The upcoming satellite missions MAP and Planck will measure the spectrum of
fluctuations in the Cosmic Microwave Background with unprecedented accuracy. We
discuss the prospect of using these observations to distinguish among proposed
models of inflationary cosmology.Comment: To appear in the proceedings of the XXXIIIrd Rencontres de Moriond,
  "Fundamental Parameters in Cosmology", 17-24 January, Les Arcs, France. 6
  Pages, 2 figures, uses moriond.st

Dodelson, Scott

Kinney, William H.

Kolb, Edward W.

English

arXiv

Kinney, W. H.

Dodelson, S.

Kolb, E. W.

UNT Digital Library

F Fermi National Accelerator LaboratoryFERMILAB-Conf-98/102-ACosmic Microwave Background Measurements as a Way toDiscriminate Among Inflation ModelsW.H. Kinney, S. Dodelson and E.W. KolbFermi National Accelerator LaboratoryP.O. Box 500, Batavia, Illinois 60510April 1998´Published Proceedings of the 32nd Rencontres de Moriond: QCD and High-Energy HadronicInteractions, Les Arcs, France, March 22-29, 1997Operated by Universities Research Association Inc. under Contract No. DE-AC02-76CH03000 with the United States Department of EnergyDisclaimerThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any oftheir employees, makes any warranty, expressed or implied, or assumes any legal liability orresponsibility for the accuracy, completeness, or usefulness of any information, apparatus,product, or process disclosed, or represents that its use would not infringe privately ownedrights. Reference herein to any specic commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the United States Government or any agencythereof. The views and opinions of authors expressed herein do not necessarily state or reectthose of the United States Government or any agency thereof.DistributionApproved for public release; further dissemination unlimited.Fermi National Accelerator LaboratoryFERMILAB{Conf{98/102{AApril 3, 1998Operated by Universities Research Association Inc. under contract with the United States Department of EnergyCosmic Microwave Background Measurements as aWay to Discriminate Among Ination ModelsWilliam H. KinneyNASA/Fermilab Astrophysics CenterFermi National Accelerator Laboratory, Batavia, IL 60510Scott DodelsonyNASA/Fermilab Astrophysics CenterFermi National Accelerator Laboratory, Batavia, IL 60510Edward W. KolbzNASA/Fermilab Astrophysics CenterFermi National Accelerator Laboratory, Batavia, IL 60510, andDepartment of Astronomy and Astrophysics, Enrico Fermi InstituteThe University of Chicago, Chicago, IL 60637The upcoming satellite missions MAP and Planck will measure thespectrum of uctuations in the Cosmic Microwave Background with unprece-dented accuracy. We discuss the prospect of using these observations to dis-tinguish among proposed models of inationary cosmology.Electronic mail: kinneyw@fnal.govyElectronic mail: dodelson@hermes.fnal.govzElectronic mail: rocky@rigoletto.fnal.gov1 IntroductionThere has been great interest recently in the topic of parameter reconstruction fromthe Cosmic Microwave Background (CMB). Most of the emphasis in the literature is onthe question of determining cosmological parameters such as the density 0, the Hubbleconstant H0, or the value of the cosmological constant . The purpose of this talk isto take a dierent approach and apply the machinery of CMB parameter estimation tothe subject of inationary model building.[1] We assume generic features of the universeconsistent with ination (a at universe, vanishing ), but allow parameters sensitiveto ination to vary. The goal is to determine how well we will be able to distinguishbetween popular ination models using upcoming CMB experiments, in particular theall-sky satellite missions MAP and Planck.The parameters that are of interest for ination are the ratio of tensor to scalaructuation amplitudes measured at the quadrupole r  Ctensor2 =Cscalar2 , and the spectralindex of scalar uctuations n. Fixed parameters are the density of the universe 0 = 1,and the present vacuum energy  = 0. Other parameters are allowed to vary, and we plotthe expected sensitivity of NASA's MAP satellite[2] and the ESA's Planck Surveyor[3]as ellipses projected onto the r   n plane.2 Parameterizing InationInation in its most general sense can be dened to be a period of accelerated expansionof the universe, during which the universe evolves toward homogeneity and atness. Thisacceleration is typically due to the energy density of the universe being dominated byvacuum energy, with equation of state p '  . Within this broad framework, manyspecic models for ination have been proposed. We limit ourselves here to models with\normal" gravity (i.e., general relativity) and a single order parameter for the vacuum,2described by a slowly rolling scalar eld  (the inaton). These assumptions are notoverly restrictive { most widely studied ination models fall within this category, includ-ing Linde's \chaotic" ination scenario, ination from pseudo Nambu-Goldstone bosons(\natural" ination), dilaton-like models involving exponential potentials (power-law in-ation), hybrid ination, and so forth. Other models, such as Starobinsky's R2 modeland verions of extended ination, can, through a suitable transformation, be viewed interms of equivalent single-eld models.The slow-roll approximation, in which the eld evolution is dominated by drag fromthe cosmological expansion, is consistent if both the slope and curvature of the potentialare small, V 0; V 00  V . This condition is conventionally expressed in terms of the\slow-roll parameters"  and   M2P l4H 0 ()H ()' M2P l16V 0 ()V ()2; (1)and ()  M2P l4H 00 ()H ()' M2P l8"V 00 ()V ()  12V 0 ()V ()2#: (2)Slow-roll is then a consistent approximation for ;   1. The parameter  can in fact beshown to directly parameterize the equation of state of the scalar eld, p =   (1  2=3),so that the condition for ination of accelerating expansion is exactly equivalent to  < 1.To match the observed degree of atness and homogeneity is the universe, we requiremany e-folds of ination, typically N ' 50.Ination not only explains the high degree of large-scale homogeneity in the universe,but also provides a mechanism for explaining the observed inhomogeneity as well. Themetric perturbations created during ination are of two types: scalar, or curvature pertur-bations, which couple to the stress-energy of matter in the universe and form the \seeds"for structure formation, and tensor, or gravitational wave perturbations, which do not3couple to matter. Both scalar and tensor perturbations contribute to CMB anisotropy.Scalar uctuations can be quantitatively described by perturbations PR in the intrinsiccurvature scalarP1=2R (k) =1pHMP lpk 1=dH: (3)The uctuation power is in general a function of wavenumber k, and is evaluated when agiven mode crosses outside the horizon during ination k 1 = dH. Outside the horizon,modes do not evolve, so the amplitude of the mode when it crosses back inside the horizonduring a later radiation or matter dominated epoch is just its value when it left the horizonduring ination. The spectral index n is dened by assuming an approximately power-lawform for PR withn  1  d ln (PR)d ln (k)' 1  4+ 2; (4)so that a scale-invariant spectrum, in which modes have constant amplitude at horizoncrossing, is characterized by n = 1. Instead of specifying the uctuation amplitudedirectly as a function of k, it is often convenient to specify it as a function of the numberof e-foldsN before the end of ination at which a mode crossed outside the horizon. Scalesof interest for current measurements of CMB anisotropy crossed outside the horizon atN ' 50, so that PR is conventionally evaluated at PR (N = 50). Similarly, the powerspectrum of tensor uctuation modes is given byP1=2T (kN) =1pHMP lN=50: (5)The tensor spectral index isnT =  2 =  2PTPR: (6)Note that nT is not independent of the other parameters PR, PT , and n. Equation (6) isknown as the consistency condition, and is true only for single-eld models. (In models4with multiple degrees of freedom, Eq. (6) weakens to an inequality.) We then have threeindependent parameters for describing ination models: PR, PT , and n. Equivalently,we can use normalization PR and the two slow-roll parameters  and . This will proveto be the more convenient choice for categorizing models. For comparing to observation,it proves convenient to use a dierent set of three parameters: normalization at thequadrupole Qrms PS (equivalent to PR), spectral index n, and tensor-to-scalar ratio r.The ratio r of tensor to scalar modes measured at the quadrupole is[4]r  CTensor2CScalar2= 13:7; (7)so that tensor modes are negligible for  1. It is a simple matter to transform betweenslow-roll parameters ,  and the parameters n and r.3 Inationary ZoologyEven with the restriction to single-eld, slow-roll ination, the number of models in theliterature is large. It is convenient to dene a general classication scheme, or \zoology"for models of ination. we divide models into three general types: large-eld, small-eld,and hybrid, with a fourth classication, linear models, serving as a boundary betweenlarge- and small-eld. A generic single-eld potential can be characterized by two inde-pendent mass scales: a \height" 4, corresponding to the characteristic vacuum energydensity during ination, and a \width" , corresponding to the characteristic change inthe eld value  during ination. The height  is xed by normalization, so the onlyremaining free parameter is the width . Dierent classes of model are distinguished bythe value of the second derivative of the potential, or, equivalently, by the relationshipbetween the values of the slow-roll parameters  and . These dierent classes of modelshave readily distinguishable consequences for the CMB. Figure 1 shows the r   n planedivided up into regions representing the large-eld, small-eld and hybrid cases, describe5Figure 1: The parameter space divided into regions for small-eld, large-eld and hybridmodels. The linear case is the dividing line between large- and small-eld.in detail below.3.1 Large-eld models: 0 <   Large-eld models are potentials typical of \chaotic" ination scenarios, in which thescalar eld is displaced from the minimum of the potential by an amount usually of orderthe Planck mass. Such models are characterized by V 00 () > 0, and 0 <   . Thegeneric large-eld potentials we consider are polynomial potentials V () = 4 (=)p,and exponential potentials, V () = 4 exp (=). For the case of power-law ination,V () / exp (=), the slow-roll parameters are =  / (MP l=)2 ; (8)and r = 7(1  n). This result is often incorrectly generalized to all slow-roll models, butis in fact characteristic only of power-law ination. Note that we have a one-parameterfamily of models, parameterized by the width . For chaotic ination with a polynomial6potential V () / p, the slow-roll parameters are =pp+ 200=pp  2: (9)Again we have r / 1  n, so that tensor modes are large for signicantly tilted spectra.Unlike the case of the exponential potential, the scale  drops out of the expressions forthe observables, and the models are parameterized by the discrete exponent p.3.2 Small-eld models:  < 0 < Small-eld models are the type of potentials that arise naturally from spontaneous sym-metry breaking. The eld starts from near an unstable equilibrium and rolls down thepotential to a stable minimum. Small eld models are characterized by V 00 () < 0and  < 0 < . The generic small-eld potentials we consider are of the form V () =4 [1  (=)p]. In all cases, the slow-roll parameter  is very small, so that tensor modesare negligible. The cases p = 2 and p > 2 have very dierent behavior. For p = 2, theslow-roll parameter  is /  (MP l=)2; (10)so that the width of the potential    must be of order MP l for the spectral indexn ' 1+2 to satisfy observational constraints. As in the case of the exponential potential,the p = 2 small-eld case is a one-parameter family of models. For p > 2, /  p  1p  2(11)is independent ofMP l=, so that   MP l is consistent with a nearly scale-invariantscalar uctuation spectrum.3.3 Hybrid models: 0 <  < The hybrid scenario frequently appears in models which incorporate ination into super-symmetry. In a hybrid ination scenario, the scalar eld responsible for ination evolves7toward a minimum with nonzero vacuum energy. The end of ination arises as a result ofinstability in a second eld. Hybrid models are characterized by V 00 () > 0 and 0 <  < .We consider generic potentials for hybrid ination of the form V () = 4 [1 + (=)p] :The eld value at the end of ination (and hence N=50) is determined by some otherphysics, and we treat N=50 in this case as a freely adjustable parameter. The slow-rollparameters are then related as=2 (p  1)pN=50p(12)The distinguishing feature of many hybrid models is a blue scalar spectral index, n > 1.This corresponds to the case  >> , or N=50  . Because of the extra freedom tochoose the eld value at the end of ination, hybrid models form a two-parameter family.There is, however, no overlap in the r   n plane between hybrid ination and othermodels.3.4 Linear models:  =  Linear models, V () / , live on the boundary between large-eld and small-eldmodels,with V 00 () = 0 and  =  .4 Parameter Estimation from the CMBObservations of the CMB don't directly measure r and n. What is actually measured isanisotropy in the temperature of the CMB as a function of angular scale. It is convenientto expand the temperature anisotropy on the sky in spherical harmonics:T (; )T0=1Xl=0lXm= lalmYlm(; ) (13)where T0 = 2:726K is the average temperature of the CMB today. Ination predictsthat each alm will be Gaussian distributed with mean zero and variance Cl  hjalmj2i.8For Gaussian uctuations, the set of Cl's completely characterizes the uctuations. Thespectrum of the Cl's is in turn dependent on cosmological parameters { 0, H0, B, andso forth. The dependence on parameters is complicated, and a spectrum of Cl's for agiven set of cosmological parameters is calculated by numerically evaluating a Boltzmannequation[6]. Given a set of observational uncertainties Cl in the CMB spectrum, thegoal is to determine observational uncertainties in n and r. Given a set of parametersfig, assume a true set of values true. The Fisher information matrix, given byij =Xl@Cl@i=true1(Cl)2@Cl@j=true; (14)characterizes the accuracy with which the parameters i can be measured, with thetypical error in the ith parameter being of orderp( 1)ii. The parameters we marginalizeover are the spectral index n, the tensor to scalar ratio r, the normalization Qrms PS,baryon fraction B, and the Hubble constant H0. Fixed parameters are the matterfraction 0 = 1 and the cosmological constant  = 0. We consider central values forthe parameterstrue	= fn; r;Qrms PS;B;H0g = f0:9; 0:7; 18K; 0:08; 50g : (15)The expected uncertainty in the Cl's for a given experiment can be estimated as[7]Cl =r22l + 1Cl + 2pixelpixel exp l22beam; (16)where 2pixel is the rms pixel nose, pixel is the area of a pixel in steradians, and 2beamis the width of the beam, which is assumed to be gaussian. For MAP, we take beam =0:4250:3 and 2pixelpixel = (35K)2 (0:3)2. For Planck, we take beam = 0:4250:167and 2pixelpixel = (3K)2 (0:167)2. Figure 2 shows the generic ination models plottedin the r   n plane with typical 2 error ellipses from MAP and Planck.9Figure 2: Ination models plotted in the r   n plane, with 2 error ellipses from MAPand Planck.5 ConclusionsIt is evident from Figure 2 that MAP will allow at least rough distinction betweenlarge-eld, small-eld, and hybrid models. Planck, on the other hand, will allow forthe beginnings of real precision work in constraining ination models. Planck will becapable, for instance, of distinguishing the exponent p in a chaotic ination model V () /p. David Lyth has presented arguments that an appreciably large r is theoreticallydisfavored.[5] We nish by noting that these arguments will be put to observational testwithin the next few years!AcknowledgmentsWe thank Uros Seljak and Matias Zaldarriaga for use of their CMBFAST code. Thiswork was supported in part by DOE and NASA grant NAG5-7092 at Fermilab.10References[1] For details and additional references, see S. Dodelson, W. H. Kinney, and E. W.Kolb, Phys. Rev. D 56, 3207 (1997).[2] http://map.gsfc.nasa.gov/[3] http://astro.estec.esa.nl/SA-general/Projects/Planck/[4] M. S. Turner, M. White and J. E. Lidsey, Phys. Rev. D 48, 4613 (1993).[5] D. H. Lyth, Phys. Rev. Lett. 78, 1861 (1997).[6] U. Seljak and M. Zaldarriaga, astro-ph/9603033.[7] L. Knox, Phys. Rev. D 52, 4307 (1995).11

Cosmic microwave background measurements as a way to discriminate among inflation models

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Cosmic Microwave Background Observations as a Way to Discriminate Among Inflation Models

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