Tackling non-linearities with the effective field theory of dark energy and modified gravity

We present the extension of the effective field theory framework to the mildly non-linear scales. The effective field theory approach has been successfully applied to the late time cosmic acceleration phenomenon and it has been shown to be a powerful method to obtain predictions about cosmological observables on linear scales. However, mildly non-linear scales need to be consistently considered when testing gravity theories because a large part of the data comes from those scales. Thus, non-linear corrections to predictions on observables coming from the linear analysis can help in discriminating among different gravity theories. We proceed firstly by identifying the necessary operators which need to be included in the effective field theory Lagrangian in order to go beyond the linear order in perturbations and then we construct the corresponding non-linear action. Moreover, we present the complete recipe to map any single field dark energy and modified gravity models into the non-linear effective field theory framework by considering a general action in the Arnowitt-Deser-Misner formalism. In order to illustrate this recipe we proceed to map the beyond-Horndeski theory and low-energy Horava gravity into the effective field theory formalism. As a final step we derived the 4th order action in term of the curvature perturbation. This allowed us to identify the non-linear contributions coming from the linear order perturbations which at the next order act like source terms. Moreover, we confirm that the stability requirements, ensuring the positivity of the kinetic term and the speed of propagation for scalar mode, are automatically satisfied once the viability of the theory is demanded at linear level. The approach we present here will allow to construct, in a model independent way, all the relevant predictions on observables at mildly non-linear scales.Comment: 19 pages. Sec IV and Appendix B added. Matches JCAP versio

Deviations from General Relativity in Cosmology and Astrophysics

Two of the most prominent challenges of Modern Cosmology are the recent late-time accelerated expansion of the Universe and Dark Matter (DM). DM is of fundamental importance in the process of structure formation at galactic, extragalactic and cosmic scales. It seems to dominate down to small galactocentric radii, as highlighted by the galaxies rotation curves and on cosmic scale, it is well known that the spatial distribution of galaxies is biased with respect to the underlying DM distribution. This relation is called ``bias''. Part of this thesis is devoted to the investigation of the DM issue. In particular, we study the DM density profile in the Orion dwarf galaxy. This galaxy is a good candidate to understand the physics of DM as in general, the kinematics of dwarf galaxies is dominated by this dark component. Moreover, due to the availability of high precision data, it becomes crucial to understand accurately the bias relation, so we elaborate on the Lagrangian bias, when the initial mass fluctuation field is considered Gaussian and the bias is local. It is well known that the \Lambda CDM has been very successful in accounting for current cosmological data, although it suffers from some outstanding problems, such as the small value of \Lambda, DM problems on small scales, early Universe shortcomings and the lack of a correct scheme to quantize General Relativity. These lead people to propose and investigate alternative models, which are based on deviations from General Relativity at cosmic scales. The bulk of the present thesis is devoted to the study of gravity theories, which consider an extra scalar degree of freedom (DoF), in order to modify the gravitational interaction at large scales and account for the late time acceleration. In particular, we develop the gradient expansion formalism in order to explore the phenomenology associated with the non-linear derivative interactions of the most general scalar tensor theories that lead to second order field equations. This approach is very useful to probe on super horizon scale the Inflation scenario. Finally, in the quest of a model independent parametrization for gravity theories, the effective field theory formalism has been applied to the phenomenon of cosmic acceleration. It is developed using a perturbative approach in which an extra scalar DoF appears only at the level of perturbations. We investigate the viability of background functions by means of a thorough dynamical analysis. In conclusion, we present the implementation of this framework into CAMB/CosmoMC creating, what we dubbed, EFTCAMB/EFTCosmoMC. These codes will allow to test gravity theories with the most recent data releases

An Extended action for the effective field theory of dark energy: a stability analysis and a complete guide to the mapping at the basis of EFTCAMB

We present a generalization of the effective field theory (EFT) formalism for dark energy and modified gravity models to include operators with higher order spatial derivatives. This allows the extension of the EFT framework to a wider class of gravity theories such as Horava gravity. We present the corresponding extended action, both in the EFT and the Arnowitt-Deser-Misner (ADM) formalism, and proceed to work out a convenient mapping between the two, providing a self contained and general procedure to translate a given model of gravity into the EFT language at the basis of the Einstein-Boltzmann solver EFTCAMB. Putting this mapping at work, we illustrate, for several interesting models of dark energy and modified gravity, how to express them in the ADM notation and then map them into the EFT formalism. We also provide for the first time, the full mapping of GLPV models into the EFT framework. We next perform a thorough analysis of the physical stability of the generalized EFT action, in absence of matter components. We work out viability conditions that correspond to the absence of ghosts and modes that propagate with a negative speed of sound in the scalar and tensor sector, as well as the absence of tachyonic modes in the scalar sector. Finally, we extend and generalize the phenomenological basis in terms of $\alpha$-functions introduced to parametrize Horndeski models, to cover all theories with higher order spatial derivatives included in our extended action. We elaborate on the impact of the additional functions on physical quantities, such as the kinetic term and the speeds of propagation for scalar and tensor modes.Comment: 36 pages, matches published version, typos correcte

A de Sitter limit analysis for dark energy and modified gravity models

The effective field theory of dark energy and modified gravity is supposed to well describe, at low energies, the behaviour of the gravity modifications due to one extra scalar degree of freedom. The usual curvature perturbation is very useful when studying the conditions for the avoidance of ghost instabilities as well as the positivity of the squared speeds of propagation for both the scalar and tensor modes, or the St\"uckelberg field performs perfectly when investigating the evolution of linear perturbations. We show that the viable parameters space identified by requiring no-ghost instabilities and positive squared speeds of propagation does not change by performing a field redefinition, while the requirement of the avoidance of tachyonic instability might instead be different. Therefore, we find interesting to associate to the general modified gravity theory described in the effective field theory framework, a perturbation field which will inherit the whole properties of the theory. In the present paper we address the following questions: 1) how can we define such a field? and 2) what is the mass of such a field as the background approaches a final de Sitter state? We define a gauge invariant quantity which identifies the density of the dark energy perturbation field valid for any background. We derive the mass associated to the gauge invariant dark energy field on a de Sitter background, which we retain to be still a good approximation also at very low redshift ($z\simeq 0$). On this background we also investigate the value of the speed of propagation and we find that there exist classes of theories which admit a non-vanishing speed of propagation, even among the Horndeski model, for which in literature it has previously been found a zero speed. We finally apply our results to specific well known models.Comment: 22 page

The role of the tachyonic instability in Horndeski gravity

The tachyonic instability is associated with the unboundedness of the Hamiltonian from below and results in an unstable low-$k$ regime. In the cosmological exploration of modified gravity, it is seldom taken into account, with more focus given to the popular no-ghost and no-gradient conditions. The latter though are intrinsically high-$k$ statements. Here we combine all three conditions into a full set of requirements that we show to guarantee stability on the whole range of cosmological scales. We then explore the impact of the different conditions on the parameter space of scalar-tensor gravity, with particular emphasis on the no-tachyon one. We focus on Horndeski gravity and also consider separately the two subclasses of $f(R)$ and Generalized Brans Dicke theories. We identify several interesting features, for instance in the parameter space of designer $f(R)$ on a $w$CDM background, shedding light on previous findings. When looking at the phenomenological functions $\Sigma$ and $\mu$, associated to the weak lensing and clustering potential respectively, we find that in the case of Generalized Brans Dicke the no-tachyon condition clearly cuts models with $\mu\,,\,\Sigma>1$. This effect is less prevalent in the Horndeski case due to the larger amount of free functions in the theory.Comment: 9 pages, 4 figures - accepted version by JCA

Growth of non-linear structures and spherical collapse in the Galileon Ghost Condensate model

We present a detailed study of the collapse of a spherical matter overdensity and the non-linear growth of large scale structures in the Galileon ghost condensate (GGC) model. This model is an extension of the cubic covariant Galileon (G3) which includes a field derivative of type $(\nabla_\mu\phi\nabla^\mu\phi)^2$ in the Lagrangian. We find that the cubic term activates the modifications in the main physical quantities whose time evolution is then strongly affected by the additional term. Indeed, the GGC model shows largely mitigated effects in the linearised critical density contrast, non-linear effective gravitational coupling and the virial overdensity with respect to G3 but still preserves peculiar features with respect to the standard $\Lambda$CDM cosmological model, e.g. both the linear critical density contrast and the virial overdensity are larger than those in $\Lambda$CDM. The results of the spherical collapse model are then used to predict the evolution of the halo mass function, non-linear matter and lensing power spectra. While at low masses the GGC model presents about 10% fewer objects with respect to $\Lambda$CDM, at higher masses for $z>0$ it predicts 10% ($z=0.5$)-20% ($z=1$) more objects per comoving volume. Using a phenomenological approach to include the screening effect in the matter power spectrum, we show that the difference induced by the modifications of gravity are strongly dependent on the screening scale and that differences can be up to 20% with respect to $\Lambda$CDM. These differences translate to the lensing power spectrum where qualitatively the largest differences with respect to the standard cosmological model are for $\ell<10^3$. Depending on the screening scale, they can be up to 25% on larger angular scales and then decrease for growing $\ell$. These results are obtained for the best fit parameters from linear cosmological data for each model.Comment: 15 pages, 11 figures, 2 tables. Figures 8, 9 and 10 updated; new figure 11 added. Extended discussion in Sections 6 and 7. Matches the PDU journal versio