta longitud esta´ndar evolu-ciona con la expansio´n del universo y permite medir la tasa de aceleracio´n, que concuerdacon los datos de supernovas.La interpretacio´n de estos datos se han realizado dentro de un paradigma esta´ndard,en el que la gravedad esta´ descrita por la teor´ıa de la Relatividad General, el universo esespacialmente homogeneo e iso´tropo a grandes escalas y la aceleracio´n es causada por unaConstante Cosmolo´gica. Las expectaciones teo´ricas sobre el valor de dicha constante son su-periores al valor observado en muchos ordenes de magnitud. Adema´s de la aceleracio´n, elmodelo esta´ndar necesita de otros elementos no convencionales como la materia oscura yla inflacio´n cosmolo´gica para ser fenomenolo´gicamente satisfactorio. El escepticismo sobre el1The Brooklyn Follies (2005)xiResumen y Conclusionesvalor de la constante cosmolo´gica, unido a las promesas de nueva f´ısica hechas por la cos-molog´ıa moderna, han favorecido la propuesta de una gran cantidad de modelos alternativos.E´stos pueden clasificarse segu´n las hipo´tesis que violan dentro del paradigma esta´ndar, y susprediccions se pueden comparar con observaciones cosmolo´gicas para determinar su viabil-idad. Esta Tesis aplica este programa en diferentes casos, para validar los fundamentos delModelo Cosmolo´gico Esta´ndar y acotar posibles desviaciones de las suposiciones fundamen-tales. Los resultados incluyen el estudio de modelos sin Constante Cosmolo´gica basados endesviaciones de la homogeneidad, parametrizaciones de las ecuaciones cosmolo´gicas y exten-siones de la teor´ıa de la Relatividad General.Modelos inhomogeneos no Copernicanos: Desde el punto de vista de nueva f´ısica, laposibilidad ma´s sencilla consiste en reducir el grado de simetr´ıa de la me´trica. Postulando unespacio-tiempo esfe´ricamente sime´trico alrededor de nuestra galax´ıa, es posible reinterpretarla aceleracio´n como un espejismo, causado por las diferentes tasas de deceleracio´n, que de-penden de la distribucio´n de materia en funcio´n de la distancia. En este tipo de modelos esnecesario que nuestra galaxia este´ situada en un lugar especial del universo: cerca del centrode una regio´n con menor densidad de materia. Cuando la edad del universo y la proporcio´nentre materia bario´nica y oscura son independientes de la coordenada radial, estos modelosrepresentan el crecimiento por colapso gravitacional de una perturbacion con una amplitudinicial pequen˜a. Estos casos particulares esta´n descritos por el perfil de la distribucio´n demateria y la normalizacio´n del ritmo de expansio´n, y se conocen modelos Lemaˆıtre-Tolman-Bondi (LTB) adiaba´ticos.En el cap´ıtulo 3 se estudian modelos LTB adiaba´ticos con un perfil de tipo GBH, loscuales no son compatibles simultaneamente con observaciones geome´tricas de supernovas y laescala de oscilaciones acusticas (BAO). Al igual que en el modelo esta´ndar, la escala de BAOevoluciona localmente con el factor de escala, lo que produce una diferencia entre su valoren la direccio´n radial y angular, as´ı como una dependencia con el radio. Este u´ltimo efectohace que la misma inhomogeneidad que permite emular la aceleracio´n, estire de maneradiferente la escala de BAO en diferentes puntos, haciendola ma´s grande cerca de nuestragalaxia (donde hay menos materia y el universo se expande ma´s) que a distancias mayores.Los nuevos datos a alto deplazamiento al rojo z & 0,5 permiten explorar ambos reg´ımenes yhacen incompatibles ambos observables. Este efecto ocurre en todos los modelos adiaba´ticosindependientemente del perfil de materia. La comparacio´n con observables geome´tricos proveeun nuevo argumento a favor de la aceleracio´n del universo y en contra de este tipo de modelos,complementario a las medidas del ritmo de expansion en distintas e´pocas y el efecto SunyaevZeldovich cine´tico.La discrepancia entre los modelos LTB adiaba´ticos sin Constante Cosmologica y losobservables geome´tricos puede emplearse para acotar modelos ma´s generales. El estudio demodelos tipo LTB con Constante Cosmolo´gica permite determinar el grado de inhomogenei-dad compatible con las observaciones en casos ma´s realistas. Tambie´n es posible reducir lasrestricciones del modelo, considerando modelos no adiaba´ticos (con tiempo desde el Big-Bang o fraccio´n de materia´ oscura/radiacio´n sobre bariones dependiente de la posicio´n),desviaciones desde la posicio´n central o incluso me´tricas ma´s generales. Este programa sebeneficiar´ıa en gran medida de la inclusio´n de observables dina´micos, tales como el efectoSachs Wolfe integrado o las distorsiones en espacio de desplazamientos al rojo.xiiModificaciones fenomenolo´gicas: El estudio de modelos basados en parametrizacioneses u´til para simplificar el analisis de ideas teo´ricas que son dificiles de tratar o no poseenuna formulacio´n bien definida. Esta posibilidad se ha explorado en el Cap´ıtulo 4 para elcaso de modificaciones de las ecuaciones de Friedmann basadas en consideraciones sobre laspropiedades termodina´micas del espacio tiempo. Esta aproximacio´n tiene la desventaja deque la obtencio´n de las ecuaciones depende de una serie de hipo´tesis que son va´lidas en elmarco de Relatividad General, pero que aun no se han comprobado en estos escenarios. Apesar de carecer de una descripcio´n completa y autoconsistente, este t´ıpo de estudio permiteexplorar nuevos mecanismos f´ısicos de manera sencilla, y los resultados son validos en general,para cualquier mecanismo subyacente que introduzca las mismas modificaciones a nivel delas ecuaciones. Por otro lado, la alternativa de emplear una aproximacio´n autoconsistentebasada en un modelo concreto tiene la ventaja de que permite explorar los mecan´ısmos f´ısicossubyacentes en una variedad de situaciones, limitadas por consideraciones computacionales.Modelos ma´s alla´ de Relatividad General basados en me´tricas disformes: Elhecho de que la aceleracio´n del universo suceda en escalas en las que no existe una validacio´nindependiente de la teor´ıa de la Relatividad General sugiere una revisio´n en profundidad desus fundamentos. Una posibilidad de extender la teor´ıa de Einstein es mediante la aplicacio´nde una transformacio´n disforme en algu´n sector de la accio´n, dada por los gradientes deun campo escalar. Esta prescripcio´n se emplea´ sobre un escalar cano´nico en el Cap´ıtulo 5 ysobre el Lagrangiano de materia en el cap´ıtulo 6. Las relaciones disformes generalizan las bienconocidas transformaciones locales de escala (o transformaciones conformes) permitiendo unadistorsio´n de la estructura causal en la metr´ıca transformada con respecto a la inicial. Estetipo de relaciones son las ma´s generales posibles entre la me´trica gravitatoria y la de materiacompatibles con convariancia general, y aparecen la descripcion efectiva de ciertas teor´ıascon dimensiones adicionales extensas.Las transformaciones disformes permiten construir modelos de energ´ıa oscura en losque el campo escalar pierde su energ´ıa cinetica, contribuyendo a la expansio´n del univer-so u´nicamente mediante su potencial, que actua como una Constante Cosmolo´gica efectiva.Esto sucede por la tendencia del campo a evitar dina´micamente una singularidad en la met-rica disforme, y funciona tanto en el caso de la auto-interaccio´n del propio campo (modelode quintaesencia disforme) como en el caso de un acoplo con la materia oscura (modelo demateria oscura disformemente acoplada), que fueron estudiados en el caso en el que no hayinteraccio´n conforme. El campo de quintaesencia disforme produce u´nicamente efectos indi-rectos en la formacio´n de estructura, y su comportamiento se asemeja mucho a una ConstanteCosmologica para valores altos de las pendientes de los potenciales, siendo dificil distinguirambas posibilidades observacionalmente. Sin embargo, el modelo de materia disformementeacoplada experimenta una fuerza adicional mediada por el campo escalar. Aunque es capazde reproducir la expansio´n cosmolo´gica en el l´ımite homogeneo, esta fuerza produce efectosimportantes en la formacio´n de estructura a gran escala, tornandola incompatible con lasobservaciones. Afortunadamente, existe mucha libertad en la teor´ıa para producir modelosviables, en los que las modificaciones sean menos dramaticas pero permitan distinguir lasdiferentes alternativas.Las transformaciones disformes tambie´n permiten relacionar entre si teor´ıas aparente-mente disconexas. En ciertos l´ımites, el Lagrangiano de quintaesencia disforme ma´s general sereduce a una serie de modelos de campos escalares no acoplados. Cuando la me´trica disformexiiiResumen y Conclusionesentra en el Lagrangiano de materia y el sector gravitatorio corresponde a Relatividad Gen-eral, se dice que la teoria resultante esta´ en la descripcio´n de Einstein. Es posible invertir latransformacio´n conforme para eliminar el acoplo directo entre el campo escalar y la materia,dando lugar a la descripcio´n de Jordan. En ella, el campo escalar se acopla cine´ticamenteal escalar de curvatura, y aparecen una serie de te´rminos en segundas derivadas del campo,que sin embargo no introducen inestabilidades en la teor´ıa. Este tipo de Lagrangianos conauto-interacciones no lineales en derivadas son capaces de producir el “effecto de Vainshtein”,que alisa los gradientes del campo escalar cerca de objetos masivos, ocultando la interaccio´nadicional.Las teor´ıas disformemente acopladas en la descripcio´n de Einstein tambie´n son capacesde evitar los efectos de la fuerza adicional en entrornos no-relativistas con una densidad deenerg´ıa elevada, tales como el Sistema Solar. En estos casos, existe un l´ımite bien definido enel que la densidad de materia tiende a infinito (al contrario que en el caso de un acoplo con-forme), y en el que la evolucio´n del campo escalar se vuelve independiente de la distribucio´nde materia y los gradientes espaciales del campo. Este “mecanismo de camuflaje disforme”,que suprime la fuerza adicional, es debido a la mezcla de te´rminos cine´ticos entre campoescalar y los grados de libertad acoplados. Su accio´n hace muy dif´ıcil detectar los efectosinducidos por el acoplo disforme mediante medidas de feno´menos gravitatorios, mientras quepermite que e´ste tenga consecuencias considerables a nivel cosmolo´gico.La equivalencia entre teor´ıas disformemente acopladas y teor´ıas con interacciones no lin-eales en derivadas sugiere que el mecanismo de camuflaje disforme y el efecto de Vainshteinson en realidad manifestaciones del mismo feno´meno, pero vistas a la luz de las descripcionesde Einstein y Jordan, respectivamente. Esta equivalencia tambie´n abre una nueva ventanapara el analisis de este tipo de teo´rias, ya que las ecuaciones en la descripcio´n de Einstein sesimplifican considerablemente con respecto a la descripcio´n de Jordan. Asimismo, hay otrosaspectos de las teor´ıas disformemente acopladas que merecen ser considerados en detalle. Es-tos incluyen el estudio de la estabilidad de la teor´ıa, modelos ma´s generales (e.g. an˜adiendo unacoplo conforme) y otras situaciones en las que puedan producirse signaturas caracter´ısticas.Discusio´n: El estudio de diferentes modelos es beneficioso para la salud de la ciencia emp´ıri-ca, ya que e´sta no puede demostrar la veracidad de una hipo´tesis, sino u´nicamente refutarsus alternativas. La comparacio´n de modelos que desaf´ıan las suposiciones habituales con lasobservaciones permite validar los fundamentos del Modelo Cosmolo´gico Esta´ndar y explorarlos l´ımites de su validez. Este programa tambie´n permite la identificacio´n de las hipo´tesis quese asumen implicitamente en las distintas etapas del analisis, y permite encontrar ejemplos enlos que e´stas no se cumplen, facilitando el disen˜o de estrategias que favorezcan una compara-cio´n entre teor´ıas y datos ma´s independiente y menos sesgada. Esto no solo es necesario paraobtener resultados fiables, sino que es necesario para comprobar las distintas hipo´tesis delModelo Cosmolo´gico Esta´ndar de manera lo ma´s independiente pos´ıble. Por u´ltimo, el estudiode teor´ıas alternativas a menudo aporta ejemplos que permiten entender mejor el paradigmaconvencional, enriqueciendo el conocimiento que se tiene sobre e´l y las tecnicas disponiblespara su analisis. Estas consideraciones sera´n importantes en las pro´ximas de´cadas, cuando elincremento en la cantidad y calidad de las observaciones cosmolo´gicas valide de manera au´nma´s so´lida los fundamentos del Modelo Cosmolo´gico Esta´ndar o nos sorprenda con nuevosdatos que indiquen la necesidad de revisarlo.xivPrefaceYou are pooped and demoralized. Whywouldn’t you be? Of course it is exhausting,having to reason all the time in a universewhich wasn’t meant to be reasonable.Kurt Vonnegut2Astronomy and cosmology are arguably the oldest sciences, and perhaps the secondoldest professions in history. After many millenia floating in the darkness of mysticism andspeculation, cosmology has finally become an empirical and predictive discipline. The effortof thousands of scientists, from astronomers to theoretical physicists, has carved a StandardCosmological Model (SCM) that is able to explain a large and increasing set of phenom-ena. The price to pay is the inclusion of three mysterious elements with no conventionalexplanation: inflation to account for the large scale homogeneity and initial perturbations ofthe universe, dark matter to enhance the formation of cosmic structure, and a CosmologicalConstant to propel the latter stage of accelerated expansion.The Standard Cosmological Model is undoubtedly beautiful. The elegance of the Gen-eral Relativistic geometric description is supplemented by the symmetry of the large scalemetric. Despite its simplicity, this scheme seems able to accurately describe the coarse grainedevolution of the universe on the largest scales throughout cosmic history. Yet, many ques-tions regarding the nature of the new elements remain unanswered: What is the mechanismbehind cosmic inflation and the properties of the dark matter particle(s)? Is cosmic acceler-ation driven by Einstein’s Cosmological Constant? If so, why is its value so small comparedto particle physics energy scales but still large enough to be observable? If not, how is itsvalue suppressed and what is the true mechanism for the acceleration?The Standard Cosmological Model has been extremely successful in explaining a broadset of observables with just a handful of parameters. The evidence supporting cosmic ac-celeration has grown since the first decisive data from type Ia Supernovae (SNe) pointingtowards the existence, and a number of independent observables agree now on the detailsof this paradigm. Measurements of the Cosmic Microwave Background (CMB) temperatureanisotropies support the spatial flatness of the universe. When combined with the determi-nation of the local expansion rate, it follows that a smooth component with negative pressure2Breakfast of Champions (1973)xvPrefacemust dominate the cosmic energy budget. Observations of the Large Scale Structure distri-bution confirm this picture. The formation rate of cosmic structures through gravitationalcollapse bounds the amount of clustering matter to be well below what is necessary for theuniverse to be spatially flat. The same physics that produces anisotropies in the CMB im-prints a characteristic Baryon Acoustic (BAO) scale in the LSS distribution, which can bemeasured by galaxy surveys. It provides a standard ruler that tracks the cosmic expansion,further supporting the paradigm of cosmic acceleration.Cosmology has made the promise of new physics. It not only supports its existence,but also provides means to validate or refute the different scenarios. As in a Gold Rush,myriads of alternatives to the Cosmological Constant have been proposed and studied inrecent years. These models necessarily modify some sector of the Standard Cosmologicalmodel. It is possible to mimic the observed acceleration without the introduction of newexotic elements, by postulating a metric which is inhomogeneous on large scales. This canbe done only at the price of living in a special location within the cosmos: the center of avery large, spherical and underdense region. Another possibility is to extend the Einstein-Hilbert action for General Relativity (GR), allowing the presence of scalar fields or evenmore dramatic modifications of the gravitational theory. Scalar fields can be uncoupled fromother components, in which case they contribute to the universe dynamics only through theirenergy and pressure density. They may alternatively interact directly with all or some of thematter species, becoming potentially detectable through experiments in our Solar Systemand the study of the formation of Large Scale Structure.The present Thesis explores models belonging to each of the categories previously de-scribed. A spherically symmetric but inhomogeneous cosmology is found to be discrepantwith observations. The matter inhomogeneity necessary to explain the luminosity-redshiftrelation of type Ia SNe is incompatible with the BAO scale imprinted on the galaxy distribu-tion, because the less dense region near our galaxy expand faster than the more dense onesfar away, producing a space dependent stretch of the cosmic ruler. Several departures fromGeneral Relativity are also explored. Models based on ideas about the thermodynamicalproperties of space-time are studied by the modifications of the background cosmologicalequations that they produce. Theories beyond GR with a full and self consistent descriptionare also considered. They are based on the addition of a scalar degree of freedom, which is in-troduced by means of a disformal tranformation the metric in some sector of the Lagrangian.The disformal transformation generalizes the well known local rescaling of the metric (con-fomal transformation), and allows one to establish connections between different theories,including the definition of new physical frames in which to study their consequences. Thisprescription also allows the construction of models for the cosmic acceleration in which thefield is may couple to matter directly or not. In the coupled case, the scalar field mediatesa new interaction that affects the growth of structure. However, a disformally coupled fieldbecomes insensitive to the matter distribution in high density, non-relativistic environments,and the additional force is hidden from gravitational experiments in this regime.Cosmology is not a beauty contest. A model should not be favored because of beingsimpler, more elegant, or widely accepted. Its validity should depend on the comparisonbetween its predictions and the available data. In order to grant a democratic and fair trial,this comparison should be performed in the least biased possible way. However, cosmic justiceis not always blind. Model dependent assumptions are sometimes made in the analysis of rawdata, and certain simplifications which might be compatible with the standard assumptionsxvimight not be fulfilled in alternative scenarios. The study of the different alternatives helpsto identify these flaws and make the process of comparison between theory and observationsas model independent as possible.Honest empirical science can not prove a model right, but rather disprove its alternatives.Studying the predictions of different models and comparing them with observations provides away to validate the foundations of the Standard Cosmological Model and quantify departuresfrom the constituent assumptions. This model based approach is complementary to modelindependent methods: while the latter allow for a general study of the data, which mightreveal new phenomenological features, the former allows one to find and study concretephysical mechanisms that produce these phenomena. The study self-consistently formulatedmodels further allows the exploration of a given physical set-up in distinct regimes, withincomputational limitations.This considerations will become increasingly important as the cosmological wealth andquality of data increase. It is the observational effort what has awarded physical cosmologyits present status, and this effort will continue during the following decades. Keeping inmind alternatives to the Cosmological Constant, or any of the standard assumptions, is agood way to be prepared for new discoveries. Hopefully, forthcoming data might allow us totest the foundations of the Standard Cosmological Model with upgraded independence fromeach other. Finally, the study of alternative theories often provides a better understandingof the standard paradigm, as different shades might be appreciated in the taste of everydayfood after a trip to an exotic land.OutlineChapters 1 and 2 introduce the fundamental concepts involved in the study of models forcosmic acceleration and their comparison to observations. Section 1.1 presents the hypothesesbehind the Standard Cosmological Model. Arguments to pursue the study of alternativesare given in Section 1.2. Section 1.3 gives an overview of the non-standard scenarios. Theobservational probes of cosmology are classified based on the type of information that theyprovide in Section 2.1. The remaining of Chapter 2 presents the most relevant ones for thesubsequent analysis.Several cosmological models that violate the standard assumptions are studied andcompared with observations in the following Chapters. The possibility of large scale inho-mogeneity and the violation of the Copernican Principle is considered in Chapter 3. Afterintroducing the LTB inhomogeneous models, the effects of the inhomogeneities on the BAOscale are detailed in Section 3.2, and used in Section 3.3 to rule out profiles with space in-dependent Big-Bang. Chapter 4 studies phenomenological modifications of the Friedmannequations (inspired by the entropic gravity proposal) and their observational consequences.Chapter 5 explores theories in which a disformal transformation is performed on a canoni-cal scalar field Lagrangian, including connections between Dark Energy Lagrangians and athorough study of the cosmological implications of the disformal quintessence model. Theuse of disformal transformations is extended in Chapter 6, where it is allowed to enter thegravitation and matter sectors. Sections 6.1 and 6.2 contain the derivation of the equationsfor the theory, and explore the connections between the disformally coupled models and otherscalar-tensor theories. Their cosmology is explored for a simple, Disformally Coupled DarkMatter model, both in the homogeneous approximation and including cosmological pertur-xviiPrefacebations, in Sections 6.3 and 6.4. The viability of extending the coupling to visible matter viaa disformal screening mechanism is explored in Section 6.5.The main results and a global discussion are presented in Chapter 7. Several techni-cal results and the description of auxiliary tools have been included in the Appendices A(statistical analysis and Markov Chain Monte Carlo algorithm), B (Boltzmann code to solvecosmological equations), C (equations for k-essence scalar fields) and D (disformal relations).xviiiAcknowledgementsLuckily, I have plenty of reasons to be thankful and many people to express gratitude to,not only for helping me in the course of this Thesis, but also for their support and theunforgettable moments that we have spent together.First of all, I would like to thank Juan Garcia-Bellido, not only the careful reading ofthis manuscript, which has greatly contributed to improve it towards the final version, butalso for being so generous with his time and sharing his knowledge and physical intuition withme. It was him who awoke my interest for cosmology in the first place, during his inspiringlectures at the UAM, and supported me since the very first time. It has been nothing butthe greatest pleasure to be able to discuss and work with him during the last years.I am also very thankful to my other advisors. Pilar Ruiz-Lapuente has always beenextremely supportive, patient, and generous, and I would not have been able to do any ofthis if it had not been for her. I enjoyed working with Tomi Koivisto and learned a lotfrom him through his careful explanations, the patient review of my results and the longdiscussions we held in Heidelberg, Utrecht, Oslo, Barcelona and Madrid, often extendingover many drinks. I am especially grateful to David Mota for sharing his vision and givingme very good advice in many occasions, as well as for devising several interesting projects inwhich I have worked.Cosmology has allowed me to meet many wonderful people, who have made these fouryears incredibly fun and enriching. Dani G. Figueroa, my “older brother”, has always beena source of inspiration and guidance, and certainly someone who knows where the fun is andhow to get there. I’ve also had great times with my other “siblings” David Alonso and AliciaBueno-Belloso. All of them have taught me valuable lessons. I can not go any further withoutthanking the people who I’ve been meeting everywhere, those who I can not associate to anyplace in particular: Edoardo Carlesi, who shares the curse of being one of the few CMBeasyusers left in the world, Federico Urban, the best chef in Norway and the most patient hikingpartner, Jose Beltran, for his good advice and all the inspiring conversations about life andthe Cosmological Constant, and Lluis Galbany for his great sense of humor.During a very important part of my thesis I was working at the Institut de Ciencies delCosmos (ICC) and the Departament d’Astronomı´a y Meteorolog´ıa (DAM) in the Universityof Barcelona (UB) with a grant associated to the Institut de Estudis Espacials de Catalunya(IECC). I would like to acknowledge the support received from these institutions and all thepeople whom I met there. I especially appreciated the company of my office mates AndreuFont and Hector Gil-Marin (which was very valuable in our windowless room) as well as theother graduate students at DAM.There have been some serious traveling these past years, and I’ve been very fortunate tolive and work in several great places. I really enjoyed my stays at the Institutt før TeoretiskxixAcknowledgementsAstrofysikk in the University of Oslo, which has become one of my favourite cities in theWorld. Being there, I had the pleasure to meet a lot of excellent people, including Nico-laas Groeneboom, Kristin Mikkelsen, Sandro Scodeller, Iain Brown, Hans Winter, ClaudioLlinares, Tuva och Ingeborg. Tusen Takk! During my time at the Institute of TheoreticalPhysics in Heidelberg, I had the luck to meet Annika, Nico, Sarah, Richard, Mattias andother nice people who made me feel at home. I am also thankful to Marc Kamionkowski forhis kind hospitality at the John Hopkins University during my visits to Baltimore.One of the strokes of luck I experienced was being able to attend the Cape TownInternational Cosmology School, thanks to the kind invitation and support of Bruce Bassett.Although it only lasted for two weeks, what I learned and experienced there has had aconsiderable impact on me as a scientist and a person. It was a real pleasure to work withSean February and Pranjal Trivedi in the voids project, and I’m really looking forward towork with them again (on that or anything else). And of course I will not forget the funtimes I spent with Alvaro de la Cruz-Dombriz, Bryan Nord, Danielle Wills, Michelle Knights,Ze Fonseca and the other people I met there.The time at the Instituto de F´ısica Teo´rica in Madrid has been very enriching. I wouldlike to thank my friends from the Master’s, Pablo Soler, Javi Abajo, Ana Rodriguez-Sa´nchezand Bryan Zaldivar, for for all the great times, past, present and future. I’m also verythankful to Chabe, Roxanna and the other people in charge of administration for being sokind and helpful. My undergraduate years at the Universidad Auto´noma de Madrid weresimply great, and it was a pleasure to study physics there. I would like to thank ArkadiLevanyuk and Andres Cano, who awoke my curiosity for physics and showed me the spiritof critical thinking, during their unforgettable lectures in mathematical methods. I’m alsograteful to the great teachers I had in high school, including Miguel Angel Ortega, PalomaMontero, Concha Mozo, Jesus Bran˜a and Javi Holgado, as well as Alex and Guillermo fromprimary school.Many other people need to be properly thanked for the great moments and interestingconversations I had with them, both within and beyond science. I shall not write furtherwithout at least mentioning Jaime Alberdi, Santi Bario, Peri, Jose Frutos, Javi Ramos,Emilio Franco, Zongalo, Pablo Merodio, Buti, Marcos Diaz, Alex Molpercebes, Rogu¨a, AnaMartinez, Fer de Juan, Hector Martinez, Javi-Teresa, Micky Knife, my cousins Olvido, Felix,Marta and Marcos, Florian und Lidia, Stefan and Michael Friedel, Elsa, Rube´n, Dieguito,Marta and Tris, as well as Teo, Rosi, Marcos Ana, Ramo´n y Charo and Fa´tima, who havereally become a “non-official family” to me. Mentioning them without any details is almostan offense, and I hope to find the time to express my gratitude to them in person.Very specially, I would like to thank my parents for their invaluable support and un-derstanding as well as for teaching me the most valuable lessons I’ve learned in life. Mygratefulness naturally extends to my uncles and cousins Jesus, Mari Luz, Julia, Jesu, Manueland Javi, for their endless generosity and their hospitality in Barcelona. My sister Ana, forthe great moments and conversations, advice and patience, deserves a very warm mentionhere. Finally, I would like to thank my dear Lindsey because, among all the things I’mgrateful to her, she was the one who made me fall in love with science again.MiguelxxNotation and AcronymsThe present conventions have been adopted throughout the following Chapters, unless oth-erwise specified. The metric signature is (−,+,+,+), and the Riemann curvature is definedas in Eq. (1.1). Coordinates are labeled by greek indices xµ where µ = 0 refers to thetime coordinate and latin indices to spatial coordinates only. Sum over contracted indices isimplicitly assumed and units in which the speed of light c is set to unity are used.Partial derivatives with respect to a variable are denoted with commas, L,φ ≡ ∂L/∂φ.Partial derivatives w.r.t. coordinates are indicated either by commas or by the partial deriva-tive sign φ,µ ≡ ∂µφ ≡ ∂φ/∂xµ, using either the coordinate or its index. A semi-colon willindicate a covariant derivative φ;µν = ∇ν∇µφ. Derivatives with respect to time coordinatesare indicated with a dot A˙(r, t) ≡ ∂A/∂t. t refers to cosmic time, τ to conformal timeand both satisfy dt = a(τ)dτ . In a function of one variable, a prime will denote a deriva-tive A′(φ) ≡ A,φ, while for functions of space-time, a prime will indicate a derivative withrespect to the radial coordinate A′(r, t) = ∂A/∂r. Fourier transforms are defined so that∂if(t, ~x) = ikif˜(t,~k). δD, δ(n)D denote the Dirac delta function and its n-dimensional general-ization. Barred quantities are constructed out of disformal metrics of the form given by Eqs.(6.1, 5.3) (Appendix D contains a compilation of relations that apply to disformal metrics).The acronyms used throughout the text are summarized in Table 2 below for conve-nience:Name ReferenceBAO Baryon Acoustic Oscillation(s) 2.4.1BBN Big Bang Nucleosynthesis 2.2CC Cosmological Constant Λ 1.1.5CDM Cold Dark Matter 1.1.4CGBH Constrained GBH 3.1.1Cl Angular power spectrum (2.7, 2.9)CMB Cosmic Microwave Background 2.3DA Angular Diameter Distance (2.1)DC Disformal Coupling 6.2.2DCDM Disformally Coupled Dark Matter 6.3.1DE Dark Energy 1.3.2DL Luminosity Distance (2.2)DM Dark Matter 1.1.4dof degree of freedomEF Einstein frame 6.2, (1.25)xxiNotation and AcronymsName ReferenceEH Einsten-Hilbert (action) (1.2)eos equation of stateFRW Friedmann-(Lemaˆıtre)-Robertson-Walker (1.5)GBH Garcia-Bellido Haugbølle matter profile 3.1.1GR General Relativity 1.1.1Gpc Gigaparsec (3.09 · 1025m)H0 Hubble rate today [km/Mpc/s]h Normalized Hubble rate today [100 km/Mpc/s]ISW Integrated Sachs-Wolfe effect 2.3JF Jordan frame 6.2, (1.23)Λ Cosmological Constant 1.1.5lhs left hand sideLSS Large Scale Structure 2.4LTB Lemaˆıtre-Tolman-Bondi 1.3.1MCMC Markov Chain Monte Carlo AMG Modified Gravity 1.3.3Mpc Megaparsec (3.09 · 1022m)OCGBH Open Constrained GBH 3.1.1P (k) Power spectrum (2.17)rhs right hand sideSM Standard Model (of particle physics)or Standard Matter 1.1.4SCM Standard Cosmological Model 1SNe Type Ia Supernovae 2.5WMAP Wilkinson Microwave Anisotropy Probewx Equation of state of the x componentwrt with respect toz redshiftΩx cosmic fraction of the x- component today (1.8)Table 2: Acronyms used in the text.xxiiPart IFoundations1Chapter 1The Standard Cosmological ModelOne should always look for a possiblealternative, and provide against it. It is thefirst rule of criminal investigation.Arthur Conan Doyle1Recent years have witnessed enormous advance in the quantitative understanding ofcosmology and the establishment of a Standard Cosmological Model. In addition to knownforms of matter, Einstein’s General Relativity and an ansatz for the space-time metric (spa-tially homogeneous and isotropic), two mysterious elements need to be added in order toaccount for the observations. These give the name to the standard, ΛCDM model: ColdDark Matter (CDM) to explain the formation and dynamics of cosmic structures and aCosmological Constant (Λ) to account for the dimming of distant supernovae [11, 12].The steadily increasing precision and wealth of data and the surprising findings call fora revision of the hypothesis made in the construction of the Standard Cosmological Model.Dropping the large scale homogeneity assumption allows one to reinterpret the observedacceleration as a mirage, allowing in principle for a universe with zero Cosmological Constant[8]. Other alternatives have been proposed in which the Cosmological Constant is substitutedby a more general form of dark energy (DE) that fuels the acceleration [9]. Finally, the lastmajor path to address the dynamics of the cosmic acceleration is by postulating modificationsof General Relativity acting only on the largest cosmological scales [7].Comparing these alternative models against cosmological observations does not onlyprovide potential hints to test new physics: it is itself a way to validate the assumptions onwhich the standard paradigm is formulated. These assumptions will be presented in Section1.1. A number of reasons to consider alternatives will be discussed in 1.2, and an overview ofthe possible modifications that deal with the problem of cosmic acceleration in Section 1.3.1The Return of Sherlock Holmes (1905)3Chapter 1: The Standard Cosmological Model1.1 Ingredients and PreparationAs a logical construction, the standard cosmological model requires six hypothesesStandard Model = GR + FRW + Initial Conditions + SM + CDM+Λ1. General Relativity: As gravity is always an attractive interaction, it dominates onmacroscopic distances and will be crucial for cosmological phenomena.2. Friedman-Robertson-Walker metric: GR is a metric theory governed by nonlinear par-tial differential equations. A simple ansatz for the metric is required, which in thestandard case is based on maximal spatial symmetry.3. (Inflationary) Initial Conditions: The paradigm of cosmic inflation is able to provideinitial conditions for the perturbations around the background metric, explains theobserved value of the spatial curvature and further supports the choice of the metric.4. Standard Matter: The known forms of matter and their cosmological effects have to beaccounted for, notoriously baryions (nucleons and electrons), photons and neutrinos.5. Cold Dark Matter: Structure formation requires the presence of a form of matter doesnot interact with light, usually assumed to be a new, weakly interacting particle species.6. Cosmological Constant: The observed acceleration of the universe can be explained ina simple way by the presence of an energy density that does not evolve in time.Assumptions 1 and 4 are an extrapolation to the cosmological realm of good old earthlyphysics, assumption 2 is a model simplification justified a posteriori and assumptions 3, 5and 6 require the introduction of new physics, for which so far there is only cosmological and(in the case of dark matter) astrophysical support.1.1.1 General RelativityThe theory of General Relativity (GR) provides a very elegant framework in which thegravitational interactions can be related to the geometry of the spacetime [13, 14]. Thefundamental degree of freedom is the space-time metric that determines the physical distancesds2 = gµνdxµdxν , which is treated as a fully dynamical variable. The theory relates thedynamics of the metric to the matter through the curvature of the space-time, encoded inthe Riemann tensorRαβµν ≡ ∂[µΓαν]β + Γαγ[µΓγν]β (1.1)where Γαµν ≡ gαλ(gα(µ,ν) + 12gµν,λ) is the torsion free (Γαµν = Γανµ) connection that gives ametric compatible (∇αgµν) covariant derivative ∇µvν = ∂µvν + Γνµαvα. Besides the metricformulation, to fully specify the dynamics an action for the metric has to be provided. Itturns out that the simplest, nontrivial, theory that can be constructed out of gµν workssuperblyS = SEH + Sm =∫d4x√−g R16piG+ Sm , (1.2)41.1. Ingredients and Preparationwhere an unspecified matter sector has been included, g is the determinant of the metric,and R ≡ gµνRααµν is the Ricci scalar. The above equation is known as the Einstein-Hilbertaction. The variation wrt the metric lead to Einstein equationsGµν ≡ Rµν − R2gµν = 8piGTµν . (1.3)The lhs is known as the Einstein tensor. The source term, the energy momentum tensor isdefined out of the variation of the matter action, which is covariantly conserved by virtue ofthe Bianchi identitiesTµν ≡ −2 δSmδgµν, ∇µTµν = 0 . (1.4)1.1.2 Metric Ansatz: The Copernican PrincipleEinstein equations (1.3) are a set of ten coupled, non-linear, partial differential equationsthat are very difficult to solve unless some simplifying assumptions are made. Knowing thatstatic metrics are not a solution in the presence of matter, the simplest possible guess is to tryto model the “spatial average” and consider the evolution of the mean cosmic density. Moreprecisely, one can assume that the metric is formed by a foliation of maximally symmetricspatial Sections, i.e. homogeneous and isotropic.2 This choice gives the Friedman-Lemaˆıtre-Robertson-Walker metricds2FRW= −dt2 + a(t)2 dr21− kr2 + a(t)2r2dΩ2 , (1.5)where dΩ2 ≡ dθ2 + sin(θ)dφ2 is the solid angle element, and the scale factor a(t) encodesdynamics of the spatial Sections, i.e. the expansion of the universe. The above metric has tobe regarded as a large scale, averaged limit of the true universe metric. Inhomogeneities andanisotropies can be introduced as small perturbations within this picture, making the modelmore predictive on the intermediate scales, at which the perturbations remain sufficientlysmall.The assumption of isotropy can be justified by the observations of the Cosmic MicrowaveBackground tempearture fluctuations. After removing the dipolar component (which is likelydue to the peculiar motion of our galaxy) the remaining fluctuations deviate at the orderof 10−5 from the mean and do not show evidence of a preferred direction.3 The case forhomogeneity is trickier, as it is possible to postulate a spherically symmetric universe, whichwould seem isotropic if we are located very close to the center of symmetry, as it will beexploited to construct inhomogeneous cosmological models without a Cosmological Constant.Rather than waiting for observational validation, the assumption of homogeneity is invokeda priori by means of the Copernican Principle, i.e. the statement that our location in theuniverse is not be special (e.g. the center of a large, symmetric region).There are two independent Einstein equations for the homogeneous metric 1.5, knownas the Friedmann equations: (a˙a)2=8piG3ρ− ka2, (1.6)2It is further necessary to specify a global topology for the space-time manifold, for which it is assumedthat spatial sections are simply connected. Although the choice of a non-trivial topology does not influencethe local gravitational dynamics, it implies the existence of periodicity scales in the universe that violateglobal isotropy. Such scales must be very large r & 24Gpc in order to fulfill by CMB observations [15].3Some claims of prefered directions in the CMB have been made and disputed, e.g. references [16, 17].5Chapter 1: The Standard Cosmological Modela¨a= −4piG3(ρ+ 3p) . (1.7)Here ρ = T 00 and pgij = T ij are the energy and pressure density (1.10). It is conventionalto define the Hubble factor as H = a˙/a and the fraction of a cosmic component today asΩi =ρiρ0, (1.8)where ρ0 =3H208piG is the critical density, defined in terms of the Hubble rate today H0.1.1.3 Inflation: Symmetry and Initial ConditionsThere is overwhelming evidence that the universe is not homogeneous on sufficiently smallscales. In order to give an accurate cosmological description, it is necessary to considerdeviations from the perfect FRW metricds2PT = ds2FRW + δgµνdxµdxν . (1.9)In the standard picture, these departures are very small at early times and become increas-ingly significant as cosmic structures develop by the action of gravity. For a broad range ofscales and most of the cosmic evolution, it is sufficient to consider only linear corrections tothe Einstein equations, computed using the above perturbed metric to first order in δgµν .The CMB shows that these fluctuations had a very small amplitude amplitude ∼ 10−5 duringthe recombination epoch.In order to explain the observed degree of isotropy in the cosmic microwave backgroundradiation, it is postulated that the universe underwent a period of accelerated expansion atvery early times, known as cosmic inflation. If the inflationary period lasts long enough,it allows for opposite regions in the sky to be causally connected. The fact that they haveapproximately the same temperature can be then seen as a natural consequence of localphysics. As inflation magnifies an initially tiny region into the whole observable universe,it has the additional benefit of diluting away any preexisting feature, such as cosmic relicsor initial inhomogeneities. Hence, it provides further support for the choice of a maximallysymmetric metric to describe the universe expansion.In addition to homogenize the universe, inflationary models source initial perturbationsin the metric by amplification of quantum fluctuations to macroscopic scales. Each infla-tionary model provides a set of initial conditions for the perturbation equations that wouldeventually give rise to the large scale structure of the cosmos through gravitational collapse.Due to the empirical success of these predictions, inflation is often considered an ingredientof the standard cosmological model, even in the absence of a preferred inflationary model.The physical predictions for inflation are:F Negligible Curvature: Despite it is not a component in itself, it can be quantised interms of Ωk =kρ0a2. CMB observations combined with H0 and BAO yield a valuecompatible with the inflationary prediction |Ωk| < 0.005 [18].F Adiabatic, Gaussian and nearly scale invariant initial perturbations: Although thedetails depend on the inflationary model, the CMB typically constraints departuresfrom this scheme [19].61.1. Ingredients and Preparation1.1.4 Standard and Dark MatterEinstein Equations describe how the space time responds to the distribution of matter andenergy. The matter content in the universe can be modeled through an energy-momentumtensor of the perfect fluid formTµν = (ρ+ p)uµuν + pgµν , (1.10)where uµ is the four velocity and ρ, p the energy and pressure densities respectively. Thetime component of the covariant energy momentum conservation (1.4) applied to a FRWspacetime yields the continuity equation:ρ˙+ 3H(ρ+ p) = 0 . (1.11)For most of the cosmic evolution, the dominant fluids can be treated as barotropic, for whichp = wρ. This allows the integration of the conservation equationρ = ρ0a−3(1+w) = (1 + z)3(1+w) . (1.12)The second equality here relates the scale factor to the redshift, which relates the increase inwavelength of photons due to the cosmological expansion as 1 + z ≡ λr/λe = a(tr)/a(te) (i.e.the ratio between the received and the emitted wavelength).The relevant matter species for the dynamics of the late universe are the following(quoted abundances correspond to reference [20])F Baryons: The term is used generically for electromagnetic interacting, stable fermions.Its value can be inferred from the primordial deuterium abundance and the CMBΩb = 0.0225± 0.0006h−2.F Photons: The CMB is the best blackbody observed ever. Its measured temperatureT = 2.7255± 0.0006K translates into a value Ωγ = 2.47 · 10−5h−2.F Neutrinos: Although the amount of cosmological neutrinos is not directly observed,their properties affect the formation of LSS. Current bounds on neutrino masses are5.4 ·10−4 < Ωνh2 < 3.0 ·10−3, where the upper bound is cosmological and the lower oneis obtained in terrestrial neutrino oscillation experiments. The neutrino mass translatesto Ωνh2F Cold Dark Matter: To correctly explain the formation of cosmic structure, there mustbe a significant amount of non electromagnetically interacting matter which was non-relativistic (i.e. cold) at the onset of galaxy formation ΩCDM = 0.112 ± 0.006h−2.Dark matter is also essential to explain the CMB anisotropy pattern, as it is able tocluster before recombination and form potential wells. Other observations, such as therotational curves of galaxies and the gravitational lensing by galaxy clusters provideindependent evidence of its existence.1.1.5 The Cosmological ConstantThe final ingredient is the Cosmological Constant (CC), i.e. a constant term in the Einstein-Hilbert action (1.2) introduced to explain the acceleration of the cosmic expansionSEH+Λ =∫d4x√−g 116piG(R− 2Λ) , (1.13)7Chapter 1: The Standard Cosmological Modelthat modifies Einstein EquationsRµν − R2gµν + gµνΛ = 8piGTµν . (1.14)The CC term can be incorporated into a redefinition of the energy momentum tensor in therhs. Its effect then is equivalent to a perfect fluid with TΛµν ∝ gµν , with negative pressurep = −ρ and constant energy density ρΛ ≡ Λ/(8piG). Hence, the Cosmological Constantrepresents a source of energy that does not dilute with the cosmic expansion, exactly theproperties one would expect from the energy stored in the vacuum [21].Such a term would be unobservable in a theory based in a static space-time for whichonly energy differences are observable. In GR, the CC couples to the determinant of themetric in (1.13), thus affecting dynamics of the space-time. In particular, plugging p = −ρ inthe second Friedmann equation (1.6) with k = 0 shows that it produces a constant positiveaccelerationa¨a=8piG3ρΛ > 0 . (1.15)That is, the effect of the Cosmological Constant on the universe expansion is opposite to theone caused by standard, gravitating matter. Current observations suggest that the Cosmolog-ical Constant is the dominant component in the cosmic energy budget, with ΩΛ = 0.73±0.03[19].1.2 The Need to Go BeyondIn this Section I will review some of the arguments to go beyond the standard cosmologicalmodel, emphasizing those related with the Cosmological Constant as the source of cosmicacceleration.1.2.1 Theoretical IssuesThe standard cosmological model relies on three pieces of new physics: inflation, dark matterand the Cosmological Constant. Considering alternative paradigms might help to obtain abetter understanding of the model and search for possible connections between the differentmysterious elements. In this light, the idea of inflation provides the most compelling reasonto study alternatives to the Cosmological Constant. Inflationary scenarios require a periodof accelerated expansion similar to the one produced by a Cosmological Constant, but at amuch higher energy scale. On the other hand, inflation has to end in order to evolve into auniverse filled with matter, and hence the inflationary epoch can not be due to a CosmologicalConstant term.4 The fact that there might have been an acceleration mechanism at work inthe very early universe different than Λ suggests that a similar mechanism might be at workin the late universe.The Cosmological Constant also faces two naturalness problems related to its actualvalue when interpreted as a vacuum energy and due to its time evolution.The lack of acompelling theory for quantum gravity indicates that General Relativity has to be revised,at least in the high-energy/small-scale regime. The modifications introduced in the pursue of4In addition, the running of the spectral index for primordial perturbations ns = 0.963±0.014 [19] indicatesa departure from a “perfect Cosmological Constant” (ns = 1) at the time when the perturbations on the scalesprobed by the CMB were seeded.81.2. The Need to Go Beyonda renormalizable theory of gravity might yield some information on the value and propertiesof Λ, or whatever mechanism behind cosmic acceleration.The Cosmological Constant ProblemSince the Cosmological Constant is equivalent to a constant energy density in the action, itseffective value receives different contributions. The most problematic of these is the expectedfrom quantum fields in the vacuum [22]. The standard, quantum mechanical harmonic os-cillator (or any other bound system) has a energy proportional to the frequency ~ω even inthe absence of excitations. As a free quantum field is described as a collection of infiniteharmonic oscillators, its energy in the vacuum state is infinite in principle, although it canbe renormalized to an arbitrary values by the addition of a suitable counter term Λ0. Em-ploying dimensional regularization to preserve Lorentz invariance and removing the divergentcontribution, the resulting vacuum energy associated to a scalar field of mass m is [23]∆Λeff =Gm48pilog(m2µ2), (1.16)where µ is a renormalization scale. Although the contribution is dependent on the choice of µ,the above expression shows how the corrections are sensitive to the mass scales present in thetheory. The details are different for other types of field, but the conclusion is similar: Theyall produce an effective contribution to the total Cosmological Constant which is genericallymuch larger than the true value, i.e. Λ is unstable against quantum corrections (technicallyunnatural).In addition, the Cosmological Constant receives a classical contribution from the fieldpotentials, due to the phase transitions predicted within the standard model of particlephysics, e.g. electroweak symmetry breaking. At very early times where the temperature ofthe universe is T & mH ≈ 125GeV, the potential for the Higgs field has a single minimum, andhence the field fluctuates around a unique vacuum state. When the electroweak transitionoccurs, the Higgs potential is deformed into a Mexican hat form, the central extremumbecomes unstable and the vacuum state is displaced towards a degenerate set of minima.Besides being responsible for the masses of the W and Z bosons, this process displaces theminimum of the Higgs potential by ∆ρEW = v4/4 ∼ O(100GeV)4, where v is the vacuumexpectation value of the field and λ is the quartic self coupling [23]. A constant term can beadded to set the value of the potential energy to zero, but only before or after the transition.According to General Relativity, this offset would produce the same effect as a (perhapsnegative) CC with a value much larger than the observed one ∆ρ ∼ 1055ρ0. Similarly, alower energy phase transition within quantum chromodynamics would induce an imbalancein the effective Cosmological Constant of order ∼ (100MeV)4, which is also too large, with∆ρ ∼ 1045ρ0.When one considers all the terms that contribute in the same way as the CosmologicalConstantΛtotal = Λ +∑fields∆Λeff + 8piG∆ρPT + 8piGρQCD , (1.17)it turns out that the different contributions have to cancel with exquisite precision to yieldthe observed value today. This fine tuning is known as the Cosmological Constant problem.There are important caveats to these arguments. The quantum corrections have beencomputed assuming a perturbative expansion in the fields. More importatnly, the whole9Chapter 1: The Standard Cosmological Modelcomputation is performed in a fixed background metric and does not take into account theeffects of gravity, which might be crucial to address purely gravitational effects such as theacceleration induced by constant terms in the action. Although the CC problem should be ad-dressed within a consistent framework for quantum gravity, general covariance suggests thatlocal physics can be described by the special relativity in a sufficiently small region, offeringsome support to the above argument. If the CC problem can not be solved within quantumgravity, it might be pointing towards the existence of a yet unknown principle or mechanismthat keeps its value small or zero, and which may have observational consequences.5The Coincidence ProblemThe energy densities of matter and Cosmological Constant are similar today, but their relativescaling, ρΛ/ρm ∝ a3, implies that Λ was completely negligible in the cosmological past andwill be absolutely dominant in the future. If we think of the Cosmological Constant as aninitial condition set in the early universe (when the energy density was very high comparedwith the current one), it seems very unlikely that it acquires a value comparable to matter atthe present cosmological epoch, when galaxies and other large scale structures have formed.Had the Cosmological Constant been a couple orders of magnitude higher, the universe wouldbe empty of large scale structure. But if it had been just a couple orders of magnitude smallerit would be hardly measurable.The lack of an explanation for why the Cosmological Constant becomes relevant at thepresent epoch is known as the coincidence problem. Although this issue is somehow aesthetic,this piece fine tuning might be pointing towards something more complex. The situation isanalogous to the inflationary paradigm. Without inflation, the observed value of the spatialcurvature Ωk ∼ 0 would seem oddly fine tuned. The inflationary paradigm is able to givea natural explanation for the current bounds and solve the (aesthetic) problem of the smallspatial curvature, on top of the more severe problems that lead to the idea in the first place.1.2.2 Observational ReasonsEven if the standard model turns out to be correct and the only remaining task is to un-derstand the details, the only way to prove it as a physical theory is by disproving thealternatives. This provides an explicit way to test the assumptions behind the standard cos-mological model, while searching potential signatures of new physics. In addition, exploringalternative scenarios helps to improve the model independence of cosmological probes, andthus their reliability.Regardless of its empirical success, the cosmological constant is a very stiff model. Iffurther data becomes incompatible with a non-evolving cosmological constant (even if theevolution is tiny) the cosmological constant would be ruled out and an alternative theorybecomes necessary. There are actually some observations that seem at odds with the standardparadigm, which are summarized below. Although these measurements are disputed due toa lack of statistical significance or systematic effects, they might be the first outliers pointingtowards necessary modifications.5An argument of symmetry might explain both a vanishing and a small value for Λ, in a similar way assymmetries set certain masses to zero (e.g. the photon by gauge symmetry) or protect their value againstquantum corrections (e.g. fermions and broken chiral symmetry). More generally, it is conceivable that atheory of quantum gravity may be pathological if Λ > 0 and Λ < 0, in which case Λ = 0 would be a consistencyrequirement.101.2. The Need to Go BeyondModel Independence of Cosmological ProbesThe standard cosmology presented in the previous pages is by far the best worked out modelin the market.6 It is so widely accepted that many observational analyses implicitly includesome of the model hypotheses in the treatment of raw data. For example, many cosmologicalprobes of large scale structure require assuming a fiducial cosmology in order to convertredshifts and angles into physical separations, and sometimes effects are tested (or evencalibrated) against numerical simulations.Sometimes, even the quantities used to relate theory and observations are meaningless ina non standard cosmology. Examples of this are the A(z) = DV (z)√ΩmH20/z function [24],that characterizes the Baryon Acoustic Oscillations feature measured in large scale structuresurveys, or the CMB shift parameter (2.12). They incorporate a factor√Ωm that is usefulto eliminate degeneracies of the model. Although practical for standard cases, this extradependence invalidates A(z) as a way to constrain models for which ρm(z) does not evolve as(1 + z)3 (e.g. coupled quintessence or inhomogeneous models for which ρm(z) = ρm(r, t)∣∣z).On the other hand, using the ratio between the sound horizon at the drag epoch (whichsets the BAO feature initially) and a distance to that redshift makes the measurement mucheasier to interpret in more general scenarios.Testing non-standard cosmologies is helpful to realize which quantities make the moresense as physical effects and make observational probes as model-independent as possible.The analysis of raw data should be performed without assuming fiducial cosmological modelsand aim at the extraction of physical effects that are both observable and possible to com-pute and interpret within any given cosmology. Chapter 2 introduces several cosmologicalobservables, emphasizing the model independence of the different probes.Observational OutliersIt is important to bear in mind that the certain sets of evidence pointed towards the existenceof mysterious elements in the standard model even before the evidence from supernovae,CMB and large scale structure observations was solid or even existed. Claimed problems ofΛCDM on large scales are the large scale velocity flows, alignment and lack of power of lowmultipoles in the CMB angular power spectrum and large scale alignment in the QSO opticalpolarization data. Other potentially problematic features are related to small scale structureformation. These include the tension between dark matter simulations and observationswith regard to both the density profiles of dark matter halos and for the number of predictedsubstructures inside a given host halo, the baryonic Tully-Fisher relation and the constantgalactic surface density.7 The seemingly discrepant measurements might be due to statisticalfluctuations, systematic measurement effects or imprecise theoretical models. However, theyare useful to keep in mind, for they might be pointing towards a failure of one or more of thestandard hypotheses.6For example, non-linear structure formation within the SCM has been worked out using different pertur-bative schemes (e.g. linear and higher order perturbation theory, Lagrangian and renormalized perturbationtheory) and numerical N-body simulations involving increasing resolution. Although some progress has beenmade, the level of understanding is considerably lower for alternative models.7The reader is referred to reference [17] for an overview.11Chapter 1: The Standard Cosmological Model1.3 Exotic Tastes and Alternative RecipesIn an effort to provide phenomenologically richer and theoretically more pleasant alternativesto Λ, a vast amount of proposals have been considered [7, 9, 25–27]. These alternatives usuallyappeal to an unknown principle to set the total CC to zero,8 and provide an alternativemechanism to accelerate the universe, which necessarily requires an additional modificationof one of the assumptions behind the standard model:1. Allowing for large scale inhomogeneity - See Section 1.3.1 and Chapter 3.2. Extending the Einstein-Hilbert action, Eq. (1.2)2.1. Adding a new (dark) energy component - See Section 1.3.2 and Chapter 5.2.2. Modifying the gravitational sector or the coupling to matter - See Section 1.3.3and Chapters 4, 6.Other possibilities that aim to explain supernovae dimming have been proposed but will notbe considered here. These include as axion-photon mixing [31, 32], gray dust and super-novae evolution. Although these models can easily explain the supernovae observations, theygenerally fail when other probes are taken into account.An additional approach is to consider parameterizations that extend the theory in cer-tain directions and can be compared with observations, regardless of the physical originof the modifications. The simplest example is the constant equation of state dark energymodel wCDM, which allows one to test the allowed departures from a Cosmological Con-stant w = −1. This approach allows one to test consistency relations and allowed departuresobservationally. More sophisticated approaches have been proposed in order to encompassvery general modifications in a series of free functions (e.g. the Post-Friedmann approach[33]). Choosing a parameterization (e.g. for w(z)) has the problem of pre-selecting a veryrestricted subset of the space of functions that define the theory. More generally, one cantry to reconstruct the free functions that define the cosmological model (either fundamen-tal or parametric) without any a-priory restriction. This has been done in the context ofcosmic acceleration using genetic algorithms [34] and Gaussian random processes [35]. Pa-rameterizations can capture physical features in a model independent way, but going beyondparameterizations is necessary to have complete, self consistent models that can be exploredin different regimes.The alternatives are described in the following subsections, focusing on the contextrelevant for the results presented in Chapters 3-6. For a further details and a broader contextthe reader is referred to the excellent reviews on inhomogeneous cosmologies, dynamical darkenergy and alternative theories of gravity by Clarkson [8], Copeland et al. [9] and Clifton etal. [7] respectively.1.3.1 Inhomogeneous ModelsThe cosmological information is gathered through photons from distant sources (e.g. super-novae or galaxies), that were emitted at the time corresponding to a given redshift z andarrive from a given direction nˆ. Hence, most cosmological probes are only able to explore a8This requires solving the old cosmological constant problem. Several attempts in this direction have beenproposed [28–30]121.3. Exotic Tastes and Alternative Recipesthree dimensional submanifold (within the four dimensional space-time), that is, the light-cone spanned by the past null geodesics converging to our galaxy at the present time. If theuniverse is homogeneous on the observed scales, this information can be used to infer thetime evolution of the cosmological model. But in general, observations based on direct lightcan not tell apart time evolution from spatial variation.The level of CMB isotropy suggests that there is little dependence on the directionof observation nˆ on very large scales. It is then justified to consider spherically symmetricbut inhomogeneous metrics. If the pressure component can be neglected, the result is theLemaˆıtre-Tolman-Bondi metricds2LTB= −dt2 + A′2(r, t)1− k(r) dr2 +A2(r, t) dΩ2 , (1.18)which contains the FRW metric (1.5) as the particular case of no radial dependence A(r, t) =a(t)A0(r), k(r) = k. The above metric is able to explain the supernovae observations with-out a Cosmological Constant if the matter content represents an underdense region of sizecomparable to the Hubble radius (∼Gpc) and our galaxy is located very close to the centerof symmetry r = 0. The apparent acceleration is a spatial effect in these models: the lowermatter density around our galaxy decelerate the expansion less at low redshift, while thehigher matter density far away increases the deceleration at higher redshift.Living at the center of an large, underdense region seems unnatural for two reasons. Thefirst one is the choice of a special location (which is contrary to the Copernican Principle), andthe second one is that so large and spherically symmetric voids are very unlikely to form outof the standard initial conditions that inflationary models typically predicts (in terms of thesize and the amplitude of the initial perturbation). However, the LTB metric is compatiblewith the Big Bang theory (i.e. very homogeneous universe) if the local expansion rate H0(r)is chosen to make the age of the universe space independent and the matter componentsare adiabatically mixed.9 In that case, the LTB metric describes the growth of an initialperturbation with arbitrarily small amplitude at early times.On the other hand, any judgment about our underlying cosmological model should relyon observations rather than aesthetic criteria. The modifications on the matter distributionnecessary to fit the supernovae data also alter the predictions for several other cosmologicalobservables, which can then be used to test the model. Indeed, combinations of differentdata sets can be used to rule out inhomogeneous cosmological models without dark energyin four different ways:1. Tension between local expansion rate and the CMB.2. Isotropy of the CMB for off-center observers, as constrained from the kinematic Sunyaev-Zel’dovich effect. This probe is very powerful, and can be used to rule out models withinhomogeneous Big-Bang time [36].3. Tension between SNe and BAO due to the effects of the inhomogeneous expansion.4. Tension between SNe and galaxy ages [37].9Once the symmetry is reduced, it is possible to extend the inhomogeneity to other degrees of freedom,e.g. baryonic/dark matter ratio or space dependent age of the universe.13Chapter 1: The Standard Cosmological ModelChapter 3 deals with the details of this type of models and their comparison to obser-vations. The results related to the third method are presented together with the referencesto the work where methods one and two are described.Effects from Standard InhomogeneityAlthough they are not considered in the present thesis, there are two effects related to theinhomogeneity of the universe that might be relevant for the dark energy problem. The firsthas to do with the interpretation of observations assuming a homogeneous cosmology. Effectsfrom small scale inhomogeneities, e.g. the lensing of supernovae light by cosmic structures,might affect the recovered value of the cosmological parameters and should in principle betaken into account [27, 38]. The second effect has to do with the non-linear nature of GeneralRelativity and the process of averaging to obtain the FRW cosmological description. Sincethe Einstein tensor contains higher powers of the metric, its average is different than theEinstein tensor computed out of the average metric, i.e. 〈Gµν [gµν ]〉 6= Gµν [〈gµν〉]. If the non-linear corrections were strong enough to explain the cosmic acceleration, they would furtherprovide a solution to the coincidence problem by linking the acceleration to the epoch in whichnon-linear structures form on small scales. However, the issue of whether these effects canactually modify the background dynamics significantly is still an open question (cf. [39, 40]and references therein for different points of view).1.3.2 Dynamical Dark EnergyIf the universe is homogeneous and general relativity holds, the second Friedmann equation(1.7) shows that cosmic acceleration only requires that the total energy-momentum satisfiesp < −ρ/3. It is therefore possible that a new kind of cosmological fluid causes the accelerationas long as the strong energy condition is not satisfied [41, 42]. For example, a canonical scalarfieldSφ = −∫d4x(12gµνφ,µφ,ν + V (φ)), (1.19)becomes equivalent to a constant term if the field derivatives (first term above) are negligiblewith respect to the potential.In the context of cosmology, such a situation is known asslow roll. Scalar fields have been used copiously to build models of inflation and late timeacceleration. The theory described by the above action is usually known as quintessence andis described by the Klein-Gordon equation (with an arbitrary potential)2φ− V ′(φ) = 0 FRW−−−→ φ¨+ 3Hφ˙+ V ′(φ) = 0 . (1.20)Despite the arbitrariness in the choice of V , quintessence models have been fairly wellcharacterized. The energy momentum obtained from the above action ρ = X+V , p = X−Vrestricts the equation of state to w ≥ −1, with the equality corresponding to the case of acompletely homogeneous and non-evolving field (e.g. at the minimum of a potential). Onecan further distinguish between models thawing out, in which the field is initially frozen by theHubble friction, and starts evolving towards higher values of w when V ′ ∼ Hφ˙, and modelsfreezing in, when the equation of state is evolving towards w ∼ −1 [43]. For certain choices ofthe potential, scalar fields are able of tracking the dominant energy component, contributinga fraction of early dark energy during most of the cosmic history [9, 44–47]. Tracking modelshave been used to alleviate the coincidence problem. Their presence throughout the whole141.3. Exotic Tastes and Alternative Recipescosmic history increases the Hubble rate at all times, indirectly reducing the clustering rate ofordinary matter. Other than early dark energy, quintessence fields affect very little the growthof cosmological perturbations and are very difficult to distinguish with large scale structureobservations. The speed of sound for field perturbations is equal to the speed of light and itsmass is of the order of the Hubble constant. Therefore these fields cluster negligibly excepton scales comparable to the Hubble radius H(t)−1, which produces a small modification onthe large scale CMB (see Section 2.3). In addition, the potential can be chosen to satisfy anyexpansion history (with w ≥ −1), making the general case very unpredictive in the absenceof further requirements on V .More general theories than (1.19) can be considered, as can be described by the gener-alized k-essence actionSK =∫d4xL(X,φ) , (1.21)where X ≡ −12gµνφ,µφ,ν is the standard (or canonical) kinetic term.10 K-essence fields canhave considerably different features than their quintessence cousins (e.g. negative equation ofstate, variable speed of sound). Many theories derived from the above action are pathologicalor unstable in some sense (i.e. introduce ghost degrees of freedom or superluminal signalpropagation).Chapter 5 explores a generalized k-essence scalar field derived from a disformal relation(1.27) out of which a set of known dark energy models can be derived in some limit.Coupled Dark EnergyIt is also possible to consider theories in which the scalar field modifies the gravitational or thematter sector in the action. This will generally produce a theory of modified gravity (see nextSection), unless the coupling is restricted to just part of the matter Lagrangian, explicitlyviolating the weak equivalence principle. That case represents an interaction between thescalar field that provides dark energy and a particular speciesS =∫d4x(√−gR16piG+ Lφ(X,φ) + Lc(gµν , φ, ψc) + Lu(gµν , ψu)), (1.22)where ψc/u denote the coupled/uncoupled matter. An interesting case is given by a scalarfield coupled only to dark matter [48, 49] or neutrinos [50]. Since these species can notbe used for gravitational experiments, such models can feature an additional, scalar fieldmediated force with cosmological implications and evade the precision tests of gravity in theSolar System. On the other hand, connecting the dynamics of dark energy to other cosmicconstituents offers an interesting possibility to try to alleviate the coincidence problem.The paradigmatic example is the scalar field conformally coupled to neutrinos, for whichthe field only enters the matter Lagrangian through a multiplicative factor of the metrictensor Lc(A(φ)gµν , ψc) [50, 51]. This coupling modifies the scalar field and neutrino energyconservation equations with a term A(φ)(ρ − 3p) that vanishes as long as the neutrinos arerelativistic (p = ρ/3). The coupling is then used to trigger a slow roll phase in the scalarfield when the neutrinos become non-relativistic. Since the neutrino mass has a similar order10Even more general theories free of pathologies can be considered (see equation 1.28). However, theygenerally introduce modifications in the gravitational sector and will hence be described in the next Section.Appendix C summarizes the most important equations for the theory described by the action (1.21).15Chapter 1: The Standard Cosmological Modelof magnitude to the energy scale of the observed Cosmological Constant, these mass varyingneutrinos dark energy models provide a solution to the coincidence problem by relating bothscales. Unfortunately, the extra interaction introduced has considerable impact on largescale structure formation and addressing the viability of the model requires considering thenon-linear regime [52, 53].Chapter 6 examines the implications of generalizing the conformal coupling. As a poten-tial application, the cosmological implications of a purely disformal coupling to dark matterare described in Sections 6.3.1, 6.3.1 and 6.4.2.1.3.3 Modified GravityFinally, the dark universe paradigm calls for a revision of Einstein theory. The great empiricalsuccess of GR has been to explain gravitational phenomena on the Solar System and compactobjects like neutron stars and pulsars. The mysterious elements of the standard cosmologicalmodel are introduced to alter the dynamics on scales for which there is little or no independentprobe of the workings of gravity. It remains possible that cosmic acceleration is due to amodification of gravitational dynamics manifest on cosmological scales.It might seem at first that the action for General Relativity (1.2) is rather arbitrary,considering the enormous freedom available in the construction of geometric scalars. How-ever, there are essential limitations in the choice of the theory of gravity. Ostrogradski’sTheorem [54] states that there exists a linear instability in any non-degenerate theory whosefundamental dynamical variable appears in the action with higher than 2nd order in timederivatives. By constructing the Hamiltonian for this type of theories, the theorem showsthat it is not bounded from below and therefore it accepts configurations with arbitrarilylarge negative energy [55, 56]. It is possible to bypass Ostrogradski’s result by consideringdegenerate theories, i.e. those in which the highest derivative term can not be written as afunction of canonical variables. The dynamics of this type of theories is described by secondorder equations of motion, even while the action contains higher derivative terms.The gravitational action is severely constrained by Ostrogradski’s Theorem becausethe curvature tensors do contain second derivatives of the metric. Lovelock’s Theorem [57]states that the only action of the form∫d4xL[gµν ] (i.e. based on a local scalar Lagrangiansolely depending on the metric tensor and its derivatives) which gives rise to second orderequations of motion is of the Einstein-Hilbert form with a Cosmological Constant (1.13).The introduction of other geometric invariants not only changes the character of the theorysubstantially, but may render it unstable. The alternative theories can be classified as waysout of Lovelock’s Theorem [7]:T1: Add additional degrees of freedom other than the metric.T2: Work in a space with dimensionality different from four.T3: Accept higher than second derivatives of the metric in the field equations.T4: Give up either rank (2,0) tensor field equations, symmetry of the field equations underexchange of indices, divergence-free field equations or locality.1111Certain non-local theories with an infinite number of derivatives provide another way around Ostrograd-ski’s Theorem. These and other T4 theories will not be discussed further here.161.3. Exotic Tastes and Alternative RecipesSome theories may fall into different categories within the above classification. High derivativetheories (T3) that are acceptable in light of Ostrogradski’s Theorem can usually be expressedas a second order theory with extra degree(s) of freedom (T1). This includes the f(R) theoriesin which the Einstein-Hilbert action includes an arbitrary function of the Ricci scalar. Thesetheories can be shown to be safe [55] through its equivalence to a scalar tensor theory witha Lagrangian density√−g (φR+ U(φ)) (plus a kinetic term for φ), where U(φ) depends onthe functional form of the original f(R). Certain higher dimensional theories (T2) can alsobe expressed in terms of additional degrees of freedom (T1) at an effective level.Scalar-Tensor Bimetric TheoriesThe addition of scalar degrees of freedom has been extensively exploited to build modificationsof gravity. Metric theories already include a scalar d.o.f. in addition to the propagatingspin-2 tensor. This scalar mode is non-dynamical if the action has the EH form, but somealternative theories make it active, which is the reason why f(R) can be expressed as a healthyscalar-tensor theory. In addition, scalars do not introduce preferred directions and mediateuniversally attractive interactions.12 The paradigmatic example of scalar modifications arethe old school scalar-tensor theories∫d4x√−g(h(φ)R16piG+X − V (φ) + LM (gµν , ψ)), (1.23)where the field has been redefined as to yield a canonical kinetic term X = −12∇µφ∇µφ. Bya field dependent local rescaling of the metric or conformal transformationgµν → h−1(φ)gµν , (1.24)it is possible to remove the field dependent coefficient from the Ricci scalar. The resultingtheory reads ∫d4x√−g(R16piG+ X˜ − V (φ˜) + LM (h−1(φ˜)gµν , ψ)), (1.25)where φ → φ˜ has been redefined in order to keep the canonical form of the scalar field La-grangian [7]. Theories related by metric transformations of the form (1.24) are said to beexpressed in different conformal frames. The examples above show two particularly interest-ing choices, the Jordan frame (Jf) (1.23) in which matter couples to a single rank-2 tensorand the Einstein frame (Ef) (1.25) in which the gravitational sector has the Einstein-Hilbertform but matter couples to an effective metric (1.24) which explicitly depends on the scalarfield.The conformal equivalence between the theories (1.23) and (1.25) motivates consideringmore general bi-metric theories for which the effective metric in the gravitational and mattersectors is allowed to be different.13 An interesting case is when the gravitational sector is ofthe Einstein-Hilbert formSEH−BM =∫d4x(√−gGR [gGµν]−√−gMLm (gMµν , ψ)) , (1.26)12Theories involving vectors have been considered as candidates for cosmic acceleration (e.g. [58, 59]) andmodified gravity that attempt to avoid Dark Matter (TeVeS) [60] or spontaneously break Lorentz invariance(Einstein-Aether) [61], among other applications.13Although the term bi-metric often refers to theories with two fundamental rank 2 tensors (tensor-tensortheories) [7], it will be used here in the sense described in the text (e.g. Eq. 1.26 and its extensions).17Chapter 1: The Standard Cosmological Modelbut the relation between the two metrics is arbitrary gGµν 6= gMµν . An advantage of this choiceis the ease to identify in which limits GR can be recovered, which greatly simplifies theconstruction of phenomenologically viable scenarios. Bekenstein [62] argued that the mostgeneral relation between the two metrics that respects general covariance and involves asingle scalar degree of freedom has the formgMµν = A(φ,X)gGµν +B(φ,X)φ,µφ,ν . (1.27)This expression is known as a disformal relation, and it generalizes the well known conformaltransformation (1.24) by the addition of a term depending on the gradients of the scalar field.Unlike in the conformal case, the above relation establishes different causal structures for thetwo metrics whenever B 6= 0. The class of dark-energy models obtained from a canonicalscalar field constructed using a disformal metric are studied in Chapter 5. In Chapter 6 theanalysis is generalized to bi-metric theories (1.26) in which matter is directly coupled to ametric of the form (1.27).Conformally related scalar-tensor bi-metric theories can be classified into the dark en-ergy type and self accelerating (cf. [63]). Viable cosmological theories have to provide accel-eration in the Jordan frame metric, where cosmological observations are implicitly performed.Theories of the dark energy type are accelerated also in the Einstein frame. Cosmology inthe Ef is described by Friedmann equations, and therefore acceleration implies the existenceof an energy component with p < ρ/3 in this frame. Therefore, those can be thought ofas the generalization of the Dark Energy models introduced in Section 1.3.2. On the otherhand, self accelerating theories are those in which the acceleration occurs only in the Jordanframe. These theories imply that there is no effective cosmological constant in Einstein frameand the acceleration is entirely due to the conformal factor. Although this classification isconceptually useful, it does not contemplate the possibility of relating theories by means ofdisformal transformations, as will be explored in Section 6.2.One may consider the generalization of Lovelock’s Theorem in the case of a scalar tensortheory, i.e. what is the most general local action that produces second order equations ofmotion. Following Bekenstein, the matter metric can be allowed to have a conformal as well asa disformal contribution (1.27). For the gravitational sector, it is the Horndeski Lagrangian.The most general gravitational sector was first derived by Horndeski [73] and has receivedconsiderable attention in recent years [30, 64, 74–78]. It is given by the Horndeski LagrangianLH =5∑i=2Li . (1.28)The different pieces can be written (up to total derivative terms that do not contribute tothe equations of motion) as [76]L2 = G2(X,φ) , (1.29)L3 = −G3(X,φ)2φ , (1.30)L4 = G4(X,φ)R+G4,X[(2φ)2 − φ;µνφ;µν], (1.31)L5 = G5(X,φ)Gµνφ;µν − 16G5,X[(2φ)3 − 3(2φ)φ;µνφ;µν + 2φ ;ν;µ φ ;λ;ν φ ;µ;λ]. (1.32)181.3. Exotic Tastes and Alternative RecipesTheory G2 G3 G4 G5 gMµνGeneral Relativity Λ 0M2p2 0 gµνQuintessence (1.19) X + V (φ) 0M2p2 0 gµνGeneral k-essence‡ (1.21) K(X,φ) 0 M2p2 0 gµνOld school Scalar-Tensor- Jordan Frame (1.23) X˜ + V (φ˜) 0 h(φ)M2p2 0 gµν- Einstein Frame (1.25) X + V (φ) 0M2p2 0 h−1(φ)gµνCovariant Galileon§ [64] c1φ− c2X c3M3XM2p2 − c4M6X 3c5M9X2 A(φ)gµνKinetic GravityBraiding [65, 66] K(X,φ) G(X,φ)M2p2 0 gµνPurely KineticGravity [67] X 0M2p2 −λ φM2p gµνDBI Probe Brane† [68] λγ−1 −M35 γ−1M24 −βM25m2γ2 gµνDisformally CoupledScalar [1, 69] X + V (φ) 0 0 0 Agµν +Bφ,µφ,ν‡ See Table 5.1 for an assortment of k-essence models constructed using disformal relations.§ The usual Galileon [70] is recovered in the absence of curvature. The analysis of these theories oftenpostulates a conformal coupling betwen the matter and the field (a.k.a. conformal Galileon).† References [71, 72] provide generalizations constructed within the probe brane scheme.Table 1.1: Horndenski projection of Modified Gravity and Dark Energy theories. HereM2p = (8piG)−1, X = −12φ,µφ,µ and γ = 1√1−2X is a brane Lorentz factor.On top of a generalized k-essence (1.29), the remaining pieces (1.30-1.32) fix the allowedhigher derivative dependences, which rely on the anti-symmetric structure of φ;µν termsto cancel higher derivatives in the equations of motion. Note that a constant term in G4produces Einstein gravity. One can then consider the scalar-tensor theorySH−BM =∫d4x(√−gLH +∑i√−gM,iLM (gM,iµν , ψ)), (1.33)as the most general case, where different matter metrics gM,iµν have been allowed for thedifferent components (e.g. different coupling to dark matter and baryons is allowed) andno particular Frame has been prescribed. Table 1.1 contains several examples that can bedescribed within the Horndenski Lagrangian, including some particular cases studied in thefollowing Chapters. It has been written based on the work of De Felice, Kobayashi andTsujikawa [76], who considered cosmological perturbations in the quasi-static approximation19Chapter 1: The Standard Cosmological Modelfor a number of these theories. Table 1.1 includes some further examples and does notimplicitly restrict the theory to the Jordan frame.The higher derivative terms have been found to arise in the effective description of severalhigher dimensional theories.14 G3, G4 and G5 are the covariant generalizations of the third,fourth and fifth Galileon terms [70]. The third order Galileon term appears in the decouplinglimit of the five-dimensional Dvali-Gabadadze-Porrati model (DGP) [79], and the other twowere proposed as an extension. This theories posses Galilean invariance φ→ φ+ c+ bµxµ inflat space-time, a symmetry that grants beneficial properties such as the non-renormalizationof the coupling by quantum effects [80]. This type of theories were also obtained in a unifiedway from extra-dimensional models in which matter is confined to a 3+1 dimensional braneembedded in a larger bulk space [68, 71, 72, 81–83].15 In the case of a single extra dimension[68] the terms obtained out of the brane tension, and extrinsic, Ricci and Gauss-Bonnetcurvature fall into G2, G3, G4 and G5 when projected into the Horndenski’s Lagrangian(1.28). In models with large extra dimensions, the dynamics should interpolate between thesmall scales in which the physics is effectively four dimensional and the large scales in whichthe existence of a bulk space becomes manifest. The Vainshtein effect described below allowsthis interpolation by hiding the scalar mediated force within a certain radius from pointsources, i.e. it provides a screening mechanism for the modifications to General Relativityon small scales.Screening mechanisms are central in the construction of gravitational theories that havesignificant cosmological implications while remain consistent with precision gravity tests.They provide a way to keep the scalar degrees of freedom inactive in environments which aresimilar to the Solar System. These mechanisms are usually analyzed in a frame where there isa conformal coupling between the field and matter, in which the scalar force is proportionalto the spatial gradient of the field. Screening mechanisms can be classified as kinetic orpotential:• Kinetic screening mechanisms rely on the kinetic terms of the field to hide the scalarforce. In the Vainshtein mechanism [84] higher derivative self interactions (κ1, κ3, κ8 6=0) do not allow large gradients around point sources. The spatial derivatives of thefield are suppressed within a certain Vainshtein radius from the source. In the simplestcase (G2 = X,G3 = m−2, G4 = M2p /2, G5 = 0) it readsrV ≈(2GMm2)1/3≈ 10Mpc[(H0m)2 M1015M]1/3, (1.34)where M is the point source mass and M is the mass of the Sun.A form of kinetic screening can also be achieved if the scalar field coupling to matterinvolves a disformal part, in which case the higher derivative terms of the field equa-tion acquire an energy density dependence [(A− 2BX)gµν −BTµν ]φ;µν , and the fielddecouples in the limit ρ→∞. The consequences of the disformal screening mechanismare explored in Section 6.5.14As these theories should accept an effective, four dimensional representation in certain circumstances, thegeometric origin of these terms is plausibly what renders them viable in light of Ostrogradski’s Theorem.15The action for this type of theories is constructed using geometric scalars computed out of the inducedmetric g¯µν = gµν + pi,µpi,ν , which has the same form as the disformal relation (1.27) with A,B = 1. The fieldpi represents the position of the brane in this scheme. See Section 6.2 for a discussion.201.3. Exotic Tastes and Alternative Recipes• Potential screening mechanisms rely on the interplay between the coupling and thepotential to hide the scalar force. When mediated by a scalar particle, the potentialassociated to the scalar force acquires the Yukawa formΦs(r) =αre−mφr , (1.35)where α is the intensity of the coupling and mφ is the mass of the scalar, that determinesthe range of the interaction. Its value is given by the second derivative with respect tothe field of the effective potential Veff (φ) = V (φ) +A(φ)ρ where a conformal couplingto matter has been assumed.The Chameleon mechanism [85, 86] requires a decreasing potential and an increasingcoupling (e.g. V = M5/φ and A ∝ eβφ/Λ). It relies on the environment dependent massof the scalar to grow large in dense environments, so that massive objects contributeto the scalar force only through a thin shell close to the surface. The Symmetronmechanism [87, 88] postulates a degenerate potential with two minima and a couplingthat vanishes if φ = 0 (e.g. V ∝ (φ2 − φ20)2 and A = 1 + φ2/Λ2). If the energy densityis low, the field sits at ±φ0 and the coupling is active. If the density is high, thesymmetry is restored in the effective potential and φ ∼ 0. Therefore, screened objectsonly contribute to the force through the matter close to their surface.The cosmology and local gravitational effects of alternative theories of gravity featuringscreening mechanisms has been extensively studied (e.g. [89–95]). Theories endowed withmechanisms of the potential type are bound to field ranges m−1φ . 1Mpc in order to keep theSun screened from the scalar force [63], and therefore modifications to structure formationoccur only on non-linear scales.The results on Modified Gravity and Dark Energy models presented in Chapters 5 and6 concentrate on theories which involve disformal relations (1.27) in their construction. Adisformally constructed canonical scalar field Lagrangian encompasses many proposed darkenergy models for different choices of the free functions, as it is explored in Section 5.1.1. Theexistence of two causal structures in this type of theories makes it easy to construct models forcosmic acceleration. In a FRW background, the two metrics are related by g¯00 = −1 +Bφ˙2,so the field enters into slow roll whenever Bφ˙2 approaches unity. This has been explored indetail for disformal quintessence (Section 5.2) and canonical quintessence disformally coupledto matter (Sections 6.3.1 and 6.4.2).Disformal relations also simplify the analysis of certain higher derivative theories, in-cluding those of the DBI form, by providing an Einstein frame description in which thegravitational sector is standard but matter appears disformally coupled to the field. Thistransformations are explored in Section 6.2, where different frames are presented. Disformallycoupled theories also provide a screening mechanism that hides the modifications of gravityin high density environments, as it is explored in Section 6.5. Based on the equivalence tosome higher derivative theories, this is likely to be the Vainshtein mechanism in disguise,which appears different in the Einstein frame description. Disformal relations offer hence anew way to construct and analyze scalar field extension of the Einstein-Hilbert action thatare relevant for the dark energy paradigm.21Chapter 1: The Standard Cosmological Model22Chapter 2Observational Probes of CosmologyReality is not always probable, or likely. Butif you’re writing a story, you have to make itas plausible as you can, because if not, thereader’s imagination will reject it.Jorge Luis Borges1The first compelling evidence in favor of the cosmic acceleration came from the studyof high redshift type Ia supernovae (SNe), standard candles of similar luminosity which ap-peared fainter than they would in a spatially flat, matter dominated universe [11, 12]. In thefollowing years, further information obtained from other probes started confirming the stan-dard picture described in the previous Chapter. The Wilkinson Microwave Anisotropy Probe(WMAP) [16, 18, 19] together with several ground based experiments [96–98] measured theanisotropies in the Cosmic Microwave Background (CMB) with great precision. Their dataprovides accurate information of the initial perturbations that seeded the cosmic structuresthat formed through gravitational collapse, and have proven very useful to constraint cos-mological parameters within the standard cosmological model and its extensions. Galaxysurveys such as SDSS [24, 99–102] and WiggleZ [103, 104] have produced very detailed mapsof the Large Scale Structure (LSS) of the universe, including the signature of the BaryonAcoustic Oscillation (BAO) scale in the matter correlation function. This finding provides astandard ruler, whose length can be measured at different cosmological epochs and used totrack the expansion of the universe.The precision and consistency of these and other probes allows to place the standardcosmological model under increasingly stringent tests, which would hopefully allow to estab-lish the validity of the hypotheses as independently as possible from one another. This greateffort aimed at observationally determining the physical laws behind cosmic acceleration willcontinue over the following decades. The Planck satellite will release even more detailedmaps of the cosmic microwave background [105] and the Dark Energy Survey (DES) will1Quoted in Worldwide Laws of Life: 200 Eternal Spiritual Principles (1998) by John Templeton.23Chapter 2: Observational Probes of Cosmologyprobe the distribution of cosmic LSS during the following years [106]. This internationaleffort will continue with the Euclid satellite mission [107, 108], which will map the wholeextragalactic sky as deep as to reach redshifts of 2, before the accelerated expansion be-gan. These galaxy surveys will determine the BAO scale with exquisite precision at differentcosmological epochs. In addition, their multi probe design makes them capable to extractfurther cosmological data such as SNe, weak gravitational lensing, clusters of galaxies, thatwill place even stronger constraints on departures from large scale homogeneity and GeneralRelativity.The most relevant cosmological probes for the present Thesis are introduced in thisChapter. Section 2.1 gives a general classification of the observables and the type of knowledgethat can be gained from them. Sections 2.2 to 2.6 give an overview of the effects and findingsthat can be used to constraint cosmological models, emphasizing the underlying assumptionsintroduced in their analysis and how they can be made model independent. The discussionis necessarily brief. More details can be found in the recent review by Weinberg et al. [10]and the references included in each Section.2.1 Geometry and DynamicsCosmological probes can be classified into geometric and dynamical (see Table 2.1). Geo-metric probes are based on background quantities and measure the zero order expansion, i.e.the predicted quantities to be compared with observations depend only upon the elementsof the unperturbed metric, at least to a good enough approximation.2 A classical example isthe angular diameter distanceDA(z) =l⊥θ∣∣∣∣O(z)=11 + z∫ z0dz′H(z′), (2.1)where the first equality is observational, i.e. the ratio of the (assumed) known physical lengthin the transverse direction l⊥ of an object at redshift z and the measured angle it subtends inthe sky θ (assumed small), and the second equality is the prediction for a flat FRW cosmology.Another useful geometric probe is the luminosity distance:DL(z) =√L4pi F∣∣∣∣∣O(z)= (1 + z)∫ z0dz′H(z′). (2.2)Here the first equality is the observational value, related to the ratio between the (assumedknown) luminosity L and the observed flux F of an object at redshift z. The second equalitygives the value within a flat FRW cosmology. The angular diameter distance enters theanalysis of the BAO scale, as measured in the CMB and galaxy surveys (Sections 2.3 and2.4.1), and the luminosity distance is compared with supernovae data (Section 2.5).Geometric probes can also be applied to inhomogeneous cosmologies rather straight-forwardly. The equivalent of the angular diameter and luminosity distances for this typeof models is presented in equations (3.18,3.19), and the necessary considerations to use theBAO scale are made in Section 3.2 in Chapter 3. Current data allow to rule out LTB modelswith homogeneous Big-Bang and Λ = 0 using geometric probes only (Section 3.3).2That means, geometric probes depend only on the scale factor and the Hubble rate in FRW cosmologies,perhaps through integrals or derivatives. In LTB models the cosmological probes may depend on the two scalefactors (or expansion rates), which are different in the radial and angular direction due to the inhomogeneity.242.1. Geometry and DynamicsType Probe Redshift Range SectionGeometric Type Ia SNe 0.01− 1.5 2.5Local expansion (H0) . 0.1 2.5CMB priors z∗ 2.3.1, Ref. [19]BAO scale from LSS 0.1− 0.8(3) 2.4.1Primordial Nucleosynthesis ∼ 1010 2.2Dynamical CMB primary ∼ z∗ 2.3CMB secondary 0→ z∗ 2.3CMB kSZ 0− 1 2.4.2, Ref. [109]LSS power spectrum 0.1− 1(3) 2.4LSS Lensing 0− 1 2.4.2LSS Clusters 0− 1 2.4.2Dynamical Local Gravity Tests 0 2.6, Ref. [110]Table 2.1: Summary of cosmological probes. The different observables have been classifiedinto geometric and dynamical. z∗ is the recombination redshift, numbers in parenthesis referto expected limits for future surveys. See reference [10] for details.Dynamical probes are those which crucially depend on the departures from the back-ground metric on cosmological scales. A common coordinate choice to describe metric per-turbations is the conformal-Newtonian gaugeds2CN= a2(τ)[−(1 + 2Φ)dτ2 + (1− 2Ψ)d~x2] , (2.3)which has been written in terms of the conformal time τ =∫a(t)dt. The gravitationalpotentials Φ,Ψ are sourced by the perturbed energy momentum tensor δTµν(~x, τ), which isusually expressed in terms of dimensionless variables such as the density contrastδ(~x, τ) ≡ ρ(~x, τ)〈ρ〉(τ) − 1 . (2.4)Cosmological observables sensitive to departures from homogeneity have the potential toprobe the gravitational collapse and the formation of cosmic structures. This information isessential for non-standard cosmological models, as it potentially allows to distinguish mod-ifications from gravity from non-interacting dark energy models with the same backgroundhistory. Of particular interest are those effects that can be computed within linear pertur-bation theory. Non-linear effects are more difficult to connect with the models on trial, andthe computational tools that allow their exploration have been considerably less developedfor non-standard cosmologies than for ΛCDM.One of the challenges of cosmological model testing is to determine to what extenteach cosmological probe is affected by dynamical effects. Some probes, such as the BAOscale in LSS and CMB distance priors are intrinsically dynamical. But since the physicalquantities involved in the comparison can be reliably computed using just the background25Chapter 2: Observational Probes of Cosmologyinformation, they are considered geometric observables. No probe is purely geometric, asthere will always be dyanamical effects if sufficient precision is achieved (e.g. through peculiarmotions). However it is a useful classification to bear in mind.2.2 Primordial NucleosynthesisThe first event that yields direct information about the early universe involves the formationof the first composite nuclei, a period known as primordial nucleosynthesis or Big-BangNucleosynthesis (BBN) [111]. At the onset of BBN, weak interactions keep protons andneutrons in equilibrium, slightly favoring protons due to their lower mass mn −mp = 1.293MeV. The formation of composite nuclei is dominated by two body collisions at the BBNtemperature scale, and hence deuterium 2H is the first step in the chain of nuclear reactions.The binding energy of deuterium ED = 2.224 MeV is too low for it to be stable untilthe universe cools down to T . 0.1MeV. This is due to the high photon number densitynb/nγ = η10 · 10−10, which allows a significant population in the high energy tail of thedistribution. When the temperature is low enough, the available neutrons combine withprotons to form deuterium, and most of it is quickly processed into heavier elements, with amajor amount ending in the form of 4He.The predicted abundances can be compared with observations in low metallicity sys-tems, which have not been significantly affected by stellar nucleosynthesis. The measuredabundances can be used to determine certain cosmological parameters and allow the de-termination of the baryon to photon ratio independently of the CMB. Deuterium and 4Hemeasurements agree well with expectations, but 7Li observations lie a factor 3-4 below theBBN+WMAP prediction, a mismatch known as the “cosmic lithium problem” [20, 112, 113].BBN is also sensitive to the expansion rate at early times and can be therefore be used toconstraint the number of relativistic degrees of freedom Neff [20] as well as non-standard sce-narios, such as a possible time variation of the Newton’s constant [114], the amount of earlydark energy [115], heavy particle decays or the variation of fundamental couplings [116, 117].2.3 Cosmic Microwave Background RadiationSimilarly to the formation of the first nuclei, no neutral atoms can form in the early universeuntil the temperature is sufficiently low. The electromagnetic interacting species are tightlycoupled and perturbations propagate in the baryon-photon plasma as sound waves . Neutralhydrogen starts forming when the temperature drops below T . 1 eV and the universe startsbecoming transparent as light is able to travel without encountering charged particles. Thisepoch is called recombination and occurred at z ≈ 1090. The photons emitted then haveredshifted and diluted since recombination and can be seen today in the form of a CosmicMicrowave Background (CMB) with an average temperature ≈ 2.73 K. Several experimentshave measured anisotropies of the CMB radiation, whose study is very useful to constraintcosmological models.The line of sight integration method provides a very direct way to study the physicaleffects behind the CMB anisotropies [118]. For a spatially flat universe and adopting theconformal Newtonian gauge (2.3), the linear temperature anisotropy Θ ≡ δT/T due to theFourier mode k and projected on the multipole l today can be written asΘ(s)l (k, τ0) =∫ τ00dτS(s)T (k, τ)jl(k(τ0 − τ)) , (2.5)262.3. Cosmic Microwave Background Radiationwhere µ ≡ nˆ · ~k/|k|. The integrand is a convolution of the l-th spherical Bessel function jland the scalar sourcesS(s)T (k, τ) ≡ g(Θ0 + Φ +v˙bk+Π4+3Π4k2)+ e−κ(Φ˙ + Ψ˙)+ g˙(vbk+3Π˙4k2)+3g¨Π4k2. (2.6)Here dots denote derivatives wrt conformal time τ , κ(τ) =∫ τ0τ neσTadτ′ is the optical depth(determined from the free electron fraction ne and the Thompson scattering cross section σT )and g ≡ −κ˙e−κ is the visibility function.3 In addition to the gravitational potentials Φ,Ψ,the sources include the temperature monopole, the baryon velocity vb and the polarizationtensor Π, all of them depending on τ and k. Besides simplifying the computation, the line ofsight approach allows the identification of different contributions to the total anisotropy. Theterms proportional to the visibility function (which peaks around the recombination epoch)are known as primary anisotropies, since they are imprinted at z ∼ 1090 when the photonsdecouple:• Primary monopole Θ0. The anisotropy is proportional to the local photon energydensity fluctuation at the last scattering surface.• Sachs-Wolfe effect Φ. Photons are redshifted as they climb up the initial potential wellswhen they are emitted.• Dopler shift induced by baryon velocity vb, v˙b.• Polarization effects Π, Π˙, Π¨. Although these effects are typically small, they are neces-sary for an accurate description of the anisotropies.The term proportional to e−κ is active since last scattering. This gives rise to a secondarycontribution• Integrated Sachs-Wolfe (ISW) effect Φ˙ + Ψ˙. This term represents how the photonstraveling from the last scattering surface are affected by the potential wells they en-counter. If the potential wells were static, the blueshift gained when entering would becompensated by the redshift necessary to escape. But if the potentials are evolving intime, there is a net contribution on the temperature anisotropy.Gravitational lensing of the CMB provides an additional secondary contribution to the tem-perature fluctuation [119, 120]. However, lensing effects require a bending of the photontrajectories due to inhomogeneities, which are already an linear effect. Hence they constitutea higher order correction to (2.6).The angular power spectrum is obtained by the convolution of the temperature anisotropy(2.5) with the initial power spectrum of the metric perturbationC(s)l = (4pi)2∫dkk2PΦ(k)∣∣∣Θ(s)l (k, τ0)∣∣∣2 . (2.7)Here PΦ(k) is defined for the potential in an analogous way to the matter power spectrumpresented below (2.17). Similar expressions exist for the polarization and the tensor sources.3There is also an equation for the polarization spectrum, featuring different sources. However, dark energymodels do not typically affect the CMB polarization, unless there is some specific coupling to photons.27Chapter 2: Observational Probes of CosmologyTo compare with observations, it is useful to project the observed temperature anisotropyinto spherical harmonicsΘobs(nˆ) =∞∑l=0l∑m=−lalmYlm(nˆ) , (2.8)so that the coefficients alm encode the dependence on the direction of observation nˆ. In astatistically isotropic universe the mean vanishes 〈alm〉 = 0. If the distribution is Gaussian,it is fully characterized by the variance〈almal′m′〉 = C(obs)l δll′δmm′ (2.9)The obtained value of C(obs)l is the quantity to be compared against equation (2.7). It ispossible to compute non-Gaussian correlations (e.g. 〈al1m1al2m2al3m3〉), an information thatcan be related to the primordial non-Gaussianity induced by inflation and has been boundedby CMB experiments [19, 121].One of the greatest advantages of the CMB anisotropies is that they can be computedfor any cosmological model, provided that there is a reliable linear perturbation theory. Manydark energy models only modify the late time physics substantially, leading to an even simplerimplementation. In this cases, the most significant departures come from the low-l end ofthe spectrum, for which there are less projections m to obtain the angular power spectrum(2.9). Therefore, the CMB is mostly useful to detect significant departures at late times (e.g.the ISW effect), but is limited when it comes to precision measurements of the low redshiftdynamics behind cosmic acceleration. Pre-recombination effects such as the presence of earlydark energy are still tightly bound by the CMB measurements [122].2.3.1 CMB Distance PriorsUnderstanding the CMB physics requires considering the dynamics of the perturbationsall the way since the early universe. However, the reason behind the acoustic oscillationsthat gave rise to the peaks is simple and can be understood in geometric terms. Initialperturbations in the baryon-photon fluid propagate a finite distance until the recombinationepoch in which photons decouple. The total length traveled by a sound wave depends on thespeed of sound cs =1√3(1 + 3ρb4ργ)−1/2. It is known as the sound horizonrs(z∗) =∫ (1+z∗)−10cs(a)a2H(a)da . (2.10)The recombination redshift z∗ can be calculated using fitting formulae [123] or the full back-ground history, which is recommendable in non-standard cases (e.g. in the presence of earlydark energy). The sound horizon is a coordinate distance. To obtain the physical, measurabledistance, it is necessary to multiply by the scale factor at the time of interest a(t) = 1/(1+z∗).Comparing the physical sound horizon at recombination with the angular diameter distanceallows one to define the CMB acoustic scalelA = (1 + z∗)piDA(z∗)rs(z∗), (2.11)282.4. Large Scale Structurewhich is inversely proportional to the angle subtended by the sound horizon observed todayθ∗. The acoustic scale is a good observable, in the sense that it can be easily defined in anystatistically isotropic or spherically symmetric cosmology.In certain situations it is useful to compare dark energy models against a reduced set ofparameters instead of the full angular power spectrum. This allows to include informationfrom the CMB in the analysis without computing cosmological perturbations. One possibilityis then to use the WMAP distance priors lA, z∗, R [19] where in addition to the alreadydescribed quantities the shift parameter is defined asR =√ΩmH20 (1 + z∗)DA(z∗) . (2.12)The use of these parameters is valid for FRW models containing matter, radiation, curva-ture, dark energy, three neutrinos with minimal mass, and standard initial conditions. Itsuse is convenient when the model does not affect the clustering of matter significantly (e.g.quintessence) or the equations for the perturbations are not well known (e.g. phenomenologi-cal modifications of the Friedmann equations). On the other hand, it misses important effectsthat may occur in modified gravity and the values of z∗, R can not be used to constraint inho-mogeneous models.4 The shift parameter R is also difficult to interpret in models where theevolution of matter is non-standard (e.g. coupled quintessence). A more model-independentapproach to geometric CMB analysis that is valid for inhomogeneous cosmologies can befound in reference [124].2.4 Large Scale StructureThe gravitational pull exerted by inhomogeneities is responsible for the growth of cosmic largescale structures. An overdense region attracts the matter around it, which in turn enhancesthe gravitational attraction on its surroundings. By the action of gravitational instability, thematter is always transfered from the underdense to the overdense regions, and the initiallyquasi-homogeneous universe becomes populated by collapsed non-linear objects and emptyregions (voids). The Hubble expansion acts as a friction term and inhibits the growth ofperturbation modes which are larger than the Hubble radius at any given time. Therefore,smaller perturbations start growing earlier.The clustering on cosmological scales is dominated by the dark matter component withzero pressure. In the hydrodynamic limit, it follows the Euler and continuity equation [125]δ˙m(~x, τ) +∇ · [(1 + δm(~x, τ))~u(~x, τ)] = 0 , (2.13)~˙u(~x, τ) +H(τ)~u(~x, τ) + ~u(~x, τ) · ∇~u(~x, τ) +∇ΦN (~x, τ) = 0 , (2.14)where u(~x, τ) is the velocity, H(τ) ≡ a−1da/dτ is the conformal Hubble rate and δm isdefined as in (2.4) . On sub-Hubble scales the Newtonian potential ΦN is given by thePoisson equation∇2ΦN (~x, τ) = 4piGρm(τ)δ(~x, τ) . (2.15)Equations (2.13,2.14) are both non-linear. Their general study requires sophisticated tech-niques such as renormalized perturbation theory, halo model or N-body simulations [126–128].4A method to use simplified CMB constraints in inhomogeneous models is explained in Section 3.3.1.29Chapter 2: Observational Probes of CosmologyFor linear perturbations δ, ~u 1 on sub-Hubble scales and assuming zero vorticity ∇×~u = 0,the above equations can be written in Fourier space as a single, second order equationδ¨m(k) + 2H(τ)δ˙m(k) = 4piGρ(τ) δm(k) . (2.16)The density contrast δm(~k) can be computed for a given cosmological model usingperturbation theory. Similarly to the CMB, it is necessary to define statistical quantitiesin order to compare theoretical predictions and observations. The matter distribution ischaracterized by the power spectrum〈δ(~k) δ(~k′)〉 = (2pi)3Pδ(k) δ(3)(~k − ~k′) , (2.17)where δ(~k) is the Fourier transform of the density contrast (2.4).5 The use of the abovepower spectrum to compare with observations is common. However, projecting the observedredshifts and angular locations of galaxies into physical coordinates ~x (which is necessary toFourier transform) requires the assumption of a fiducial cosmology.A model independent approach based on the use of directly measurable quantities wasproposed recently[129, 130]. Instead of real or Fourier space, the density contrast is computedas a function of the redshift z and the direction of observation nˆδz(nˆ, z) ≡ ρ(nˆ, z)〈ρ〉(z) − 1 =N(nˆ, z)〈N〉(z) +V (nˆ, z)V (z). (2.18)As a consequence of their observable nature, all the terms in the above equation are them-selves gauge invariant. In order to compare with observations, (2.18) can be projected intoan angular power spectrum C(δ)l (z,∆z) for different redshift separations or smearing, and thesame procedure can be applied to the data. This is analogous to CMB analysis, except thatLSS surveys are three dimensional and have the additional redshift dependence. The studyof different angular multipoles and redshift separations allows to disentangle the differenteffects contributing to δz. The disadvantage of this method is the finite amount of statisticsavailable for each possible correlation. In practice, current surveys take best statistical ad-vantage by condensing all the information in a Fourier power spectrum (2.17) for differentredshift slices, although that introduces a spurious, model dependent (θ, nˆ) → ~x conversionin the analysis.Galaxy surveys measure galaxies, not the underlying matter density. A galaxy is avery non-linear object, whose formation and evolution depend on its environment in a verycomplex way. Therefore, galaxies are biased tracers of the underlying matter distribution,i.e. their density contrast is such thatδg ≈ bgδm + bg(2)δ2m + · · · (2.19)where the galaxy bias coefficients can in principle depend on time, scale and galaxy type.Non-linear terms have been included schematically. Although they will not be consideredfurther, realistic bias models should include some non-linear part to distinguish overdenseand under-dense regions. Another observational effect is caused by the peculiar velocitiesof galaxies, which affects the cosmological redshift in a systematic way - galaxies moving5One can also consider higher order statistics, e.g. 〈δδδ〉, which may be sourced by non-linear gravitationalclustering or be related to primordial non-Gaussianity [121].302.4. Large Scale Structuretowards an overdense cosmic structure will appear redshifted if they are in front of it (wrtthe observer) and blueshifted if they are behind. This makes a spherical distribution in realspace (e.g. a cluster of galaxies) look squeezed when observed by a survey in (z, nˆ), an effectknown as Redshift Space Distortions (RSD). Taking linear galaxy bias and RSD into accountallows to relate the galaxy and matter power spectra [131]Pg(k, µ) = (bg + fµ2)2Pδ(k) , (2.20)where µ = nˆ ·~k/|k| is the angle between the Fourier mode and the line of sight direction andf ≡ d log δ(a)d log a≡ Ωm(a)γ(a) , (2.21)is the growth factor. The second equality defines an useful parameterization in terms of thegrowth index γ(a), that encapsulates the effect of the background matter on the clustering.Its value is very close to a constant and equal to 6/11 in ΛCDM universes, while modifiedgravity models predict different values. Equation (2.20) can be used to determine f fromthe angular dependence of the galaxy power spectrum. A compilation of growth factorobservational data and the predictions can be found in reference [132].Non-standard cosmological models can affect considerably the formation of structure.Theories of modified gravity typically incorporate an additional interaction that increases theforce and enhances gravitational instability. In certain cases, the growth factor (2.21) mightacquire an additional scale dependence at the linear level [133]. Models in which dark energycouples to dark matte do induce an additional bias between the dominant matter componentand the baryons. These features are explored in Section 6.4.2 for the case of disformallycoupled scalar fields.The case of inhomogeneous cosmologies has been considerably less explored due to thetechnical difficulties it poses. These include the lack of a decomposition of linear perturbationsinto scalar, vector and tensor, together with the need to solve partial differential equationsdepending on (r, t) insted of ordinary ones [134]. Numerical studies of the growth of structurein LTB spacetimes have been performed using non-linear numerical simulations [135, 136]and linear perturbation theory [137]. Some consequences of these studies are described inSection 3.2.2.2.4.1 The Baryon Acoustic Oscillation ScaleThe large scale distribution of matter also reflects the propagation of perturbations in baryon-photon fluid before recombination as a sequence of oscillations superimposed on the powerspectrum. Equivalently, it can be seen as a peak in the two point correlation function, whichis related to the power spectrum by a Fourier transform. The BAO scale provides a statisticalstandard ruler that allows to reconstruct the expansion history of the low redshift universeusing information from galaxy surveys. As the BAO signal is suppressed by the baryonto matter ratio, its determination requires large survey volumes in order to gain statisticalsignificance. Although it has the same origin as the CMB angular scale (2.10), the LSS-BAOscale is fixed by the sound horizon at the drag epoch td, which is the time in which thebaryons decouple [138]. The high photon to baryon ratio keeps the latter coupled longer,and hence the drag redshift zd is lower than the recombination one z∗.The BAO scale is intrinsically three dimensional (like any other LSS quantity and unlikethe CMB), and can be measured at different redshifts. Thus, it is possible to observe it31Chapter 2: Observational Probes of Cosmologyparallel to the line of sight (radial BAO) through the correlation in redshift, in addition to theangular correlation (angular BAO). The comparison between the theoretical and predictedBAO scales is presented in detail in Section 3.2 for LTB models, in which new effects arisedue to the inhomogeneity. The relevant equations for the projected BAO length in angle andredshift (3.37, 3.40) can be straightforwardly applied to a homogeneous cosmologyθBAO =rs(zd)(1 + z)DA(z). (2.22)∆zBAO = H(z)rs(zd) (2.23)In order to gain statistical significance, the BAO scale is usually obtained from the rawdata by projecting it into spatial coordinates (assuming a fiducial model) instead of using amodel independent extraction in the variables z, nˆ (2.18).6 Due to historical reasons and theimpossibility to disentangle the different directions after the projection, it is customary toquote the data in terms of the geometric mean of the angular and redshift correlationdz ≡(θ2BAO∆zBAOz)1/3=rs(zd)((1 + z)2DA(z)2 z H(z)−1)1/3. (2.24)The denominator in the second equality is known as the geometric mean distance DV . Theangular correlation is squared to account for the two dimensions in the solid angle. Similarlyto the product of the correlation in the different distances, it is possible to consider the ratioof the characteristic scales as seen in redshift and angular space. The process of determiningwhether the δz, θ fulfill the geometric relationships predicted by the FRW metric is called anAlcock-Paczynski test. Section 3.2.5 describes it briefly in the context of LTB universes.Dynamical EffectsThe BAO scale is a paradigmatic example of a geometric cosmological probe. The charac-teristic length remains constant in coordinate space and tracks the background expansion toa very good approximation, since dynamical effects depend on the density variations within∼ 150 Mpc spheres and the universe is very well described by linear perturbation theory onsuch large scales. In ΛCDM the BAO scale is constant in coordinate space due to the lackof scale dependence of subhorizon perturbations δk(t) = D(t)δk(t0) (2.16) up to non-lineareffects. This preserves the shape of the power spectrum and the two point correlation func-tion, hence making BAO a geometric probe. The first dynamical effects enter non-linearly,broadening the shape of the peak and systematically reducing the BAO scale at the subpercent level [142–146]. Other systematics degenerate with BAO evolution might be causedby scale dependent bias or non-gaussian effects.The evolution of the BAO scale has received considerably less attention in the caseof non-standard cosmologies.7 Sherwin and Zaldarriaga [146] argue that the BAO shift isan intrinsically non-linear effect, as combines the fact that overdense regions expand lessand contribute more to the two-point correlation function. However, linear perturbationsbecomes scale dependent in many alternative theories of gravity δk(t) = D(t, k)δk(t0), and6This was necessary when the data was scarce. Current galaxy surveys allow for independent determina-tions of the angular [139] and radial [140, 141] BAO scales7The discussion here is restricted to the case of modified gravity models. The geometric status of BAOmeasurements in inhomogeneous cosmologies is discussed in Sections 3.2.1 and 3.2.2.322.4. Large Scale Structure6dF SDSS WiggleZ Carnero et al.z 0.106 0.2 0.35 0.44 0.6 0.73 z 0.55dz 0.336 0.1905 0.1097 0.0916 0.0726 0.0592 θBAO 3.90◦∆dz 0.015 0.0061 0.0036 0.0071 0.0034 0.0032 ∆θBAO 0.38◦C−1ij =4444 0. 0. 0. 0. 0.0. 30318 −17312 0. 0. 0.0. −17312 87046 0. 0. 0.0. 0. 0. 23857 −22747 105860. 0. 0. −22747 128729 −599070. 0. 0. 10586 −59907 125536 . (2.25)Table 2.2: BAO data. The first six data points are volume averaged and correspond toTable 3 of [103]. Their inverse covariance Matrix is given by (2.25), and was obtained fromthe covariance data in reference [103] in terms of dz. The last point corresponds to an angularmeasurement given in [139].dynamical effects may occur within this approximation. Theories featuring additional forcesacting on distances ∼ 150 Mpc (e.g. Vainshtein screened models) might enhance the shift ofthe BAO scale with respect to the standard model.Current BAO DataThe current BAO data span the redshift interval 0 − 0.8. The dz data at lower redshiftobtained by SDSS and 6DFGS [101, 147, 148] is complemented with the points provided bythe WiggleZ collaboration [103]. An additional point obtained from SDSS DR7 catalog in therange [0.5− 0.6] has been extracted as a purely angular correlation [139]. It is independentof the other SDSS measurements at z = 0.2, 0.35. All dz data points are summarized inTable 3 of reference [103]. However, the inverse covariance matrix for the data is provided interms of the variable A(z), which can not be interpreted easily in non-standard cosmologies,as discussed in Section 1.2.1. In order to retain the covariances between the different datapoints and analyze the models in terms of a meaningful variable, the inverse covariance matrixwas obtained in terms of dz using the correlation matrix. The data are summarized in Table2.2, and can be also seen in Figure 3.5 compared to several homogeneous and inhomogeneousmodels.The likelihood is assumed to be Gaussian and given byχ2BAO =∑i,j(di − d(zi))C−1ij (dj − d(zj)) +(θBAO(0.55)− θ0.55BAO)2∆θ2BAO, (2.26)where the indices i, j are in growing order in z, as in Table 2.2.The choice of data is convenient because it covers the redshift range z ≤ 0.8 with aregular spacing and the correlations are known (2.25). Other available BAO scale determi-nations (e.g. references [100, 102, 149–151]) would add points at intermediate redshifts with33Chapter 2: Observational Probes of Cosmologysimilar error bars and unknown covariances, and therefore we expect they will not increasethe precision of the constraints. Determinations of the radial BAO scale [140, 141] are par-ticularly interesting to constrain inhomogeneous models [152, 153] due to the distinct radialrescaling factor (3.35), see Section 3.2.4. Nonetheless, they are not included here due to thelack of knowledge about the correlations with other data points.2.4.2 Other LSS ProbesThe previous description has introduced some possible ways to use Large Scale Structureto study cosmology. However, there are other probes that test the dynamical properties ofcosmological models, including weak gravitational lensing and galaxy cluster counts.Weak Gravitational LensingOne of the consequences of general relativity is the deflection of light by matter. The tra-jectories of photons traveling over cosmological distances are affected by the intermediatematter distribution, which acts similarly to a lens in geometric optics. It produces a changein the perceived size (magnification) and shape (shear) of background objects.8 Magnifica-tion is difficult to use as a cosmological probe, because it requires knowing the intrinsic sizeof the objects under study with significant precision.Observing the correlations of shapes within a background set of galaxies allows one toreconstruct the intermediate density distribution. It is then possible to define a shear powerspectrum analogous to (2.17). Unlike galaxy surveys, this method is directly sensitive tothe total matter density, and therefore overcomes the difficulties of galaxy bias. Its directdependence on the gravitational potentials is also very valuable to test theories of modifiedgravity, for which the Poisson equation (2.15) may be altered [154]. However, systematiceffects are important and technically very challenging. These include the difficulties in theshape measurement of the galaxies and the existence of intrinsic alignments between them.Clusters of GalaxiesClusters of galaxies form out of the high density fluctuations present in the early timesperturbations, and their abundance at a given redshfit can be used to constraint cosmologicalmodels [155]. Cluster abundance studies rely on the comparison between the predicted spacedensity of massive halos to the observed space density of clusters, which can be identifiedvia optical, X-ray, or CMB observables that should correlate with halo mass. Their numberdensity is sensitive to a combination of the perturbation normalizations and the matterenergy density, and this dependence can be used to constraint cosmological parameters. Astheir formation depends on the dynamics of structure formation, they can be used to probealternative theories of gravity. However, clusters are non-linear objects and their use ascosmological probes in non-standard scenarios is challenging.Besides galaxies and Dark Matter haloes, clusters of galaxies also contain a large amountof very hot, ionized gas. This gas affects the energy distribution of the CMB photons byinverse Compton scattering processes (energy transfer from an electron to a photon), aneffect known as the thermal Sunyaev-Zel’dovich effect. Clusters can indeed be detected bydistortions in the photon spectrum induced in the CMB. The relative motion of the cluster8Some systems produce strong gravitational lensing effects, such as the occurrence of multiple images of agiven source. Cosmological constraints usually rely on more subtle modifications (weak lensing), which canbe studied statistically over large survey volumes.342.5. Type Ia Supernovaew.r.t. the CMB rest frame affects the magnitude of this effect, a contribution known as thekinematic Sunyaev-Zel’dovich (kSZ) effect. The kSZ effect induced by the relative motionis also produced in any other situation in which the CMB is not isotropic in the clusterreference frame. This property allows to use bounds on the kSZ effect to probe the largescale homogeneity of the universe [36, 156].2.5 Type Ia SupernovaeThe Type Ia supernovae (SNe) are extremely intense explosions that can be observed up tohigh redshift z ∼ 1.5. They are believed to occur when the mass of a white dwarf surpassesthe Chandrasekhar limit, either because of matter accretion or through merging with acompanion star. At this point the electron degeneracy pressure is not able to compensatethe gravitational instability, leading to a collapse and subsequent explosion as the nucleardegeneracy pressure is reached. The existence of this triggering mechanism sets the commonenergy scale, which can be sharpened through a phenomenological calibration procedure.The corrected SNe luminosity converts them into reliable standard candles, which can beused to measure cosmological distances as a function of redshift.The dimming of distant supernovae constitutes a solid probe of dark energy models inthe interval 0.01 . z . 1.5. The difference in magnitude between each observed supernovaeat redshift zi and the theoretical expectation given by (2.1) isµth(zi)− µobsi = 5 log10(DL(zi)1Mpc)+ 25− µ0 − µobsi ≡ ∆µi − µ0 , (2.27)where the last equality defines the quantity ∆µi, used for the observational constraints. Thequantity µ0 is proportional to the intrinsic luminosity of the supernova explosions. Sinceits value is unknown theoretically, it will be allowed to take arbitrary values. Observationaldeterminations of µ0 will be used to constrain the local expansion rate, as will be describedin Section 2.5.1.In addition to the intrinsic luminosity, two other unknown quantities are necessaryto calibrate the supernovae measurements and obtain a standard candle. These are thestretch (the duration of the supernovae explosion) and the color (to account for dust ex-tinction), which are assumed to be universal and introduce linear corrections on all theSNe µobsi = µB,i − µ0 + α(si − 1)− βci. These factors are calibrated by assuming an FRW-ΛCDM model, and should in principle be allowed to vary if the cosmology changes [157, 158].However, the result of the analysis should not vary significantly in models which produce lu-minosity distance curves very similar to the standard model. Considering the covariancematrix between the supernova data should also be beneficial, since it adds information aboutthis calibration procedure by taking into account the covariance between supernovae withsimilar color and stretch. Of course, the case would be different in more radical cases, e.g. ifthe luminosity distance has features or sudden variations.Supernova are a geometric probe to a very good approximation, granting very directinformation on the luminosity distance (2.2). However, there are two ways in which cosmologymay affect the SNe distance determinations dynamically:• Lensing of SNe light induced by cosmic structures, which can induce a systematicmagnification. Although lensing is included into the budget of known errors, it is done35Chapter 2: Observational Probes of Cosmologyin a simplified and not completely model independent way [27, 159]. Similarly, SNeredshift determinations are affected by peculiar velocities.• SNe evolution. If supernovae that originated in older galaxies are intrinsically fainter(or brighter) than their younger counterparts in a way not corrected by stretch andcolor, then the cosmological parameter estimate would be biased.9Fortunately, spectroscopic analyses show that the supernovae properties have little depen-dence on their age. These and other systematic uncertainties involved in the use of SNe ascosmological probes have been extensively studied, and the modern wealth and quality ofdata allow for a consistent study of this systematic error sources and a variety of cross checksthat validate their use as standard candles [10, 160–162]. Furthermore, the use of comple-mentary probes (e.g. BAO distance determinations) is able to rule out the models that relyon variable supernovae properties or direct effects on propagating light (e.g. axion-photonmixing).Current Supernovae DataMost of the observational constraints presented in the following chapters employ the Union2supernovae compilation [163], which consists in 557 SNe redshift-magnitude measurementsafter corrections for color and shape. The likelihood is computed using the covariance matrixincluding systematic errors Cij− 2 logLSNe = χ2SNe =∑i,j(∆µi − µ0)C−1ij (∆µj − µ0) . (2.28)This result depends on the actual value of the intrinsic luminosity µ0. Since it is unknown,the likelihood has to be maximized for each model with respect to µ0 for each model underconsideration [162]. Expanding the above expression and substituting back the value of µ0such that ∂(χ2)/∂µ0 = 0, gives the optimal likelihood for each modelχ2SNe =∑i,j∆µiC−1ij ∆µj −(∑i,j C−1ij ∆µj)2∑i,j C−1ij. (2.29)The Union2 SNe dataset can be seen in Figure 3.4 compared with both homogeneous andinhomogeneous models.2.5.1 Local Expansion RateRecasting the expression for the luminosity distance (2.27) in units of H0µth(zi) = 5 log10 (H0DL(zi)) + 25− µ0 − 5 log10(H0[Mpc−1]) . (2.30)it is possible to see that the intrinsic luminosity µ0 is degenerate with the Hubble constantfor homogeneous models, for which cosmic distances (2.1, 2.2) only depend on it througha global H−10 factor. On the other hand, inhomogeneous cosmologies introduce additional9That might happen if different processes producing type Ia SNe (merging vs accretion) occur at differentrates depending on the age of the universe. More exotic cases (e.g. time variation of G or other fundamentalconstants) might involve a variation of the Chandrasekhar mass Mc ∼M3p/m2H ∝ G−3/2.362.6. Local Gravity Testsscales and allow for more involved dependences [164], and hence the determination of thelocal expansion rate requires the knowledge of the intrinsic supernovae luminosity. Since thedependence of the expansion rate with redshift in an arbitrary cosmology can be differentthan in the ΛCDM case (even for low redshifts z < 0.1) it is necessary to devise a modelindependent method.A recent measurement of the local expansion rate using Ia type supernovae yields avalue H0 = 73.8± 2.4kms−1Mpc−1 [165]. The supernovae intrinsic luminosity was measuredusing over 600 Cepheid stars from eight nearby galaxies in which type Ia supernovae havebeen observed. The Cepheids are calibrated comparing their luminosity to three differentdistance estimates: 1) the geometric distance to NGC 4258 as obtained from water masersorbiting its central black hole, 2) trigonometric paralaxes to Cepheid stars in the Milky Wayand 3) relating the distance to the Large Magellanic Cloud obtained from eclipsing binaries.The local expansion rate is obtained by finding the best fit for a fiducial FRW model withΩM = 0.3, ΩΛ = 0.7 to 253 low redshift type Ia supernovae (z < 0.1) using the measuredintrinsic luminosities. In particular, the quoted value of H0 is the average of the valuesobtained from the three different calibrations.In order to reproduce the method outlined above and provide a fair comparison in anarbitrary cosmology, the constraints on the model are implemented through the supernovaeluminosities rather than the model parameter Hin. The value and the error in the luminositywere obtained by comparing the fiducial model fixing H0 = 73.8 and 73.8±2.4 to the Union2data in the range z < 0.1 (195 SNe) and finding the value of µobs0 that gives the best fit,using equation (2.32) below. The result isµobs0 = −0.120± 0.071 . (2.31)The “predicted” intrinsic luminosity that can be compared to the observation is the best fitµ0 found using the Union2 data for the model under investigationµbf0 =∑i,j C−1ij ∆µj∑i,j C−1ij, (2.32)using the distance modulus and the inverse covariance matrix of the data (see previous Sectionand equations (2.27,2.28). The associated likelihood is assumed to be Gaussianχ2H0 =(µbf0 − µobs0 )2∆µ02. (2.33)2.6 Local Gravity TestsBesides its elegant description of gravity as the dynamics of the space-time geometry, themajor strength of General Relativity is to correctly predict a variety of phenomena that occurin the Solar System and other astrophysical systems at low redshift [110]. Although not trulycosmological probes, local tests of gravity are very useful to constrain departures from GRsuch as additional scalar forces.Classical gravity tests may involve null or time-like geodesics, which allow the explo-ration of different properties. The perihelion precession of Mercury and the precise Lunarlaser ranging measurements of the Earth-Moon system are two examples involving massive37Chapter 2: Observational Probes of Cosmologybodies. Tests based on the trajectory of null geodesics include the gravitational redshift,the deflection of light (including gravitational lensing) and the time delay of a signal passingnear the Sun. Laboratory tests of gravity can explore the existence of short range interac-tions, e.g. of the Yukawa type (1.35), using torsion pendulum experiments. Current boundslimit their coupling strength on scales larger than 0.056 mm [166]. These measurements canachieve the precision necessary to detect small field effects due to the good control of sys-tematic effects and the possibility of collecting considerable statistics. Gravity has also beentested in the strong field regime outside the Solar System, through the observation of binarypulsars (i.e., rotating neutron stars emitting a beam of radio noise). Several systems havebeen observed in which the orbital period decreases according to the energy emitted in theform of gravitational waves, in great agreement with the General Relativistic expectation.Many of the aforementioned tests have the advantage of being able to directly translatein to bounds of the Parameterized Post Newtonian (PPN) approximation coefficients [110].The theoretical values can also be computed within alternative theories of gravity in termsof the model parameters, allowing a very neat and direct comparison between theory andexperiments. For recent review on gravitational experiments and current bounds on PPNparameters, see Ref. [20].38Part IIResults39Chapter 3Large Scale Homogeneity andNon-Copernican Void ModelsThere is nothing more natural than toconsider everything as starting from oneself,chosen as the center of the world; one findsoneself thus capable of condemning the worldwithout even wanting to hear its deceitfulchatter.Guy Debord1Recent years have witnessed enormous advance in the quantitative understanding ofcosmology and the establishment of a Standard Cosmological Model. Its construction isgrounded on the general relativistic description of space-time with the usual Einstein equa-tions. The Ansatz for the space-time is a spatially homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric, chosen to satisfy the generalized Copernican Principle, orCosmological Principle. In addition to the known particles (baryons, photons and neutri-nos), two mysterious elements need to be added in order to account for all the observations.These give the name to the standard, ΛCDM model: Cold Dark Matter (CDM) plus a cos-mological constant (Λ), the last one necessary to explain the dimming of distant supernovae[11, 12]. With the standard choice of the metric, the supernovae data imply that the universeis currently undergoing a phase of accelerated expansion.The situation changes when the Cosmological Principle Hypothesis is dropped. As thesupernovae we observe occur in our past lightcone, the changes in luminosity we interpret astime evolution might be due to spatial variations if the universe is not homogeneous. If theinhomogeneity represents an underdensity with a size comparable to the Hubble radius andour galaxy is located near its center, supernovae observations can be successfully accountedfor without the introduction of new physics. This type of so-called “large void models”1Panegyric (1989)41Chapter 3: Large Scale Homogeneity and Non-Copernican Void Modelscan be described by a spherically symmetric Lemaˆıtre-Tolman-Bondi (LTB) metric, andhave been studied as an alternative to the standard ΛCDM scenario [38, 40, 157, 167–181].Many different aspect of these alternative models have been considered over the past years,including observational constraints [27, 36, 152, 153, 156, 158, 164, 182–192], the growthof perturbations [134–137, 193, 194], and the physics of the Cosmic Microwave Background(CMB) [109, 195–204]. See reference [8] for a recent review.If the time to Big Bang and the baryon-mater ratio are independent of the location,the LTB type of metric represents the gravitational growth of an adiabatic perturbationfrom an initially quasi-homogeneous state [193]. Although a Gigaparsec-sized void is difficultto reconcile with the standard inflationary paradigm, it might still be possible through theproduction of large non-perturbative inhomogeneities associated with the stochastic natureof the inflaton evolution [205]. The only philosophical problem associated with this typeof models is the requirement of being located very close to the center in order to preservethe great degree of isotropy observed in the CMB [196], but ultimately, only cosmologicalobservations can tell us about the geometry and distribution of the cosmos and our positionin it, provided that this question is meaningful.The aim of this work is to analyze LTB models in the light of the most recent cosmologi-cal data. The Hubble Space Telescope has made a precise measurement of the local expansionrate [165] that challenges this type of void models, which typically require a low value to fitthe CMB power spectrum [153, 187]. The determination of the Hubble parameter relies onthe calibration of distant supernovae using Cepheid variable stars and the subsequent fit toa fiducial ΛCDM model in the low redshift range. Although applying priors directly on H0is fine for homogeneous cosmologies, LTB universes can have a very different evolution inthe relevant redshift range. Therefore, our analysis is based on the intrinsic Ia supernovaluminosity instead of the value of the Hubble parameter.Further restrictions on this model are obtained from the scale of Baryon Acoustic Os-cillations (BAO) and its evolution in an inhomogeneous cosmology by including the mostrecent BAO data up to redshift z = 0.8 provided by the WiggleZ collaboration [103] andCarnero et al. [139]. In a FRW cosmology, the BAO scale is space-independent and theconstraints it yields arise from providing an independent measurement of cosmic distancesrelative to a standard ruler, with an initial length determined by the physics of the earlyuniverse. Once baryons decouple, the dominant effect on the observed physical scale is to bestretched by the expansion of the universe. However, in the less symmetric LTB cosmologythe initially constant BAO scale grows differently: the physical scale acquires an additionalradial dependence and is stretched differently in the longitudinal and transverse direction,due to the existence of two expansion rates.Adding information about the BAO scale at higher redshift reduces considerably theroom for its value in the early universe. If the depth of the void is chosen to fit the supernovaeluminosity distances, the inhomogeneous expansion produces a mismatch between the BAOscale at low and high redshift, posing a new problem for these models. Adding informationof the CMB increases the discrepancies by restricting the initial acoustic scale, but the con-straints are independent of the particular values, i.e, depend only on the geometric propertiesof the model regardless of the calibration of the standard rulers and candles. In particular,these constraints are independent of the primordial power spectrum, a critical assumptionnecessary to rule out large void models using the tension between the CMB and the localexpansion rate [158] and are therefore complementary to these.423.1. Lemaˆıtre-Tolman-Bondi ModelsSection 3.1 describes the general LTB void models, giving the corresponding Einstein-Friedmann equations, as well as the standard solution in absence of pressure. In a subsectionthe adiabatic assumption is introduced, i.e. that the time since Big Bang is space indepen-dent, and thus the model only depends on a single function, the inhomogeneous matter profileΩM (r). This is chosen to have the GBH parameterization, which is also presented. Section3.2 presents in detail the evolution of baryonic features in terms of free-falling trajectories ofthe background metric and compute the BAO observables. Section 3.3 describes the cosmo-logical data used to analyze the model, including a method to use the supernova luminosityto constrain the local expansion rate. It also contains the results from the comparison of themodel and the data and describe the tensions between the different datasets, as well as theresult of different model comparison criteria. The conclusions are summarized in Section 3.4,discussing the generality of the results and stating several modifications that might renderthe model viable.3.1 Lemaˆıtre-Tolman-Bondi ModelsThe LTB model describes general spherically symmetric space-times and can be used as atoy model for describing voids in the universe [206–208]. The starting point is the generalmetricds2 = −dt2 +X2(r, t) dr2 +A2(r, t) dΩ2 , (3.1)where dΩ2 = dθ2+sin2 θdφ2. Units in which c = 1 will be assumed in the folowing. Assuminga spherically symmetric matter source with negligible pressure, Tµν = −ρM (r, t) δµ0 δ0ν , the(0, r) component of the Einstein equations, Gtr = 0, sets the form of X(r, t). The resultingcosmological metric becomesds2 = −dt2 + A′2(r, t)1− k(r) dr2 +A2(r, t) dΩ2 , (3.2)with an arbitrary function k(r) playing the role of the spatial curvature parameter. Theother components of the Einstein equations read [173, 182, 184]H2T + 2HTHR +k(r)A2+k′(r)AA′= 8piGρM , (3.3)2H˙T + 3H2T +k(r)A2= 0 , (3.4)where dots and primes denote ∂t and ∂r respectively, and we have defined the transverse (i.e.in the angular direction) and radial Hubble rates asHT ≡ A˙/A, and HR ≡ A˙′/A′ . (3.5)The reduced Hubble rate can be defined as usual: HR/T ≡ 100hR/TKm/Mpc/s. It is alsouseful to consider the normalized shearε ≡ HT −HRHR + 2HT, (3.6)i.e. the difference between the radial and transverse expansion weighted by the total expan-sion [152]. This variable provides a local quanitfication of the departures with respect tohomogeneous cosomologies, e.g. to characterize the growth of structure [136].43Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models0 2 4 6 8 100.20.40.60.81.0r @Gpc DWM H r LΡH r ,t 0 L Ρ¥H t 0 LΡH r ,t10 L Ρ¥ H t10 LHT H r ,t 0 LHL H r ,t 0 LFigure 3.1: Physical parameters in the LTB model. The density contrast at t(z = 0), t(z =10) (red) shows the evolution from an initially less inhomogeneous state, and differs fromthe function ΩM (r) (black dotted). The expansion rates in the radial (purple dashed) andtransverse (blue dashed) directions differ the most where the void is steeper. The profileshown has R = 2.5,∆R = 1.Integrating (3.3) yieldsH2T =F (r)A3− k(r)A2, (3.7)in terms of another arbitrary function F (r). Substituting it into the first equation givesF ′(r)A′A2(r, t)= 8piGρM (r, t) , (3.8)where ρM (r, t) is the physical matter density. Since F (r) is time-independent, one can chooset = t0 and compute the integrated matter density in a comoving volume today, V = 4pir3/3,as ρ¯(r) = 1V∫ r0 4pir′2dr′ ρM (r′, t0) , and construct with it the ratio ΩM (r) ≡ ρ¯(r)/ρ¯c(r), whereρ¯c(r) = 3H20 (r)/8piG is the critical density in that volume [152].The boundary condition functions F (r) and k(r) are specified by the nature of theinhomogeneities through the local Hubble rate, the integrated mass ratio and the local spatialcurvature,F (r) = H20 (r) ΩM (r)A30(r) = 8piG∫ r0dr′r′2ρM (r′, t0) , (3.9)k(r) = H20 (r)(ΩM (r)− 1)A20(r) , (3.10)where functions with subscripts 0 correspond to present day values, A0(r) ≡ A(r, t0) andH0(r) ≡ HT (r, t0). With these definitions, the (position dependent) transversal Hubble ratecan be written as [173, 182]H2T (r, t) = H20 (r)[ΩM (r)(A0(r)A(r, t))3+ (1− ΩM (r))(A0(r)A(r, t))2], (3.11)443.1. Lemaˆıtre-Tolman-Bondi Modelsand we fix the gauge by setting A0(r) = r. For fixed r and ΩM < 0 the above expression isequivalent to the Friedmann equation, and has an exact parametric solution in terms of thevariable η:A(r, t) =ΩM (r)2[1− ΩM (r)] [cosh(η)− 1]A0(r) , (3.12)H0(r)t =ΩM (r)2[1− ΩM (r)]3/2[sinh(η)− η] . (3.13)Very good approximate solutions can also be found by Taylor expanding around an Einsteinde Sitter solution [184].In addition to the solution of Einstein Equations (3.12,3.13) it is necessary to obtainthe coordinates on the lightcone as a function of redshift. For light traveling along radial nullgeodesics, ds2 = dΩ2 = 0 yieldsdtdr= ∓ A′(r, t)√1− k(r) , (3.14)which, together with the redshift equation [182, 208],d log(1 + z)dr= ± A˙′(r, t)√1− k(r) , (3.15)allows us to write a parametric set of differential equations, with N = log(1 + z) being theeffective number of e-folds before the present time,dtdN= −A′(r, t)A˙′(r, t), (3.16)drdN= ±√1− k(r)A˙′(r, t), (3.17)where the equations are integrated with the initial condition r(0) = 0, t(0) obtained from(3.12,3.13) for r = 0, A = A0.The angular diameter distance is given by the dΩ element of the metric evaluated onthe lighctone, and is related to the luminosity distance by the redshift due to photon redshiftand time dilationDA(z) = A(r(z), t(z)) , (3.18)DL(z) = (1 + z)2A(r(z), t(z)) . (3.19)The dynamics of the LTB metric in the only-matter approximation discussed aboveare fully specified by the two functions ΩM (r), H0(r) independently of the type of matterpresent, as long as it exerts no pressure. But by dropping the symmetries of the FRW modela spherically symmetric but inhomogeneous mixture of baryonic and dark matter can beaccommodated. A possible parameterization in terms of the total matter density would befb(r) ≡ ρb(r, t)ρm(r, t), (3.20)where there is no time dependence because the energy density of baryons and dark matterevolves identically at late times.Before explaining the choice of matter profile and the physical restrictions on the model,let us briefly summarize the approximations used throughout this work45Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models• Spherical symmetry as given by (3.2) and perfectly central location of our galaxy atr = 0, t = t0 as initial conditions for the lightcone integration (3.16).• Radiation energy and pressure neglected as a source of the expansion (3.11).• Early time and large radius FRW limit of the model, necessary to compute the BAOscale (Section 3.2) and the relative locations of the CMB peaks (Section 3.3.1).• Perturbations of the LTB metric neglected. The evolution of the BAO scale from earlytimes is studied by analyzing the geodesics of the background metric (3.2) (Section3.2).3.1.1 The Adiabatic GBH ModelGeneral LTB models are uniquely specified by the two functions k(r) and F (r) or equivalentlyby H0(r) and ΩM (r), but to test them against data it is necessary to parameterize thefunctions, so that a finite dimensional space is analyzed. In this Chapter we will use theGBH model [184] to describe the matter profile in terms of a reduced number of parameters.In addition to the choice for the free function ΩM (r), we further impose that the time to BigBang is space-independenttBB(r) = H0(r)−1 1√ΩK(r)√1 +ΩM (r)ΩK(r)− ΩM (r)√Ω3K(r)sinh−1√ΩK(r)ΩM (r) = t0 , (3.21)where ΩK(r) = 1−ΩM (r). The above expression can be obtained from integration of (3.11)[182, 184] or by solving for A = A0(r) in (3.12,3.13). This condition reduces the functionalfreedom associated to H0(r) to a single normalization constant H0, that is related to theoverall age of the universe.Additionally, we require that there are no large scale baryonic isocurvature modes, i.e.the baryon fraction (3.20) is constant. This type of voids can be regarded as the gravitationalcollapse of a large scale, adiabatic and spherically symmetric perturbation which has a smallamplitude at early times. Its adiabatic nature is related to the fact that there is only onefunctional degree of freedom that sets the shape of the remaining free functions (ΩM (r) inour case, which in turn fixes H0(r) and the baryon fraction).The above conditions give a relation between H0(r), ΩM (r) and fb(r), and hence con-strain the models to one free function. Our chosen model is thus given by [184]ΩM (r) = Ωout +(Ωin − Ωout)(1− tanh[(r −R)/2∆R]1 + tanh[R/2∆R])(3.22)H0(r) = H0 1ΩK(r)− ΩM (r)√Ω3K(r)sinh−1√ΩK(r)ΩM (r) = H0 ∞∑n=02[ΩK(r)]n(2n+ 1)(2n+ 3), (3.23)fb(r) = fb = constant (3.24)where the second equation follows from (3.21), and the third by demanding constant baryonto matter ratio. This paremeterization was introduced to lower the shear around the voidwall (e.g. with respect to Gaussian profiles) and allow a unified description of cuspy and flatcentral regions [184]. Each void model is specified by the following parameters:463.2. The Baryon Acoustic Scale in LTB Universes• Ωin: Matter/curvature fraction at the center of the void (equations 3.8-3.10). As deepervoids produce more fictitious acceleration, this parameter plays a major role in theconstraints presented in Section 3.3.• Ωout: Asymptotic (r → ∞) matter/curvature fraction in which the inhomogeneousregion is embedded. Two possibilities will be considered separately depending on theasymptotic curvature of the universe:-CGBH: Flat Ωout = 1, as suggested by inflationary physics.-OCGBH: Open Ωout ≤ 1, which allows a better fit to the CMB.• ∆R: Slope of the inhomogeneity. Smaller values of ∆R produce steeper profiles andincrease the shear (3.6).• R: Shape of the void. ∆R R describes an inhomogeneity with a central plateau ofapproximately constant density, while ∆R R produces a cuspy central region.• H0: Expansion rate normalization that determines the Big Bang time (3.21).• fb: Baryon fraction over the total matter content. Its value affects the pre-recombinationphysics and determines the value of the BAO scale and CMB peak locations.The choice of the constrained model is important because, in our gauge, void modelswith an inhomogenous Big Bang would contain a mixture of growing and decaying modes, andconsequently the void would not disappear at early times, making them incompatible withthe Standard Big Bang scenario [193]. By restricting ourselves to adiabatic LTB models thecentral void is reduced to an insignificant perturbation in an otherwise homogeneous universedescribed by an FRW metric, both at large distances and early times. This requirement,together with the condition of constant baryon fraction, ensures the space-independence ofthe early BAO scale, which is a key part of the present analysis.3.2 The Baryon Acoustic Scale in LTB UniversesInhomogeneous cosmologies stretch the BAO scale differently than their homogeneous cousins.There are three potential effects to be addressed when computing the BAO scale in LTB mod-els at the background level:1. Inhomogeneous expansion: The matter distribution will source the expansion of theuniverse in a position dependent way. Therefore, there will be a radial dependence ofthe physical scale in addition to the time dependence.2. Anisotropic expansion: In general the expansion rate in the radial and transverse di-rection will be different HT 6= HR, resulting in two different BAO scales lT 6= lR, asseen by a central observer. The relation between this effect and the Alcock-Paczynskitest will be discussed in Section 3.2.5.3. Radial coordinate drift: Displacements in the radial direction are not a symmetry of theLTB metric and the free-falling baryon features are not ensured to remain at constantr. In Section 3.2.1 we show that this effect does not occur for timelike geodesics.47Chapter 3: Large Scale Homogeneity and Non-Copernican Void Modelsçççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççìøõõõõõõõô-2 0 2 4 6 8 10024681012r @Gpc Dt@GyrDt H z =0 Lt H z =1Lt H z =100 LWe are hereconstant tconstant ΡFigure 3.2: The LTB model with space-independent Big Bang. Black lines representhypersurfaces of constant time t = t(z) for z = 0, 1, 100 (dashed) and constant densityρ(r, t) = ρ(r∞, t(z)) (thick, continuous). The filled areas show schematically the regions inwhich the void can be considered homogeneous (early times and large radii). Red constantlines represent our lightcone, where the coordinates of SNe (blue circles on the left) and BAO(green triangles on the right) observations has been added. Vertical dotted lines correspondto the geodesic worldlines of our galaxy (red) and the BAO fiducial locations (green). Thephysical BAO scale at different z is obtained from the asymptotic value (represented by thefilled green triangle), extrapolated to early times, for which the universe is approximatelyhomogeneous (horizontal green dotted line) and evaluated at the lightcone coordinates usingthe LTB metric (see Section 3.2 for the details).Possible effects from higher order corrections will be discussed at the end of Section 3.2.2.Our approach to predict the BAO scale in LTB models with space-independent BigBang time relies on the homogeneity properties of the metric at large radius (i.e. the profileflattens) and early times (i.e. the Big Bang time and the baryon fraction are independentof the position). Relaxing this assumptions would require a more careful treatment whichgoes beyond the scope of this work. We first analyze the evolution of the BAO scale inthe inhomogeneous cosmology by following the geodesics of the LTB metric (Section 3.2.1).Afterwards, the asymptotic physical scale computed in the limit r R, ∆R using the fittingformulae is extrapolated to a suitable early time te at which the void is just a negligibleperturbation. The obtained value can be then projected to the coordinates of observation483.2. The Baryon Acoustic Scale in LTB Universesr(z), t(z) using the previous results (Section 3.2.3). Finally, the physical scale is related tothe observed quantity dz quoted by the galaxy surveys (Section 3.2.4). The procedure issketched in Figure 3.2.The derivation we are presenting avoids using certain concepts that might be equivocalwhen used for inhomogeneous cosmologies. In particular, we will only use redshift as acoordinate on the lightcone or in the asymptotic region when the FRW limit can be applied.We will also avoid the term “comoving” and will refer to “coordinate” instead, as well as payspecial attention to distinguishing physical distances and relative coordinate separations.3.2.1 Free-falling Scales in the LTB MetricThe propagation of sound waves in the baryon-photon fluid present in the early, expanding,universe leaves an imprint at a characteristic length that will be observable in the late universeas a peak in the correlation function of galaxies [24, 138]. When the universe becomes neutral,baryon-photon interactions become effectively negligible and the baryonic overdensities startbehaving as free-falling test bodies. We can therefore analyze the relative separation of theinitial baryon clumps and the galaxies they will form by following the geodesics of the LTBmetric: xµ(τ) = {t(τ), r(τ), θ(τ), φ(τ)}, where x˙µ = dxµdτ . The BAO scale can be tracedsimply by following two nearby trajectories with an initial separation equal to the baryonacoustic scale at a sufficiently early time.The transverse evolution is the simplest. Since rotations are an isometry of the LTBmetric, the momentum in the angular directions is conserved and trajectories with φ˙, θ˙ = 0initialy will remain at constant angular coordinates. This can be readily seen from thegeodesic equation for the θ coordinateθ¨ + 2A′Ar˙θ˙ + 2A˙At˙θ˙ = 0 , (3.25)for which θ(τ) = θ0 is a solution. Its stability follows by demanding timelike, slow geodesicsfor which t˙ r˙ and noting that the second term is positive in an expanding universe (A˙ > 0,HT > 0) and acts as a friction against the angular velocity θ˙. Therefore, initial angularseparation ∆θ is conserved in coordinate space and the associated, transverse physical scalecan be obtained integrating the angular element of the metric lTphys = A(r, t)∆θ.Since shifts in the radial direction are not an isometry of the LTB metric, r˙ is notautomatically conserved and the determination of the radial acoustic scale requires a morecareful treatment. The geodesic equations for a trajectory with φ˙, θ˙ = 0 aret¨+A′A˙′1− k(r) r˙2 = 0 , (3.26)r¨ +[k′(r)2(1− k(r)) +A′′A′]r˙2 + 2A˙′A′t˙r˙ = 0 . (3.27)Similarly to the angular case, a particle initially at rest at some early time te, r˙(te) = 0,will remain at constant radial coordinate location r(τ) = r(te). Timelike, non-relativistictrajectories with t˙ r˙ are again stable in an expanding universe due to the longitudinalHubble friction term A˙′/A′ > 0, and geodesics will remain at constant coordinate separations49Chapter 3: Large Scale Homogeneity and Non-Copernican Void Modelsat different cosmic epochs.2 The physical distance in the r direction can be obtained simplyby integration using the radial element of the metric lRphys =∫ √grrdr ≈ A′/√1− k∆r.To summarize, since coordinate locations are conserved in geodesic evolution, we canprovide the following relation for freely falling, physical scales in the LTB metric at differenttimes t, te in the transverse and longitudinal directionslTphys(r, t) =A(r, t)A(r, te)lTphys(r, te) , (3.28)lRphys(r, t) =A′(r, t)A′(r, te)lRphys(r, te) . (3.29)3.2.2 BAO Scale Evolution Beyond Zero OrderThe above analysis so far has dealt with the differences between homogeneous and inhomo-geneous models at the zero order level. As it was discussed in Section 2.4.1, the BAO scalein homogeneous, GR cosmologies is slightly shifted towards lower values due to non-lineareffects on matter clustering. The lack of this effect to linear order is a consequence of thescale independence of the growth function for each Fourier mode.In inhomogeneous universes the situation is different, since the lack of symmetry pro-duces the failure of the perturbation decomposition and the treatment of scalar perturbations(in the metric and matter density) have to be considered together with vector and tensorperturbations [134, 193]. The vorticity (vector) component is subdominant because the LTBmetric is rotationally invariant, but the scalar potential is sourced at linear order by a termproportional to the background shear traced with the tensor perturbations. Fortunately, forthe GBH profiles considered here, the background shear is below 5% and these contributionswill be subdominant with respect to the much larger effect of the inhomogeneous expansion(see Figure 3.3 and Section 3.2.4). February et al. recently presented the numerical com-putation of the BAO scale in LTB models within linear theory, and considering only scalarperturbations [137]. They found a shift on the BAO scale at the percent level, which isnevertheless considerably smaller than the departures induced by the inhomogeneous andanisotropic expansion discussed above.Further support for the assumption of a constant BAO scale in coordinate space isprovided by N-body numerical studies. Alonso et al. [135] run simulations in which inhomo-geneous matter profiles are implemented through an initial underdensity of Gpc size. Theirresults show that the (local) matter density contrast grows with the scale factor in a wayanalogous to that of an open universe with a value of the matter density ΩM (r) correspond-ing to the appropriate location r, showing an effective decoupling between the small scaleclustering and the evolution of the void. Corrections from the large scale inhomogeneity areproportional to the local shear weighted by a factor O(1) [136], and are hence small for theprofiles allowed by observations.3.2.3 The Physical BAO Scale at Early Times and on the LightconeThe solutions of the LTB metric with space-independent Big Bang represent an inhomogene-ity that grows due to gravitational instability out of a very homogeneous state. For typical2The situation would considerably change if Γrtt was different from zero in the LTB metric. It might beas well possible to devise profiles for which the first term in (3.27) overcomes the second for sufficiently rapidgeodesics, i.e. high r˙/t˙, but this is not the case for the models under study.503.2. The Baryon Acoustic Scale in LTB Universesvoids at t(z = 100) the physical density contrast ρm(r, t)/ρm(r∞, t) is of order 1%, whileat t(z = 1000) it shrinks to ∼ 0.1%. As the baryionic features develop between t ∼ 0 andt ∼ t(z = 1000), it is a good approximation to consider that the physics responsible forrecombination and the origin of the baryon acoustic scale are indistinguishable from theircounterparts in homogeneous cosmologies. It will be therefore assumed that for models withspace-independent Big Bang the physical BAO scale is isotropic and coordinate independentat early times on constant time hypersurfaces.3lBAO(r(z), te) ≈ lBAO(r∞, te) . (3.30)The early time BAO scale can be obtained from the asymptotic value at different timesusing equation (3.28) or (3.29). In the r → ∞ limit the universe is indistinguishable froma FRW cosmology, and we can compute the BAO scale using the fitting formulae providedby Eisenstein and Hu [138] in terms of the asymptotic values of the matter density andbaryon fraction. These effective values are obtained by projecting the LTB parameters onthe lightcone at a very high redshift ze ≈ 100, for which 1) HT ≈ HR and the universe isapproximately homogeneous on a constant t = t(ze) hypersurface, 2) the point r(ze) is awayfrom the inhomogeneous region and 3) the radiation contribution is still negligible. Thesevalues are given byΩeffm =ρ(r∞, t0)3H2T (r∞)≈ Ωout , (3.31)Ωeffb ≈ fb Ωout , (3.32)Heff0 =2HT (ze) +HR(ze)3√Ωeffm (1 + ze)3 + (1− Ωeffm )(1 + ze)2. (3.33)Here Heff0 can be understood as “rewinding” the LTB value of the average expansion rate2HT (ze)/3 +HR(ze)/3 using the FRW asymptotic value of Ωm.The fitting formulae give a comoving scale in a FRW universe which coincides with thephysical value at t = t0 for the usual definition of the scale factor in the asymptotic FRWmetric a(t0) = a0 = 1. The corresponding scale at t0 is valid in the limit r →∞, but can berelated to the radial and transverse physical scales using (3.28,3.29,3.30). For a point locatedon the past lightcone of the central observer, the values arelTBAO(z) ≡ ξT (z)lBAO(r∞, t0) =A(r(z), t(z))A(r(z), te)A(r∞, te)A(r∞, t0)lBAO(r∞, t0) , (3.34)lRBAO(z) ≡ ξR(z)lBAO(r∞, t0) =A′(r(z), t(z))A′(r(z), te)A′(r∞, te)A′(r∞, t0)lBAO(r∞, t0) . (3.35)The first equalities define a transversal and longitudinal rescaling factors (see Figure 3.3),which reduce to (1 + z)−1 in the homogeneous limit.3A more physical criterion would be to consider constant density hypersurfaces. Although the difference isof order 0.1% in models with space-independent Big Bang, it might render helpful to generalize the treatmentof BAO for profiles with general H0(r).51Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models0.0 0.2 0.4 0.6 0.8 1.01.01.11.21.31.4zH1+zLΞHzLGeom . AverageH1+zL H ΞT2ΞL L1 3Tran sverserescalin gLongitudin alrescalin gFigure 3.3: Effects on the LTB metric on the BAO scale with respect to their FRW analogue.The difference between the transverse (blue, dashed) and longitudinal (purple, dot-dashed)factors accounts for the anisotropy of the scales in the angular and transverse direction,respectively. The geometric averaged rescaling factor (black solid) is used for volumetricBAO determinations through the quantity dLTBz = (1 + z)ξ(z) dFRWz (3.43). Note that allthree curves coincide in r = 0, since the void is locally isotropic at the center.3.2.4 Comparison with the Observed BAO ScaleThe BAO scale can be extracted from the galaxy correlation function measured in galaxysurveys. What is actually observed is a combination of the angular correlation θBAO and thecorrelation in redshift space ∆zBAO [209]. In order to compare the models with observations,we need to relate the isotropized correlation measured by the surveysdz =(θ2BAO∆zBAOz)1/3, (3.36)to the physical scales computed in the previous Section.4The angular correlation can be readily obtained from the definition of the angulardiameter distance as the ratio between a known (transverse) length and the angle it subtendsθBAO =lTBAO(z)DA(z). (3.37)The redshift correlation can be related to the radial coordinate separation by means of theredshift equation (3.15)∆zBAO =∫ r(z)+∆rBAO/2r(z)−∆rBAO/2dzdrdr ≈ (1 + z)A˙′(r(z), t(z))√1− k(r) ∆rBAO , (3.38)4The present discussion follows Section 4.6.3 of Biswas et al. [187]. Our result (3.41) has the same formas their equation (4.48) after several coefficients cancel out. However, their computation assumes the BAOscale to be given by the local values of ΩM (r),ΩB(r), H0(r) instead of obtaining them from the asymptoticFRW value and the factor (1 + zrec) is taken to be given by the volume element comparison on the worldlineof constant r = r(z) instead of by the asymptotic value r →∞.523.2. The Baryon Acoustic Scale in LTB Universeswhere in the second equality the integrand has been assumed to be constant.5 Similarly, thecoordinate characteristic scale ∆r is given in terms of the physical scale through an integral∆rBAO ≈√1− k(r)A′(r(z), t(z))lRBAO(z) . (3.39)Both equations relate the physical correlation with the redshift correlation∆zBAO = (1 + z)HR(z)lRBAO(z) (3.40)Constructing the geometric mean (3.36) using (3.37,3.40) is straightforward:dLTBz =(HRz(1 + z)1DA(z)2)1/3ξ(z) l(r∞, t0) , (3.41)where the scale conversion arising from (3.34,3.35) has been introduced in the factor ξ(z) ≡(ξR(z)ξ2T (z))1/3, given byξ(z) =(A′(r(z), t(z))A′(r(z), te)A′(r∞, te)A′(r∞, t0))1/3(A(r(z), t(z))A(r(z), te)A(r∞, te)A(r∞, t))2/3, (3.42)using a suitable early time te = t(z ∼ 100) to convert the scale as described in the previousSection. Note that due to the FRW limit, the ratios of the factors computed at r∞ can beexpressed as redshift factors a(te)/a0 = (1 + ze)−1.Equation (3.41) can be easily related to the usual expression for dzdLTBz = (1 + z)ξ(z)l(r∞, t0)DV (z)= (1 + z)ξ(z) dFRWz . (3.43)in terms of the usual volume distanceDV (z) =((1 + z)2DA(z)2 zHR(z))1/3. (3.44)Relation (3.43) absorbs the effects of the inhomogeneous rescaling in the BAO observations,which pick up a factor (1 + z)ξ(z) with respect to the FRW case. The difference betweenthe two rescaling factors accounts for the anisotropy between the transverse and longitudinalBAO scales, while their redshift dependence is a consequence of the inhomogeneity. Botheffects are shown in Figure 3.3. Note also that there will be an additional difference becauseof the modified relations between the angular diameter distance (related to the transverseexpansion) and the longitudinal expansion rate entering the geometric mean distance (3.44).3.2.5 The Alcock-Paczynski Effect in LTB ModelsThe Alcock-Paczynski (AP) effect [210] is the geometric distortion of spherical objects dueto cosmological expansion, since distances in the radial direction away from an observer are5This approximation is justified because the void variation scale is much larger than ∆rBAO. The exactresult ∆z = z(r + ∆rBAO)− z(r −∆rBAO) can be obtained by inverting r(z).53Chapter 3: Large Scale Homogeneity and Non-Copernican Void Modelsdetermined in redshift space, while transverse distances are seen as angular separations inthe sky. This motivates the definition of the dimensionless distortion factor:fAPFRW(z) ≡ ∆z∆θ= DA(z)HR(z)(1 + z) . (3.45)In a homogeneous universe, the above relation can be tested against spherical (or sphericallydistributed) objects for which ∆z and ∆θ are measured.6 This technique has been used toconstrain cosmological models [104, 151, 211].In LTB models with space-independent Big Bang time, initially spherical distributionsare intrinsically distorted due to the local shear, as discussed in Section 3.2.1. Using theangular and redshift projection of physical distances for the inhomogeneous models given byequations (3.37, 3.40), the analogue of the AP relation is modified by the ratio of the radialand transverse rescaling factorsfAPLTB(z) =ξR(z)ξT (z)DA(z)HR(z)(1 + z) =ξR(z)ξT (z)fAPFRW(z) . (3.46)The values of these factors in both directions can be seen in Figure 3.3. As the universeexpands faster in the transverse than in the radial direction, the distortion factor has a lowervalue than in FRW models, on top of the different relation between DA(z) and HR(z).The distortion factor (3.46) is sensitive to cosmic shear (3.6) through the ratio of thetransverse and radial rescaling (e.g. steeper profiles enhance the asymmetry). In the limit ofzero background shear, the ratio of rescaling factors tends to one, and the only difference w.r.t.FRW comes from the different relation between the angular diameter distance and the radialHubble rate. Therefore, the information one obtains from the AP effect is complementaryto the geometric mean distance given by Eq. (3.43), which only depends on the expansion,i.e. the product of the rescaling in the three spatial directions, and is unable to tell apartξR from ξT . Therefore, the AP effect is not only able to distinguish FRW from LTB models,but could eventually allow to observationally discriminate between different LTB profiles.3.3 Analysis and ResultsIn this Section the LTB models described in Section 3.1 will be compared to observations.The present analysis relies on the interplay between the cosmic distances obtained by type Iasupernovae and the distances and rescaling constraints from the baryon acoustic oscillationscale. SNe can be regarded as a standard candle and BAO as a standard ruler, which suffersadditional effects due to the inhomogeneity (as described in Section 3.2). The measurementof the local expansion rate and the CMB peaks are also considered, their effect being toprovide a calibration for the standard candles and rulers, respectively. However, the mainresult is independent of this calibration.3.3.1 Observational DataThe models were compared against several datasets, which are described below. The BAO,SNe and H0 data have been described in Chapter 2. A simplified method to implement CMB6In practice, the AP test is difficult to perform due to dynamical effects such as the Redshift SpaceDistortions caused by cosmic structures, which induce peculiar velocities that affect the redshift in a systematicway.543.3. Analysis and Resultsconstraints in LTB models without computing cosmological perturbations is presented at theend of this Section.Type Ia SupernovaeThe dimming of distant supernovae constitutes a solid probe of void models in the interval0.01 . z . 1.5, as the luminosity distance depends on all the parameters of the modelin a nontrivial way. One should in principle consider variations on the color and stretchparameters when fitting the model [157, 158]. However, the result of the analysis should notvary significantly since the LTB models we are considering usually give luminosity distancecurves very similar to the standard model. The present analysis also includes the covariancematrix between the supernova data, which adds information about this calibration procedureby taking into account the covariance between supernovae with similar color and stretch. Thesupernovae data will be used as presented in Section 2.5.Local Expansion RateThe dependence of the expansion rate with redshift in a LTB cosmology is in general verydifferent than in the ΛCDM case, even for low redshifts z < 0.1. In order to reproduce themethod used in [165] and provide a more fair comparison, we implement the constraints on themodel using supernovae luminosities rather than the model parameter Hin. The constraintslabeled as H0 are implemented as described in Section 2.5.1.Baryon Acoustic ScaleAlthough the use of BAO to constrain LTB models has raised some criticism [27, 164], we willrely on our results from Section 3.2 showing that the baryonic features remain at constantcoordinate positions to a good approximation and relating the transverse and radial BAOscales at different redshifts to the asymptotic values. The data used are those presentedin Section 2.4.1 and summarized Table 2.2. These measurements are given in terms of thevariable dz, which in our model is computed as (3.41), or alternatively (3.43), (3.44). Theadditional data point involving the angular correlation is to be compared to the theoreticalθBAO prediction (3.37).Cosmic Microwave BackgroundThe cosmic microwave background radiation in LTB models has been actively investigated[199–203], as it constitutes the most solid piece of evidence for statistical isotropy and themost powerful tool in cosmological constraints. The obtention of precise constraints fromthe CMB is beyond the scope of this work, where we focus only on geometric aspects of themodel. Therefore and only a relatively simple analysis based on the location of the first peakswill be employed, in order to give an idea of the effects of calibrating the standard rulers.This method yields weaker constraints than using the whole WMAP data and the spectracomputed in linear perturbation theory, WMAP distance prior R, la, z∗ [19] (See Section 3.3and Figure 3.7) or other model independent determinations [124].If our galaxy is located very close to the center of the void, the radiation coming fromthe CMB will be highly isotropic and therefore well described by the angular power spectrumCl, with no direction dependence. As usual, it will display a characteristic pattern of peaksand troughs located at multipoleslm = (m− φm) lA , (3.47)55Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4343638404244zDL@MagDLFRWCGBHOCGBHwFRW0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.6-0.4-0.20.00.20.40.6zDL-DLHLL@MagDFigure 3.4: Supernovae data and luminosity distance. The gray points correspond to theUnion2 compilation [163] used in the computation of the likelihood. The black points cor-respond to a binning of the same data using the covariance matrix, and are intended onlyfor visual aid. Color lines correspond to the minimum χ2 models described in Section 3.3,rescaled with the optimal value of µ0, as described in the text.0.1 0.2 0.3 0.4 0.5 0.6 0.70.050.100.150.200.250.300.350.40zd zLFRWCGBHOCGBHwFRW0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.03-0.02-0.010.000.010.020.030.04zd z-d zHLL6dF SDSSCarnero et al .WiggleZFigure 3.5: BAO data and dz for the best fit models described in Section 3.3 and residualswith respect to ΛCDM. The point at z = 0.55 has been converted from angular to volumedistance by means of a fiducial model.563.3. Analysis and Resultswhere integer values of m label the peaks, half integer values correspond to troughs, and φmare corrections that depends on the details of the cosmology before the recombination epoch.The overall factor is fixed by the CMB acoustic scalelA = piDA(z∗)rs(z∗)(1 + z∗)−1, (3.48)determined by the ratio between the observed angular diameter distance until recombinationand the sound horizon at that epoch. Further information on the cosmological parameterscan be obtained by considering the relative heights of the acoustic peaks compared to thefirst one Ha = Cla/Cl1 .The decoupling epoch occurs at an early time when the universe is very homogeneousand the primary anisotropies are produced on our past lightcone at a radius much larger thanthe size of the void r(z ∼ 1100) R. In this case the pre-recombination physics is effectivelythe same as in a homogeneous universe, and we can assume that the relative peak positions(m− δφm) and heights Ha are those of a FRW universe with the effective asymptotic valuesof the LTB model discussed in Section Section 3.2.3.7 On top of modifying these asymptoticparameters, the only effects from the void will be to shift the peaks by varying the acousticscale lA through the angular diameter distance DA(z∗). Our analysis neglects secondarycontributions such as the integrated Sachs Wolf effect on the lower multipoles or the actionof gravitational lensing, which affects the relative heights of the peaks. Furthermore, we willassume no radiation contribution to the angular distance to recombination.8In order to compare the theoretical predictions and the observations we will follow theapproach described by Marra and Pa¨a¨kko¨nen [189]. The corrections to the peak locations φmand heights depend on the effective parameters through the ratio of matter-radiation densityand recombination and the physical baryon density Ωmh2, as well as the spectral index nsthat characterizes the power spectrum of primordial perturbations. Note that relaxing thecommon assumption of a nearly scale invariant primordial spectrum considerably reducesthe tension between CMB and the local expansion rate [158]. Accurate fitting formulae interms of these quantities are provided in reference [123] for the recombination epoch z∗ andthe sound horizon rs(z∗), reference [214] for lm with m = 1, 32 , 2, 3 and reference [215] for therelative height Ha of the second and first peak a = 2, 3. Figure 3.6 shows the location of thepeaks reconstructed using this method.The total likelihood is given byχ2CMB =∑m∈{1, 32,2,3}(lobsm − lLTBm)22σ2lm+∑a∈{2,3}(Hobsa −HLTBa)22σ2Ha, (3.49)where the positions and heights of the peaks are those matching the WMAP 7 year best fitmodel. As in reference [189], we have taken the errors to be of 1% for the position of thefirst peak and 3% for the remaining parameters. It is important to note that this likelihoodanalysis is very simplified and its main aim is to provide an insight on how the informationfrom the CMB helps to sharpen the BAO constraints by fixing the initial size of the standardruler.7Variations in the effective CMB temperature have not been considered because for the profiles underconsideration (compensated voids) there is no significant departure from T0 = 2.725K [176, 189].8See references [201, 212, 213] for discussions on radiation in the context of LTB models.57Chapter 3: Large Scale Homogeneity and Non-Copernican Void Modelsæææ æààà àììì ìòòò òæàìò200 400 600 800 1000100020003000400050006000lC lLFRWwFRWCGBHOCGBHæ æææà àààìì ììò ò òò200 400 600 800 1000-0.04-0.020.000.020.04lDClClFigure 3.6: CMB spectrum. Black dots and error bars correspond to the binned WMAP7data and the red line is the CMB only best fit for ΛCDM [18]. The color points are thereconstructed positions of the peaks using the method described in Section 3.3.1 for theminimum χ2 models (Section 3.3). For visualization aid, the WMAP best fit height has beenassumed for the first peak, which is equivalent to a normalization, and the first through dueto the lack of a fitting formula (The residuals in H3/2 are due to the propagation of l3/2).Although the formulae do not exactly recover the values computed in linear perturbationtheory, they fall within the assumed errors (1% for l1, 3% for the rest). Note that theLTB models require values (fb ≈ 0.7, ns ≈ 0.6 that depart considerably from the standardmodel (Section 3.3). More precise constraints taken into account the full spectrum wouldconsiderably lower the quality of the fit.3.3.2 MCMC AnalysisIn order to constrain the parameter space and address the viability of the different models,we run several Markov Chain Monte Carlo (MCMC) analysis using a modified version ofthe publicly available code CMBEasy [216], which includes the integration of the coordinatesover the lightcone and the computation of the cosmological observables in the LTB model.The data sets employed are described in Section 3.3.1. CMBEasy’s built-in MCMC driverestablishes the convergence of the chains through the test of Gelman and Rubin [217], whichdetermines the length of the burn-in sequence and freezes the step-size, which is a necessarycondition for the convergence of the MCMC algorithm [218] (see Appendix A). In addition,the chains were monitored manually to ensure a proper sampling of the parameter space.Additionally to the CGBH and the OCGBH models described in Section 3.1.1, a ΛCDMmodel and a wCDM model with constant equation of state were studied using the samedata. Separate runs were performed for each of the displayed contours corresponding tothe constraints of the separate sets (CMB, BAO, SNe) as well as the combined constraintsH0+BAO+CMB+SNe. For the inhomogeneous models the additional combinations H0+SNeand BAO+CMB were considered, which combine the information of standard candles/rulerstogether with their calibrations (as opposed to the SNe/BAO-only). All the runs used flatpriors on the model parameters, which are given in Table 3.1.The results from the combined constraints can be seen in Table 3.2. Figures 3.7, 3.8,3.9 and 3.10 show the two-dimensional marginalized likelihood contours obtained from the583.3. Analysis and ResultsFRW ModelsH0 [Mpc/km/s] ΩM ΩΛ −w 100fb ns30− 90 0.05− 0.8 0− 1.2 0− 5 1− 25 0.05− 1.3GBH-LTB ModelsHin [Mpc/km/s] Ωin Ωout R [Gpc] ∆R [Gpc] 100fb ns30− 90 0.01− 0.5 0.1− 1 0− 5 0.5− 5 1− 25 0.05− 1.3Table 3.1: Priors on the model parameters used in the MCMCs. In order to facilitate thecomparison between the two LTB models, for the CGBH profile we have fixed the value ofH(r = 0) ≡ Hin at the center of the void instead of the more obscure parameter H0. Ωoutand w are only varied in the OCGBH and wCDM models.individual and combined data sets. Our discussion starts by considering the homogeneousreference models. Then the results for the inhomogeneous CGBH and OCGBH profiles willbe addressed, and the goodness of fit of the different models compared using different criteria.3.3.3 Homogeneous ModelsFor ΛCDM the recovered parameters are in good agreement with previous results. However,the region compatible with CMB data (first plot in Figure 3.7) is broader than usual aroundthe flatness line. This lack of precision is caused by the partial use of the CMB data (i.e.only the peaks instead of the whole Cl spectrum), together with the broad parameter priorsallowed. When combined with other measurements, it also affects the recovered value of thecurvature Ωk = 0.003+0.015−0.025, which is still very close to flat but has larger error bars thanusual. In a computation taking into account the full WMAP7 data, a deviation with respectto the measured values of the peak positions would also displace many of the intermediatepoints and cause a more dramatic decrease of the likelihood, leading to tighter bounds.The weakness of the CMB constraints is reflected again in the recovered values for thewCDM model (second and third plots in Figure 3.7). In this case all the parameters exceptH0, which is independently constrained by the nearby expansion rate, depart considerablyfrom the standard ones. These allow for lower values of the baryon fraction and the spectralindex, which in turn increase the matter fraction and decrease the dark energy content. Avery dramatic consequence of the weakness of these constraints can be seen in the recoveredvalue of the curvature Ωk = 0.04 ± 0.02, two sigma away from flatness. The low value ofΩΛ ≈ 0.62 is then compensated with an anomalously low equation of state w ≈ −1.34.99The discrepancies disappear when the WMAP7 distance prior R, z∗, lA [19] are used instead of the peakpositions (black, unfilled contours in Figure 3.7), recovering Ωk ≈ 0 and w ≈ −1. These quantities have beenvery accurately determined by the WMAP collaboration using the full CMB spectrum, and are able to breakthe degeneracies in the model (e.g. the baryon fraction). Although they are considerably more precise thanthe CMB peak information we used (described in Section 3.3.1), the WMAP distance prior can not be directlyapplied to inhomogeneous models (e.g. LTB models with decoupling redshifts z∗ & 1110 considerably higherthan the standard value z∗ = 1091.3± 0.9 can yield a good fit [199]).59Chapter 3: Large Scale Homogeneity and Non-Copernican Void ModelsΛCDM ModelH0 [Mpc/km/s] ΩM ΩΛ 100fb nsAll - Min χ2 70.7 0.28 0.72 17 0.97Marginalized 70.3+1.7−1.5 0.27±0.03 0.73± 0.05 17± 0.04 0.99+0.06−0.09wCDM ModelH0 [Mpc/km/s] ΩM ΩΛ w 100fb ns72.8 0.32 0.66 −1.26 12 0.8773.5 ± 2.3 0.33 ± 0.04 0.64 ± 0.06 −1.26+0.17−0.22 0.10+0.03−0.02 0.82+0.08−0.06asymptotically flat Constrained GBH model (CGBH)Hin Ωin R [Gpc] dR [Gpc] 100fb nsAll - Min χ2 66.4 0.21 0.02 2.78 7.7 0.74Marginalized 66.0±1.4 0.22± 0.04 0.18+0.64−0.18 2.56+0.28−0.24 7.7± 0.4 0.74± 0.03BAO+CMB 61.6±2.4 0.32+0.06−0.04 3.92+0.48−3.71 2.76+0.50−0.88 7.8± 0.8 0.73± 0.04SNe+H0 74.0±2.6 0.07± 0.04 1.95+1.22−1.82 3.19+1.63−1.66 - -asymptotically Open Constrained GBH model (OCGBH)Hin Ωin Ωout R [Gpc] dR [Gpc] 100fb ns71.8 0.21 0.87 0.30 1.48 6.3 0.6771.1± 2.8 0.22±0.04 0.86± 0.03 0.20+0.87−0.19 1.33+0.36−0.32 6.2± 0.5 0.68± 0.0363.8+4.2−2.8 0.35±0.06 0.98+0.02−0.11 0.72+2.5−0.67 1.79± 0.89 6.8± 0.9 0.69+0.05−0.0373.4+3.1−2.1 0.06±0.04 0.89+0.09−0.25 0.80+1.66−0.74 1.63+2.04−0.79 - -Table 3.2: Parameters from the MCMC including H0+SNe+BAO+CMB discussed Section3.3. The first lines correspond to the minimum χ2 models, while the second lines correspondsto the best fit model with one sigma errors after marginalizing over the remaining parameters.Note that since the LTB models do not give good fits to the data the errors are apparentlyvery small (see Figures 3.7, 3.8, 3.9 and 3.10). For the sake of comparison, in the case of theLTB models the results from the separate fits using BAO+CMB and SNE+H0 have beenadded (third and fourth lines).603.3. Analysis and ResultsΛCDM ModelwCDM ModelFigure 3.7: One and two sigma regions for the marginalized likelihood function correspond-ing to the ΛCDM and wCDM homogeneous model as obtained from BAO (green), CMB (Or-ange) and SNe (blue). Gray contours are the combined constraints H0+BAO+CMB+SNe.Note that the CMB compatible regions are much broader than usual due to the simplifica-tion of the method. Black unfilled lines in the wCDM plots correspond to using the WMAPdistance prior R, z∗, lA [19] combined with H0+BAO+SNe or individually (shown only in theΩΛ − Ωm plane), which recover the standard results.61Chapter 3: Large Scale Homogeneity and Non-Copernican Void ModelsNote also how the BAO and SNe contours span a similar region in both cases. This is aconsequence of them being determined by measurements of standard rulers and candles witharbitrary calibration, over a comparable redshift interval (due to the new BAO data providedby the WiggleZ collaboration, up to z ∼ 0.8). Since in FRW both datasets depend only onthe same distance-redshift relation and are consistent with each other, they yield basicallythe same information and the recovered regions overlap.3.3.4 Inhomogeneous ModelsDiscrepancies between the different datasets are encountered for both models regarding thematter content and the expansion rate at the center of the void, as can be seen in Figures3.8, 3.9, 3.10. This becomes particularly clear for the Ωin − hin plane.The most severe problem for both models is the existing tension regarding the value ofΩin determined from BAO and SNe, which differs by roughly 3σ for the two models (see figures3.8, 3.9). For example, the asymptotically flat model (CGBH) the BAO-only 1D marginalizedlikelihood yields Ωin = 0.28+0.06−0.05(1σ), a much higher value than determined by SNe Ωin =0.07± 0.04(1σ). This discrepancy is showing how the distance redshift relation necessary toexplain the supernovae dimming is incompatible with the stretch of the standard ruler insidethe void due to the inhomogeneous rescaling discussed in Section 3.2. The low value of Ωinnecessary to fit SNe observations increases the expansion rate and therefore stretches theBAO scale considerably near the center, making it incompatible with the observed values athigher redshift. This is a purely geometric discrepancy, valid for arbitrary calibration of thestandard candles and rulers, and is completely independent of the dynamics originating thecharacteristic length. Figures 3.4, 3.5 show how the best fit models represent a compromisethat fails to fit both datasets at low redshift, where the BAO rescaling (Fig. 3.3) is largest.Naturally, the tension becomes more dramatic when the SNe data are compared tothe BAO+CMB combination, because the CMB effectively reduces the allowed range of theinitial BAO scale by constraining Hin, fb and Ωout. The independence of the constraints onthe initial BAO scale is the reason why the asymptotically open model (OCGBH) does notease the tension between BAO and SNe. This is partly because the apparent freedom gainedfrom allowing Ωout to vary does not provide essentially different values of the asymptotic BAOscale, already ensured by the freedom in fb. The other reason is that neither SNe nor BAO-only constraints seem to depend on the value of Ωout. We can regard the dashed green andblue contours in Figures 3.8, 3.9, 3.10) as the purely geometric constraints for arbitrary valuesof the standard rulers/candles, while the filled green and blue contours would correspond toadding priors to those calibrations.The asymptotically flat model (CGBH) also shows a tension between the value of Hindetermined by H0+SNe and CMB+BAO, both being discrepant at 3σ. Note that the tensionis manifest even using very simplified CMB data, although these yield much looser constraintsthan the full Cl spectrum. In the asymptotically open model, the additional freedom achievedallows one to recover agreement by reducing the value of Ωout, yielding a concordant valuefor Hin from the different datasets. However, the tension would reappear in more a thoroughanalysis including the full CMB power spectrum data, which typically require hin ∼ 0.4−0.5[153]. This increase in the local expansion rate also reduces the age of the universe, which isproportional to H0(r)−1. Although it has not been explicitly accounted for in the MCMC,Figure 3.11 shows how models with a higher expansion rate enter in tension with the limits623.3. Analysis and ResultsFigure 3.8: CGBH model. One and two sigma regions for the marginalized likelihoodfunction as obtained from BAO (green), CMB (Orange) and SNe (blue). The filled contoursrepresent the combined constraints using SNe+H0 (blue) and BAO+CMB (green). The blacklines correspond to the combined data BAO+CMB+SNe+H0.63Chapter 3: Large Scale Homogeneity and Non-Copernican Void ModelsFigure 3.9: OGBH model. One and two sigma regions for the marginalized likelihoodfunction as obtained from BAO (green), CMB (Orange) and SNe (blue). The filled contoursrepresent the combined constraints using SNe+H0 (blue) and BAO+CMB (green). The blacklines correspond to the combined data BAO+CMB+SNe+H0.643.3. Analysis and ResultsFigure 3.10: Continuation of Figure 3.9.on the age of the universe obtained from Globular Clusters [219], posing yet another difficultyfor this type of models.The recovered values of the baryon fraction and the spectral index, mainly determinedby the CMB, are much lower than in standard cosmologies. This trend agrees with previousstudies in which compensated voids are constrained using the full CMB [176]. We note thatthese features are not relevant to our discussion because 1) the main results are geometricand do not depend on the details of the CMB physics and 2) the obtained values of fb, nsrely on a simplified treatment of the CMB peaks.The best fit models turn out to be rather cuspy, as can be seen in the preference towardsR ≈ 0 Gpc in both the asymptotically flat and open voids. Since the fit is not very good andthe individual observations are not very restrictive by themselves (including the combinationsBAO+CMB, SNe+H0), this feature might well be due to a compromise between the differentdatasets, and could be related to the fact that cuspy voids achieve better resemblance to an65Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models10 11 12 13 14 150.00.20.40.60.81.01.2tbb @Gyr DLL max95%ExcludedLFRWCGBHOCGBH wFRWFigure 3.11: Age of the universe for the best fit models. The curves correspond to marginal-ization of the MCMC chains over the (homogeneous) Big Bang time (note that it is a derivedquantity rather than a parameter varied in the exploration). The gray area shows the regionexcluded by the age of Globular Clusters in the Milky Way.accelerating universe at low redshift [171]. In this case, the size of the void is given by thesteepness of the inhomogeneity ∆R, which acquires a value ∼ 2.5 Gpc in the flat case but asmaller value ∼ 1.5 Gpc in the asymptotically open case. Again, the individual data sets donot yield significant enough information about the value of R,∆R. Other than the smallnessof the asymptotically open void and the better agreement it gives on the value of H0, thereis no significant difference between the two models.3.3.5 Model ComparisonWe now proceed to compare the different models under several criteria. The tensions betweenthe different datasets (Figures 3.8, 3.9 and 3.10) and the poor fit to SNe and BAO (Figures3.4 and 2.2) will be reflected in poorer figures with respect to the homogeneous models.Furthermore, inhomogeneous models will be additionally penalized because they have a largernumber of parameters. Table 3.3 shows the χ2 values associated to the different observationsand the total values, as well as the result of the different model comparison criteria discussedbelow.The standard frequentist analysis of parameter estimation, given a set of data, is notvery useful for model selection, since it is difficult to compare models with different numberof parameters. For instance, the usual method of comparing minimum χ2 per effective degreeof freedom normally misses the point and is not very decisive, as can be seen in the veryclose values achieved by the different models. Other methods to decide which model givesthe best description include various Information Criteria which additionally penalize modelsdescribed by more parameters. Such include the (corrected) Akaike Information Criterion(AIC) [220] and Bayesian Information Criterion (BIC) [221], given byAIC = χ2min + 2k +2k(k − 1)N − k − 1 , (3.50)BIC = χ2min + k lnN , (3.51)663.3. Analysis and ResultsCGBH OCGBH ΛCDM wCDMUnion SNe 539.94 539.06 530.70 530.40Hubble µ0 6.97 0.38 2.17 0.146dF 5.35 4.73 0.35 0.09SDSS 0.73 0.04 1.29 1.24WiggleZ 0.65 1.20 0.93 0.63Carnero et al. 0.78 0.12 0.61 0.34Total BAO 7.51 6.09 3.18 2.30Peak positions 0.87 0.30 0.96 0.07Peak heights 1.13 0.11 0.24 0.04Total CMB 2.00 0.41 1.20 0.11Total χ2 +19.89 +9.40 536.56 -3.62# free parameters 6 7 5 6χ2/d.o.f. 0.985 0.968 0.948 0.943Akaike IC (3.50) +22 +13 546.6 -1.6Bayesian IC (3.51) +26.2 +15.7 568.3 +2.7Bayes factor (3.53) +10 +6 282.2 +2.6Table 3.3: χ2 contributions to the maximum likelihood models as found by the MCMCswith H0+BAO+CMB+SNe and results from different model comparison criteria discussedin Section 3.3.5. The values χ2 as well as the model comparison criteria are given show thetotal value for the ΛCDM best fit, while the other models are given relative to those (minusvalues are favoured w.r.t. the concordance model, while positive values are disfavoured). TheBayes factor is given by the difference in − logE (3.53).where k is the number of free parameters of a given models and N the number datapointsused in the constraints. The Bayesian evidenceE(D|M) =∫du L(D|u,M)pi(u,M) , (3.52)is given by the integral of the likelihood L(D|u,M) over the values of the model parametersu allowed by the priors pi(u,M). The computation of the Bayesian evidence is difficult ingeneral, therefore, we will use a simple expression which can be obtained provided that thelikelihood is a single isolated peak, far from the edges of the prior ranges [184]− lnE = − lnLmax + lnA+n∑iln(umaxi − umini ) , (3.53)where A is the normalization of the likelihood, and [umini , umaxi ] is the range of parameter uiallowed in the MCMC exploration (assuming a flat priors), i = 1 . . . n. Moreover, for the67Chapter 3: Large Scale Homogeneity and Non-Copernican Void Modelscase of a Gaussian likelihood,L(u) = A exp[− 12uTC−1u], (3.54)we find A = (2pi)−n/2/√detC, where C is the covariance matrix and xi = ui − u¯i. It isclear that whenever the prior ranges are too big for the likelihood, the Bayesian evidence ispenalized. An estimate of the covariance matrix can be obtained assuming that the obtainedparameters are independent from each other. In that situation, the covariance matrix isgiven by the square of the one sigma allowed ranges in each parameter and the determinantbecomes detC =∏i σ2i , where we take the average between the upper and lower bounds givenin Table 3.2. The determinant computed using the eigenvalues of the covariance matrix usedin the MCMC sampling yields similar results.The bottom part of Table 3.3 shows similar values for the homogeneous models, but thepreferred one depends ultimately on the chosen criterion. It is interesting to note that theBayes factor favors the simpler ΛCDM with a difference of 2.6, despite the slightly better χ2fit of the wCDM model. This difference is due to the presence of an additional parameter, w,together with the large prior postulated for it, [−5, 0], relative to the 1σ region, ∆w ≈ 0.2.The logarithm of the Bayes factor for the LTB models w.r.t. to the fiducial ΛCDM is 10and 6, for the asymptotically flat and open GBH inhomogeneous models respectively, due tothe bad fit and the larger parameter range explored. Although the asymptotically open modelyields a better Bayes factor, even with an additional parameter, both are strongly disfavoredaccording to Jeffreys’ scale, since the difference in the logarithm of the Bayes factor is higherthan 5. The Akaike and Bayesian Information Criteria also prefer the homogeneous model,the rejection being significantly stronger for the asymptotically flat case due to the extratension in the local expansion rate. Increasingly accurate data have significantly worsenedthe fits of inhomogeneous universes, which were compatible just few years ago (cf. reference[184]).3.4 DiscussionIn this Chapter we presented new constraints on inhomogeneous Lemaˆıtre-Tolman-Bondimodels in the light of the most recent cosmological data, focusing on profiles of the GBHtype with a space-independent Big Bang and baryon fraction. The inclusion of higher redshiftBAO data together with type Ia supernovae allows one to reject the models based only onBAO and SNe, independently of other observational data such as the CMB. Additionally,a model independent method to constraint the local expansion rate through a prior on thesupernovae luminosity was introduced.The physical BAO scale at early times was computed in terms of the asymptotic valueand then projected to different redshifts using the background LTB metric. This method isjustified due to the existence and stability of constant coordinate geodesic solutions, whichare expected to be followed by baryonic overdensities in position space. In addition to thetime evolution, the BAO scale is shown to become inhomogeneous and anisotropic due tothe different expansion rates in the radial and transverse directions. The dependence of theobserved BAO scale on both the cosmic distances and the evolution of the scale factor leadsgenerically to different predictions than pure distance indicators such as SNe. The departureis largest near the center of the void, precisely because there is less matter to slow down683.4. Discussionthe expansion that drives the growth of the BAO scale. Ultimately, the difference betweenthe two distances can be regarded as a concrete realization of more general tests of theCopernican Principle [222].The addition of BAO data at higher redshifts increases considerably their constrain-ing power in this type of models because they help to fix the asymptotic value. The resultrepresents a new drawback for this type of models, as the value of the local matter densityΩin & 0.2 preferred by BAO is about 3σ apart from the value Ωin . 0.18 found using Super-novae, as can be inferred from Figures 3.8 and 3.9. The tension between the two datasetspersists when asymptotically open models are studied, and worsens when the informationfrom the CMB is added, since it constraints the parameters involved in the acoustic scaledetermination (fb, h,Ωout). Asymptotically flat LTB models show an additional tension re-garding the value of the local Hubble rate when CMB and BAO are combined. AllowingΩout ≤ 1 relaxes this incompatibility, but we expect it to re-emerge in a more detailed anal-ysis of the CMB including all scales and secondary contributions such as the ISW effectand gravitational lensing. Additionally, larger values of the expansion rate might render theuniverse too young to account for the ages of stars in globular clusters. The adiabatic GBHmodels fail to simultaneously fit the data, and a Bayesian analysis shows that they are ruledout at high confidence.The above results were obtained for a particular choice of the matter profile. However,the difficulties of the model are manifest in the determined value of matter contrast at thecenter of the void, while the remaining parameters are poorly constrained by individualdatasets. The departure between cosmic rulers and candles becomes most severe at thecenter of the void, and we expect that Ωin ≡ ΩM (0) captures this tension regardless of otherfeatures. Since Ωin can be defined for any LTB model independently of the parameterization,it is reasonable to expect this result to hold for all large void models with space-independentBig Bang and baryon fraction. Nonetheless, we have to keep in mind that the SNe andBAO constraints depend on all the parameters through the distance determinations and theevolution of the BAO scale up to a certain redshift, and therefore different shapes for theprofile might soften the tension between the two datasets.10Similarly to the dependence of CMB constraints with the primordial power spectrum[158], it is conceivable that fine tuned initial perturbations could be used to reconcile the BAOobservations with SNe in adiabatic voids. However, such conditions would not only need toprovide an enhanced scale to explain the observed feature in galaxy correlation, but also tohide the actual BAO scale that would naturally form due to the existence of a preferred scale(the sound horizon at the recombination epoch). On top of this challenging task, the fakeBAO scale should be shorter near the center to compensate the inhomogeneous growth andfit the observations, therefore requiring some amount of radial dependence that would be atodds with the (quasi) homogeneous initial state.The effects of the inhomogeneity on the BAO scale are unavoidable. Even if a differentvoid profile might yield a slightly better fit, more precise data e.g. from future surveys suchas Euclid [108, 223] will eventually be able to distinguish adiabatic LTB models from thehomogeneous case regardless of the shape of the inhomogeneity. Together with the remainingobservational problems for large void models with space-independent Big Bang, this sets the10Recently, genetic algorithms were used to analyze the sensitivity of LTB model profiles to the data withoutthe introduction of a parameterization [34]. The results were not conclusive with respect to the shape of thevoid profile.69Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24120140160180200Wm h²rsHzdL@MpcD fb=0.01fb=0.21CGBH best fit H fb=7%LFigure 3.12: FRW, coordinate baryon acoustic scale rs(zd) as a function of the physicalmatter density Ωmh2 for different values of the baryon fraction. The mass of the baryons actslowering the speed of sound in the baryon-photon fluid, and therefore increasing their amountreduces the resulting acoustic scale, which is related to the sound horizon. An LTB modelwith a higher baryon fraction near the center might render the BAO and SNe observationscompatible by lowering the value of dz ∝ rs(zd) near the center of the void (see Figure 3.5).stage for abandoning the adiabatic assumption. A scenario with space dependent Big Bangtime would require more careful considerations on the origin of the BAO scale to account forthe early time inhomogeneity, but it is still possible that the freedom gained from decouplingH0(r) from ΩM(r) renders BAO and SNe observations compatible, although the modulationof the Hubble rate is restricted by the local and asymptotic values, fixed by SNe luminositypriors and the CMB.11A simpler possibility to reconcile SNe and BAO would be to allow for large scale baryonisocurvature modes, and induce a radial dependence on the early time BAO scale througha non-constant baryon to matter ratio fb(r), a possibility that has been explored as a wayto explain the observed abundances of primordial nuclei and attempt to solve the primordiallithium problem [180]. Figure 3.5 suggests that lowering the local value near the center of thevoid would give a nicer fit to the observations, since the value of dz ∝ rs/DV is proportionalto the physical acoustic scale. A higher baryon fraction acts reducing the speed of sound ofthe baryon-photon fluid, therefore shortening the sound horizon that determines the observedBAO scale (see Figure 3.12). Adding more baryons at the center of the void would providethe necessary freedom to compensate for the inhomogeneous expansion and render the modelphenomenologically viable, although more involved and less appealing.12To summarize, we have shown how the BAO scale, acting as a standard (but evolving)ruler and the supernovae explosions, acting as standard candles, lead to different predictionsin an inhomogeneous universe, which are disfavored by current data. The conclusion ofthe analysis is that the use of purely geometric probes, that only recently have become11Models with arbitrary H0(r) also predict a too large kinematic Sunyaev-Zel’dovich effect [36, 203]. Itmight be still possible to avoid these constraints by including an additional baryon to photon profile η(r) [8].12Baryon isocurvature modes are severely constrained by CMB observations [19, 224]. However, theseconstraints apply to scales much smaller than the size of the void.703.4. Discussionsufficiently constraining, is able to rule out the whole class of adiabatic LTB models. This isindependent of other dynamical constraints, like those coming from the kinematic Sunyaev-Zel’dovich effect or the integrated Sachs-Wolfe effect, which in the near future can be usedto definitely rule out all inhomogeneous models without dark energy. The present resultsare also relevant for observationaly constraining more general inhomogeneous models [225]including some recent proposals that incorporate dark energy [226–229].71Chapter 3: Large Scale Homogeneity and Non-Copernican Void Models72Chapter 4Phenomenological Modifications:Entropic GravityPlay! Invent the world! Invent reality!Vladimir Nabokov1Phenomenological models are a common way to observationally test consistency rela-tions within the standard model. The most known example is the constant equation of statedark energy model wCDM. It is not intended to represent a physically realistic scenario, yetit is useful to encapsulate possible departures form the standard model with the addition ofa single parameter. Useful parameterizations should be able to interpolate between differenteffects, e.g. evolving/static dark energy or scale dependence of gravitational clustering. Inaddition, they should contain the standard model as a particular point of the parameterspace. Testing them against observations is then a simple way to regard the data and theirsensitivity to departures from the standard paradigm, and the significance of such departureswhenever present.A different but related idea is to use phenomenological models to describe certain physi-cal situations that are difficult to consider in full detail, or for which a complete, self consistentdescription does not exist. Hopefully, one can then be able to predict certain physical phe-nomena and use them to test the underlying principles in a given context. This Chapterundertakes the study of recent ideas about the internal structure of the space-time, namelythat gravity is an entropic force of thermodynamical origin. Modifications of gravity canbe introduced within this paradigm only phenomenologically, due to a lack of a Lagrangianformulation. The Friedmann equations corresponding to these models have been obtained byseveral authors from the Newtonian force, following an argumentation that yields the rightresult in the case of General Relativity. Using this as a starting point, the modified FRWevolution is used to constraint these scenarios.1Look at the Harlequins! (1974)73Chapter 4: Phenomenological Modifications: Entropic Gravity4.1 Gravity and ThermodynamicsThe notion of gravity as an emergent force has been contemplated for a long time [230]. Thederivation of gravitational field equations from thermodynamics by Jacobson [231] supportsthis, and has lead to further substantial hints of evidence for the idea [232]. Recently, theproposal was put forward that gravity is a thermodynamic phenomenon emerging from theholographic principle [233]. It was argued that the Newton’s law of gravitation can beunderstood as an entropic force caused by the change of information holographically storedon a screen when material bodies are moving with respect to the screen. This is described bythe first law of thermodynamics, F∆x = T∆S, connecting the force F and the displacement∆x to the temperature T of the screen and the change of its entropy, ∆S. T can be thenidentified with the Unruh temperature without referring to a horizon. Newton’s second lawfollows if the entropy increase is ∆S = 2pim∆x, m being a particle mass and ∆x the distanceto a holographic screen. More to the point, assuming equipartition of the energy [234] givenby the enclosed mass, Newtonian gravitation emerges.Cosmology has been also considered in this framework [235, 236]. As is well known,the Friedman equation can be deduced from semi-Newtonian physics. Thus it ensues fromthe above arguments as shown by reference [237]. Reference [233] has also inspired modifi-cations to the cosmic expansion laws. The purpose of the present Chapter is to uncover theimplications of such modifications. Two approaches are investigated. One set of correctionsto the Friedmann equations is motivated by the possible connection of the surface termsin the gravitational action to the holographic entropy. Reference [238] noted that (at thepresent level of the formulation) this is equivalent to introducing sources to the continuityequations, implying non-conservation of energy. In another approach, the derivation of theFriedmann equation as an entropic force from basic thermodynamic principles is generalizedby taking into account loop corrections to the entropy-area law [239]. The result is corrob-orated by its concordance with previous considerations [240], and is consistent with energyconservation though still lacks a covariant formulation. While it might seem preliminary toinvestigate in detail the predictions of these models whose foundations, at the present stage,are rather heuristic, we believe it is useful to explore their generic consequences. Knowingthe possible form of viable extensions to our standard Friedmannian picture along the linesof reference [241] can shed light on the way towards more rigorous derivation of the effectiveentropic cosmology, and on the prospects of eventually testing these ideas by cosmologicalobservations.An encouraging result in this respect is that viable cosmologies in realizations of thequite different approaches we focus on, possess an identical expansion rate (in the simplestbut relevant setting of a universe filled by a single fluid dominated cosmology). It is alsointeresting that at high curvatures this expansion rate reduces to a constant. Thus not a bigbang singularity, but instead inflation is found in the past. However, we find that the highercurvature terms, motivated by quantum corrections, are not viable as they lack a consistentlow-energy limit. Furthermore, we impose bounds on the unknown parameters of the modelsby considering the scale of inflation, big bang nucleosynthesis (BBN) and from the modifiedbehavior of dark matter in the post-recombination universe.Although only phenomenologically motivated, an interesting case is a monomial correc-tion to the area-entropy law. Such can result in acceleration without dark energy with a goodfit to the data. We perform a full Markov Chain Monte Carlo (MCMC) likelihood analysis744.2. Modifications from Surface Termsexploiting astronomical data from baryon acoustic oscillations [242], supernovae [163] andcosmic microwave background [19]. A slightly closed universe turns out to be preferred bythe data, unlike in the standard ΛCDM model. We consider also the evolution of pertur-bations, which is determined uniquely if the Jebsen-Birkhoff law is valid. A characteristicfeature is then the growth of gravitational potentials in conjunction with the modified growthof overdensities.The surface term approach is discussed in Section 4.2. In Section 4.3 we consider theimplications of a specific form of area-entropy law motivated by quantum gravity, and inSection 4.4 we explore a phenomenological power-law parametrization of this law, includinga discussion on the perturbation evolution 4.5. Finally, the results we obtained are conciselysummarized in Section 4.6.4.2 Modifications from Surface TermsEasson, Frampton and Smoot recently argued that extra terms should be added to the ac-celeration equation for the scale factor. This was discussed from various points of view, inparticular it was conjectured that the additional terms can stem from the usually neglectedsurface terms in the gravitational action. Present acceleration of the universe [243] and infla-tion [244] were proposed to be explained by the presence of these terms without introducingnew fields. This is obviously an exciting prospect warranting closer inspection.Slightly different versions of the acceleration equation were introduced in both of theabove mentioned papers. The following parametrization of the two Friedmann equationsencompass all those versions and their combinations:2H2 =8piG3ρ+ α1H2 + α2H˙ + 8piGα3H4 , (4.1)H˙ +H2 = −4piG3(1 + 3w)ρ+ β1H2 + β2H˙ + 8piGβ3H4 .The six coefficients αi, βi are dimensionless for all i = 1, 2, 3. The extrinsic curvature at thesurface was argued to result in α1 = β1 = 3/2pi and α2 = β2 = 3/4pi and quantum correctionsin nonzero β3 [243, 244]. The equations imply thatdHdN= H(8piGc2H2 − c1), (4.2)where N = log(a) is the e-folding time andc1 ≡ (3− α1)(1 + w)− 2β12− α2(1 + 3w)− 2β2 , (4.3)c2 =α3(1 + 3w) + 2β32− α2(1 + 3w)− 2β2 . (4.4)Note that c1 is proportional to the lower order corrections, and c2 is proportional to the higherorder contributions ∼ H4. The information lost by having only one differential equation in(4.2) should be compensated by imposing boundary conditions from (4.1) to its solutions.2We will not adress the case in which the two equations degenerate to one by a particular choice of theparameters.75Chapter 4: Phenomenological Modifications: Entropic GravityIn the general case of multiple fluids, the model does not uniquely determine how to theEoS (equation of state) w evolves. The reason is that the two equations (4.1) result in onlyone (non)conservation equation for the total density, and we have no unique prescription forhow the relative densities behave if the total density consists of a mixture of fluids. Fromthe viewpoint of reference [238], the source terms for the individual fluids are undetermined.However, the most relevant special case of a single-fluid dominated universe allows an exactsolution where these ambiguities are absent.4.2.1 Single FluidIn the case that w is a constant, Eq.(4.2) can be easily solved:H2(a) =c18piG[c0(aa0)2c1+ c2] , (4.5)c0 =132pi2G2(1 + w)ρ0. (4.6)We chose the integration constant c0 in such a way that when the entropic corrections vanish,we recover the standard Hubble law. Although there are six unknown factors in (4.1), thecosmological implications are rather unambiguous and can be encoded in the two numbers c1and c2. Thus it is both feasible and meaningful to constrain them, despite our ignorance of atheoretical prediction. From the form of H in (4.5), it is also transparent that as a→ 0, wehave a de Sitter solution and the model indeed predicts inflation. As the scale factor grows,(nearly) standard evolution is recovered: so it is also simple to see the present versions ofthe model do not provide dark energy. At early times, w = 1/3, we can obtain constraintsfrom BBN, and from the inflationary scale by estimating the amplitude of fluctuations. Boththe scaling modification c1 and the constant term c2 can be bounded. At late times, w = 0,we can obtain constraints at least from the modified scaling law for dust, c1. Let us firstconsider the early universe constraints before adding a cosmological constant to obtain thepresent acceleration.Let us consider the constraints from the early universe. During radiation domination,Eq. (4.5) becomesH2 = 8piGρ12c1a2(c1−2) + 32piG2c2ρ. (4.7)From this we see that the variation effective Newton’s constant isδGeff/G ≈ (12c1 − 1)(1− 4c1 log a)− 16pi2G2c1c2ρ . (4.8)This variation can be bounded by requiring successful BBN. For instance, reference [114]derived that δGeff = 0.09+0.22−0.19. The radiation energy density is given byρ = g∗pi230T 4 , (4.9)where we use for the number of effective relativistic degrees of freedom g∗ at the nucleosyn-thesis temperature T ∼ 1 MeV the value g∗ = 10.75. Plugging in the numbers, we obtain− 3.5 · 10−3 < 2− c1 < 1.1 · 10−3 . (4.10)764.2. Modifications from Surface Terms− 2 · 1084 < c2 < 6 · 1084 . (4.11)Because of the tremendous hierarchy between the Planck and the BBN scale there is avery poor constraint on the high curvature corrections c2. This can be also written as2√piGc1/42 < 457.6GeV−1 by restoring the dimensions.If inflation is considered to be driven by the entropic corrections, we can estimate theirmagnitude from the amplitude of perturbations observed in CMB. The amplitude of thespectrum of quantum fluctuations of massless fields is expected to be given by the ratio〈δφδφ〉 = 8piGH2∼ 10−10 , (4.12)where is the slow-roll parameter and the right hand side is determined from observations.The spectral index as determined from observations gives ∼ O(0.01). Since at early timesequation (4.5) predicts (nearly) exponential expansion with the Hubble rate 8piGH2 = c1/c2,and we know that c1 must be of order one, successful generation of observed fluctuationsfrom entropic inflation suggest that c2 ∼ 1012. This estimate is much more tentative thanthe previous, since it depends on the detailed physics of inflation.4.2.2 Adding a Cosmological ConstantAcceleration at late times can be obtained by the addition of a Λ-term. Then (4.2) generalizestodHdN= H(8piGc2H2 − c1)+ c3ΛH, (4.13)c3 =1 + w2− α2(1 + 3w) + 2β2 , (4.14)and is solved byH2 =c14piGc2+√32piGc2c3Λ− c2116piGc2tanh(√32piGc2c3Λ− c21(N −N0))(4.15)The form of the Friedmann equation above is completely different than in the usual case. Thisis due to the nonlinearity stemming from the presence of higher curvature corrections. At lowcurvatures, their effect does not disappear as the most naive expectation would be and thelimit c2 → 0 of Eq.(4.15) is ill defined. The de Sitter solution is recovered at asymptoticallylate times, but the preceding evolution may not approximate standard cosmology and matterdomination is lost. To cure this, we suggest removing the higher curvature corrections (c2 = 0and consider two scenarios, one with a standard cosmological constant but modified matterscaling, and another one in which matter non-conservation is absorbed by the cosmologicalconstant, which is allowed to evolve in time.Prescription I - Constant Λ:In the pressence of a constant Λ term the behavior of the Hubble rate is just what is expectedfrom Eq.(4.5). Now (4.13) is solved byH2(a) =8piG3ρ0a−2c1 +13− 21+wβ1Λ . (4.16)77Chapter 4: Phenomenological Modifications: Entropic Gravity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 0.012L/L mδ c1MCMCFigure 4.1: Constraints on the modified equation of state for CDM and baryons scaling lawby using SNe, BAO and CMB distance priors. Radiation has been assumed to follow theusual scaling. Only very slight deviations from the usual scaling law are allowed.The integration constant ρ0 corresponds to the renormalised energy density at a = 1. Simi-larly, the cosmological constant is slightly ”dressed”. The conclusion is that the observableeffect to the expansion is the modified scaling of matter density.The modified scaling law of dark matter can be used to impose tight bounds from thelate universe observations. This has been explored in reference [245], who derived constraintson the EoS for dark matter, taking into account experimental data both on the backgroundand on the perturbations. Adopting the prescription where the Newton frame sound speedvanishes we can translate the result into our case as:− 8.78 · 10−3 < 2c1 − 3 < 1.86 · 10−3 . (4.17)for 99.7% C.L. bounds. Reference [245] took into account the full CMB and LSS data.However, as we cannot deduce the perturbation evolution in these models unambiguously, itis useful to consider constraints ensuing solely from background expansion. It turns out thatby including the latest data on SNeIa, BAO and CMB, the reached precision is only slightlylower. The result is shown in Fig. 4.1 and corresponds to the bounds− 17.28 · 10−3 < 2c1 − 3 < 20.50 · 10−3 . (4.18)at 99.7% C.L.Prescription II - Dynamical Λ: One can also consider the case that matter continuityequation is not violated. Then the Λ-term must be responsible for the non-conservation in aconsistent system. The Hubble law can be derived analogously to the above cases and onereadily finds that it now has the formH2 =8piGρ3(1 + 2(α1−β1)3(1+w)) + Λ0a2(α1−β1) . (4.19)784.3. Modifications from Quantum Corrections to the Entropy-area LawSo the Λ-term acquires a dynamical behavior. In case α1 − β1 < 0 this would help with thecoincidence problems, since one could consider initial large values for Λ, which has dilutedto the presently observed scale. Note that this is different from usual dark energy approach,where Λ is tuned to zero (or in any case to an even smaller than the value consistent withobservations) and then a new dynamical component is added to explain the acceleration.Reference [246] has also derived this result, which can be equivalently arrived at by imposingonly the second Friedmann equation in (4.1). The first one then follows by integration, andthe dynamical Λ can be viewed as an integration constant.In this scenario, the BBN constraint for the effective gravitational constant gives− 0.62 < β1 − α1 < 0.20 . (4.20)From WMAP7 measurements on the equation of state of dark energy, combined with othercosmological data [19], we obtain an even tighter bound,− 0.05 < β1 − α1 < 0.11 . (4.21)This is in qualitative agreement with reference [246], where the entropic corrections werebounded with the CMB acoustic scale.4.3 Modifications from Quantum Corrections to the Entropy-area LawThere is evidence from string theory and from loop quantum gravity that the two leadingquantum corrections to the area entropy-law are proportional to the logarithm and the inverseof the area [247, 248]. Reference [239] derived the Friedmann equation from an underlyingentropic force taking into account quantum corrections to the entropy formula. We slightlygeneralize their final result (Equation (25) in [239]) by allowing multiple fluids (labeled i)with constant equations of state wi and a cosmological constant, and we also allow for spatialcurvature:H2 +ka2=∑i[8piG3− 8β(1 + 3wi)G29(1 + wi)(H2 +ka2)− 2γ(1 + 3wi)G33pi(5 + 3wi)(H2 +ka2)2]ρi+Λ3[1− βpiG(H2 +ka2) log a+γ4pi2G2(H2 +ka2)2]. (4.22)As seen in the previous Section, there is a problem recovering the usual evolution atlow curvature if we include the high curvature correction proportional to γ. This can be seeneasily also in this case. Defining the shorthand notationsSβ ≡ −8βG29∑i1 + 3wi1 + wiρi − βGΛ3pilog a , (4.23)Sγ ≡ −2γG23pi∑i1 + 3wi5 + 3wiρi + γGΛ12pi2, (4.24)the solutions for the Hubble rate may be written asH2 +ka2=1− Sβ2GSγ±√1− 12piG2ρSγ2GSγ, (4.25)79Chapter 4: Phenomenological Modifications: Entropic Gravitywhere the total matter density is denoted by ρ =∑ρi. It is obvious that in the limit wherethe corrections tend to zero, we do not recover standard cosmological evolution. Thus thehigher order corrections here suffer from a similar problem as we encountered in the previousSection.Therefore we set γ = 0 and consider only effect of the leading logarithm correction tothe entropy, proportional to β . The solution for the Hubble rate can then be written as,neglecting the cosmological constant,H2 +ka2=8piG3ρ[1 +8βG29∑i(1 + 3wi1 + wi)ρi]−1. (4.26)Thus, the corrections occur near the Planck scale. If β is large enough this can supportinflation since the RHS tends to a constant when matter is relativistic and wi = 1/3 for allspecies i. Again, we can constrain this from the effective G at BBN. It is interesting to notethat the form of the entropic Friedmann equation assumes the same form as in the previousSection, where the derivation was quite different, the underlying physical assumptions leadingto a (non-)conservation and apparently different form of the force law.From (4.26), the effective variation of the Newton’s constant is now given byδGN/G =11 + 4βG23 ρBBN− 1 ≈ −43βG2ρ . (4.27)Using (4.9) for the radiation density at nucleosynthesis and proceeding analogously to Section4.2.1, we find that the BBN bound on the magnitude constant β is |β| . 2.7 ·1035, translatingto|√8piGβ| < 0.0042 1/GeV . (4.28)Again it is clear that BBN is not efficient to constrain the corrections. Furthermore, sincethe scale of inflation is below the Planck scale, we have to consider very large values of β.However, from considerations of loop quantum gravity and string theory the natural valuefor β is of order one. Considering such values, inflation takes place at the Planck scale, wherewe cannot trust the perturbatively entropy-area law, expected to hold only at the limit oflarge horizon size.4.4 Dark Energy from a Generalized Entropy-area LawIn the following we consider the possibility of infrared modifications to the large-scale behaviorof gravity. Such can ensue from corrections to the S ∼ A relation that grow faster than A.Among such is the volume correction that scales as ∼ A3/2. Interestingly, corrections of thistype imply, within the entropic interpretation of gravity, a modified Newton’s law which mayexplain the galactic rotation curves without resorting to dark matter [249]. This motivatesus to study whether we may generate modified gravity at the largest scales in such a waythat we would avoid the introduction of dark energy field or a cosmological constant.For this purpose, we consider the area-entropy law of the formS =A4`2P+ s(A) , (4.29)804.4. Dark Energy from a Generalized Entropy-area Lawwhere the function s(A) represents the quantum corrections and `2P = G~/c3. We assumethat A = QN , where Q is a constant to be determined, and that the entropy changes by onefundamental unit (corresponding to unit change in the number of bits on the screen withradius R) when ∆r = η~/(mc), r being the comoving radial coordinate. Then the first law ofthermodynamics together with the equipartition of energy leads to the modified Newtonianlaw of gravitation3F = − Q2c3Mm2pikB~ηR2(1`2P+∂s∂A)= −GMmR2(1 + 4`2P∂s∂A), (4.30)where in the second equality we made the identification Q2 = 8pikBη`4P . Let us furtherassume the power-law corrections(A) =4piσn(ar`P)2n∼ An . (4.31)This type of parametrization for entropic gravity effects has been recently considered by otherauthors [251]. Taking into account that in the cosmological context the active gravitationalmass is given by the Tolman-Komar mass (M = 4pi3 (ρ+ 3p)a3r3 for a FRW metric), and thatthe R = ar = 1/√H2 + k/a2, we obtain the Friedmann equationH2 +ka2=8piG3∑i[1 + σ(1 + 3wi)1 + 3wi − 2n(1`2P (H2 + ka2))n−1 ]ρi . (4.32)Not surprisingly, the possible infrared corrections, n > 1, are precisely those which couldbe significant in cosmology at late times. The nonperturbative form of s(A) is of courseis unknown, but the volume correction is known to be given by n = 3/2, so n > 1 is notsomething to exclude a priori.In the spatially flat case, if the energy density is dominated by a fluid with the EoS wand the corrections dominate over the standard term in (4.32), the expansion is described bythe effective EoSweff =1 + wn− 1 . (4.33)Thus a matter dominated universe accelerates given n > 3. With larger n, the effective EoSis more negative, but phantom expansion can be achieved only when w is itself negative. Theexact evolution of weff including the effects of possible spatial curvature is shown in Figure4.2.In order to obtain the bounds on the parameters arising from the modified Friedmannequation, a suitably modified version of CMBeasy [252] was employed together with a MCMCcode, taking into account astronomical data from baryon acoustic oscillations [242], super-novae [163] and cosmic microwave background [19]. The results are displayed in Figure 4.3and Table 4.1. Due to the geometric nature of the modifications, a possible curvature of thespatial Sections was allowed. This revealed a preference towards slightly closed universes,3In Ref.[250] it was instead assumed that the number of bits is directly proportional to entropy, which is notcompatible with our assumption A = QN . From the former assumption follows instead F = −GMm/(R2 +`2P s(A)/pi).81Chapter 4: Phenomenological Modifications: Entropic Gravity-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.1 1weffln(a) Ωm = 0.2 Ωm = 0.3 Ωm = 0.4Figure 4.2: Effective equation of state for the generalized entropy-area law DE. The linescorrespond to an open, flat and closed universe with ΩS = 0.7 and n = 2.5. Equation (4.33)would give w = −0.6 for this case.All BAO CMB SNeh 0.68± 0.02 − 0.58± 0.06 −n 3.8± 0.7 > 2.9 − > 3.0ΩS 0.69± 0.02 − 0.71± 0.12 0.76± 0.21Ωm 0.29± 0.02 0.27± 0.05 0.33± 0.13 0.39± 0.10Ωk 0.012± 0.005 − −0.04± 0.05 −0.16± 0.24Table 4.1: Maximum likelihood values and 1 sigma error bars from the constraints of Section4.4. Note that Ωk = 1− Ωs − Ωm is a derived parameter.which might be due to the appearance of k in the r.h.s. of (4.32). Relatively lower values ofn are favoured with respect to higher ones because higher values reproduce a total equationof state which is too close to −1. Note also the existence of degeneracies in the individualdatasets, which are broken by the combined constraints.Table 4.2 shows the results of model comparison with ΛCDM including the Bayesianand the Akaike criteria given by −2 logLmax+p log d and −2 logLmax+2p respectively, withp the number of free parameters and d the number of experimental data points. Eventhougha cosmological constant is favoured in all cases, the values of χ2 are very similar and mostof the difference in these cases is due to the additional parameter n in the entropy-correctedmodel.824.4. Dark Energy from a Generalized Entropy-area LawFigure 4.3: Bounds on parameters for the generalized area-entropy law DE using Sne (blue),CMB (orange), BAO (green) and the three combined datasets (gray). Note that closeduniverses (Ωm + ΩS > 1) are slightly preferred in this model. Entropic corrections gaveχ2S = 533.28, slightly higher than the value obtained for a similar MCMC for ΛCDM withχ2Λ = 532.34.S(A) Λχ2 533.28 532.34χ2/d.o.f 0.956 0.952Bayesian 558.61 551.31Akaike 541.28 538.32Table 4.2: Model comparison according to different criteria.83Chapter 4: Phenomenological Modifications: Entropic Gravity4.5 On the Evolution of PerturbationsLue, Roman Scoccimarro and Starkman [253] have shown that by assuming the Jebsen-Birkhoff theorem [254] it is possible to deduce the evolution of the spherical overdensities ina dust-filled universe given the background evolution. It is not clear to us whether the entropicgravity obeys the Jebsen-Birkhoff theorem. Therefore we did not include the constraints fromperturbations into the likelihood analysis of Section 4.4.We will take this approach, which was first developed to study DGP and related models,as a first approximation to gain insight into the clustering of matter in entropic cosmology.The perturbation evolution equation can be derived by tracking the surface of a star in aSchwarzchild metric embedded in the background of FRW, where the expansion is givenby some gravity theory deviating from Einstein’s GR. At linear order, the density contrastevolves asδ¨ + 2Hδ˙ =(2H˙ +H¨H)δ . (4.34)If the same background expansion is due to a smooth dark energy component, the growth ofperturbations is governed byδ¨ + 2Hδ˙ = 4piGρMδ . (4.35)The difference is thus only the source term in the RHS of Eq.(4.34) due to the clumpinessof the effective fluid, whereas with smooth dark energy only the matter density acts as agravitational source in the RHS of (4.35). This also determines the behavior of the metricperturbations, which can be probed by various observations, in particular weak lensing andISW. Following reference [253] one may then find that∇2a2Φ = H˙δ , (4.36)∇2a2Ψ = −(2H˙ +H¨H)δ , (4.37)showing that the entropic corrections can source the temporal and spatial potentials differ-ently, similarly to the effect of anisotropic stress. This is an interesting prediction as it allowsto distinguish the possible entropic origin of acceleration from other dark energy models (e.g.quintessence).As an example, we consider the power-law parametrization (4.31) presented in theprevious Section. The growth rate f of the perturbations can be defined asf ≡ d log δd log a. (4.38)Asymptotically, when the effective EoS is given by (4.33), the growing solution to the evolu-tion equation (4.34) correspond to growth ratef =32n, (4.39)844.5. On the Evolution of Perturbations 1e-09 1e-08 1e-07 1e-06 1e-05 500 1000 1500 2000 2500 3000 3500 4000 4500|2H’+ H’’/H| , 4pi G ρmt [Mpc] n = 2 n = 4 n = 6 n = 10 Λ 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 500 1000 1500 2000 2500 3000 3500 4000 4500δ entro / δΛt [Mpc] n = 2 n = 4 n = 6 n = 10 0.4 0.6 0.8 1 1.2 1.4 0.1 1d log( δ ) / d log( a )z n = 2 n = 4 n = 6 n = 10 Λ -2e-06-1e-06 0 1e-06 2e-06 3e-06 4e-06 5e-06 6e-06 7e-06 8e-06 0.1 1k2/a2 (Φ - Ψ) entro [Mpc-2 ]z n = 2 n = 4 n = 6 n = 10Figure 4.4: Growth of structure in entropic cosmologies. Top left: Coefficient of δ in ther.h.s of (4.34) (thick lines) and (4.35) (thin lines) for some values of n. Top right: Timeevolution of δ normalized to the ΛCDM value. Bottom left: Evolution of the growth factor(4.38) obtained using the modified equation (4.34) (thick lines) and the usual one (4.35)(thin lines). Data points correspond to Table 1 in Ref. [132]. Bottom right: Anisotropicstress caused by the modified gravitational potentials (4.36,4.37). All lines correspond to flatuniverses with Ωm = 0.3, ΩS = 0.7.which dominates at late times if n > 9/4.4 As n increases, the decay of perturbations becomesless rapid, and when n → ∞, the background is de Sitter and the matter density is frozen.In dark energy cosmologies described by (4.35), the solution with growing δ is absent andthe transition behavior, which is relevant to observations, is different. The growing solution(4.39) corresponds asymptotically to Φ,Ψ ∼ ap, where p = 2 − 9/(2n) if 0 < n < 9/4 andp = 0 if n > 9/4. Thus we expect that the gravitational potentials will tend asymptoticallyto a constant value if the present acceleration stems from entropic nature of gravity.Numerical solutions for the full evolution of perturbations (4.34, 4.36, 4.37) are plottedin Figure 4.4. Table 4.3 shows the numerical values of the growth factor (4.38) at a = 1 and50, together with the limit previously described. Discrepancies occur for the cases with lown, which have less negative e.o.s. and need longer time to reach the limit. The growth of4The other solution f = 3/n− 2 is a decaying mode for the observationally allowed values of n, See Table4.2.85Chapter 4: Phenomenological Modifications: Entropic Gravityn a = 1 a = 50 a→∞2 0.326 -0.374 -0.54 0.390 -0.359 -0.3756 0.460 -0.246 -0.2510 0.533 -0.149 -0.15Table 4.3: Numerical and asymptotic values of the growth factor f .inhomogeneities can be enhanced or damped depending on the value of n. The weakeningof gravity responsible for the acceleration is reflected on smaller scales in the lower value ofthe r.h.s. factor of (4.34) responsible for gravitational instability. However, that term canbecome larger than 4piGρm around the beginning of the acceleration due to the variation of Hthrough its time derivatives.5 This effect can easily overcome the weakening of gravity if n islarge enough (more pronounced transition), leading to higher values of δ. The enhancementeffect is also responsible for the resulting anisotropic stress due to the same factors appearingin (4.34) and in (4.37).4.6 DiscussionRather than trying to guess the fundamental underlying mechanism describing the micro-structure of space time, this Chapter has described some approaches to address simple modelsof modified gravity inspired by the entropic gravity proposal. These models were arguedto introduce parametric modifications of the Friedmann equations, which can have beenconstrained through their effects on the universe expansion at early and late timesWe found that the higher order curvature corrections, motivated by quantum effects,lead to an inflationary graceful exit problem and thus can be excluded, at least in the sim-plest scenarios. In the surface term approach of Section 4.2, we identified the parametercombinations (4.3) that can be constrained by observations. There c1 quantifies the lowerorder and c2 the higher order contributions. We obtained−17.28 · 10−3 < 2c1 − 3 < 20.50 · 10−3 ,√|8piGc2| < 0.02 1/GeV .from late and early universe constraints, respectively. It was also observed that in two quitedifferent approaches, the higher order corrections lead to a similar expansion law that predictsinflation in the early universe. However, the introduction of a cosmological constant changesthe character of the Friedmann equation changes and matter domination is severely modifiedif Λ, c2 6= 0 simultaneously. In an alternative prescription assuming c2 = 0 and retaining thecosmological matter conservation laws, an estimate for the parameters can be given asβ1 − α1 = 0.02± 0.08 . (4.40)5Ref. [253] argue that slower growth of inhomogeneities follow from the acceleration condition. However,they consider a Friedmann equation which is locally H2 ∼ ρn which is inequivalent to (4.32).864.6. DiscussionIn the quantum corrected approach discussed in Section 4.3, the leading logarithm correctionto the entropy-area relation was shown to be constrained by BBN in a similar way, and thefollowing inverse area correction was the cause of the graceful exit problem.In addition, we studied a phenomenological power-law correction to the entropy formula.The addition of a term ∝ An can produce accelerating expansion in the late universe forcertain values of n. Combining the available data to bound the power of the correction, weobtainedn = 3.8± 0.7 . (4.41)We may go significantly further if the evolution of spherical metrics can be argued to dependonly on the amount of enclosed matter, as it is the case within General Relativity. Then thefeatures of linearized structure are captured by the three equations (4.34,4.36,4.37), whichpredict a growth factor and evolution of the gravitational potentials that is different fromthe general relativistic case.The origin of the thermodynamical properties of space-time remains mysterious. Evenin that case, simple modified theories of gravity based on the entropic gravity proposal canbe motivated, constructed and explored to a certain extent using cosmological information.Certain modifications, such as postulating a different entropy-area law can provide mecha-nisms for dark energy and generically have observable consequences, although there are anumber of assumptions that are necessary to extract predictions from the theory. The studyof background evolution and comparison with geometrical probes was possible by derivingFriedmann equations from the Newtonian force. Similarly, a prescription for the growthof structure could be obtained only after assuming the Jebsen-Birkhoff theorem. Althoughthese methods are valid for General Relativity, there is no guarantee that they will remaincorrect in more general theories, and the lack of a complete formulation makes it difficult tocheck these assumptions, if possible at all. Eventually, a wholesome approach to describe thespace-time degrees of freedom would be necessary to properly address the full set of predic-tions. Only this may ultimately validate or rule out the concept of entropic gravity and itspossible variations.87Chapter 4: Phenomenological Modifications: Entropic Gravity88Chapter 5Standard Matter and Scalar Fields:Disformal QuintessenceYou can’t just have stuff that is free andescapist, you have to have stuff that isconfrontational as well. You need stuff that ismystical but you need the realism too.Irvine Welsh1Cosmological observations reveal that a mysterious constituent with negative pressure,so called Dark Energy (DE), accounts for about 70% percent of today’s mass energy budgetand is causing the expansion of the universe to accelerate [19]. A major challenge in presentday cosmology is to discover the physical nature of Dark Energy. The quest to find its pre-cise nature, whether as an exotic field or a modification of General Relativity, is ongoing andmany possibilities have been explored [9, 25]. The phenomenologically simplest CosmologicalConstant is however theoretically problematic [23]. Quintessence provides a dynamical alter-native to the static cosmological constant. It can track the background energy density, morenaturally resulting in similar orders of magnitude for the Dark Energy and matter energydensities today. However, some mechanism is necessary to end the scaling era and initiatethe acceleration. Several possibilities have been considered, such as introducing a suitablebump in the potential [255], coupling the field to the Gauss-Bonnet term [256, 257], couplingthe field to other matter components [49, 258], considering non-canonical Lagrangians [259],or introducing several scalar fields [260]. In the following, a disformal relation [62] is appliedfor this purpose.Disformal relations have frequently appeared in gravitational theories alternative to darkmatter, such as TeVeS [60] or other covariant Modified Newtonian Dynamics formulations[261]. This is necessary to produce gravitational lensing, as light has then to propagate on adifferent causal structure than gravity. Disformal relations have also been used to construct1Interview by Alan Black for 3:AM Magazine, 200489Chapter 5: Standard Matter and Scalar Fields: Disformal Quintessenceinflationary models [69], and are indeed the key to varying speed of light theories that attemptto solve the horizon problem [262, 263]. They generalize the conformal transformations (i.e.local rescalings of the metric), which are well known and have a number of applications,such as relating the simplest scalar-tensor theories to General Relativity with a scalar fieldentering the matter sector [264]. One can then consider the features of the more generaltransformation as a model building tool and to examine extended sets of relations betweendifferent theories (cf. Chapter 6).Here a preliminary step is made by exploring one of the simplest possible scenarios:a canonical scalar field self-coupled through a disformal metric. This allows us to uncovermany relations between seemingly disconnected Dark Energy models. If only the potentialterm is considered, one recovers a tachyon or a Chaplygin gas. When the kinetic term isincluded, some other previously considered models can be recovered in certain limits, aswe will discuss in Section 5.1. The rest of the Chapter is devoted to the study of a simplepossibility, disformal quintessence, which can act as a viable DE model. Section 5.2 is devotedto study of this model, and contains the main results of this paper. We describe in detailthe background evolution of this model in Section 5.2.1. Particular attention is given tothe transition mechanism that provides an exit from the scaling era. The details of thistransition depend on the two parameters of the model, and thus they can be constrained bycosmological data. The evolution of perturbations is considered in Section 5.2.2 in order tocompute the CMB and matter power spectra. Armed with these solutions,a Markov ChainMonte Carlo analysis of the model is performed in Section 5.2.3, combining different geometricand dynamical data. A broad range of the parameter space is found to be compatible withobservations, but the region corresponding to a shallow slope of the exponential potential orthe disformal factor can be observationally excluded. The model is discussed in Section 5.3.5.1 Dark Energy from the Disformal RelationA generic scalar field Lagrangian minimally coupled to gravity is a function of two variables,L = L(φ,X) . (5.1)It can be expanded as a series in the kinetic term X = −12gµνφ,µφ,νL = L(φ, 0) + L,X(φ, 0)X + 12L,XX(φ, 0)X2 + . . . (5.2)The first term corresponds to the potential and the second term equals one if canonicallynormalized. The coefficient of the third term has units 1/M2 and is suppressed by a factor(H/M)2 for cosmological applications. Unless the prefactor is huge, this term is negligiblein the late universe, and a canonical scalar field should be an adequate description of almostany natural case. However, this can be avoided if L is chosen such that the coefficients inthe formal series (5.2) are suitably amplified, e.g. by their dependence on the field.Let us consider the following generalization of the conformal transformationg¯µν = A(φ)gµν +B(φ)φ,µφ,ν (5.3)defined by a scalar field φ. Equation (5.3) constitutes the most general relation between met-rics that preserves causality and covariance and is defined through φ and its first derivatives905.1. Dark Energy from the Disformal Relationonly [62]. Note that the relation (5.3) becomes meaningful only after we have specified thecoupling to gravity. Here it will be considered that the gravitational sector is given by theEinstein-Hilbert term of the unbarred metric, and matter couples to the unbarred metric aswell.Throughout the present Chapter, quantities computed or constructed using the metric(5.3) are denoted “barred” or “disformal”. Quantities constructed using gµν are denoted as“unbarred”, and do not depend on the scalar field. The dependence on the scalar field willbe also omitted from the conformal and disformal factor A,B ≡ A(φ), B(φ).5.1.1 Disformal Dark EnergyThe simplest theory where (5.3) appears to be relevant isSΛ¯ = −∫d4x√−g¯Λ = −∫d4x√−gA3/2√A− 2BXΛ , (5.4)where the relation between the metric determinants (5.11) has been used. Thus the cosmo-logical term of a disformal metric is a tachyon in a constant potential. Computation of theenergy-momentum tensor (C.2) w.r.t. the unbarred metric reveals that ρ = A5/2 Λ√A−2BXand p = A3/2Λ√A− 2BX. The above action therefore corresponds to a cosmic componentobeying the equation of statep = −A4(φ)Λ2ρ. (5.5)If the conformal factor A is constant, the equation of state (5.5) corresponds to Chaplygingas, which has been applied in cosmology in attempts to unify dark matter and Dark Energy[265]. Our result shows that this rather exotic fluid can be obtained in an extremely simpleway from a cosmological constant term by taking into account a possible disformal relationto the gravitational metric.The Λ term describes the potential of the tachyon, and therefore it seems reasonable toconsider it as a field dependent term V (φ). A straightforward generalization of this schemeis to include also a kinetic term. In the case of canonical kinetic term we obtain the modelSφ =∫d4x√−g¯(ε2g¯µνφ,µφ,ν − V (φ)). (5.6)The sign of the kinetic term ε is kept arbitrary, as it does not complicate the analysis. Thecase ε = 1 was considered in [266] as a variation of the quintessence scenario. The effectof the disformal relation is then to end the scaling era and begin the accelerated expansion.The canonical field in the disformal metric assumes the rather unappealing form in terms ofthe unbarred metricLφ = A32(εX√A− 2BX +√A− 2BXV (φ)). (5.7)For a derivation of this expression and a variation of the model, see Section 5.1.2. Then,expansion in the kinetic energy followsLφ = −A2V − AX(ε−BV ) − 12BX2(2ε−BV ) (5.8)− B2X32A(3ε−BV ) + 5B3X44A2(2ε−BV ) + · · ·91Chapter 5: Standard Matter and Scalar Fields: Disformal Quintessenceε B(φ) V (φ) Model Remarks0 0 Λ Cosmological constant w = −11 0 V (φ) Quintessence w > −1, [41, 42]-1 0 V (φ) Phantom quintessence w < −1, [267]0 > 0 ±Λ (Anti-)Chaplygin gas Eq. 5.4, [265]0 1 V (φ) Tachyon condensate [268, 269]-1 A(φ) 0 K-essence‡ A = B, [259, 270]-1 eβφ 0 Dilatonic ghost‡ [271]-1 - - Disformal phantom1 > 0 V (φ) Disformal quintessence [266], Section 5.2‡ Assuming that the expansion (5.8) is consistent to second order in X.The equivalence is at the level of the Lagrangian rather than dynamically,when the solutions to the equations are considered.Table 5.1: Disformal Dark Energy models obtained from the action (5.6). The first threemodels do not use a disformal transformation. The second set contains examples which havebeen thoroughly studied in the literature. Disformal quintessence will be considered furtherin this work, while the phantom version is left for further studies. All models are purelydisformal (A = 1) unless explicitly mentioned. A modification to obtain further variations isbriefly discussed in subsection 5.1.2.Due to the form of the expansion, all terms after first order in X are of the form XnBn−1A2−n[pnε−qnBV ] with pn, qn rational numbers. The argument following (5.2) can be circumventedif the coefficients are adequately enhanced so that the expansion is not valid anymore. Onepossibility is to choose an exponential form for the disformal function B with A = 1.As it has been shown, the disformal relation has a very simple connection with theChaplygin gas. It turns out that the theory given by (5.6,5.7) can encompass a broad rangeof Dark Energy models with non-minimal kinetic term through different choices of ε, B,V and A. Some variations are shown in Table 5.1. Since it contains derivatives of φ, theinclusion of the disformal metric g¯µν makes the scalar field dynamical even in the absence ofa canonical kinetic term. Disformal quintessence will be discussed in depth in section 5.2.5.1.2 A Variation of the ModelBy contraction with (5.3), the inverse disformal metric can be shown to beg¯µν =1A(gµν − BA− 2BXφ,µφ,ν). (5.9)Then we see thatX¯ =XA− 2BX . (5.10)925.2. Disformal QuintessenceThe determinant of the disformal metric is (for a method to obtain the result see AppendixC of Ref.[60])√−g¯ = A2√1− 2BAX√−g . (5.11)Using these formulas the form (5.7) follows immediately.One may also consider an alternative prescription for the canonical field (5.6). Thekinetic term is written there is terms of the inverse metric as X¯ = −12 g¯µνφ,µφ,ν . Thealternative formulation employs the metric in the combination Xˆ = −12 g¯µνφ,µφ,ν . Thisis a consistent but not the minimal prescription, since depending on the viewpoint, it isequivalent to 1) mixing the two metrics (since the derivative indices in (5.11) are raised withthe unbarred metric), considering non-canonical field (as the unbarred metric can be barredby introducing field combinations) or defining the kinetic energy in terms of differentials withrespect to one-forms (and not the coordinate vector as usually). Now Xˆ = X(1−2BX), andin terms of the matter metric, this Lagrangian corresponding to this model isLˆφ = A32√A− 2BX [X(1− 2BX)− V (φ)] . (5.12)We shall not consider this variation further here.5.2 Disformal QuintessenceWe will now explore the model given by action (5.6) and (5.7) with ε = 1. For the sakeof simplicity, we will restrict to the purely disformal case A = 1. The dynamical equationsderived in 5.2.1 and 5.2.2 hold for any choice of B and V , but we will rather focus on anexponential dependenceB = M−4p eβ(φ+φx)/Mp , (5.13)V = M4p e−αφ/Mp , (5.14)for concreteness. The model parameters are chosen so that at early times the disformalfeatures are negligible and the field behaves as normal quintessence. The form for V ensuresthe existence of scaling solutions in this regime if α > 2 [272]. The exponential form for Ballows the disformal features to become relevant without introducing a new scale.5.2.1 Background EvolutionFor a flat FRW metric, the energy and density pressure in the Einstein frame read [266]ρ =1√L(εφ˙22L + V)(5.15)p =√L(εφ˙22L − V)(5.16)where the lapse functionL ≡ g¯00g00= 1−Bφ˙2 , (5.17)93Chapter 5: Standard Matter and Scalar Fields: Disformal Quintessencemeasures the relative time flow in the barred metric (5.3) relative to the unbarred one.2 Thedisformal factorD = Bφ˙2 , (5.18)is a measurement of the deviation with respect to canonical quintessence.The field equation can be written in a form analogous to a single harmonic oscillatorwith coefficients that depend on φ, φ˙φ¨+ 3FMHφ˙+PM = 0 . (5.19)The analogs of mass, friction and potential terms can be written in terms of the disformalfactor and the lapse asM = ε(1 + 12D) + LBV , (5.20)F = L[ε(1− 12D) + LBV ] , (5.21)P = [ε34φ˙2 +12LV ]B,φφ˙2 + L2V,φ . (5.22)In addition to the disformal factor (5.18) and the lapse (5.17), the dimensionless disformal-potential factor BV offers yet another quantitative estimate of the disformal propertiesBV = e[(β−α)φ+βφx]/Mp . (5.23)We assume that the field has reached the tracking solution with before the time of BigBang Nucleosynthesis. Then the field contributes a constant fraction of the energy density(Early Dark Energy) which depends on the effective equation of state w and the slope of theexponential potential α [9]Ωede =3γ2(1 + w) . (5.24)This behavior lasts while D 1 and BV 1. If the field is tracking the matter density andD ≈ 0, then V ∼ 12 φ˙2 and both conditions hold simultaneously. The disformal factor evolvesasBφ˙2 = 2βα[ΩφρM−4p]1− βα ∝ a3(1− βα ) , (5.25)and will be a growing function of time only if β > α, a condition necessary to push the fieldout of the attractor. The parameter φx produces a shift in the transition time and hence Ωφis a monotonous growing function of it. In the following we will discuss the model in termsof α, β and Ωφ.When the disformal factor Bφ˙2 becomes of order one, the time slows down in theunbarred metric g¯00 = −1 + Bφ˙2 (in which φ lives), and the field is pushed towards a slowroll phase (as seen in the unbarred metric). This causes the equation of state to becomenegative and source the acceleration of the universe. Figure 5.1 displays the effects of thetransition in wφ for different values of β/α. High values lead to a rapid slow down of thefield and a more negative equation of state while low values are associated with gradual2The lapse function defined here is different than the one appearing in the ADM formalism in GeneralRelativity [273].945.2. Disformal Quintessence-1-0.8-0.6-0.4-0.200.2 0 2 4 6 8 10wφzβ=80αβ=12αβ=3αβ=1.5α0.20.40.60.81 0 2 4 6 8 10c s2zβ=80αβ=12αβ=3αβ=1.5αFigure 5.1: Disformal quintessence dependence on β/α as a function of redshift (α = 10and Ωφ = 0.7). Left: Equation of state for the field. Higher values of β/α produce sharpertransitions while low values lead to smoother ones.. Right: Speed of sound squared (seesection 5.2.2), equivalent to F/M≈ 1−D.and longer transitions in which the field preserves a significant velocity at later times. Theevolution of the disformal factor is similar regardless of the parameters, but becomes moreclear for higher values of β/α (See Figure 5.2). The transition is associated with a maximumand a rapid fall towards a certain value, followed by a slower reduction. The existence ofa maximum follows from the role of D in the disformal metric, which implies that D < 1at any time. Otherwise the metric would run into a singularity and many quantities wouldblow up, notoriously the energy density (5.15). Hence a slow down of the field is thereforeexpected whenever D approaches unity.No similar bound can be put on the potential disformal factor, which grows exponentiallywith φ and can overcome the small velocity and reach large values after the freeze out. Itcan be seen how the change of slope in BV roughly corresponds to the transition due to theslow down of the field. This factor represents the dominant contribution to M, increasingthe inertia term in the field equation 5.19. Despite the disformal factor is restored to verysmall values after the transition, the dynamics of the field remain affected by the large valueof BV .If β α, the disformal factor is much smaller than one, except for some interval aroundthe freeze-out transition. An expansion to first order in D can be hence used to gain someintuition about the model. Let us regard the terms in (5.19) in the light of this expansion.For the friction term we get a simple correction with respect to the usual quintessence case(F/M = 1)FM = 1−D +O(D2) . (5.26)Taking into account the exponential dependence on the field, the expansion of the potential95Chapter 5: Standard Matter and Scalar Fields: Disformal Quintessence10-310-210-1 0.01 0.1 1 10Da100 101 102 103 104 BVβ=80αβ=12αβ=3α β=1.5αFigure 5.2: Disformal functions D, BV for different β/α as a function of the scale factor(α = 10 and Ωφ = 0.7). Before the transition both lines evolve together. The disformalfactor D < 1 corresponds to the lower lines. The bounce avoids a metric singularity in (5.3).The disformal-potential term BV corresponds to the lines that cross above one. This termdominates M after the transition. Note that the evolution has been extrapolated to a = 10.yieldsPM =V ′1 +BV[1 + 3D 1− 2BV2(1 +BV )]+B′φ˙21232 φ˙2 + V1 +BV+O(D2) . (5.27)The first term reduces to the usual V,φ at early times, but it is suppressed by the BV factor inthe denominator at late times (the expression in brackets remains of order one at all times).The second term represents the purely disformal contribution to the potential and is equallysuppressed by BV at late times. At early times, it is suppressed by B′φ˙2, which is smalluntil the transition by construction. With the exponential choice of the functions (5.13, 5.14)B′ = βMpB and V′ = − αMpV have opposite signs and therefore the two terms push the fieldin opposite ways. The second, disformal force term is hence able to decelerate the field whenit becomes non-negligible.Neglecting fine details from (5.27), and within our exponential model the “force” actingon the field will flip sign whenD ∼ αβ. (5.28)The contribution from the disformal factor can be much larger than V,φ if β α, which isthe ultimate cause for the transition to occur so rapidly. Since it is proportional to D ∝ φ2,it decreases as the field slows down and eventually the usual term will come to dominatethe force when (5.28) is fulfilled again (Figure 5.2 shows that the disformal factor stabilizesapproximately around this value). As the transition happens, BV increases and stronglysuppresses the force term acting on the field. The later and softer slow down would then bedriven by the friction term (5.26), which does not suffer from this suppression.965.2. Disformal Quintessence 0.001 0.01 0.1 1 1 10 100β/α1+w0α = 3 α = 10α = 30α = 50Figure 5.3: Field equation of state at present time as a function of β/α for different valuesof α. There is a clear trend towards wφ − 1 + α/β unless α or β/α are very low. Modelswith early dark energy deviate from this trend because the transition is not complete whenΩφ ≈ 0.7 is reached.It is possible to compute the equation of state from equations (5.16, 5.15) to first orderin φ˙2/V and Dw ≈ −1 +D + 12φ˙2/V ≈ −1 +D , (5.29)where in the last equality potential domination was assumed. according to (5.28) we expectthat w0 + 1 ∼ α/β. Figure 5.3 confirms this trend for a wide range of parameters. For largeramounts of early Dark Energy (low α) or slow transitions (low β/α) the freeze out does notfinish before the required Ωφ is reached. The properties of the accelerated expansion aretherefore linked to the quotient β/α. Apparently the only role of α is to regulate the amountof early Dark Energy and therefore shift the transition time to compensate the time intervalnecessary to achieve Ωφ today.5.2.2 Cosmological PerturbationsThe perturbations were studied in the synchronous gauge [274]. The relevant equations arethe same as in Ref. [275] were p is given by the Lagrange density (5.7). With the defini-tions from the previous section, we can write the energy density and velocity perturbationintroduced by the disformal field asδρφ = L−5/2[Mφ˙ ˙δφ+ Pδφ], (5.30)(ρ+ p)θφ = − FL5/2k2aφ˙δφ , (5.31)where δφ ≡ φ(~x, t)/φ(t). The scalar field equation obeyed by such perturbations can bewritten asδ¨φ+[3H +M˙M]˙δφ+[ FMk2 +m2φ]δφ+12FM h˙kφ˙ = 0 . (5.32)97Chapter 5: Standard Matter and Scalar Fields: Disformal QuintessenceHerre hk is the interaction with gravity introduced through the covariant derivative and thefield mass m2φ generalizes the V,φφ term. It is given in terms of the background valuatedpartial derivatives of (5.7) asm2φ =1M[(φ¨+ 3Hφ˙)p,Xφ + p˙,Xφφ˙+ p,φφ]. (5.33)As in the homogeneous limit, the smallness of the disformal factor ensures no significantdeparture from the canonical quintessence case at early times.The speed of sound characterizes the propagation of fluctuations within the field. Forthe purely disformal case it is given by [266]c2φ =FM = L1 +BV L −D/21 +BV L+D/2 . (5.34)The allowed values of BV and D bound its value between 0 and 1, see Figure 5.1. It isvery close to one before the transition, as for canonical quintessence. Therefore, the growthof perturbations at the tracking stage will be strongly suppressed, and by the time of thetransition to slow roll the Dark Energy overdensity will be many orders of magnitude belowthe matter contrast. As for usual quintessence, the finite amount of early Dark Energy willincrease the expansion rate and damp the formation of structure in all epochs. This departureof matter domination is reflected in a slope in the Newtonian gravitational potentials and aslight enhancement of the first acoustic peaks on the CMB.The transition affects the terms involved in (5.30-5.32). It flips the sign of P at freezeout and increases its value several orders of magnitude. This variation is compensated by areduction of δφ, which mantains the velocity and energy density perturbations at a similarlevel, as can be seen in Figure 5.4. The speed of sound drops below unity for a certain timeand then stabilizes with a value that depends on β/α, as can be seen in Figure 5.1. Althoughthis enhances the clustering properties of the disformal field, the inhomogeneities in the fluidcan grow no faster than the matter ones (for which the speed of sound is zero) and are verysuppressed w.r.t. matter inhomogeneities. Therefore, they are completely subdominant.The main departures from ΛCDM are caused by the different equation of state andtransition times, i.e. by purely background physics. Lower values of β/α render the transitionslower, but a longer expansion period is required to achieve the same density fraction. Botheffects shift the angular scales of the CMB and affect the Integrated Sachs Wolfe signal.Higher values of β/α mimic the Λ behavior at late times. If the value of α is also high (i.e.negligible early DE) matter domination is recovered and disformal quintessence becomes verysimilar to the standard model.5.2.3 Observational ConstraintsA Markov Chain Monte Carlo simulation was used to obtain constraints on the parameterspace of the theory by comparing with WMAP 7 year power spectrum data [19], supernovaefrom the Union dataset [162], SDSS DR7 baryon acoustic peak position [242] and matterpower from the SDSS Luminous Red Galaxies (LRGs) sample [99]. The uncertainty in thegalaxy bias factor was treated by choosing the highest likelihood value in the range (1, 3)within a linear model. This choice accounts for the uncertainty associated while avoiding toolarge bias factors. Spatial curvature was set to zero in all the models. The upper bounds inparameter space α ∈ (3, 20) and β/α ∈ (1.5, 30) were chosen by sensitivity considerations,985.2. Disformal Quintessence891011121314 0.1 1P dekτ0 = 100β=80αβ=12αβ=3α1213141516 0.1 1P dekτ0 = 10β=80αβ=12αβ=3α 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.1 1Φ + Ψakτ0 = 100Λ β=80αβ=12αβ=3αβ=1.5α 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0.1 1Φ + Ψakτ0 = 10Λ β=80αβ=12αβ=3αβ=1.5αFigure 5.4: Large scale perturbations for α = 10. Top: Dark energy power spectrumalong the scale factor (arbitrary units). Bottom: Gauge invariant potentials for the samek−modes.l(l+1)/2pi C l α = 10Λ β = 80 αβ = 12 αβ = 3 α β = 1.5 α0.010.1110 100 1000lP(k)α = 10Λ β = 80 αβ = 12 αβ = 3 α β = 1.5 α0.110.0001 0.001 0.01 0.1 1rel. dev.k/h [Mpc-1]Figure 5.5: CMB (left) and matter power (right) spectra for different values of β/α (α = 10)and relative deviation with respect to Λ. The initial amplitude of the perturbations has beenchosen to fit the first peak.99Chapter 5: Standard Matter and Scalar Fields: Disformal QuintessenceFigure 5.6: Likelihood contours for disformal quintessence. Shaded regions correspond tothe 1 and 2-σ obtained by the full MCMC analysis while the black lines correspond to arestricted analysis using only information from the background expansion (SNe, BAO andCMB distance priors). The constraints in α and β/α correspond respectively to high fractionsof early Dark Energy and slow disformal transitions towards accelerated expansion.since very little departure from Λ can be observed for large values. The lower bounds ensurethe existence of matter attractor solution for the field (α > 3), disformal transition (β > α)and avoid computational difficulties. The simulation was performed through a modification ofthe publicly available Boltzmann code CMBeasy [216, 218]. Four chains were run adding upto 20870 accepted models. An additional MCMC using only background information (SNe,BAO and CMB distance priors as described in section 5 of [19]) was run for comparison. Theresults are displayed in Table 5.2 and Figure 5.6.Observations are compatible with a large patch of parameter space in the α, β/α planeand confirm the preference towards higher values of both parameters where the Λ limit canbe achieved. Constraints are tighter on α due to the cumulative effect that early Dark Energyexerts in the growth of perturbations, as can be inferred by a comparison between the runsusing the complete and the restricted sets of observations (top plot in Figure 5.6). Thisdifference is likely to arise when the allowed bias factor fails to compensate the effect of earlyDark Energy to produce a good fit to LRGs data, as well as due to the effect in the third andfurther acoustic peaks, as seen in Figure 5.5. The constraints on the properties of acceleratedexpansion (i.e. on β/α) do not improve significantly in the full analysis because the relevantfeatures affect mostly distance information, encoded in SNe, BAO and CMB angular shiftmeasurements. Other features such as the integrated Sachs-Wolfe effect are significant onlyfor very low values of the parameters and do not introduce relevant differences in the regionof interest.1005.3. DiscussionParameter Maximum Likelihood 68% C.L.100Ωbh2 2.23 ±0.05Ωmh2 0.135 ±0.004h 0.690 ±0.014α 15.9 & 8.5β/α 17.2 & 4.9τ 0.086 ±0.015ns 0.971 ±0.014Amp. 2.91 ±0.015Table 5.2: Results from the Monte Carlo Markov Chain analysis. All the usual cosmologicalparameters lie within one sigma from the WMAP 7 constraints presented in [19]. Amplitudeis given by ln(1010As)− 2τ .5.3 DiscussionDisformal generalizations of the conformal transformation have found several applicationsin cosmology, particularly in the frameworks of gravitational alternatives to dark matterand varying light speed alternatives to inflation. In this Chapter it has been shown thatthe disformal relation can be also very prolific in generating alternative explanations for thecosmic acceleration and provide new connections between models. Here we considered thecanonical scalar field action where the metric is replaced by the disformal one, i.e. we madethe substitution gµν → g¯µν in the Lagrangian. Different choices of the free functions allowedto recover several Dark Energy models proposed in the literature, as can be seen in Table5.1.Disformal quintessence arises in a very simple way as an application of this substitution,which results effectively in a non-standard self-interaction of the scalar field. This self-interaction causes the field to accelerate the universe given exponential functions of thefield and under the condition α/β < 1 in (5.13, 5.14). Acceleration can be achieved formodels in which the Lagrangian parameters are of order Mp, due to the shift symmetry ofthe exponential functions. Moreover, the exponential potential allows scaling solutions, sopractically no tuning whatsoever of initial conditions for the field is required. The modelevolution was studied considering the homogeneous limit and linear perturbations. Thedependence of the background dynamics on model parameters allows us to put constraints onthem. These become more stringent if we take into account the evolution of linear structures,although the effects are subtle and mostly geometric in nature (i.e. depend on the backgroundproperties of the model). A Markov Chain Monte Carlo simulation was used to obtainconstraints on the parameter space of the theory by comparing with WMAP 7 year data,SNe from the Union dataset, baryon acoustic peak position and LRGs power spectrum fromSDSS. Small values of α can be ruled out because of the effect of early Dark Energy. Onthe other hand, small values of β/α result in slow transition to acceleration, which is alsodisfavored by the data. However, when both α and β/α are sufficiently large (the precise101Chapter 5: Standard Matter and Scalar Fields: Disformal Quintessencelimits shown in Figure 5.6), it becomes very difficult to distinguish the model from ΛCDM.These results are a first step in the exploration of the disformal relation as a way toconnect different theories and build cosmological models. Next Chapter discusses the roleof the disformal relation in Dark Energy scenarios by the introduction of couplings to otherforms of matter, using the disformal prescription gµν → g¯µν . It turns out that disformalrelations allow to relate different scalar tensor theories, and a mechanism similar to thescalar field self coupling is able to initiate a slow roll phase when it couples to non-relativisticmatter. Furthermore, such a coupling is in agreement with local gravitational tests by meansof a disformal screening mechanism.102Chapter 6General Relativity and ScalarForces: Disformal CouplingThe only true voyage of discovery would benot to visit strange lands but to possess othereyes, to behold the universe through the eyesof another, of a hundred others, to behold thehundred universes that each of them beholds,that each of them is.Marcel Proust1In the standard ΛCDM model of cosmology, the universe at the present day appearsto be extremely fine tuned. The energy scale of the Λ component is about 10−30 times themost naive expectations of the theory. Generalizations of Λ in the form of dynamical scalarfields have been proposed, whose time evolution could more naturally result in the observedenergy density today [9]. High energy physics generically predicts an interaction between ascalar degree of freedom and other forms of matter, which in turn could help to explain whythe field becomes dynamically important at the present epoch. There are myriad variationsof such models, but in most of them the coupling can be effectively described by a field-dependent mass of the coupled particles. Those Yukawa-type couplings can be motivated bya conformal relation to scalar-tensor theories, which includes also the f(R) class of modifiedgravity [7].However, for any other type of gravity modification, the relation between the matterand gravitational metric will be non-conformal. When given by a scalar field φ, the disformalrelation can be parametrized asg¯µν = A(φ)gµν +B(φ)φ,µφ,ν , (6.1)where commas denote partial derivatives. Considering the most general physical case, Beken-stein [62] argued that both functions A and B may also depend upon (∂φ)2, but we will focus1In Search of Lost Time - The Captive (1923)103Chapter 6: General Relativity and Scalar Forces: Disformal Couplingon the simpler case here. Previous applications of such a relation to cosmology include vary-ing speed of light theories [276, 277], inflation [69], massive gravity [278], dark energy [3, 266],gravitational alternatives to [60, 261] and extensions of [279] Dark Matter.Such a relation can also be motivated in theories with extra dimensions, in which matteris confined to a 3+1 dimensional brane embedded in a larger bulk space [68, 71, 72, 81–83, 280–282]. The action for this type of theories is constructed using geometric scalarscomputed out of the induced metricg¯µν = gµν + piI,µpiI,ν , (6.2)where the moduli fields piI represent the coordinates orthogonal to the brane and gµν is thebrane metric prior to the embedding, necessary to describe gravity. In the case of a singleextra dimension [68], each curvature invariant is related to a generalized covariant GalileonLagrangian [64, 74] with a particular form of the Horndenski free functions in (1.28). Theusual Galileon terms [70] are then obtained by assuming a flat initial metric gµν → ηµνand taking the non-relativistic limit (i.e. lower order corrections in (∂pi)2). These theorieshave attracted considerable attention because they capture interesting features of higherdimensional models such as DGP [79], including the Vainshtein screening mechanism [84].Screening mechanisms are central to the construction of alternative theories of gravity,as they help cover the effects of the additional field in dense environments or around massive,compact objects. Modifications are then allowed to occur over cosmological scales of lengthand energy density, while the gravitational physics operating in the Solar System are closeenough to GR to satisfy the precision tests sketched in Section 2.6. To date three possibilitiesbased on the conformal coupling have been explored, namely the Vainhstein, chameleon andsymmetron mechanisms [84, 85, 87]. The chameleon mechanism relies on the high mass ofthe field in dense surroundings, and the symmetrons are screened in high ambient densitydue to their field-dependent coupling. The Vainshtein mechanism is somewhat different, asit relies on the non-linear derivative self-interactions of the scalar field to flatten its profilearound matter sources within the so-called Vainshtein radius, while keeping it active overcosmological distances.This Chapter presents results that may simplify considerably the analysis of theoriesbased on higher dimensional models. In Section 6.2 it is argued that by performing a dis-formal transformation, a theory in which the gravitational sector is standard but the mattermetric is constructed disformally (6.1) can be rewritten in a form equivalent to the fourthorder covariant Galileon Lagrangian, which arises from the scalar curvature constructed outof the effective metric (6.2) [68]. Disformally coupled theories provide an Einstein frame de-scription of certain brane-world constructions, similar to the way in which the field dependentcoefficient of the Ricci scalar can be moved from the gravitational to the matter sector in old-school scalar-tensor theories. In Section 6.2.2, the equations of motion are derived in a framein which the gravitational sector has the Einstein-Hilbert form but the matter action includesthe scalar field as prescribed by Equation (6.1). This Einstein frame description unravels anew screening mechanism in which the coupling vanishes if the field is static and the coupledmatter behaves as non-relativistic dust. The properties of the field are discussed in Section6.5, and some simple solutions are derived. Due to the equivalence between the disformallycoupled theory and a covariant Galileon, the disformal screening mechanism might just bean alternative way to regard the Vainshtein effect and study its properties.1046.1. A Test Particle in a Disformal MetricEven in the absence of a screening mechanism, long range interactions are still phe-nomenologically viable if they only couple the scalar degree of freedom to Dark Matter par-ticles, which are not involved in local tests of gravity. As a concrete example to analyze thecosmological implications of a Disformally Coupled Dark Matter model (DCDM), in which itis assumed that Dark Matter is the only species directly coupled to the field. It is introducedin Section 6.3 where its dynamics and observational compatibility at the background levelare analyzed. The intensity of the purely disformal coupling is approximately proportionalto the scalar field energy density ρφ, unlike in the conformally coupled case for which it isproportional to the coupled matter density ρm. Cosmological perturbations are analyzed inSection 6.4, including an analytic expression for the small scale effective gravitational con-stant. The simple DCDM model enhances the growth of the coupled matter density contrasttoo much to be compatible with observations, but several possibilities to render the modelviable are discussed in Section 6.4.3. Although many results would still hold if the couplingis extended to baryons, the results of explicit computations are given for the example DCDMmodel.Throughout the present Chapter, quantities computed or constructed using the metric(6.1) are denoted “barred” or “disformal”. Quantities constructed using gµν are denoted as“unbarred” and do not involve the scalar field.6.1 A Test Particle in a Disformal MetricLet us start with the simple exercise of determining the dynamics of a point-like particlewith mass m coupled to the disformal metric (6.1). A Lagrangian density for such a systemis given by √−g¯L¯p = −m√−g¯µν x˙µx˙νδ(4)D (xµ − xµ(λ)) , (6.3)where the dot means derivative w.r.t. the affine parameter λ along the trajectory x(λ) andthe correct weight for the delta function has been taken.2 The effects from the coupling canbe seen from the barred four-velocity modulus in (6.3)g¯µν x˙µx˙ν = Ax˙2 +B(φ,µ x˙µ)2 . (6.4)Distances are dilated by the conformal factor A, as usual. The disformal factor B gives anadditional direction-dependent effect proportional to the projection of the four-velocity alongthe field gradient. The equations of motion can be obtained by maximizing the proper timeof the particle along its path. The result is the disformal geodesic equationx¨µ + Γ¯µαβx˙αx˙β = 0 , (6.5)where the barred Levi-Civita connection has been assumed to be torsion-free and such thatthe metric compatibility relation holds for barred quantities, i.e. ∇¯αg¯µν = 0. It can becomputed from (6.1) and written in terms of unbarred covariant derivatives of the barredmetric in a rather compact formΓ¯µαβ = Γµαβ + g¯µλ(∇(αg¯β)λ −12∇λg¯αβ). (6.6)2The one-dimensional definition of the delta function requires that its generalization to higher dimensionscancels out the tensor density in the integrand δ(4)D (x−x0) = 1√−g∏a δD(xα−xα0 ) (e.g. in spherical coordinatesδ(3)(x) = δD(r)δD(θ)δD(φ)/r2 sin θ). Hence it does not matter whether√−g or√−g¯ is used in the integration,as long as the delta function is consistent with it.105Chapter 6: General Relativity and Scalar Forces: Disformal CouplingNo assumption about the dependence of A,B has been done to obtain the above expression,which remains valid if A,B depend on X. Note that the difference between the two connec-tions is a tensor, as expected. Appendix D.1 shows the expansion of (6.6) in terms of A,Band its derivatives, which is rather lengthy to be included here.The stress energy tensor w.r.t. the unbarred metric can be computed by variation of(6.3) with respect to gµνTµνp ≡2√−gδ(√−g¯ L¯m)δgµν= Amx˙µx˙ν√g ˙¯x2δ(4)D (xµ − xµ(λ)) . (6.7)If the gravitational metric is the unbarred one, this is the energy momentum tensor sourcingthe space-time geometry.3 This result can be used to express the particle Lagrangian interms of the energy momentum tensor√−g¯L¯p = Tp + BAφ,µφ,νTµνp = g¯µνTµνp . (6.8)The above expression gives an effective form for the coupling to matter. It shows how thekinetic term of the scalar mixes with the matter content, a very important property that liesat the heart of disformally coupled theories, including the disformal screening mechanismexplored in Section 6.5.6.2 The Zoo of Disformally Related TheoriesThe previous Section presented a simple example of a theory in which the matter Lagrangianis constructed using a disformal metric (6.1). Although no gravitational sector was specified,the simplest possibility is to assume that it is given by the Einstein-Hilbert form computedout of the unbarred metric gµν . In this case, Einstein equations retain the usual form and aresourced by the energy momentum tensor (6.7). We shall refer to disformally coupled theoriesin which the gravitational sector is standard as being expressed in the Einstein Frame (EF),in analogy with old-school scalar-tensor theories.More generally, one wishes to know what kind of theories can be constructed using twometrics that are disformally related and study the relations between them, in analogy tothe equivalence between scalar-tensor theories minimally coupled to matter and conformallycoupled theories with a standard gravitational sector but with a non-minimal coupling ofmatter to the scalar field. The starting point is a general bi-metric theory where the gravitysector has the EH form, but with unspecified forms for the gravitational and matter metricsS =∫d4x(√−gGR [gGµν]−√−gMLm (gMµν , ψ)) . (6.9)Playing with the disformal relations between gGµν and gMµν allows one to write it differentframes. Besides the Einstein frame, an obvious possibility is to consider the Jordan Frame(JF), an equivalent description where matter appears minimally coupled and all the modi-fications occur in the gravitational part of the action. But since the disformal coupling has3It is possible to write (6.7) in the perfect fluid form Tµν ≡ ρuµuν if the coupled matter four velocity andthe energy density are identified with uµ = x˙µ/√−x˙2 and ρ = mδ(4)D (xµ − xµ(λ))√x˙2g(1− BA(uµφ,µ)2)−1/2.1066.2. The Zoo of Disformally Related TheoriesEinstein (6.10)gµν→A−1gµνTTTTT))TTTTTTgµν→gµν−BAφ,µφ,νjjjjttjjjjjgµν → 1Agµν − BAφ,µφ,νConformal (6.12)gµν→A−1gµνTTTTTT**TTTTTDisformal (6.11)gµν→gµν−Bφ,µφ,νjjjjjjuujjjjjJordan (6.13)Coupling to φ Uncoupled Conformal (Agµν) Disformal (Bφ,µφ,ν)Matter Jordan Einstein, Conformal frame Einstein, Disformal frameGravity Einstein Jordan, Disformal frame Jordan, Conformal frameTable 6.1: Physical frames for disformally coupled theories. The conformal and disformalframes refer to the type of coupling present in the matter sector. The transformation ruleson top are based on the action 6.9 and given in terms of the definitions (6.10-6.13) (Notethat the transformations commute). The lower Table summarizes the type of coupling to thefield present in the different sectors depending on the frame.two parts, two more intermediate frames can be defined in which only a certain part of thecoupling enters the matter action.The four possibilities are described below and summarizedin Table 6.1, together with the transformations that provide the connections between them.For the sake of simplicity, the Einstein frame has been defined using a matter metric of theform (6.1), consistently with the notation used in most of the Chapter (and Section 6.2.2 inparticular).1. Einstein Frame:gGµν = gµν , gMµν = Agµν +Bφ,µφ,ν . (6.10)This is the formulation used throughout the rest of the Chapter and for which theequations are derived in Sections 6.1 and 6.2.2.2. Disformal Frame:gGµν =1Agµν , gMµν = gµν +Bφ,µφ,ν . (6.11)The disformal part enters the matter Lagrangian explicitly. The conformal factor entersthe gravitational sector through a coupling to R (e.g. old school scalar-tensor theories).3. Conformal Frame:gGµν = gµν −BAφ,µφ,ν , gMµν = Agµν . (6.12)The conformal part enters the matter Lagrangian explicitly and the field couples directlyto gravity, as described in Section 6.2.1.107Chapter 6: General Relativity and Scalar Forces: Disformal Coupling4. Jordan Frame:gGµν =1Agµν − BAφ,µφ,ν , gMµν = gµν . (6.13)Matter is minimally coupled to a metric and the field enters the gravitational sectorexclusively.The JF is the most convenient frame to analyze certain properties of the theory and itspredictions, as matter follows the geodesics of the simple metric gµν . The matter metric inthe remaining frames contains the scalar field explicitly, and therefore matter moves alonggeodesics that involve the field variations (6.5) in these representations. These frames arestill interesting to analyze the theory. For example, the equations simplify considerably inthe EF, just like in conformally related theories. Once these are solved, the solutions can beused to transform back to the Jordan Frame.The explicit computation of the curvature scalar for a metric which includes a disformalpart allows one to connect the theory studied in the Einstein frame with a particular sectorof the Horndenski Lagrangian (1.28). As anticipated in the introduction, the thus obtainedtheory is related to a type of covariant Galileon when expressed in the conformal or Jordanframe.6.2.1 Disformal Curvature: The Conformal FrameIt is possible to get a sense of disformally coupled theories in a different frame by applyingthe transformations sketched in Figure 6.1 to known actions. These are easy to apply to theterms involving a scalar field, as was described in Chapter 5, where it was shown that applyinggµν → Agµν +Bφ,µφ,ν to a canonical scalar field transforms it into disformal quintessence√−g(X − V )→ A3/2√−g(X√A− 2BX −√A− 2BXV), (6.14)up to the ambiguity in the definition of the kinetic term described in Section 5.1.2. Thistransformation allowed us to relate several dark energy models to disformally self-interactingscalar fields, which were summarized in Table 5.1.Considering similar relations when the transformations involve the gravitational sectorin (6.9) requires the computation of the Ricci curvature for a barred metric that includesthe scalar field as in (6.1). The starting point is the difference between the standard and thebarred connection (6.6)Kαµν ≡ Γ¯αµν − Γαµν = g¯αλ(∇(µg¯ν)λ −12∇λg¯µν)(6.15)where the symmetrization is defined as t(αβ) ≡ 12 (tαβ + tβα). The barred Riemann tensor isobtained from the usual definition, and it can be related to the unbarred one in a manifestlytensorial form in terms of (6.15)R¯αβµν ≡ ∂[µΓ¯αν]β + Γ¯αγ[µΓ¯γν]β (6.16)= Rαβµν +∇[µKαν]β +Kαγ[µKγν]β ,where anti-symmetrization is defined without the usual 12 coefficient A[αβ] ≡ Aαβ−Aβα. TheRicci scalar follows from the contractionR¯ ≡ g¯µνR¯αµαν . (6.17)1086.2. The Zoo of Disformally Related Theorieswith the inverse barred metricgµν =1A(gµν − BA− 2BXφ,µφ,ν). (6.18)Finally, the disformal Einstein-Hilbert Lagrangian density requires the barred volume element(D.4) to be covariant√−g¯ = √−gA2√1− 2BAX . (6.19)Note that no assumption has been made about the functions A,B out of which the geometricquantities (6.15-6.17) are computed. However, the general computation is very lengthy, andit is useful to adopt some simplifications.Let us focus for the time being on a theory in the conformal frame, for which thedisformal part is absorbed into the gravitational sector. The following computation assumesthus a gravitational metric of the form (6.12)gGµν ≡ g¯µν = gµν +D(φ)φ,µφ,ν , (6.20)where the disformal factor D(φ) is yet to be specified.4 The Jordan frame can be obtainedat the end of the computation by inverting the conformal transformation gµν → A−1(φ)gµνin the resulting metric and curvature objects. Since the transformation rules for curvaturetensors under conformal relations are well known [13], and Galileon-like theories usuallyretain a conformal coupling to matter in phenomenological applications [283], the Jordanframe curvature will not be computed explicitly.Let us further redefine the field to simplify the barred metric (6.20)pi ≡∫ √D(φ)dφ ⇒ g¯µν = gµν + pi,µpi,ν . (6.21)The above expression has the same form as the effective metric in probe-brane theories 6.2.It simplifies considerably the computation of the connection tensor (6.15), which now readsKαµν = g¯αλ (pi,λpi;µν) . (6.22)The barred Riemann tensor can be computed directly from (6.16), rewriting anti-symmetrizedderivatives in terms of the curvature ∇[µ∇ν]X ···αi······βj ··· = ΣiRαiλµνX···λ········· −ΣjRλβjµνX ·········λ···. Theresult is also simpleR¯αβµν = g¯αλ(Rλβµν + γ2pi;λ[µpi;ν]β). (6.23)Hereγ =1√1 + pi,αpi,α, (6.24)is a Lorentz factor that arises from the inverse metric (6.18). The barred Ricci scalar can beeasily obtained by a second contractionR¯ =(gµν − 2γ2pi,µpi,ν) [Rµν + γ2(pi;µν2pi − pi;µ;αpi;αν) ]. (6.25)4Just as in the rest of the Chapter, D has been assumed to be independent of the field derivatives X.This assumption is important in order to simplify the computations performed below. However, given theimportance of the X dependence of higher derivative terms in the Horndenski Lagrangian (1.28) and relatedtheories, it is worth considering the more general case D(φ,X), although in this case the equations wouldbecome very involved.109Chapter 6: General Relativity and Scalar Forces: Disformal CouplingThe gravitational Lagrangian√−g¯R¯ just requires multiplying by the barred volume factor(6.19) to make it covariant.The total action for the theory in the Conformal frame is obtained by adding a matterLagrangian with a conformal factor A(φ) in the matter metricSCF =∫d4x√−g[1γR− 2γpi,µpi,νRµν −A2Lm(Agµν , ψ) (6.26)+ γ((2pi)2 − pi;µνpi;αβ)+ 2γ3(pi,µpi,µαpi;ανpi,ν − pi,µpi;µνpi,ν2pi)].These results were previously obtained by de Rham and Tolley [68] in the context of higherdimensional gravity theories. The action (6.26) corresponds to a particular form of the G4term in the Horndenski Lagrangian (1.31), which reduces to the fourth covariant Galileonterm in the limit X 1, with the right non-minimal coupling to gravity. The reader isreferred to Section 5 of Ref. [68] for results on other curvature invariants and the relationsto other scalar-tensor theories. The Jordan frame expression of covariant Galileons wasconsidered by Appleby & Linder [93], where both the conformal and the disformal couplingto matter were shifted to the gravitational side of the action. The theory studied there isessentially different, as it includes all the covariant Galileon terms, but in the limit of smallX, and the authors make no attempt to obtain an Einstein frame description.6.2.2 Equations in the Einstein FrameIn this Section the equations for a disformally coupled theory which accepts an Einsteinframe description will be derived. Such a theory is given by the following actionSEF =∫d4x[√−g(R[gµν ]16piG− Lφ)−√−g¯L¯m(g¯µν , ψ)](6.27)=∫d4x√−g[R[gµν ]16piG−A2√1− 2BAX L¯m(Agµν +Bφ,µφ,ν , ψ)].The interacting matter sector√−g¯L¯m is to be constructed using the barred metric (6.1), asit has been made explicit in the second line using the explicit form of g¯µν and its determi-nant (6.19). It will be further assumed that there is no dependence on the barred metricderivatives.5 A scalar field Lagrangian density of the k-essence type Lφ = Lφ(φ,X) hasbeen also included. More general dependence on the field derivatives may be considered (e.g.Kinetic Gravity Braiding [65]), but this term gives relatively simple equations of motion, andis general enough to accommodate the both a canonical Lφ = X + V and a disformally self-interacting scalar described in Chapter 5. The matter Lagrangian may include other pieceswith different couplings, e.g. an uncoupled matter sector can be included by the addition ofa Lagrangian√−gLu constructed out of the unbarred metric.5This implies that the disformal connection (6.6) does not appear in the action. This assumption holdsfor scalar fields and gauge vectors vanishes due to the lack of indices and antisymmetry of the kinetic term,respectively. Although other fields may couple to g¯µν,λ, the assumption simplifies the equations considerablyand is common in the analysis of scalar tensor theories.1106.2. The Zoo of Disformally Related TheoriesThe stress energy tensors for both species will be further defined in terms of the con-travariant gravitational metricTµνφ ≡2√−gδ (√−gLφ)δgµν, (6.28)Tµνm ≡2√−gδ(√−g¯ L¯m)δgµν, (6.29)such that the Einstein field equations take the usual form Gµν = 8piGTµν and the totalenergy-momentum is covariantly conserved with respect to the unbarred metric by virtue ofthe Bianchi identities: ∇µ(Tµνm + Tµνφ ) = 0. However, the coupling causes that this relationdoes not occur for each component separately. Instead, only the total energy-momentum forthe field and matter is conserved and ∇µTµ(φ)ν = −∇µTµ(m)ν . The divergence of the scalarfield stress tensor can be re-casted as∇µTµ(φ)ν =(Lφ,φ −∇µ ∂Lφ∂φ,µ)φ,ν ≡ Qφ,ν , (6.30)where the interaction densityQ has been defined and is equal to the variation of the uncoupledLagrangian with respect to the field δLφ/δφ. The equation for the scalar field, δS/δφ =δLφ/δφ + δLm/δφ = 0, allows one to write the above expression in terms of the matterLagrangianQ =1√−g(∇µ∂(√−g¯L¯m)∂φ,µ−√−g¯L¯m,φ). (6.31)The coupling can be evaluated by application of the chain rule. For the specific form of thebarred metric (6.1), the variation with respect to the field yieldsδ(√−g¯L¯m)δgµν∂gµν∂g¯µν∂g¯µν∂φ=√−g2ATµνm(A′gµν +B′bµbν), (6.32)and similarly for the derivative with respect to the field gradient.After replacing these expressions in (6.31), the coupling readsQ = ∇µ(BATµνm φ,ν)−[A′2ATm +B′2Aφ,µφ,νTµνm]. (6.33)The equations for the coupled matter component and the field are then∇µTµνm = −Qφ,ν , (6.34)Mµν(φ)φ;µν + Lφ,φ − 2XLφ,Xφ = Q , (6.35)where Mµν(φ) ≡ (Lφ,Xgµν + Lφ,XXφ,µφ,ν) is the general kinetic term for the scalar. Einsteinequations Gµν = 8piG(Tµνm + Tµν(φ)) together with (6.34, 6.35) and (6.33) determine unam-biguously the evolution of matter, the scalar field and the metric. These equations naturallycontain the case of conformally coupled quintessence, where only the coupling to the trace ofenergy momentum is present in Q. Note that so far this result is general and does not dependupon the matter content as long as the matter action only depends on the field through thebarred metric (6.1) algebraically.111Chapter 6: General Relativity and Scalar Forces: Disformal CouplingScalar field equationThe first term in (6.33) contains higher derivatives of the variables Tµν;µ, φ;µν due to thekinetic mixing in the mater action, e.g (6.8). These have to be solved for in order to integratethe evolution equations (6.34, 6.35), which can be done after adopting a coordinate system.For the scalar field, it is possible to eliminate the matter derivatives in a general way bycontracting (6.34) with φ,ν and solving for φ,ν∇µTµνm . The result can be inserted back in(6.35) and rearranged asMµν∇µ∇νφ+ AA− 2BXQµνTµνm + V = 0 , (6.36)were we have defined:Mµν ≡ Lφ,Xgµν + Lφ,XXφ,µφ,ν − B TµνmA− 2BX , (6.37)Qµν ≡ A′2Agµν +(A′BA2− B′2A)φ,µφ,ν , (6.38)V ≡ L,φ − 2XL,Xφ . (6.39)The above equation can then be used instead of (6.35) to determine the evolution of the scalarfield. It displays very clearly the role of the coupling, which enters not only as a modificationto the effective potential (second term), but also in the coefficient for the higher derivativesof the field. This feature will be ultimately responsible for the screening mechanism thatdisformal models exhibit in high density regions, explored in Section 6.5.Coupling to Perfect FluidsThe analogue of (6.36) for the covariant matter conservation equation without second orderfield derivatives can not be obtained without choosing a time slicing due to the differenthigh derivative structure in both equations. Nevertheless, there is no need to do so, since wealready found a bona fide field equation (6.36) that can be integrated consistently with thecorresponding equation for matter (6.34), substituting the appropriate value of Q.6Assuming a perfect fluid in the Einstein frame Tµν = (ρ+p)uµuν+pgµν with uαuα = −1,it is instructive to project (6.34) along and perpendicular to the matter four velocity. Thisdetermines how the local law of energy conservation and the geodesic equation are modifiedby the couplinguα∇αρ+ (ρ+ p)∇αuα = Qφ,αuα . (6.40)(ρ+ p)uα∇αuµ + [gµα + uµuα] (∇αp+Q∇αφ) = 0 . (6.41)In the first equation the coupling the usual energy conservation relation, due to the energytransfered from the scalar field, which is modulated by the projection of the field gradientalong the 4-velocity. The second equation determines the departure of geodesic motion w.r.t.the gravitational metric. The first term describes the force arising from the pressure gradientand the second the additional force exerted by the scalar field. Both forces are projected intothe direction ⊥ uµ (coefficient in brackets) due to the orthogonality of the four velocity andfour acceleration.6It is possible to solve for the time derivatives of all the variables after a metric ansatz has been chosen,as will be done in the study of FRW models and cosmological perturbations, see Sections 6.3, 6.4.1126.3. Background Cosmology6.3 Background CosmologyLet us consider an application of the theory (6.27), where the scalar acts as a quintessencefiend and the disformal coupling is used to trigger cosmic acceleration. Using the Einsteinframe description the Friedmann equations have the usual formH2 +K =8piG3(ρ+φ˙22+ V ) , (6.42)H˙ +H2 = −4piG3(ρ+ 2φ˙2 − 2V ) , (6.43)but the conservation equations for matter and the scalar field have to be computed from(6.34, 6.36):ρ˙+ 3Hρ = Q0φ˙ , (6.44)φ¨+ 3Hφ˙+ V ′ = −Q0 , (6.45)were the background order coupling factor readsQ0 =A′ − 2B(3Hφ˙+ V ′ + A′A φ˙2) +B′φ˙22(A+B(ρ− φ˙2)) ρ , (6.46)after solving away the higher derivatives. In the following we restrict to flat space, K = 0.At this stage it is possible to understand the difference between the pure conformal(B = 0) and disformal (A = 1) cases by writing (6.46) in terms of the equation of state andthe scalar field energy density:Q(c)0 =A′2Aρ , (6.47)Q(d)0 ≈( B′2B(1 + wφ)− V′2V(1− wφ))ρφ +√3Mp((1 + wφ)ρtotρφ)1/2 , (6.48)where in the second line it has been assumed that Bρ 1 & Bφ˙2. This approximationis satisfied by the model under study when the coupling is active, see Figure 6.2. The lastterm in (6.48) represents the contribution from the Hubble term, which is subdominant whenthe slopes of B, V are large. The above expressions imply that the conformal and disformalcoupling between Dark Energy and Dark Matter are related to essentially different darkenergy parameterizations, where the interaction is either proportional to ρ = ρdm [49, 258]or ρφ [284, 285].6.3.1 An Example Model: Disformally Coupled Dark MatterIn what follows it will be assumed that the field is only coupled to Dark Matter, while radia-tion and baryons follow geodesics of the gravitational metric and do not feel the scalar inter-action directly. If baryons are also coupled, then the ratio ρdm/ρb remains fixed, because bothspecies feel the same effective metric.7 Postulating that the baryonic and electromagnetic7This can also be seen directly from (6.46): in the denominator of Q0, the energy density has to besubstituted by the total one ρ→ ρdm+ρb, while the multiplicative coefficient ρ would refer to each individualspecies.113Chapter 6: General Relativity and Scalar Forces: Disformal Couplingsectors are constructed out of the gravitational metric also avoids problems with precisiongravity tests and the subtleties related to the existence of different frames, hence simplifyingthe analysis of cosmological observations.To study the dynamics within a particular example, we focus on a simple DisformallyCoupled Dark Matter (DCDM) model, constructed with the following prescriptions• Dark Matter disformally coupled to a canonical scalar field, following Eq. (6.44-6.46).• An exponential parametrization for the disformal relation and the scalar field potential:B = B0eβ(φ−φ0)/Mp , V = V0e−γφ/Mp , A = 1 . (6.49)with Mp = (8piG)−1/2. The conformal factor A has been set to the trivial value in orderto focus on the novel features. Furthermore, the coupling is chosen to be negligible inthe early universe, and hence initial conditions and early evolution are not affected.• Uncoupled baryons, photons and neutrinos, which follow the usual barotropic scalingrelations ρ = a3(1+w). Zero cosmological constant.Besides being motivated from some high energy scenarios, the exponential forms (6.49) fa-cilitate the choice of natural scales for the constant prefactors by shifting the zero point ofthe field (e.g. B0 ∼ M−4p , V0 ∼ M4p , A0 dimensionless). The model set up is similar tothe uncoupled self-interacting field case described in Chapter 5. In particular, the potentialensures a tracking stage for the field and the value of φ0 is chosen to tune the transition timewhen the disformal coupling becomes relevant. Although only dark energy is affected by thecoupling, radiation and baryons are included in order to provide a more realistic description.With this set up, the evolution at early times is as in the usual exponential quintessencemodel, where the field tracks the dominant fluid component and the slope of the potential γdetermines the amount of Early Dark Energy (EDE) [9]Ωede =3γ2(1 + wm) , (6.50)which depends on the dominant matter component equation of state parameter wm. The newfeatures appear when the disformal factor Bφ˙2 becomes of order one. Then the clocks thattick for Dark Matter, g¯00 = −1 + Bφ˙2, slow down and make the effective equation of stateapproach minus unity asymptotically. The field also slows down to avoid a singularity in theeffective metric g¯µν , and the universe enters into a de Sitter stage. This natural resistanceto pathology was also observed in the disformal self-coupling scenario described in Chapter5 and references [3, 266]. The disformal coupling provides then a mechanism that triggersthe transition to an accelerated expansion. The relatively steeper the slope of the disformalfunction is, i.e. the higher the ratio β/γ, the faster the transition happens, as seen in Figures6.1, 6.2. This transition also produces a short “bump” in the equation of state, which affectsthe growth of structure.The departures with respect to the standard ΛCDM model allow the use of cosmologi-cal observation to constrain the model parameters in the homogeneous approximation usingdifferent datasets. The Union2 Supernovae compilation and the local expansion rate mea-surement were implemented as described in Section 3.3.1. The BAO scale data was also usedas described in Section 3.3.1, but computed using the Dark Matter fraction extrapolated1146.3. Background Cosmology0.1 0.2 0.5 1.0 2.0 5.0 10.01010410105101061010710108aΡ@Mpc-4 DScalar FieldCoupled MatterΓ =10 H all LΒ=40 ΓΒ=15 ΓΒ=5 Γ10- 4 0.001 0.01 0.1 1 10 100-1.0-0.50.00.5awScalar FieldCoupled MatterΓ =10 H all LΒ=40 ΓΒ=15 ΓΒ=5 ΓFigure 6.1: Background evolution of disformally coupled matter. Left: evolution of theenergy density for the field (red) and coupled matter (blue) for different choices of the couplingslope β. Right: equation of state for the field (red) and coupled matter (blue). High values ofβ/γ (solid, dashed) give a good fit to observations, while low values (dotted) do not produceenough acceleration.0.1 0.2 0.5 1.0 2.0 5.0 10.00.0010.1101000105107aΓ=10 , Β =40 ΓBΦ, t2BΡBV0.1 0.2 0.5 1.0 2.0 5.0 10.00.010.11101001000104aΓ=10 , Β =5 ΓBΦ, t2BΡBVFigure 6.2: Evolution of the dimensionless disformal factors Bφ˙2, Bρ and BV . Highervalues of β/α produce a sharper transition and lead to higher B at late times.from the drag epoch assuming the usual scaling ρdm ∝ a−3 in order to ensure the validity ofthe fitting formulae. The Cosmic Microwave Background constraints include the bound onearly dark energy obtained from the South Pole Telescope Ωφ(z∗) < 0.018 (95%C.L.) [122]and the angular scale, given by the ratio between the physical sound horizon at recombi-nation and the angular diameter distance θCMB = rs(z∗)(1 + z∗)/DA(z∗) = 0.5952(16) deg[18]. The effects of early dark energy are taken into account by using the model backgroundthermal history to compute rs, rather than a fitting formula. A prior on Ωbh2 obtained fromprimordial nucleosynthesis [286] was also included. Although this values assumed no EarlyDark Energy, the severe CMB bounds constraint the potential impact of neglecting EDE inBBN constraints.A Markov Chain Monte Carlo analysis with these datasets was performed using a mod-ified version of CMBEasy [216]. The obtained constraints are shown in Figure 6.3 and Table115Chapter 6: General Relativity and Scalar Forces: Disformal CouplingFigure 6.3: Marginalized one and two-sigma regions obtained from Supernovae (Blue), BAO(Green), CMB angular scale + early dark energy bounds (Orange), and combined (Gray).All contours included a prior on H0 from the HST and ΩbH20 from Big Bang Nucleosynthesis(see text).h = 0.738± 0.024 Ωb = 0.0382± 0.005Ωdcm = 0.263+0.033−0.022 Ωφ = 0.698+0.025−0.036λ > 18.6 β/λ > 15.6Table 6.2: Parameter constraints for disformally coupled Dark Matter. Error bars corre-spond to the one-sigma allowed ranges.1166.4. Cosmological Perturbations6.2. The slope of the potential γ determines the amount of early dark energy, and is there-fore best constrained by the CMB. The ratio β/γ determines the equation of state at latetimes. Its value is best bound by the combination of all different probes. In the homoge-neous approximation there are no higher bounds on the parameters β and γ, and the modelis completely viable with χ2disf = 538.79 versus χ2ΛCDM = 538.91 obtained from the best fitWMAP7 ΛCDM model compared to the same datasets. The lower value of the equationof state parameter for the disformally coupled matter is responsible for the higher value ofΩdm, at the expense of a lower Ωφ than in standard quintessence or cosmological constantcosmologies. These results reproduce closely the constraints on the disformal self-interactionof the (otherwise uncoupled) scalar field described in Chapter 5: For steep slopes γ andβ, the background evolution becomes increasingly similar to ΛCDM. However, the model isessentially different from ΛCDM and disformal quintessence, as is quite obvious when onelooks at the effective Dark Matter equation of state in Figure 6.1. The growth of perturba-tions studied in the next Section shows that the model is not viable as it is. A glimpse toalternative, viable models is provided in Section 6.4.3.6.4 Cosmological PerturbationsA more realistic description of the universe requires considering cosmological perturbations.The starting point is equation (6.31), which can be used to read both the disformal matternon-conservation and the field dynamical equation. Working in the Newtonian gaugeds2 = −(1 + 2Φ)dt2 + a2(1− 2Ψ)d~x2 , (6.51)the equation for the scalar field isδφ¨+ 3Hδφ˙+(k2a2+ V ′′)δφ = −δQ− 2Φ (Q0 + V ′)+ φ˙(Φ˙ + 3Ψ˙) , (6.52)while the perturbed continuity and Euler equations for the disformally coupled matter con-trast δdc = δρ/ρ (with ρ being the coupled matter component) and the divergence of itsvelocity θ = ikjT0i(m) = ikjvjρa−1 readδ˙dc +θa+Q0ρφ˙δdc = 3Ψ˙ +Q0ρδφ˙+δQρφ˙ , (6.53)θ˙ + θ(H +Q0ρφ˙)= k2(Φ +Q0ρδφ). (6.54)The coupling perturbation δQ is given in Appendix D.2 for the general case. In the case ofa purely disformal coupling, it reduces toδQ(d) = −(k2a2BρM+ (2BV ′′ −B′′φ˙2) ρ2M+(2B′(V ′ + 3Hφ˙) +B′2φ˙2(ρ− φ˙2)) ρ2M2)δφ+(1−Bφ˙2)Q0Mδdc +(B′φ˙−B(3H − ρB′φ˙)−B2(2V ′φ˙+ 3H(ρ+ φ˙2))) ρM2δφ˙+(−B′φ˙+B(6H − ρB′φ˙) + 2B2(3Hρ+ V ′φ˙)) ρφ˙M2Φ +3Bρφ˙MΨ˙ . (6.55)117Chapter 6: General Relativity and Scalar Forces: Disformal CouplingIt is a much more cumbersome combination of the fluid and field perturbations than for thepurely conformal couplingδQ(c) =12log(A)′ρδdc +12log(A)′′ρδφ . (6.56)Note that, unlike in the conformal case, the first term in (6.55) is proportional to k2 and hencethe coupling is scale dependent. This feature will be reflected in the growth of perturbationsand the power spectrum.6.4.1 Small Scale LimitTo extract the most relevant new features by analytic means, we shall consider the subhorizonapproximation. In the small scale limit, taking into account only the matter perturbationsand the gradients of the field, there is a simple expression for the perturbed interaction δQ. Inthis Newtonian limit, we further relate the field gradient to the matter perturbation throughthe field equation (6.52), which yields the simple expressionδQ(N) = Q0δdc . (6.57)Combining equations (6.53) and (6.54) together with the usual Poisson equation, we obtainthe evolution of the coupled Dark Matter overdensityδ¨dc +[2H +Q0ρφ˙]δ˙dc = 4piGeffρδdc . (6.58)In addition to an extra friction term, the source term is modulated. The last effect is capturedby defining an effective gravitational constant Geff that determines the clustering of DarkMatter particles on subhorizon scalesGeffG− 1 = Q204piGρ2. (6.59)This approximation has the same expression as the simple conformal case, although with asignificantly different functional form of the coupling Q0, which is now given by Eq. (6.46).The evolution of Geff for the disformally coupled Dark Matter example model (6.49) isshown in Figure 6.4. It is characterized by a bump at the transition, whose height increaseswith β, and a further increase when the potential becomes dominant. At the later stage, thedependence is approximately Geff/G−1 ∼ (γV/ρ)2 and yields a large value since dark energydomination requires V & ρ and γ & 18 is necessary to avoid the effects of early dark energy(6.50). This enhancement occurs on observable scales and spoils the formation of large scalestructure in this particular case. Problematic growth enhancement also occurs in conformallycoupled models that attempt to address the coincidence problem [258]. The observable effectswill be analyzed using the full perturbation δQ within the disformally coupled Dark Mattermodel. Several alternatives to render the model viable will be described in Section 6.4.3.6.4.2 Structure Formation for Disformally Coupled MatterThe linearized equations (6.52-6.55) were solved using a modified version of the Boltzmanncode CMBeasy adapted to the Disformally Coupled Dark Matter model described in Section1186.4. Cosmological Perturbations0.10 1.000.500.20 0.300.15 0.700.11101001000aGeffG-1Γ=5 H dotted L , Γ=10 H solid LΒ =15 Γ H green L , Β =40 Γ H blue LFigure 6.4: Effective gravitational constant on small scales (6.59) for different values of β, γ.The value is large at the transition due to the disformal friction term B′φ˙2, and latter dueto the contribution of the potential term BV ′ (see text and compare to Figure 6.1).6.3.1. Since matter is essentially uncoupled until z . 10 there was no need to modify theinitial conditions, which have been assumed adiabatic. Figure 6.5 shows the evolution of thedensity contrast of disformally coupled matter. The baryons, which are uncoupled in thisparticular example, are also shown for comparison. Figure 6.6 displays the power spectrafor disformally coupled matter and baryons at z = 0 for different values of the parameters.Figure 6.7 shows the CMB power spectrum and the baryon-DM bias induced by the couplingat z = 0.Besides the effect of early dark energy and late time scalar force captured in Geff , thedisformal coupling causes a considerable integrated Sachs-Wolfe effect, a fundamental biasbetween disformally coupled matter and baryons and cause large scale oscillatory featuresbeyond the BAO scale. The numerical results and the discussion are restricted to the DCDMmodel, and focus on the role of the potential slope γ, which mostly determines the late timevalue of Geff . It remains to be studied whether or not similar effects occur in viable modelssuch as the ones described in Section 6.4.3, and to what extent they might be observable bycurrent or future surveys.Early Dark EnergyBoth the baryons and the coupled Dark Matter are indistinguishable as long as the couplingis negligible. They are equally affected by the presence of early dark energy (see Figure 6.5),which produces a departure from the matter domination result δ ∝ a: EDE increases theexpansion rate without clustering, reducing the formation of structure. This effect was alsofound for the uncoupled scalar field (Chapter 5), and is most noticeable for models withhigher Ωede (e.g. γ = 4).119Chapter 6: General Relativity and Scalar Forces: Disformal Coupling1.0 10.05.02.0 3.01.5 7.00.020.050.100.200.501.001+zH1+zL∆k=0.01 Mpc-1k=0.05 Mpc-1k=0.1 Mpc-1Γ=4 , Β=20 Γ1.0 10.05.02.0 3.01.5 7.00.050.100.501.005.001+zH1+zL∆k=0.01 Mpc-1k=0.05 Mpc-1k=0.1 Mpc-1Γ=10 , Β=20 Γ0.001 0.01 0.1 1100500100050001 ´1045 ´1041 ´105k hPkCoupled MatterBaryonsΓ=4 , Β =20 Γ0.001 0.01 0.1 11001000104105106k hPkCoupled MatterBaryonsΓ=10 , Β =20 ΓFigure 6.5: Growth of disformally coupled matter and (uncoupled) baryons. Upper panel:Thick purple lines coorespond to the disformally coupled matter and thin brown lines tobaryons at the same scales. The pressence of early dark energy affects the slope at earlytimes. Afterwards, the Dark Matter perturbation starts growing rapidly, as deduced in thesmall scale approximation (see Figure 6.4). Lower panel: Evolution of the power spectrumat z = 0 (thick, solid), 0.3, 0.6, 1 (thick, dotted), 2, 4 and 10 (thin, dotted). Gray linecorresponds to a reference ΛCDM model.Late Enhanced GrowthThe growth of structure is enhanced after the transition takes place, consistently with thesmall scale approximation (6.59). The large value of Geff/G overcomes the additional frictionterm, and structures form much faster than in the standard CDM scenario. Models with lessEDE suffer a higher enhancement, because the effective gravitational constant Geff ∝ γ2 islarger and the transition occurs earlier (i.e. the field takes longer to dominate the energycontent). The effect from the bump in the effective Newton’s constant associated to thetransition is not obvious in the evolution of δ, and is subdominant with respect to potentialdomination.The enhanced growth effect is partly canceled by the early dark energy damping. Thisdegeneracy causes the relative resemblance between DCM power spectra with γ = 4 and theDM power for the standard model on small scales, but fails anywhere else. It would be worthexploring this cancellation in a more systematic way (e.g. MCMC exploration of the modelparameters). This would actually require a better understanding of the baryon bias induced1206.4. Cosmological Perturbations0.001 0.01 0.1 11001000104105106k hPkΓ=10Γ=5Γ=4LCDMCoupled DMz=00.001 0.01 0.1 1100500100050001 ´1045 ´1041 ´105k hPkBaryonsz=010- 4 0.001 0.01 0.1 1110100k hPkdisfPkLCDMDark Matter H z =0 Lrelative to LCDM10- 4 0.001 0.01 0.1 10.20.51.02.05.010.0k hPkdisfPkLCDMbaryons H z =0 Lrelative to LCDMFigure 6.6: Perturbations at z = 0 for β = 20γ and different values of γ. First line: Powerspectrum for (uncoupled) baryons and (coupled) matter. Second Line: Ratio of the powerspectra in disformally coupled models relative to ΛCDM. See Section 6.4.2 for a discussionof the different effects.by the coupling (see below). However, such an exploration is postponed for future work.Note that Fourier modes reach the non-linear regime earlier due to the enhanced cou-pling. Upon the failure of linear perturbation theory, the disformal screening mechanismexplored in Section 6.5 might hide these dramatic effects and restore the standard growth,softening the deviations on small scales. Although this seems unlikely to save the examplemodel, it might be necessary to take the effect into account to obtain a fair comparison withobservations.8Scale Dependent Growth and BiasThe power spectra show scale dependent evolution, as can be seen in the different powerspectra normalized to the corresponding ΛCDM (second line of Figure 6.6). In the standardmodel, the linear growth factor is scale independent and does not distinguish baryons fromDark Matter. For disformally coupled Dark Matter, the scale dependent growth follows fromthe k-dependent term in the perturbed coupling (6.55). This feature does not appear in8Similar enhanced growth effects have been considered in the context of quintessence conformally coupledto neutrinos, where the necessity of non-linear analysis has been pointed out [53].121Chapter 6: General Relativity and Scalar Forces: Disformal Coupling5 10 50 100 500 1000100020003000400050006000lCl10- 4 0.001 0.01 0.1 10.020.050.100.200.501.00k hbiasHz=0LP k @b DP k @dcm DΓ=10Γ=5Γ=4LCDMFigure 6.7: CMB power spectrum (left) and bias between baryons and Disformally CoupledDM (DCM) induced by the coupling (right). The enhanced growth of Dark Matter structureson small scales produces a very large ISW effect. Note that the departures are worse formodels with less early dark energy (higher γ), as derived in the small scale approximation(6.59).phenomenological coupled models, in which the growth of the coupled matter structures isenhanced, but in a scale independent way, (cf. [284]).The coupling also modifies the relation between baryonic and Dark Matter structures,since DM couples directly to the field while baryons are only indirectly affected. As baryonsare dragged into the potential wells created by the coupled matter, they follow a scale depen-dent growth pattern, delayed with respect to the dominant matter component. The resultingbias between the two species is shown in Figure 6.7. The scale dependence of the bias vanishesboth on super-horizon scales (k/h . 0.001) and the small scales (k/h & 0.1), which are welldescribed by the scale-independent Geff (6.59). The intermediate region shows the interplaybetween the scale dependent growth for the coupled matter and the baryons following thesestructures.Since galaxies form out of baryons, this fundamental bias modifies the usual DM-galaxypower relation (2.20). Such a correction needs to be taken into account when comparingthe observed power spectrum with disformally coupled models. On the other hand, othermeasurements of the matter distribution such as weak lensing would probe the structuresformed by both components, and may be used to break the degeneracy.CMB: Integrated Sachs-Wolfe EffectThe enhancement of the perturbations after the transition causes the very large IntegratedSachs-Wolfe effect appreciated in Figure 6.7, which becomes most noticeable for the modelswith higher values of γ. The model with γ = 4 gives a better fit on the l . 10 multipoles,but departs considerably in the range 10 . l . 200 due to the effect of early dark energyafter recombination. The model with less early dark energy has the opposite problem: itproduces a better fit in the intermediate range 100 < l < 200 due to the lower amount ofearly dark energy, but the ISW enhancement explodes at lower multipoles due to the highervalue of Geff . The different amounts of early dark energy have an additional effect on theCMB normalization due to the primary Sachs-Wolfe effect: by reducing the potential wells1226.4. Cosmological Perturbationsthat redshift the photons, Ωede acts increasing the height of the peaks.Oscillatory Features beyond the BAO ScaleOscillatory features can be appreciated in the coupled matter power spectrum on very largescales. These are likely created as field oscillations on scales near k ∼ H(z), which are thentransferred to the coupled component, when the coupling is active. They are most noticeablefor the models with a large early dark energy component, e.g. larger field energy density.Although it constitutes a distinctive feature of the model, the oscillations are not significantlyimprinted on the baryonic power spectrum. This, together with the large survey volumesnecessary to explore such scales would make it difficult to detect them through LSS surveys.However, the large scale oscillations would be a characteristic signature in models where thedisformal coupling is universal, in which the same effects occur to DM and Baryons.6.4.3 Viable ScenariosThe study of cosmological perturbations within the Disformally Coupled Dark Matter model(6.49) shows very drastic departures in the formation of large scale structure, that seemvery difficult to reconcile with observations. It would be interesting to obtain a more precisequantification of these discrepancies through an MCMC analysis and explore possible degen-eracies (e.g. the growth suppression from early dark energy and the enhancement from thehigh Geff). However, it is necessary to address the existence of alternative, viable scenariosalready at this stage.Luckily, the action (6.27) is very general and there is considerable room for improvementthrough different choices of the functions A,B and Lφ. There are at least two possibilities• Introduce a modulation in the disformal factor B(φ) → f(φ)B(φ), to make Q0 smallenough after the field enters the slow roll phase. This modification can render δGeffarbitrarily small, except for a relatively short time around the transition (see Figure6.1). This type of models would allow us to study the effects imprinted by the transitionbump without the problems caused by the high Geff at late times.• Constructing the field Lagrangian using a disformal metric, as in the uncoupled modelpresented in Chapter 5 and references [3, 266]. In this model the transition to slowroll would be partially driven by the scalar field Lagrangian itself, and the effects onmatter may be significantly reduced. In the minimal prescription, the matter and fieldLagrangian are constructed using the same metric (6.1) and no extra parameters areintroduced. If this model turned out not to be viable, a different disformal metric forthe field and the coupled matter would offer an alternative that is able to interpolatebetween disformal quintessence and the disformally coupled Dark Matter presentedhere (e.g. different disformal factors with B(m) = B(φ)).Other alternatives could be based on the interplay between the conformal and the disformalpart of the coupling. Viable scenarios might be exploited to alleviate the claimed problemsof ΛCDM with small scale structure formation such as the tension between Dark Mattersimulations and observations with regard to both the density profiles of Dark Matter halos andfor the number of predicted substructures inside a given host halo, the baryonic Tully-Fisherrelation, the constant galactic surface density or the large scale bulk flows (See reference [17]for a summary and references therein for further details).123Chapter 6: General Relativity and Scalar Forces: Disformal CouplingAs a final remark, let us note that the enhanced growth rushes the modes into enteringthe non-linear regime at earlier times, breaking down the perturbative approach followedhere. As it will be explained in the next Section, the disformal coupling comes equipped witha screening mechanism, that hides the effects of the additional force on dense environments.Addressing the consequences of this feature would require considering the non-linear backre-action of the field, which is not properly captured in the approximations considered so far.Chameleon-type theories also show a strengthening of the screening when non-linearities areproperly taken into account [287].6.5 The Disformal Screening MechanismIn this section we will consider the possibility of extending the disformal coupling to visiblematter, as well as other aspects of gravitational theories based on a disformal coupling inthe Einstein frame. Due to the stringent bounds on equivalence principle violations [110],some sort of screening mechanism is necessary to hide the coupling in dense environmentssuch as the Solar System. The role of a disformal coupling in the chameleon mechanism wasrecently investigated by Noller [288], who correctly noted that the disformal contribution tothe conservation equations vanishes for static, pressureless configurations. This is obviousfrom (6.34), since only the T 00 component is nonzero for dust, and when contracting thefield derivatives with the stress tensor a non-vanishing result requires time evolution of thescalar field. Therefore, addressing the effects of disformal couplings requires studying thefield dynamics in high density environments.It is possible to obtain some insight into the dynamics of the field from the analysisof the background cosmology given in Section 6.3, where it was argued that for the purelydisformal case the coupling was proportional to the scalar field energy density, cf. Eq. (6.48).This causes the existence of two regimes, a matter dominated regime in which the effects ofthe coupling are small, and a field dominated regime in which the coupled matter equationof state is modified, as can be seen in Figure 6.1. When denser regions form, the scalarfield energy density becomes insufficient to produce large effects on the matter distribution,unless the field gradients follow the matter distribution and intensify the additional force. Asit will be shown below, the kinetic mixing induced by the disformal coupling makes the scalarfield evolution insensitive to the matter distribution, as long as the matter energy density issufficiently high.For the sake of concreteness, let us restrict ourselves to a canonical scalar field coupledto a perfect fluid. The general equation (6.36) then reads(X gµν −B Tµνm )∇µ∇νφ+(A′2gµν +(A′BA− B′2)φ,µφ,ν)Tµνm −XV ′ = 0 , (6.60)where X ≡ A− 2BX < A is bounded in order to avoid a singularity in the volume elementof the barred metric√−g¯ ∝ X 1/2, cf. (6.19). The form of the field equation stronglysuggests that the dynamics of the coupled system will be different in high than in low densityenvironments. Since the energy momentum-tensor appears as a coefficient of the higherderivatives as well as in the effective potential, there is a well defined limit T 00 = ρ → ∞,in which the field equation simplifies considerably. This property will be crucial for thedisformal screening mechanism analyzed in Section 6.5.2.1246.5. The Disformal Screening Mechanism6.5.1 Pressure InstabilityThe above equation also shows that the different components of the energy momentum ten-sor can modify the character of the coefficient of the second order derivatives of the field.Equation (6.36) is a quasi-linear, diagonal second order equation [289], of the formφ;µνMµν(φ, φ,λ, Tαβ) + f(φ, φ,λ, Tαβ) = 0 . (6.61)However, its hyperbolicity relies on the signature of Mµν , which in turn depends on therelative values of the coefficients of the higher spatial and time derivatives. Focusing onthe purely disformal case A = 1 for concreteness, we see that the coefficient of the timederivatives, M00 = −1 + 2BX − Bρ, can only flip sign if B or ρ are negative (becauseof the bound 2BX < 1). However, a large enough value of the Einstein frame pressure,p > B−1 − X, may flip the sign of M ii = 1 − BX − Bp. If this happens, the equationbecomes non-hyperbolic (at least locally), challenging the good initial value formulation andturning the system unstable.Addressing the viability of the theory hence requires determining under which conditionsthe instability may occur dynamically, which in turn requires considering the evolution ofthe coupled matter components. In certain cases, the system might respond to a situationin which Bp . 1 − BX by diluting the (Einstein frame) pressure below the threshold valueor softening the spatial gradients of the scalar field. In this sense, the instability inducedby the pressure may be analogous to the potential existence of singularities in the disformalvolume element whenever g¯ ∝ A−2BX → 0. This singularity in the barred metric is avoidedby the field evolution, as it slows down whenever Bφ˙2 → A. The mechanism exploited toinduce a slow roll phase in cosmological applications (disformal quintessence and the disformalcoupling to matter described in Sections 5.2.1 and 6.3.1) is precisely this dynamical resistanceto pathology.Studying the conditions under which the pressure instability can be avoided dynamicallymight restrict the allowed functional forms of the conformal and disformal factors. In theworst case, it might spoil the disformal screening mechanism, or even completely forbid theoccurrence of a disformal coupling. Determining whether or not this is the case will be theobjective of future work. The Einstein frame pressure p will be neglected as a part of theapproximation scheme in the following analysis, implicitly assuming that the theory is wellbehaved.6.5.2 The Scalar Field in Dense, Non-relativistic EnvironmentsThe study of Solar System and laboratory tests of gravity requires considering energy densitiesthat are much higher than the cosmological average and pressure is completely sub-dominant.As a first approximation, this regime can be explored using the general scalar field equation(6.60) for a static matter distribution ρ(~x) in the limit ρ→∞. More precisely, the followingratios will be assumed to be negligibleΓµ00φ,µ ∼pρ∼ pρ(~∂φ∂tφ)2∼ XBρ∼ XBρV ′ ∼ 0 (6.62)These approximations quantify to what extent the effects of gravity and pressure are dis-regarded, the requirement of having “soft” spatial gradients relative to the time evolution125Chapter 6: General Relativity and Scalar Forces: Disformal Couplingand the fact that Bρ is large enough. This conditions will be briefly discussed at the end ofthe Section, focusing on systems in which their lack of fulfillment might lead to observablesignatures.The set of assumptions (6.62) simplifies the field equation (6.60) considerablyφ¨ ≈ − B′2Bφ˙2 +A′(φ˙2A− 12B)= − β2Mpφ˙2 , (6.63)where the first equality is general and the second applies to a purely disformal couplingwith exponential forms (6.49). The above expression departs substantially from the simpleconformal coupling, for which the ρ→∞ limit is ill-defined. Two important features of theabove equation endow the theory with the disformal screening mechanism:• The spatial derivatives become irrelevant, as they are suppressed by a p/ρ factor w.r.t.the time derivatives.9• The equation is independent of the local energy density, making the field evolutioninsensitive to the presence and distribution of massive bodies.These features ensure that the field rolls homogeneously and avoids the formation of spatialgradients between separate objects, which would give rise to the scalar force. The aboveproperties are caused by the kinetic mixing between the field and matter degrees of freedom,and lay at the core of the decoupling between both components.Let us analyze the simpler, purely disformal exponential case. The second equality ofequation (6.63) can be easily integratedφ˙(t) =Mpβ(t+Mpβφ˙(0))−1. (6.64)In this solution the field time variation is approximately constant while t Mpβφ˙(0)and slowsdown afterwards as ∝ 1/t. Since the coupling to non-relativistic matter is proportional to φ˙,stronger couplings decay earlier. It is possible to obtain an implicit solution for A = 1, ρ˙ = 0keeping the potential V , but otherwise assuming the simplifications (6.62). It is given as animplicit solutiont− t0 =∫ φφ0√B(φ′)C0 − 2V (φ′)/ρdφ′ , (6.65)where φ˙2 = C0B − 2V (φ)Bρ .10 The potential appears suppressed with respect to the energydensity. In tracking Dark Energy models, V is a decreasing function of the field and φ˙ > 0.Therefore, V is of the order of magnitude of the average cosmic density ρ0 and can be safelyneglected for high energy densities, recovering the simpler solution (6.64). As the field slows9Equation (6.63) also follows from taking the limit ρ A/B, φ˙2 in the FRW coupling density Q0 (6.45),precisely due to the absence of spatial derivatives.10Under these assumptions, Eq. (6.60) can be written as φ¨ + B′2Bφ˙2 + V′Bρ= 1Bφ˙ddt(Bφ˙2/2 + Vρ)= 0. Thesecond equality can be directly integrated, giving the constraint φ˙2 = C0B− 2V (φ)Bρ, which can be integratedagain to obtain (6.65).1266.5. The Disformal Screening Mechanismdown with time if B′/B > 0, the order of magnitude of the field time derivative is alsocosmological.One of the effects of the coupling is to modify the energy conservation equation formatter in the Einstein frame, cf. Eq. (6.40). This induces a variation of the gravitationalmass, that can be bounded by precision gravity tests. In a gravitationally bound two bodysystem, this effect is degenerate with a possible time evolution of Newton’s constant G˙G ↔ M˙M +m˙m to a first approximation, as can be argued by deriving the expression for the Newtoninanforce with respect to time. Lunar laser ranging bounds place precise bounds on this effect tothe level of G˙/G < 10−3/Gy [290]. The magnitude of energy density variation induced by adisformal coupling can be estimated as ρ˙ ≈ −(φ¨+V ′)φ˙. Assuming φ˙2 ∼ V ∼ ρ0 as discussedabove, Bρ0 & 1 and the solution (6.64), typical mass variation rates M˙/M are as small as∼ 10−6/Gy for the interstellar medium and ∼ 10−29/Gy for the average Earth density, wellbeyond the sensitivity of Lunar laser ranging.Potential SignaturesNew local, astrophysical and cosmological signatures may be found by relaxing the approxi-mations (6.62) assumed in the previous analysis. Some situations where the coupling mightbecome observable include:• Matter velocity flows: The spatial component of the matter four-velocity T 0i mixesthe time and space derivatives of the field, as non-zero velocities introduce terms propo-rional to φ;0i, φ˙ φ,i, which may source the field evolution. These effects are suppressedby a relativistic v/c factor, but they may be important in certain systems such as binarypulsars.• Pressure: Applications of the disformal coupling in the context of Dark Energy ar-guably require a value Bρ0 1, where ρ0 is the average cosmic density. Then, eventhough the pressure is usually negligible with respect to the energy density, it shouldbe easy to find systems for which Bp is also much larger than one. This might have im-portant consequences for the stability of the theory, as was briefly discussed in Section6.5.1.• Radiation: Unlike in the conformal case, the disformal coupling has non-trivial effectson ultra-relativistic fields for which T ≈ 0, cf. (6.33). Some authors have initiated thestudy of the disformal coupling in scenarios featuring radiation. Brax et al. [291] con-sidered high-precision, low-energy photon experiments, which might be able to detectthe influence of a disformal coupling on top of a conformal one. The distortions in thebaryon-photon chemical potential induced by a disformal coupling and their signatureson the CMB small scale spectrum have been studied by van de Bruck and Sculthorpe[292]. Other effects may follow if Electromagnetism is formulated in terms of the barredmetric, such as varying speed of light or modified gravitational light deflection.• Strong gravitational fields: The connection coefficient Γµ00φ,µ in the field derivativeterm is not suppressed by Bρ. It represents the effects of gravity, and was neglectedbecause it is small in most Solar System applications, since Γr00 =GMr3(r−2GM) in theSchwarzschild metric. However, this term might become relevant in strong gravitationalfields, such as the vicinity of black holes or compact objects.127Chapter 6: General Relativity and Scalar Forces: Disformal Coupling• Spatial Field Gradients: In the Bρ 1, ρ p limit, the equation for the scalar field(6.63) becomes independent of the matter content and the field derivatives. Therefore,if the field acquires a spatial modulation before reaching this limit, it will be preservedby the subsequent evolution. Spatial gradients of the field formed when the linearperturbation theory is valid would then be present today, with their actual value de-pending on the details of the transition between the perturbative (e.g. the small scalelimit considered in Section 6.4.1) and the screened regimes. Gradients of cosmologi-cal origin might seem as preferred direction effects pointing towards cosmic structureswhen analyzed in the Solar System.Spatial derivatives of the field may also be important if the field is rolling sufficientlyslow as to overcome the p/ρ factor in (6.62).These and other settings might lead to characteristic signatures and new bounds for disfor-mally coupled theories, which will be investigated in the future. It should be also possibleto obtain the coefficients of the Parameterized Post Newtonian approximation, which wouldallow a more systematic comparison to local gravity tests.6.6 DiscussionThe disformal relation provides a generalization of the conformal transformation. It hasbeen used to construct theories of modified gravity, notably those which produce non-trivialeffects on null geodesics, such as varying speed of light and gravitational alternatives to DarkMatter. It also appears in the description of branes embedded in a higher dimensional bulkspace, in which the scalar fields represent the brane position in a certain set of coordinates.The results of the present Chapter concern the subset of theories which can be expressedas General Relativity plus a matter Lagrangian, which is constructed using the disformalmetric. This provides a generalization of the old-school scalar tensor theories in the Einsteinframe: Test particles follow geodesics which explicitly involve derivatives of the scalar field,and the energy momentum of the field and coupled matter (computed w.r.t. the gravitationalmetric) is not conserved separately.It is possible to restore the theory to a Jordan frame representation by reversing thedisformal transformation, as was argued in Section 6.2.1 and computed first in Ref. [68]. Thecomputation of the Einsten-Hilbert action for a disformal metric produces a theory in whichthe field has a non-minimal derivative coupling to the Ricci scalar and in which new, higherderivative terms appear. The resulting Lagrangian density has the correct Horndenski form(1.28), and the theory is therefore described by second order equations of motion and stablein light of Ostrogradski’s Theorem. The existence of additional frames, in which only theconformal or disformal part enter the matter action explicitly, provides new tools to analyzethis type of theories and further connections between different models (e.g. a disformal theoryin the Jordan frame reduces to a 4th Galileon Lagrangian in the non-relativistic limit). Theterms arising from the inverse disformal transformation contain higher derivatives, endowingthe theory with the Vainshtein screening in the conformal and Jordan frames, which allowsthe field to cause effects on cosmological scales while remaining undetectable in the SolarSystem.The equivalence between certain higher derivative theories with conformal or minimalcoupling and disformally coupled theories with an Einstein-Hilbert gravitational sector pro-vides new means to analyze this type of models. Although the equations for disformally1286.6. Discussioncoupled theories are rather involved, they are much simpler than higher derivative Hornden-ski theories in the Jordan frame. Hence the analysis of disformal theories in the Einsteinframe can be regarded as equivalent to (at least) some scalar-tensor theories featuring theVainshtein screening mechanism. The kinetic mixing between the coupled degrees of freedommakes it necessary to solve for the higher time derivatives of the different components. Al-though this generally requires some assumptions, e.g. about the metric, a general equationwithout derivatives of the energy-momentum tensor can be obtained (6.36). Once solutionsare computed, it is possible to restore to the Jordan metric in order to interpret the resultsand compare to observations.In high density environments (as measured by the condition Bρ 1) the field doesnot feel the pressence of disformally coupled, non-relativistic matter. This provides a noveldisformal screening mechanism, which is distinct from screening mechanisms based on thefield potential (Chameleon and Symmetron). Our mechanism relies on the existence of awell defined limit ρ → ∞ in the scalar field equation, given by equation (6.63), for whichthe field evolution is independent of the matter distribution and the field gradients, up toeffects of order ∼ p/ρ, v/c. If the conformal part A is negligible, only a friction term remainsand the field coupling density (6.34) is a decreasing function of time. As it evolves belowits cosmological value (provided V ′ < 0 and B′/B > 0), the effects of the coupling aresuppressed by a factor ∼ ρ0/ρ and the theory is consistent with precision gravity tests.Potentially detectable signatures may be obtained in the presence of matter velocity flows,radiation or pressure, strong gravitational fields or gradients of cosmological origin.The disformal screening mechanism may be related to the Vainhstein effect [84], whichsuppresses the gradients of the scalar field and hides the additional force near massive sourcesdue to the higher order derivative self-interactions. In the disformal case, the screening relieson the kinetic mixing between the scalar field and the coupled degrees of freedom. Thisultimately allow the existence of a well defined ρ→∞ limit in which the field is free, or onlysubject to friction(up to conformal interactions). Hence the disformal mechanism belongs tothe kinetic screening category. Moreover, the two cases might indeed be the same physicaleffect expressed in different frames, due to the equivalence between the disformally coupledtheories and a particular higher derivative sector of the Horndenski Lagrangian.The disformal coupling offers interesting possibilities to build models for cosmic accel-eration. In the FRW approximation, the same properties that gave rise to the disformalscreening mechanism make the background coupling approximately proportional to the darkenergy density (6.48) rather than to the coupled matter energy density. This provides aconcrete realization of a class of interacting dark matter models which have been extensivelystudied using phenomenological parameterizations. The equations for linear perturbationsaround FRW contain scale dependent terms. These are absent in the pure conformally cou-pled case, and have hence the potential to distinguish the two possibilities. An analyticequation for the coupled matter perturbations was derived in the small scale limit. On top ofan additional friction term, the effect of the fifth force can be encapsulated in the definition ofan effective gravitational constant (6.59) which depends on the background coupling factor.In order to investigate the cosmological implications of a disformal coupling in a simplesetting, a Disformally Coupled Dark Matter (DCDM) example model was proposed. Thishas the advantage of avoiding the subtleties of the Einstein frame description, since grav-ity, baryons and photons share the same physical metric. Additionally, the model has theadvantage of being compatible with local gravitational phenomena without invoking the dis-129Chapter 6: General Relativity and Scalar Forces: Disformal Couplingformal screening mechanism. A DCDM model with exponential functions and no conformalcoupling (6.49) provides a dark energy model that tracks the dominant energy component atearly times. When the coupling to Dark Matter becomes active, the scalar field enters a slowroll phase in order to dynamically avoid a singularity of the disformal metric. The free pa-rameters can be constrained by observations, and the model is successful at the backgroundlevel. When perturbations are included, the DCDM model introduces a series of new effects.The effective gravitational constant for this model is too large, due to the persistence of thecoupling at late times and the domination of the energy budget by the scalar field. Thiscauses a too large enhancement of the growth factor, which affects the normalization of theDM and baryon power spectra, producing a very large ISW effect. Scale dependent effectsare reflected on matter oscillatory features on very large scales and a scale dependent biasbetween the coupled dark matter and (uncoupled) baryonic component.There is considerable freedom in the model to produce cosmologically viable scenarios.Models of the DCDM type with less dramatic growth of perturbations can be constructedby modifying the functional dependence of the disformal coupling (e.g. tuning it to becomenegligible after the transition to slow roll), the scalar field Lagrangian (e.g. constructingit with a disformal metric), or perhaps by the interplay between the conformal and thedisformal parts of the coupling. Variations or extensions of DCDM may postulate or includedisformally coupled neutrinos.Another phenomenological direction is to consider the disformal screening mechanismin detail. The results presented here considered a purely disformal coupling, monotonicallyincreasing with the field. Certainly, including a conformal coupling and more general func-tional forms is of interest. These considerations might help to avoid the instability causedby the Einstein frame pressure if Bp > 1, as it was discussed in Section 6.5.1. Once differentset-ups are formulated, it is worth to explore the observable signatures for the model byquantifying the effects outlined at the end of Section 6.5.2.The dependence of the free functions in the Horndenski Lagrangian (1.28) on the fieldkinetic term X has a very special role, as it relates the coupling to gravity to the coefficientsof the second derivative field terms. Therefore, it would be worth considering the transfor-mations between frames in the more general case in which the disformal relations are allowedto depend on X. The computation of the Ricci scalar associated to this general disformalmetric would provide the Jordan frame representation of the most general scalar-tensor the-ory that accepts an Einstein frame description. Since the equations simplify considerably inthis frame, the phenomenology of these theories would be relatively easy to address.Finally, the existence of a well behaved ρ → ∞ classical limit in the field equationsuggests that disformally coupled theories might introduce new interesting features for thephysics of gravitational singularities and other high energy regimes. The implications ofkinetic mixing for the formation of black holes or the origin of the universe is beyond thescope of the present work, but it might provide a fruitful exploration to pursue in the future.This discussion provided just a glimpse to the potential applications of the disformal relation.As a generalization of the conformal case, which was very central to the gravitation andcosmology of the 20th Century, the use of disformal transformation might provide novel waysto address the physics of the 21th Century.130Chapter 7Conclusions and OutlookPeople don’t want their lives fixed. Nobodywants their problems solved. Their dramas.Their distractions. Their stories resolved.Their messes cleaned up. Because what wouldthey have left? Just the big scary unknown.Chuck Palahniuk1The present Thesis analyzes several alternatives to the Standard Cosmological Modelthat attempt to explain the observed cosmic acceleration. The predictions of these scenarioshave been compared with current data, in order to determine which mechanism is empiricallyfavored. The simplest explanation is a Cosmological Constant. Although phenomenologicallysuccessful, it is nonetheless unsatisfying according to theoretical estimates and there are goodreasons to go beyond, even in the absence of discrepant observations. The alternatives con-sidered either drop the assumption of homogeneity on large scales or introduce modificationsin the Einstein-Hilbert action. Theories that generalize the gravitational sector introducenew degrees of freedom, either effective or fundamental. They can be roughly classified asModified Gravity or Dark Energy, depending on whether their effects on standard matterare direct or indirect. It is also possible to consider purely phenomenological modificationsof the underlying equations. Each of these possibilities has been explored in the previousChapters.The cheapest option in terms of new physics turns out to be the most difficult to rec-oncile with observations. Adiabatic inhomogeneous cosmologies without Dark Energy, whichcan be interpreted as the gravitational growth of a spherically symmetric perturbation withan initially small amplitude, are actually ruled out by current data on geometric observables.LTB models with a CGBH matter profile are unable to simultaneously fit Supernovae andBaryon Acoustic Oscillation (BAO) scale observations. In these models, the apparent accel-eration is caused by the relatively lower deceleration near the location of our galaxy (lowerdensity region), compared to the asymptotic value (higher density region). If the density1Survivor (1999)131Chapter 7: Conclusions and Outlookcontrast is enough to produce the fictitious acceleration demanded by SNe observations, thelower deceleration near the center produces an additional stretching of the low redshift BAOscale with respect to the asymptotic value, which is incompatible with current, higher red-shift data. Due to the generality of the effect, it is expected that a similar tension would befound regardless of the matter profile. This discrepancy provides further evidence againstinhomogeneous cosmological models with adiabatic initial conditions, e.g. on top of boundsfrom the kSZ effect and the expansion rate measured at different redshifts.The inhomogeneous growth of free falling scales such as the BAO correlation lengthcan be used to test inhomogeneity within more general models. It is possible to use inho-mogeneous models of the LTB type with a Cosmological Constant (or other form of DarkEnergy) to quantify the level of inhomogeneity allowed by observations. More generally, onecan further drop assumptions and consider inhomogeneity in other degrees of freedom (ageof the universe, baryon-dark matter or baryon-photon ratios), non-central locations or evenless symmetric space-times. The general program to test large scale homogeneity and theCopernican Principle requires extending the set of observables and analyzing them with-out introducing spurious assumptions. The volume-averaged BAO scale discrepancy may berendered concordant with SNe by the inclusion of Dark Energy or inhomogeneous baryonfraction. However, new data able to separate the radial and angular distances (e.g. theAlcock-Paczynski effect) can potentially break the degeneracy. This program would natu-rally profit from the inclusion of dynamical observables, such as Redshift Space Distortionsand the Integrated Sachs-Wolfe effect.The model based approach has a number of advantages with respect to phenomeno-logical descriptions of modified theories. It allows one to give a self consistent descriptionwhich is well connected to the underlying ideas. Additionally, it can be used to extractpredictions in different situations, although the possibilities are usually limited by technicalor computational difficulties. On the other hand, phenomenological models often require anumber of assumptions in order to obtain the background or perturbation equations, as itwas the case of the entropic (modified) gravity scenarios. Parameterizations also allow toexplore the inter-dependence of the assumptions (e.g. the preference towards slightly closedspatial sections by the Dark Energy model based on a Generalized Entropy-area Law). Con-crete models are complementary to phenomenological parameterizations. The former allowto study the link between observational signatures and the underlying physical mechanisms,and provide valuable inspiration for the latter.Disformal relations tend new bridges between different scalar-tensor theories of gravity.The inclusion of a scalar field generously extends the theory space from the Einstein-Hilbertto the Horndenski Lagrangian, which allows four free functions of the field and its kineticterm. The disformal transformation applied to a canonical scalar field Lagrangian minimallycoupled to gravity recovers a variety of Dark Energy models, that belong to the simplest ofHorndenski’s functions (k-essence). More generally, the disformal prescription applied to thematter sector introduces a new scalar force. If the gravitational sector is standard, thesedisformally coupled theories are the Einstein Frame representation of a theory in which thefield couples to the curvature tensors through its derivatives, together with second order(but safe) derivative terms. This Jordan Frame representation, obtained by undoing thedisformal transformation, corresponds to a particular form of the function in the HorndenskiLagrangian which corresponds to the generalized fourth order Galileon term. These theorieshave attracted considerable attention because they are endowed with the Vainshtein screening132mechanism, which hides the force mediated by the field near massive bodies.Theories based on a disformal coupling to matter come naturally equipped with a disfor-mal screening mechanism, that renders them viable in the Solar System. The kinetic mixingwith the disformally coupled degrees of freedom makes the scalar field equation independentof the matter distribution and the field spatial derivatives in high density, non-relativisticenvironments. Then the field rolls homogeneously and does not develop gradients betweenmassive bodies, hiding the scalar force. If the field is set up for cosmological applications, theeffects of the field are suppressed by the ratio between the local and cosmic average energydensity. Based on the correspondence between disformally coupled theories and generalizedGalileons, the disformal screening mechanism may be just the Vainshtein effect in disguise.The duality between the higher derivative Jordan Frame and the disformally coupled Ein-stein Frame descriptions would facilitate the exploration of certain sectors in the HorndenskiLagrangian. Just as in old school scalar-tensor theories, the Einstein Frame descriptionconsiderably simplifies the equations and is hence useful for computational purposes.Disformal relations also allow the construction of models for cosmic acceleration. Theserely on their dynamical resistance to produce singularities in the disformal metric, whichdrives the scalar field into a slow roll phase. This property was explored explicitly in twocases that avoid the subtleties of working in the Einstein Frame. Disformal quintessenceprovides a Dark Energy model compatible with current observations, as it converges to aCosmological Constant if the potentials are very steep, making it very difficult to distinguishfrom the Standard Cosmological Model. The Disformally Coupled Dark Energy models in-troduce more dramatic departures due to the direct effect on Dark Matter. The simple modelconsidered is viable a at the background level. However, the evolution of linear perturbationsis incompatible with observations, due to the strong effects on the matter power spectrumand the large Integrated Sachs-Wolfe. Fortunately, viable models may be constructed byexploring the freedom in the field Lagrangian and the coupling functions. These alternativesmight be distinguishable from other modifications of gravity and coupled Dark Matter mod-els by the additional bias between DM and baryons, as well as for the scale dependence ofthe perturbations growth, which is different from conformally coupled models.It is necessary for the health of science to bear in mind alternative models. Regardlessof its success, due respect to the Standard Cosmological Model requires the right dose ofskepticism. On the observational side, it helps to identify and avoid (or at least control)the inclusion of model dependent assumptions in the analysis of the data. Studying thephenomenological implications of alternative models and comparing them to observationsalso allows one to explicitly probe the underlying assumptions behind the standard paradigm,as well as to quantify the allowed departures. This is important, as empirical science relieson disproving alternatives rather than confirming hypotheses. Considerable knowledge canbe gained from the theoretical point of view, as the study of alternatives helps to understandand clarify the properties of the standard paradigm. These considerations will become evenmore important for the next generation of cosmological surveys, as they will either validatethe foundations of the Standard Cosmological model to higher accuracy, or open Pandora’sbox again by bringing new surprises.133Chapter 7: Conclusions and Outlook134Part IIIAppendices135Appendix AMCMC AnalysisWhen confronting cosmological models with observations, the objective is to determine whichvalues of the free parameters (our theory) are compatible with an available set of astronomicaldata (our Universe). Since measurements are not infinitely precise, the compatible set ofparameters will occupy at least a finite volume in the parameter space, which is initiallyunknown. What is known is the probability of obtaining a certain measurement for a givenmodel, which can be obtained by comparing the theoretical prediction with the experimentalvalue and its uncertainty (e.g. assuming Gaussian errors). This probability L(D|u) is knownas the likelihood of observing the data D (a vector in the space of observations) given amodel u (a vector in the space of parameters). It has to be inverted in order to obtain theposterior distribution P (u|D), i.e. the probability of u being the true model if D is observed.Both probabilities are related through Bayes theorem:P (u|D) = L(D|u)pi(u)∫ L(D|u′)pi(u′)du′ (A.1)Monte Carlo Markov Chain analysis constitute a widely used strategy to invert thelikelihood and obtain the posterior distribution. A direct grid sampling of the parameterspace requires a great deal of computational effort, since the number of points scales withexponentially with the number of parameters. On the other hand, MCMC execution timesincrease roughly linearly with the number of parameters. A Markov Chain is a stochasticsampling of the parameter space {u0, ..., un} in which the point ui only depends on ui−1. Ifthe way in which the chain is created is chosen properly, the distribution of its points willconverge to the posterior distributionDist{u0, ..., un} = P (u|D) (A.2)The sampling method used by CMBEasy is the Metropolis algorithm [293]. For a givenpoint ui, the next point is chosen as follow:1. Compute L(D|ui); the likelihood of observing D given the parameters ui.2. Propose a new parameter vector by sampling from a proposal distribution q(ui, ui+1)(see below).3. Compute the likelihood for the proposed parameter L(D|ui+1).137Chapter A: MCMC Analysis4. If the likelihood is higher Li+1 > Li, the point is added to the chain and the proceduregoes to step 2.5. If Li+1 < Li, a random variable u ∈ [0, 1] is generated. If u < Li+1/Li, the step istaken and go to 2. Otherwise go to 2, but reject the actual point ui+1.The algorithm assumes flat priors pi(u) and a symmetric proposal distribution q(ui, ui+1).Likelihoods outside the boundaries are assigned zero likelihood. An adaptive stepsize samplerhelps to the convergence and mixing of the chains. Too large stepsize will lead to many stepsbeing rejected, while too short stepsize will take a long time to sample the distribution. Thefunction q(ui, ui+1) is Gaussian, and it is estimated from the previous points in the chains.Stepsizes are taking along the principal axes of this distribution to take into account thepossible degeneracies among the parameters.The initial point in the chain is chosen randomly in the parameter space and will moveto regions of higher likelihood afterwards. Therefore, the first points in every chain do notreflect the distribution of P (u|D). In order to test the convergence and determine how manypoints to ignore, the algorithm uses a number m of chains to evaluate their mixing andconvergence. CMBeasy uses the test of Gelman and Rubin [217] to calculate the variance ofeach parameter between chains. We denote by ψij the value of one parameter at the pointj = 1, ..n of chain i, and a bar over a quantity means average over the missing indices. Thevariance between chains B and within chains W are given by [218]B =nm− 1m∑i=1(ψ¯i − ψ¯)2 , (A.3)W =1m(n− 1)m∑i=1n∑j=1(ψij − ψ¯i) , (A.4)where the sums over the point index j are taken over the last n points instead of the wholechain. The quantityR =n−1n W +1nBW(A.5)should converge to one for the stationary distribution. If, for each parameter R < 1.2, thechain can be safely considered to be sampling from the posterior distribution.138Appendix BCMBEasyCMBEasy [216] is an open source program for the computation of cosmological observablesbased on CMBFAST [118]. It is written in C++, and its object oriented source code allowsvery clean modifications of the code by simply re-implementing the desired functions in theproper subclasses. As in its precursor, most quantities are stored in splines, a very efficientstructure to contain and manipulate functions of a single variable. The actual version cancompute power and CMB spectra within seconds, including tensors and polarization. Twodifferent gauges (synchronous and Newtonian) and several quintessence models are included.A typical CMB computation is done as follows:1. The background and thermal histories are computed.2. The perturbations are propagated in Fourier space for different k modes.3. The line of sight integration (2.5) is used to compute the anisotropy spectrum seentoday.The Analyze This! [218] package provides the necessary functions to observationally con-straint models. It is based on the analizethis class, containing the routines necessary tocalculate the likelihoods of the computed model for a wide set of available astronomical data,a program designed to run the MCMCs in a OpenMPI parallel environment. All the com-putations of cosmological observables contained in this thesis were performed using suitablymodified versions of CMBeasy.139Chapter B: CMBEasy140Appendix CEquations for Generalized k-essenceThe action for the scalar is specified through a Lagrangian densitySk =∫d4x√−g L(X,φ) , (C.1)where X ≡ −12gabφ,aφ,b. With these conventions, canonical quintessence corresponds toL(X,φ) = X − V (φ). The energy momentum tensor isT(φ)ab =−2√−gδ√−gL(X,φ)δgab= L(X,φ)gab + L,X(X,φ)φ,aφ,b , (C.2)where commas denote partial derivatives. The dynamical equation for the scalar field isL,φ −∇a(L,X∂aφ) = 0 . (C.3)Background equations: For the flat FRW metric using conformal time ds2 = a2(τ)(−dτ2+dx¯2) andρφ = L,Xφ′2/a2 − L , (C.4)pφ = L , , (C.5)where f ′ = f,τ . Expanding the covariant derivative in (C.3) for an FRW metric yields thefield equationMφ′′ +H(3L,X −M)φ′ + a2L,φ + L,Xφφ′ 2 = 0 , (C.6)where M≡ L,X − L,XXφ′2/a2 is useful to define.Linear perturbations (syncrhonous gauge): ds2 = a2(τ)(−dτ2 + (δij + hij) dxidxj)(using Ma & Bertschinger’s notation [274]) the relevant terms for Einstein Equations becomeδρφ =Mφ′δφ′/a2 − (L,φ − L,Xφ(φ′/a)2)δφ . (C.7)δpφ = L,φδφ+ L,Xφ′δφ′/a2 + Lh/3 , (C.8)(ρ+ p)θφ = −L,X k2a2φ′δφ . (C.9)141Chapter C: Equations for Generalized k-essenceΣij = O(δφ2) and h = δijhij . The Equation for the field perturbations isMδφ′′ + [2HM+M′] δφ′ + 12L,Xh′φ′+[L,Xk2 + (φ′′ + 2Hφ′)L,Xφ + L′,Xφφ′ + a2L,φφ]δφ = 0 . (C.10)Since the coefficients are background quantities, we can take its derivatives M ′,L′,Xφ througha spline.142Appendix DDisformal RelationsConsider the disformal relation between two metrics, specified by the two scalar functions A,B, and a vector bµ,g¯µν = Agµν +Bbµbν . (D.1)The inverse metric can be found by contractiong¯µν =1A(gµν − γ2bµbν) . (D.2)whereγ2 ≡ BA+Bb2, (D.3)and b2 ≡ gµνbµbν ≡ bµbµ. The determinant of the barred and unbarred metrics are related√g¯g= A√ABγ2= A2√1 +BAb2 , (D.4)The above relation is derived in Appendix C of Reference [60].It is possible to write the relation of stress energy momentum tensor (associated to aLagrangian√−gL) in the two metrics by using the chain ruleTµν ≡ 2√−gδ (√−gL)δgµν=√g¯gδg¯αβδgµν(2√−g¯δ (√−gL)δg¯αβ). (D.5)By identifying the quantity in brackets as T¯µν and using (D.4), the following relation followsTµν = A3√1 +BAb2 T¯µν . (D.6)The equivalent relation with lower indices is considerably more involvedTµν =√g¯gD αβµν T¯αβ , (D.7)whereD αβµν ≡δg¯αβδgµν=1A(δαµδβν − 2γ2bαb(µδβν) + γ4bµbνbαbβ). (D.8)143Chapter D: Disformal RelationsThe inverse relations are provided below for completenessgµν =1A(g¯µν −Bb¯µb¯ν), (D.9)gµν = A(g¯µν + γ¯2b¯µb¯ν), γ¯2 ≡ B1−Bb¯2 , (D.10)D¯µναβ = A(δµαδνβ + 2γ¯2δµ(αb¯β)b¯ν + γ¯4b¯µb¯ν b¯αb¯β), (D.11)where b¯µ ≡ g¯µνbν . Note that b¯µ = bµ, b¯µ = B/(AC¯)bµ and γ2b2 = Bb¯2.D.1 Disformal GeodesicsThe expression for the disformal connection (6.6) can be expanded in terms of the functionsin the disformal metricΓ¯µαβ = Γµαβ + δµ(αlogA,β) −12logA,µgαβ +1A(φ,µB,(αφ,β) −12B,µφ,αφ,β)(D.12)−γ2Aφ,µ[A,(αφ,β) −12φ,λA,λgαβ − 2X(B,αφ,β − 12φ,λB,λφ,αφ,β)]+BA[∇(α(φ,β)φ,µ)− 12∇µ (φ,αφ,β)− γ2φ,µφ,λ(∇(α(φ,β)φ,λ)− 12∇λ (φ,αφ,β))].Here γ2 ≡ BA−2BX arises from the inverse barred metric, eq. (D.2). The first term is justthe connection of the unbarred metric, and the two following terms arise from the purelyconformal transformation involving derivatives of A. The fourth term and the second linecontain the first order derivative terms from the disformal contribution to the metric B. Thethird line shows the second order derivative terms ∇∇φ.D.2 General Disformal Coupling PerturbationsThe coupling density perturbation that enters the linear equations (6.52,6.53) in the casewhere both the conformal and the disformal parts of the coupling are relevant has the rathercomplicated formδQ = Qρδdc +Qφδφ+Qdφδφ˙+QΦΦ +QΨ˙Ψ˙ , (D.13)144D.3. Lagrangian Derivatives for Disformal QuintessencewhereQρ = ρ(1− BAφ˙2)A′(A− 2Bφ˙2) +B′φ˙2 −A(2BV ′ + 6BHφ˙)2(A+Bρ−Bφ˙2)2 , (D.14)Qφ =((1− 2BAφ˙2)A′′ +B′′φ˙2 − 2B(k2a2+ V ′′)) ρ2A(A+Bρ−Bφ˙2)−((A′A)2(A2 − 2B(2A+Bρ)φ˙2 + 2B2φ˙4) +A′(−2B(V ′ + 3Hφ˙) +B′(ρ+ 2φ˙2))+2AB′(V ′ + 3Hφ˙) +A2B′2φ˙2(ρ− φ˙2)) ρ2(A+Bρ−Bφ˙2)2 , (D.15)Qdφ = −B(A+ 2Bρ)A′A φ˙− (A+Bρ)B′φ˙+B(2BV ′φ˙+ 3H(A+Bρ+Bφ˙2))(A+Bρ−Bφ˙2)2 ρ , (D.16)QΦ =B(A+ 2Bρ)A′A φ˙− (A+Bρ)B′φ˙+ 2B(3H(A+Bρ) +BV ′φ˙)(A+Bρ−Bφ˙2)2 ρφ˙ , (D.17)QΨ˙ =3Bρφ˙A+Bρ−Bφ˙2 . (D.18)D.3 Lagrangian Derivatives for Disformal QuintessenceIn this appendix, the necessary derivatives of the Lagrangian density for disformal quintessencestudied in chapter 5 are givenL(φ,X) = X√1 + 2B(φ)X−√1 + 2B(φ)XV (φ) . (D.19)L,X = 1√L[1 +D2L +BV](D.20)L,XX φ˙2 = − DL3/2[2 +3D2L +BV](D.21)L,Xφ = − 1√L[D,φL(1 +3D4L +12BV)+B,φV +BV,φ](D.22)L,φ = − 1√L[D,φ2(XL + V)− LV,φ](D.23)L,φφ = − 1√L[(D,φ)24L(−3XL − V)−D,φV,φ + D,φφ2(−XL − V)+ LV,φφ](D.24)145Chapter D: Disformal Relations146List of Tables2 Acronyms xxii1.1 Horndenski projection of Modified Gravity and Dark Energy theories 192.1 Summary of cosmological probes 252.2 BAO data points 333.1 Parameter priors used in the MCMC for LTB and FRW models 593.2 Parameter constraints for the LTB and FRW models 603.3 LTB and FRW model comparison 674.1 Parameter constraints for generalized entropy-area law DE 824.2 Model comparison criteria for generalized entropy-area law DE 834.3 Numerical and asymptotic values of the growth factor 865.1 Disformal Dark Energy models 925.2 Parameter constraints for disformal quintessence 1016.1 Physical frames for disformally coupled theories 1076.2 Parameter constraints for disformally coupled Dark Matter 116147LIST OF TABLES148List of Figures3.1 Physical parameters in the LTB model 443.2 The LTB model with space-independent Big Bang 483.3 Effects on the LTB metric on the BAO scale with respect to FRW 523.4 Supernovae data and luminosity distance for the LTB best fit models 563.5 BAO data and dz for the LTB best fit models 563.6 CMB spectrum for the LTB best fit models 583.7 Likelihood contours for the FRW models 613.8 Likelihood contours for the CGBH model 633.9 Likelihood contours for the open-CGBH model 643.10 Likelihood contours for the Open-CGBH model (continuation) 653.11 Age of the universe for the best fit models 663.12 Initial BAO scale dependence 704.1 Constraints on the modified equation of state for CDM and baryons 784.2 Effective equation of state for the generalized entropy-area law DE. 824.3 Likelihood contours for generalized entropy-area law DE 834.4 Growth of structure in entropic cosmologies. 855.1 Equation of state and speed of sound for disformal quintessence 955.2 Disformal factors D, BV for disformal quintessence 965.3 z = 0 equation of state for disformal quintessence 975.4 Evolution of perturbations for disformal quintessence 995.5 CMB and matter power spectra for disformal quintesence 995.6 Likelihood contours for disformal quintessence 1006.1 Background evolution of disformally coupled matter 1156.2 Evolution of the dimensionless disformal factors 1156.3 Compatible regions for disformally coupled matter 1166.4 Effective gravitational constant for disformally coupled matter 1196.5 Growth of structure for disformally coupled matter 1206.6 Power spectra for disformally coupled matter and (uncoupled) baryons 1216.7 CMB power spectrum and baryon-matter bias induced by the coupling 122149LIST OF FIGURES150Bibliography[1] T. 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Probing the foundations of the standerd cosmological model

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