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LA-UR-04-8611 DO-TH-04/14 Inflationary Perturbations and Precision Cosmology Salman Habib,1 Andreas Heinen,2 Katrin Heitmann,3 and Gerard Jungman4 1T-8, The University of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Institut fu¨r Physik, Universität Dortmund, D-44221 Dortmund, Germany 3ISR-1, The University of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 4T-6, The University of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: February 2, 2008) Inflationary cosmology provides a natural mechanism for the generation of primordial perturba- tionswhichseedtheformationofobservedcosmicstructureandleadtospecificsignalsofanisotropy in thecosmic microwave backgroundradiation. In ordertotest thebroad inflationary paradigm as well as particular models against precision observations, it is crucial to be able to make accurate predictions for the power spectrum of both scalar and tensor fluctuations. We present detailed calculationsofthesequantitiesutilizingdirectnumericalapproachesaswellaserror-controlled uni- 5 form approximations, comparing with the (uncontrolled) traditional slow-roll approach. A simple 0 extension of the leading-order uniform approximation yields results for the power spectra ampli- 0 tudes, the spectral indices, and the running of spectral indices, with accuracy of the order of 0.1% 2 – approximately the same level at which the transfer functions are known. Several representative n examples are used to demonstrate these results. a J PACSnumbers: 98.80.Cq 9 1 I. INTRODUCTION tion(CMBR),sensitivetobothscalarandtensormodes, v and from measurements of the density power spectrum 0 insurveysofthelarge-scalemassdistributionintheUni- Cosmological inflation [1] is a central component of 3 verse. Ground and satellite-based observations of the the present theoretical picture of cosmology. Inflation 1 CMBR anisotropy have yielded results very consistent 1 notonlydirectlyaddressesandsolvesfundamentalweak- with the essential adiabatic and Gaussian nature of in- 0 nesses of the older Big Bang picture – the flatness and flationary perturbations, with a value of the scalar spec- 5 horizonproblems–italsoprovidesanelegantmechanism 0 for the creation of primordial fluctuations [2], essential tralindexveryclosetounity[3]. Thetensorfluctuations / are expected to be much lower in amplitude and have h for explaining the observed structure of the present-day not yet been observed. Scalar perturbations seed struc- p Universe. As the Universeinflates,microscopicquantum ture formation and hence can be measured by observing - scales are stretched to macroscopic cosmological scales. o thelarge-scaledistributionofgalaxiesandneutralhydro- Quantum vacuum fluctuations provide the initial seeds r gen, as in ongoing redshift surveys [4] and Lyα observa- t that are amplified by gravitationalinstability, leading to s tions [5]. This second set of independent measurements a the formation of structure in the Universe. provides information on smaller scales than the CMBR, : The primordial fluctuations associated with inflation v yet there is an overlap region where both measurements i are, primarily, of a very simple type. In “standard” in- X havebeen shownto be consistent(Fig.1). The observed flationary models, they arise from the fluctuations of an power spectrum is P (k) = P(k)T2(k) where P(k) is r effectively free scalar field and are hence Gaussian ran- obs a the primordial power spectrum and T(k) is the transfer domfields, completely characterizedby two-pointstatis- function for radiation or matter as appropriate. tics, such as the power spectrum. The basic task, then, is to determine the scalarandtensor perturbations (vec- As measurements continue to improve, tests of the in- torperturbationsbeingnaturallysuppressed)intermsof flationary paradigmand specific inflationary models will the associated power spectra. Power spectra from stan- become more stringent, especially if accurate measure- dard inflation models are conveniently parameterized by mentsoftherunning ofthe spectralindexareperformed aspectralindexandits (weak)variationwithscale–the and tensor perturbations are observed. In addition, an “running” of the spectral index. In addition, inflation inverseanalysisoftheobservationaldatamaythenbeat- couples the generation of scalar and tensor fluctuations; tempted,inanefforttodirectlymeasuretheinflationary if both are independently measured, “compatibility” re- “equation of state” [7]. In keeping with the remarkable lations characteristic of inflation can be put to observa- improvement of observationalcapabilities, the quality of tional test. It should be noted that complicated infla- theoretical predictions has necessarily to be addressed. tionary models can be constructed to produce baroque The central theme of this paper is to quantify and im- “designer” power spectra. Present observations do not prove the quality of theoretical predictions. require building such models. Scalar and tensor power spectra for particular infla- Observational constraints on the primordial power tion models can be calculated either by direct numerical spectra arise from measurements of temperature methods or by employing analytic or semi-analytic ap- anisotropies in the cosmic microwave background radia- proximations such as the slow-roll approach [8] or uni- 2 point-source contamination, foreground subtraction (for the CMBR), galaxy bias, systematic errors, effects from baryons [12] and neutrinos [13], and limited accuracy in theoretical computation of the present nonlinear power spectrum [14] for the matter distribution. Inthispaperwefocusontheperturbationspectrumfor single-field inflationary models. We implement a direct numerical approach as well as the slow-roll and uniform approximations,withtheaimofestablishingcontrolover the errorsassociated with each method and understanding their associated advantages and disadvantages. The maintechnicaladvancesarearobustandnumericallyef- ficientstrategyforthemode-by-modeintegration,simple andusefulerrorestimatesfortheuniformapproximation, andasimpleimprovementstrategyforpowerspectraam- plitudes for the uniform approximationat leading order. Weareabletoshowthattheimprovedleadingorderuni- form approximation leads to very good accuracy for the spectralindices and their running, as wellas for the am- plitudes of the power spectra and the ratio of tensor to scalar perturbations. For the most part, the accuracy of the results obtained is of the order of 0.1%. FIG.1: Compilationofobservationsofthematterpowerspec- The method of uniform approximation employed in trumPobs(k)takenfromTegmarketal.[6](withpermission). this paper has been presented in Refs. [9, 10]. This This particular figure assumes a flat scalar scale-invariant methodisa“uniformization”ofthewell-knownWentzel- modelwithΩm =0.28,h=0.72andΩb/Ωm =0.16,τ =0.17, Kramers-Brillouin (WKB) or Liouville-Green (LG) ap- andabias,b∗ =0.92fortheSloanDigitalSkySurvey(SDSS) proximation [15] in the presence of transition points. galaxies. The solid line is thetheoretical curve. The uniform approximation began with the work of Langer [16] and others, and was followed by the notable contributions of Olver [17, 18] which we rely on for our form approximations [9, 10]. For individual models, the analysisbelow. While this line of investigationrestedon numericalapproachmaywellbepreferable,butpowerful theanalysisofordinarydifferentialequations,equivalent approximationshavetheirownadvantages. Theyprovide asymptotic results based on an integral representation intuitionandunderstandingapplicabletoentireclassesof were given by Chester, Friedman, and Ursell [19]. The models. Provided tight error controls can be met, they uniform approximation has proven very useful in many are much faster than a mode-by-mode numerical inte- applications, e.g., chemical physics [20], the semiclassi- gration for obtaining the power spectrum. Also, both cal limit in quantum mechanics [21], and the quantum- numerical and approximate strategies can be melded to- classical transition in quantum cosmology [22]. For our getherbyfirstobtainingasetofapproximateresultsover purposes, the key advantages of the Olver uniform ap- a wide range of parameters and then spot-checking nu- proximationarethatitdoesawaywithWKB-likematch- merically. It is our view that both strategies should be ingconditions[23](suchaprocedurefailstoindicate the applied and compared against each other as they have error of the approximation), has controllederror bounds differing error modes. Finally, it should be kept in mind over the entire domain of interest, is systematically im- that the ultimate accuracy with which the primordial provable, and allows analytic simplifications in special fluctuations must be computed is limited by how accu- cases of physical interest. rately the radiation and matter transfer functions are The paper is organized as follows. Section II provides known, as well as by the errors associated with observa- thenecessarybackgroundregardingthecalculationofthe tions. Atpresent,thetransferfunctionscanbecomputed primordial power spectra and spectral indices for single to about 0.1% accuracy [11], sufficient for dealing with field inflation models. In Section III the results from the next-generationobservations. We willuse this value asa uniform approximation in leading order are given, while relative figure of merit when discussing errors below. It inSectionIVthe essentialequationsforthe slow-rollap- should be noted that the accuracy requirements for the proximationare summarized. In Section V we give a de- tensor or gravitationalwave component are not as strin- tailed description of the numerical determination of the gent as for the scalar component; the expected signal primordialpower spectra and spectral indices by solving is at large length scales, where cosmic variance becomes the mode-equations numerically, in the uniform approx- unavoidable. Finally, measurementsofthe radiationand imation, and in the slow-rollapproximation. We investi- matter power spectrum at higher values of k face issues gatethree differentexamples inSection VI andconclude suchastherapidfall-offoftheprimarysignal,secondary with a discussion of our results in Section VII. 3 II. BACKGROUND B. Mode Equations, Power Spectra, Spectral Indices, and their Running Thegenerationofperturbationsduringinflationisdue to the amplification of quantum vacuum fluctuations by The modern understanding of fluctuations generated the dynamics of the background spacetime. In order to by inflation is based on the gauge-invariant treatment calculatetheperturbationspectraforthesequantumvac- of linearized fluctuations in the metric and field quanti- uum fluctuations three steps are necessary: the dynam- ties [24, 27, 28, 29]. A particularly convenient quantity icsofthebackgroundspacetimemustbedetermined,the for characterizing the perturbations is the intrinsic cur- mode equations for the scalar and tensor perturbations vature perturbation of the comoving hypersurfaces [30], mustbesolved,andfinally,thepowerspectrathemselves ζ u/z, where u is a gauge-invariant scalar perturba- ≡ must be calculated as functions of wave-number k. We tion [24], and z a/(c H)[ H˙/(4πG)+ /a]1/2, where s ≡ − K will briefly outline these three steps in the following; for c is the sound speed and is the curvature of spatial s K detailed derivations the reader is refered to the litera- sections. Forsingle-fieldinflationarymodels,this simpli- ture [8, 24, 25, 26]. fiestoz =aφ˙/H. Thequantityusatisfiesthe dynamical equation A. Background Equations u′′ ∆u z′′u=0, (6) − − z Forsingle-fieldinflationarymodels,i.e.,wheninflation where primes denote derivatives with respect to confor- is driven by a single homogeneous scalar field, the dy- mal time and ∆ is the spatial Laplacian in comoving namical equation for the inflaton field φ is given by coordinates. It follows immediately that ζ is approximately constant in the long wavelength limit k 0. φ¨(t)+3H(t)φ˙(t)+ ∂V(φ) =0. (1) This is true during the inflationary phase as well as→dur- ∂φ ing the post-reheating era. Moreover,the Einstein equations can be used to connect the gravitational potential The dots denote derivatives with respect to physical Φ and ζ so that a computation of the power spectrum time t. The evolution of the Hubble parameter H(t) = A of ζ provides all the information needed (aside from the a˙(t)/a(t), where a(t) is the scale factor, is given by the transferfunctions)toextractthetemperatureanisotropy Friedmann equation of the CMBR. Details of this procedure can be found in Refs. [24, 31]. 8πG 1 H2(t)= φ˙2(t)+V(φ) . (2) The calculation of the relevant power spectra involves 3 2 (cid:20) (cid:21) a computation of the two-point functions for the appropriate quantum operators, e.g., Equivalently, by taking the derivative of H(t) with respect to time and inserting the equation of motion (1) ∞ dksinkr for φ we can write 0uˆ(η,x)uˆ(η,x+r)0 = P (η,k), (7) u h | | i k kr Z0 H˙(t)= 4πGφ˙2(t). (3) − the operator uˆ being written as Here and in the following we have set ~ = c = 1. d3k Eqns. (1) and (2) or (3) determine the evolution of the uˆ(x)= aˆ u (η)eik·x+aˆ†u∗(η)e−ik·x , (8) (2π)3/2 k k k k inflaton field completely. In addition, it is useful to cal- Z h i culate the number of expansion e-folds via whereaˆ , aˆ† areannihilationandcreationoperatorswith k k N˙(t)=H(t). (4) [aˆk,aˆ†k′] = δkk′, and aˆk|0i = 0 ∀k. The complex amplitude u (η) satisfies k Monitoringthenumberofe-foldsforwhichinflationlasts allows us to select observationally relevant inflationary z′′ u′′+ k2 u =0. (9) modelsandtheirparameters. Finally,theconformaltime k − z k (cid:18) (cid:19) η(t) is defined by SolvingEqn.(9)isthefundamentalproblemindetermin- 1 ing the primordial power spectrum. The corresponding η˙(t)= . (5) a(t) mode equation for tensor perturbations is given by As explained later, the uniform approximation is ex- a′′ v′′+ k2 v =0. (10) pressed naturally in terms of the conformal time. k − a k (cid:18) (cid:19) Oncethe backgroundequationsaresolvedwe canpro- ceed to the next step, the evaluation of the scalar and Once the mode equations are solved for different mo- tensor perturbations. menta k the power spectra for the scalar and tensor 4 modes are obtained via III. THE UNIFORM APPROXIMATION k3 u (η) 2 P (k)= lim k , (11) In two recent papers [9, 10] we have implemented an S kη→0− 2π2 z(η) approachbased on the uniform approximation(for a de- (cid:12) (cid:12) k3 (cid:12)(cid:12) vk(η)(cid:12)(cid:12)2 tailed description of the approximation,see Ref. [18]) to PT(k)=kηl→im0− 2π2(cid:12) a(η)(cid:12) , (12) calculate the power spectra and the associated spectral (cid:12) (cid:12) indices of primordial perturbations from inflation. Our (cid:12) (cid:12) (cid:12) (cid:12) methodleadstosimpleexpressionsforthepowerspectra where we denote the power spect(cid:12)rum fo(cid:12)r ζ by PS. and spectral indices with calculable error bounds. The The tensor power spectrum is often defined with an solutionfor the modes u caninprinciple be determined additional factor as k toarbitraryorderintheuniformapproximation. Inprac- tice,resultsaccuratetosub-percentlevelareobtainedat P (k)=8P (k), (13) h T next-to-leading order. leading to the definition of the tensor to scalar ratio as While a detailed derivation of the mode equations, powerspectra, andspectralindices with the correspond- P (k) 8P (k) ing error terms is given in Ref. [10], for completeness h T R(k)= = . (14) and to set notation, we provide a brief summary of the P (k) P (k) S S main equations. In addition, we give an expression for Thegeneralizedspectralindicesforthescalarandten- the scalar to tensor power spectrum ratio R(k) with the sor perturbations are defined to be corresponding error term and explain how to compute the error terms in the uniform approximation in an ef- dlnP (k) ficient way. A new result, shown here, is that a simple S n (k) = 1+ , (15) S dlnk improvementoftheleadingorderuniformapproximation dlnP (k) leadstoverygoodaccuracy,genericallybetterthan0.1%. T nT(k) = . (16) This improvement is obtained by utilizing knowledge of dlnk the next-to-leading order results without actually imple- Runningofthespectralindicesisconventionallyparame- menting them fully. terizedbythesecondlogarithmicderivativeofthe power spectra: A. The Mode Functions dlnn (k) S α (k) = , (17) S dlnk In order to solve for the scalar and tensor mode func- dlnnT(k) tions in the uniform approximation, it is necessary to α (k) = . (18) T dlnk rewrite the differential equations (9) and (10) in the following form: Numerical results for the power spectrum are often fit in the literature assuming a power-law behavior and a 1 1 small running of the spectral index around a pivot scale u′k′(η)= −k2+ η2 νS2(η)− 4 uk(η), (21) k (see e.g., Refs. [25, 32]). This fitting ansatz for the (cid:26) (cid:20) (cid:21)(cid:27) ∗ power spectrum is where ν2 =(z′′/z)η2+1/4, and S k nfit+12αfitlnkk∗ 1 1 P(k)=Afit(cid:18)k∗(cid:19) , (19) vk′′(η)=(cid:26)−k2+ η2 (cid:20)νT2(η)− 4(cid:21)(cid:27)vk(η), (22) the fitting parametersbeingAfit, nfit andαfit. The spec- where ν2 = (a′′/a)η2 + 1/4. The shift of 1/4 in the tral index is evaluated at the pivot scale: definitioTn of ν2 and ν2 is necessary in order to have a S T convergent error control function [10]. In the following n (k )=1+n . (20) S ∗ fit we will describe the analysis for the scalar modes and present only the final results for the tensor modes. Therunningisparameterizedbyα . Thisfittingformis fit The general solution for u (η) is a linear combination not self-consistentsince it is not possible to have strictly k ofthetwofundamentalsolutionsu(1)(η)andu(2)(η),viz., constant n and non-zero α given the above definitions. k k This inconsistency manifests itself by the uncontrolled growth of errors away from the pivot scale. uk(η)=A(k)u(k1)(η)+B(k)u(k2)(η), (23) Inthefollowingwewilldescribehowtoobtainapprox- imate solutions for the scalar and tensor power spectra independent of the order of the approximation. To fix andthespectralindicesandtheirrunning,assumingthat the coefficients A(k) and B(k), a linear combination the evolution of the background quantities is known. of u(1)(η) and u(2)(η) must be taken so that u (η) = k k k 5 e−ikη/√2k in the limit kη . With proper normal- Inasimilarfashiontheresultsforthe tensormodesv k →−∞ ization, the solution for u is, in leading order, can be derived. We have k uk,1,≶(η)=rπ2CfS1,/≶4(k,η)gS−1/4(k,η)[Ai(f≶)−iBi((f2≶4))], vk,1,>(η)=kηl→im0−−iCr2ν−Tη(η)exp(cid:26)23[fT,>(k,η)]3/(23(cid:27)2), with where fT,> indicates that we replace gS(k,η) with g (k,η) and η¯ with η¯ in Eqn. (25). T S T 3 η¯S 2/3 Having obtained expressions for the scalar perturba- f (k,η) = dη′[ g (k,η′)]1/2 ,(25) S,≶ ∓ ±2 ∓ S tions uk andthe tensor perturbationsvk we cannow de- (cid:26) Zη (cid:27) rive the corresponding scalar and tensor power spectra. ν2(η) g (k,η) = S k2. (26) S η2 − B. The Power Spectra One part of the solution is valid to the left of the turning point η¯ , defined as the solution to k2 =ν2(η¯ )/η¯2, S S S S TheexpressionforthescalarpowerspectrumP (k)as and the other part is valid to the right of the turning S defined in Eqn. (11) in the uniform approximation with point (η η¯). The uniform approximation allows us to ≥ the corresponding error term is calculate bounds on the errors. We write uk,≶(η)=uk,1,≶(η)[1+ǫk,1,≶(η)], (27) PS(k) = kηl→im0− 2kπ32 uk,z1(,>η)(η) 2|1+ǫk,1,>(η)|2 (cid:12) (cid:12) wukh,1e,r≶e btheeyoenrdrorleatedrimngeonrcdaeprs.ulaAtessdtehreivceodntinribduettiaoinl tino = kηl→im0−P1,S((cid:12)(cid:12)(cid:12)k) 1+ǫPk,(cid:12)(cid:12)(cid:12)1,S(η) , (33) Ref. [10], the error term is bounded by (cid:2) (cid:3) with √2 ǫ (η) [exp(λ ( )) 1] ǫP =2Re ǫ + ǫ 2, (34) | k,1,≶ | ≤ λ { Vη,α E − k,1,S k,1,> | k,1,>| +[exp(λVβ,η(E))−1]}, (28) where P1,S(k) denotes the powerspectrum for the scalar perturbations in the leading order approximation. We where ( )denotesthetotalvariationoftheerrorcontrol cannowsubstitute either the fullexpressionforu given V E k function (η). Anumericalestimateshowsλ 1.04[18]. in Eqn. (24) or the LG form from Eqn. (31). E ≃ The error control function reads Using the LG expression for u we have k η 1 d2 1 E(η) = Zη¯S(cid:26)|gS|1/41dη′2 (cid:18)|g5S|g1/41(cid:19)/2 P1,S(k)=kηl→im0− 4kπ32|z(1η)|2νS−(ηη)exp(cid:26)34[fS,>(k,η)](33/52)(cid:27), + | S| dη′. (29) 4η′2 g 1/2 − 16f 3 with the error term for the power spectrum given in | S| | S,≶| (cid:27) Eqn. (34). The calculation for the tensor power spec- The kη 0− limit defines the region of interest for trum follows along the same lines, yielding → calculatingpowerspectra andthe associatedspectralin- dices. Inthis region,the1/η2 pole dominatesthe behav- P (k)= lim k3 1 −η exp 4[f (k,η)]3/2 , ior of the solutions and the Airy solution goes over to 1,T kη→0− 4π2|a(η)|2νT(η) (cid:26)3 T,> (cid:27) the LG form leading to simple expressions for the spec- (36) tralindices[10]. Theregionofinterestliestotherightof withtheerrorterminthesameformasinEqn.(34)with the turning point where the argument of the Airy func- the substitution S T. → tions becomes large. This allows the approximation of The tensor to scalar ratio, R(k), is given by the Airy functions in terms of exponentials, leading to 8P (k) R(k)= 1,T 1+ǫR , (37) C 1 2 P (k) k,1 u (η) = g−1/4(k,η) exp [f (k,η)]3/2 1,S k,1,> √2 S 2 −3 S,> (cid:0) (cid:1) (cid:20) (cid:26) (cid:27) with the error term 2 iexp [fS,>(k,η)]3/2 . (30) 1+ǫP − 3 ǫR = k,1,T 1. (38) (cid:26) (cid:27)(cid:21) k,1 1+ǫP − k,1,S For computing the power spectra in the kη 0− limit, → only the growing part of the solution is relevant: In the case of power-law inflation, where ν is constant, the error is identically zero, as in this case ν = ν . In S T u (η)= lim iC −η exp 2[f (k,η)]3/2 . other words, the ratio of tensor to scalar perturbations k,1,> kη→0−− r2νS(η) (cid:26)3 S,> (cid:27) for power-law inflation is exact already at leading order (31) in the uniform approximation. 6 C. Estimate of the Error Bound The natural question arises as to whether or not it is possible to utilize these results to improve the leading Although we can calculate the error bound for the order expressions without recourse to full computation powerspectrafromthe generalexpressionsin Eqns.(28) of the sub-leading approximations. For most viable in- and (29), it is convenient to have simpler estimates for flationary models, ν varies slowly and corrections from the errors. the derivatives of ν are sub-dominant. The full next-to- We begin by considering the case of constantν, where leading order machinery may not be required if ν is suf- the k-independent error bound for the power spectrum ficiently well-behaved. In this section we implement this in leading order of the uniform approximation is [18] idea and derive improved leading-order results for the power spectra. Note that, at leading order, the spectral 1 λ 1 ǫP 2√2 + + + , (39) indexisexactforconstantν,thusthemainimprovement | 1|≤ 6ν 72ν2 36√2ν2 ··· tobeexpectedisintheamplitudeofthepowerspectrum. (cid:18) (cid:19) the generic ν denoting either of ν or ν . This bound is We begin with an argument from our previous S T rigorousanduseful,thoughtheprefactorisnotoptimally work[10],whichwasusedtounderstandwhytheleading- sharp for the case of constant ν. orderresultforthespectralindexwasmuchmoreprecise Suppose now that ν(η) varies slowly with time. Fix than estimated bounds for the power spectrum such as k and consider the value of ν(η) at the turning point given in Eqn. (40) would seem to indicate. The error is η¯(k), defined by kη¯ = ν(η¯). Given the slow variation in fact dominated by an amplitude prefactor which has of ν(η), this value ν¯(k)−is a slowly varying function of k. only a subdominant contribution to the spectral index. By expanding the expression for the error control func- We split the scattering potential in the form tion of Eqn. (29) locally around the turning point, we 1 1 obtain what is in effect a derivative expansion for the ν2(η) =ν¯2 +ν2(η) ν¯2, (42) error term. The leading term in this expansion, which − 4 − 4 − is free of derivatives, has the same form as the expres- and choose the η-independent but k-dependent constant sionaboveforconstantν,thoughitnowcarriesthe mild ν¯(k) to be the value of ν(η) at the turning point. As k-dependence of the variable ν case, arguedpreviously[10],thissplitting allowsustoidentify 1 λ 1 twoseparatecontributionstothetotalerrortermforthe ǫP 2√2 + + + . | k,1|≤ 6ν¯(k) 72ν¯2(k) 36√2ν¯2(k) ··· power spectrum, (cid:20) (cid:21) (40) ǫP =ǫ¯+ǫ˜. (43) This bound is not meant to be rigorous, since higher or- k,1 der terms in the derivative expansion are not included. Thetermǫ¯arisessolelyfromtheultra-localcontribution However, it is effective and useful in the case of slowly in the derivative expansion of ν, and by explicit calcula- varying ν(η). For further discussion of local approximation it is known to be of the form tions ofthis type,see alsoSectionIIIE2. InRef. [10]we have extensively discussed how a non-constant ν gives ǫ¯ = [Γ∗(ν¯)]2 1 (44) rise to a non-vanishing and k-dependent error for the − spectral index. 1 1 31 139 = + + . 6ν¯(k) 72ν¯2(k) − 6480ν¯3(k) − 155520ν¯4(k) ··· D. Improvement of the Leading Order Result The remaining error term satisfies an integral equation of the form considered by Olver [18], with a reduced in- homogeneity; explicit calculation leads to the rigorous In this section we present a new improvement for bound the leading-order uniform approximation. In previous work [10] we derived next-to-leading order results for uk ( ¯) and vk and the corresponding power spectra and spec- |ǫ˜|≤ V E(−¯)E [1+O(|ǫ¯|)]|ǫ¯|, (45) tral indices. The resulting expressions contained several V E integrals, which are tedious to evaluate. On the other where is the full error control function, ¯ is the error hand, the results at next-to-leading order for the case E E control function for the case of constant ν = ν¯(k), and of constant ν turned out to be very simple. In essence, () indicates total variation as before. higher order terms in the uniform approximation gen- V · In the generic case of slowly varying ν(η) this expres- erate a prefactor which occurs in the Stirling series for sion clearly shows that ǫ˜is significantly reduced in com- Γ(ν) [33] parison to ǫ¯. Therefore we are motivated to absorb the 1 1 139 Γ∗(ν) 1+ + + , (41) ultra-local contributions from higher order corrections ≡ 12ν 288ν2 − 51840ν3 ··· into the power spectrum, leading to an improved first- which we know a-priori to be present in the case of con- order expression stant ν. This all-orders prefactor improves the normalization of the power spectrum dramatically. P˜ (k)=P (k)[Γ∗(ν¯)]2. (46) 1 1 7 This improvement applies to both the scalar and tensor The error for the spectral index only arises from the power spectra, with ν (η) and ν (η) respectively. Since k-dependent part of the error in the power spectrum. S T typical values of ν 2 occur for both scalar and tensor Therefore,theerrorinthespectralindexissensitiveonly ∼ fluctuations,thetermsontherighthandsideofEqn.(46) to the time variation of ν . To estimate this error, the S correspondtoamplitudecorrectionsoforder10%,0.35%, spectral index as written in Eqn. (15) can be expressed 0.06%, and 0.006%, respectively. These can be viewed via the leading order power spectrum in the following primarily as local normalization corrections. form: To complete the discussion, we can obtain a non- dlnP dǫP rigorous estimate for the size of ǫ˜, again using a deriva- n (k) 1+ 1,S +k k,1,S S tive expansion in to isolate the leading local contribu- ≃ dlnk dk tions. InthisexpaEnsion,theleadingderivative-freeterms ≡ n1,S(k)+ǫnk,1,S, (49) in ¯ must cancel, and therefore the leading term is E −E with proportional to the first derivative of ν(η). After some calculation we find dǫP ǫn k k,1,S, (50) 3 ν¯′ k,1,S ≡ dk ǫ˜ [1+ (ǫ¯)] ǫ¯ | | ≤ 2 k O | | | | where we estimate ǫP by the leading order error term (cid:12) (cid:12) k,1,S (cid:12)1 (cid:12) dlnν¯(k) 1 in Eqn. (42), i.e., by ǫ¯as given in Eqn. (44). It should (cid:12) (cid:12) (cid:12) (cid:12) 1+ , (47) benotedthatthe leadingerrorsinthe spectralindexare ≤ 4ν¯(k) dlnk O ν¯(k) (cid:12) (cid:12)(cid:20) (cid:18) (cid:19)(cid:21) proportionaltok derivativesofν¯(k),unlikethesituation (cid:12) (cid:12) wherewehaveusedt(cid:12)(cid:12)hechainr(cid:12)(cid:12)uletowritethederivative for the power spectrum [Eqn. (44)]. intermsofaderivativeofν¯(k)withrespecttokandhave The above analysis can be carried out for tensor per- usedthe explicitformforǫ¯(k). Itisunderstoodthatthis turbations inanidentical fashion,including the errores- is not a rigorous inequality, since we have neglected the timation, with the replacement νS νT. The spectral → higher order terms in the derivative expansion, but it is index for gravitationalwaves is given by well-motivated and useful for slowly-varying ν(η). This η dη′ error estimate determines the order to which the local n (k)=3 2k2 lim . (51) 1,T corrections (46) should be taken into account. For in- − kη→0−Zη¯T gT(k,η′) stance, when ν is slowly varying, most of the error can p be compensated using Eqn. (46). Examples will be en- 2. The Local Approximation counteredinSectionVI,whereweexplicitlydemonstrate the success of this improvement procedure. As is to be expected,theratioR(k)ismuchlesssensitivetothenor- Itispossibletosimplifytheexpressionsforthespectral malizationerrorcomparedtotheamplitudeofthepower indicesasgiveninEqns.(48)and(51)byapproximating spectrum. For this quantity, the error is well-estimated the integrals further. By doing so, we lose the ability to by Eqn. (47). quantify the error estimates for nS and nT but are able to write down local expressions for the spectral indices. The integrands in Eqns. (48) and (51) have square-root E. The Spectral Indices singularitiesat the turning points, i.e., at the lowerinte- grationlimits. Attheupperlimit,η goestozeroandthe integrandsvanishlinearly,therefore,assumingν (η)and We now turn to the evaluation of spectral indices and S ν (η),respectively,arewell-behaved,weexpectthemain corresponding error terms from the power spectra com- T contributiontotheintegralstoarisefromthelowerlimit. puted using the uniform approximation. We will discuss Using the knowledge that ν and ν are slowly varying, furthersimplificationsleadingtolocalexpressionsforn S T S we expand them around their turning points in Taylor and n . T series. To second order in derivatives, ν2(η) ν¯2+2ν¯ ν¯′ (η η¯ )+(ν¯′2+ν¯′′ν¯ )(η η¯ )2, (52) 1. Integral Expression for the Spectral Index S ≃ S S S − S S S S − S where a bar indicates that the quantity has to be evaluated at the turning point. For ν we obtain a similar Thedifferentiationofthepowerspectrumwithrespect T expression with the index S replaced by T. We can now to k is straightforward. As stated earlier,it is important solvetheintegralsinEqns.(48)and(51)exactlyandfind to remember that the turning point η¯ is a function of S for the scalar spectral index k since k = ν (η¯ )/η¯ where ν (η¯ ) is the value of S S S S S − | | νS(η) at the turning point η = η¯S. Using this relation, ν¯′ π we find nS(k) 4 2ν¯S 1 Sη¯S 1 (53) ≃ − − ν¯ − 2 (cid:26) S η dη′ η¯2 ν¯′2 (cid:16)ν¯′′ (cid:17) n1,S(k)=4 2k2 lim . (48) + S S (2 π)+ S(1 π) , − kη→0−Zη¯S gS(k,η′) 2 (cid:20)ν¯S2 − ν¯S − (cid:21)(cid:27) p 8 and for the tensor spectral index analogously Equivalently,forthetensorspectralindexderivedfroma slow-rollexpansionofthe localresult, Eqn.(54), we find ν¯′ π n (k) 3 2ν¯ 1 Tη¯ 1 (54) T ≃ − T − ν¯ T − 2 34 28 (cid:26) T (cid:16) (cid:17) nT(k) 2ǫ¯ 3π ǫ¯2 3π ¯ǫδ¯1, (59) η¯2 ν¯′2 ν¯′′ ≃− − 3 − − 3 − + T T (2 π)+ T(1 π) . (cid:18) (cid:19) (cid:18) (cid:19) 2 ν¯2 − ν¯ − (cid:20) T T (cid:21)(cid:27) where we have used The local approximation is a further simplification of a′′ 1 the leading order result in the uniform approximation =2a2H2 1 ǫ . (60) a − 2 andwe willtest its accuracywith three numericalexam- (cid:18) (cid:19) plesinSectionVI. Wedenotetheleadingcontributionof Comparedto results givenpreviously [9, 10], the expres- unity in the curly brackets in Eqns. (53) and (54) as ze- sions for the spectral indices given here have been fully rothorder,thesecondtermproportionaltoν¯′ asfirstor- expanded to second order in slow-rollparameters. der in the localapproximation,and the remaining terms proportional to ν′2 and ν′′ as second order. IV. THE SLOW-ROLL APPROACH 3. Spectral Indices in Terms of Slow-Roll Parameters Themostpopularanalyticmethodtoevaluateprimor- dial power spectra and spectral indices is the so-called It is possible to simplify the expressions for the spec- slow-rollapproach. The basic idea behind this approach tral indices one step further and obtain results similar is the following. Recall the expression for z′′/z in terms to the ones obtained in the slow-roll approximation to of ǫ, δ and δ as given in Eqn. (57). Assume now that 1 2 be discussed in the next section. We have investigated ǫ and δ are small and constant. If this is the case, all 1 this further approximation in detail previously [10] and terms δ vanish, since they can be written as deriva- n>1 pointed out its shortcomings. Here the results are sum- tives of ǫ and δ , and all terms of order (ǫ2) can be 1 marized for leading order only. neglected. It is then also possible to expOress a2H2 as We begin by considering the spectral index for the (1+2ǫ)/η2, leading to a simplified equation for u : k scalar perturbations as given in Eqn. (53). In order to write n in terms of slow-roll parameters which are de- u′′+ k2 A/η2 u =0, (61) S k − k fined as (see e.g., Ref. [34]) with (cid:0) (cid:1) 2 H˙ 1 φ˙ 1 dn+1φ ǫ≡−H2 = 2 H! , δn ≡ Hnφ˙ dtn+1, (55) A=2+6ǫ+3δ1 =constant. (62) This equation is immediately solved in terms of Bessel we expand ν¯S, ν¯S′η¯S, and ν¯S′′η¯S2 up to second order in functions. SincetheconstantAwillbedifferentfordiffer- these parameters. The contribution ν¯S′2η¯S2/ν¯S is already entvaluesofk (asǫandδ1 areapproximatedbydifferent offourthorderandisneglected. Usingtheexpressionfor constantsfordifferentk),foreverymomentumka“best- conformal time in slow-roll parameters given by fit power-law” result is obtained and expressions for the power spectrum P (k) and the spectral index n (k) can 1 S S η 1+ǫ+3ǫ2+2ǫδ + , (56) be derived. Tensor perturbations can be treated in the 1 ≃ −aH ··· same way, leading to expressions for P (k) and n (k). T T (cid:0) (cid:1) the relation, ν2 =(z′′/z)η2+1/4, and The terminology “slow-roll” is easy to understand: the S requirement on ǫ and δ being small and roughly con- 1 z′′ 3 1 stantadmitsinflationarymodelswithsmoothpotentials, =2a2H2 1+ǫ+ δ +2ǫδ +ǫ2+ δ , (57) z 2 1 1 2 2 leading to a phase where the inflaton field rolls slowly (cid:18) (cid:19) down the potential before inflation ends and reheating itiseasytowritedownν¯anditsderivativesintermsǫ¯,δ¯1, begins. The shortcoming of the slow-roll approach is andδ¯2(Foradetailedderivationandexplicitexpressions, that there is no simple way to improve the approxima- seeRef.[10]). Asinearliersections,thebarindicatesthat tion beyond leading order. Once the assumption that the parametersareto be calculatedatthe turningpoint. the slow-roll parameters are constant is given up, at the Insertingalltheexpressionsintothelocalapproximation next formal order of the approximation, the Bessel solu- forthescalarspectralindex,Eqn.(53),allowsustowrite tion is no longer valid [35]. Different ways for improving the spectral index in terms of slow-roll parameters: the slow-roll approximation have been suggested, e.g., [34, 36, 37, 38], but they often lead to rather involved 17 n (k) 1 4ǫ¯ 2δ¯ 8ǫ¯2 π (58) expressions for power spectra and spectral indices and, S 1 ≃ − − − 6 − (cid:18) (cid:19) more importantly, are not error-controlled. 73 11 As the slow-roll approach is convenient, widely used, 10ǫ¯δ¯ π +2(δ¯2 δ¯ ) π . − 1 30 − 1− 2 6 − and for most common inflationary models expected to (cid:18) (cid:19) (cid:18) (cid:19) 9 be a reasonable approximation, we will compare slow- fortheexactnumericalsolution,andasacompletesemi- roll results against the exact mode-by-mode integration analyticmethodinitsownright,wediscussitfirstbelow. and the uniform approximation in Section VI. For completeness, we summarize here the results for the power spectraandspectralindices followingcloselythe workof Stewart and Lyth [39]. A. Leading Order Uniform Approximation: Numerical Issues Thepowerspectrainthe slow-rollapproximationread PS(k) [1+2(2 ln2 b)(2ǫ+δ1) 2ǫ] 1. Preliminaries ≃ − − − 2GH4 , (63) × π φ˙2 We begin by addressing some technical points and in- (cid:12)aH=k (cid:12)(cid:12) 2GH2 troducing conventions. Since the conformal time is de- PT(k) [1 2(ln2(cid:12) +b 1)ǫ] , (64) fined only up to a constant, we set it to zero at the end ≃ − − π (cid:12)aH=k of inflation. If the given model does not have a natural (cid:12) where b is the Euler-Mascheroniconstant, 2(cid:12) ln2 b end to inflation this point is somewhat arbitrary and we (cid:12) 0.7296 and ln2+b 1 0.2704. − − ≃ set η to zero typically some number of e-folds after the The scalar and te−nsor≃spectral indices are given by highest mode of interest freezes out. Thepowerspectrumandthe spectralindices aretobe n (k) 1 4ǫ 2δ 2(1+c)ǫ2+ 1(3 5c)ǫδ calculatedinthe limit kη 0−, sothatthe LGapproxi- S ≃ − − 1− 2 − 1 mationcanbeused. Toav→oidaccumulationofnumerical 1 1 (3 c)δ2+ (3 c)δ , (65) error,however,these quantities should not be calculated −2 − 1 2 − 2 directly in this limit. Since the freeze-out happens soon n (k) 2ǫ (3+c)ǫ2 (1+c)ǫδ , (66) after the turning point is crossed, the computation is T 1 ≃ − − − carried out some e-folds after the turning point for each and c 0.08145. The expressions nS(k) 1 4ǫ 2δ1 respectivemode,butwellbeforetheendofinflation. The ≃ ≃ − − andnT(k) 2ǫare the well-knownfirst-orderslow-roll linear combination of both solutions to the mode equa- ≃− results. tions,intermsofAi-andBi-functions,isthencompletely In the numerical calculations for slow-roll presented dominated by the exponentially growingpart. We found below, we will approximate the solutions of the mode that carrying out the computations 4 or 5 e-folds after equations in terms of Bessel functions, while solving the therespectiveturningpointprovidedsufficientaccuracy. background equations without any further approxima- In all the computations described below, the Ai- and tions. Hence, we do not assume the strict slow-roll con- Bi-functions were calculated with the algorithm given in ditions that the slow roll parameters ǫ, δ , and δ be 1 2 Ref. [40]. constant. This is not the way the slow-roll approach is often implemented when estimating inflationary parameters: Afurtherasymptoticapproximationisutilized to solve the background equations [42, 43] employing a 2. The Mode Solutions derivative expansion in the inflationary potential V(φ). Such expansions may provide useful estimates but do The full solutions u (η) and u (η) are not not give the precision required by present and upcom- k,1,> k,1,< needed for the calculation of the power spectrum and ing CMBR observations. The reason for the inaccuracy the spectral index. Nevertheless, it is useful to calculate arising from expansions in the potential, as will be seen someofthemforselectedmomentakinordertocompare inanexample,isthattheTaylorexpansioninderivatives the leading order uniform solutions to the exact numeri- can often fail. This is not the case when the parameters cal solutions found from a mode-by-mode integration of are expressed in terms of the Hubble parameter and its thedifferentialequations. Forη <η¯wecancalculatethe derivatives. Ouraimhereistodeterminetheaccuracyof integrals appearing in f (k,η), as defined in Eqn. (25) themodesolutionsthemselves,thereforewedonotfocus < numerically via onerrorsarisingfromapproximationstothebackground equations. η¯ ηi η¯ dη′ g(k,η′)= + dη′ g(k,η′), (67) Zη (cid:18)Zη Zηi(cid:19) V. NUMERICAL IMPLEMENTATIONS p p where η is an initial value of the conformal time. In i In this section we describe the numerical implemen- the actual numerical routine we therefore have to know tation of the exact mode-by-mode integration and the the second integral before we can calculate the uniform uniformapproximationanditssimplifications. Technical solutions left of the turning point: This is achieved by details can be found in Appendix A. Since the uniform an additional run of the integrator for the background approximationis useful both in setting initial conditions equations. 10 3. Power Spectra and Spectral Indices Analytical result 0.00 Numerical result Uniform, 1. Order The power spectra for scalar and tensor perturbations pc −0.05 in leading order of the uniform approximation are given M h/ by Eqns. (35) and (36). The integrals are calculated us- 1 −0.10 4 ingatrapezoidalrulewithnon-equidistantdiscretization 1 2 0 in conformal time. As mentioned above, we avoid calcu- 0. −0.15 = lating the spectra numerically in the limit kη 0−, but k instead do so some 4-5 e-folds after the turn→ing point. ); k −0.20 u In the case of power-law inflation analytical results are m( I availablefortheleadingordercontributionstothepower −0.25 spectra (see Ref. [10]). We have checked that the power spectra numerically calculated from Eqns. (35) and (36) −0.30 −4 −2 0 2 4 are in agreement with these analytic results (see Sec- e−folds tion VIA and Table I for details). Thespectralindicesintheuniformapproximationmay FIG. 2: Imaginary (growing) part of the scalar mode func- be calculated either by numerically differentiating the tion for a power-law model with p = 11 for the mode power spectrum as described in Section IIIE for the ex- k = 0.0214hMpc−1. The solid red line is the exact ana- actmode-by-modeapproach,orbyusingtheformulaein lytical solution, the dashed-dotted blue line the result from Eqns.(48)and(51). InAppendixA2wedescribehowto thenumericalsolutionoftheexactequations,andthedashed deal with the inverse square root singularities appearing green line the uniform approximation to leading order; the in the integrals to be performed in the second case. analytical and uniform approximation results are almost on topofeachother. Thearrowshowstheturningpointandthe greenbandistheestimatederrorboundfortheleadingorder uniform approximation. B. Mode-by-Mode Numerical Integration 0.20 1. Initial Conditions and Mode Functions In order to numerically obtain the mode functions, we mustsatisfythe initialconditionrequirement,i.e.,inthe pc 0.15 M limit kη , h/ →−∞ 1 4 u (η) 1 e−ikη. (68) 021 0.10 k −→ √2k =0. k Two difficulties in imposing this formal initial condi- u|; k 0.05 | tion immediately arise. First, in any numerical solution, Analytical result the calculation must begin at a finite initial time, thus Numerical result Uniform, 1. Order for modes with small enough values of k, the condition 0.00 kη may not be fulfilled. Second, for modes at −4 −2 0 2 4 → −∞ larger k values, there are very many oscillations before e−folds the turning point is reached, naively requiring very fine time-steps if the entire temporal range must be handled FIG. 3: The absolute valueof the scalar mode function for a power-law model following Fig. 2. numerically. To circumventthese problems,we use the uniform approximation to set initial conditions in a regime where it is exponentially accurate. For each mode, we take consideredatanytime. Inthe regimeofsmallk,wefirst as initial condition the uniform approximation result at notethatasinflationhastostartsomewhereinpractical roughly 20 zeros, i.e., 10 oscillations, before the turn- numericalcalculations,this introduces a lowestvalue for ing point for that mode. The number of zeros from k, defined by the criterion that the mode should be well a given time to the turning point can be estimated by inside the Hubble length. As the initial conditions for nπ k[η¯(k) η]. Asshownbelow,thisinitializationpro- inflation are unknown, we will assume (i) that inflation ≈ − ceduresuppressesnumericalerrors,especiallyinthehigh beganwellbeforethe 55-65e-foldsnecessarytosolvethe k regime where the precision of the power spectrum and flatness and horizon problems, and (ii) by the time our thespectralindexisimprovedwithouttakingsmallerand calculations are to be performed, Eqn. (68) applies. By smallertimesteps. Inaddition,onlyasmallernumberof isolating how initial conditions are defined from possible “active” modes, i.e., the modes that are within some 20 early-time artifacts, our method of implementing initial zerosbeforetheturningpointandnotyetfrozen,needbe conditions also leads to substantial improvement in the