Observational signatures of a non-singular bouncing cosmology
PPreprint typeset in JHEP style - HYPER VERSION
Observational signatures of a non-singular bouncing cosmology
Marc Lilley
APC, Universit´e Paris 7, 10 rue Alice Domon et L´eonie Duquet, 75205 ParisCedex 13, FranceE-mail: [email protected]
Larissa Lorenz
Institute of Mathematics and Physics, Centre for Cosmology, Particle Physicsand Phenomenology, Louvain University, 2 Chemin du Cyclotron, 1348Louvain-la-Neuve, BelgiumE-mail: [email protected]
S´ebastien Clesse
Service de Physique Th´eorique, Universit´e Libre de Bruxelles, CP225, Boulevarddu Triomphe, 1050 Brussels, BelgiumandInstitute of Mathematics and Physics, Centre for Cosmology, Particle Physicsand Phenomenology, Louvain University, 2 Chemin du Cyclotron, 1348Louvain-la-Neuve, BelgiumE-mail: [email protected] – 1 – a r X i v : . [ g r- q c ] J un bstract: We study a cosmological scenario in which inflation is preceded by abounce. In this scenario, the primordial singularity, one of the major shortcom-ings of inflation, is replaced by a non-singular bounce, prior to which the universeundergoes a phase of contraction. Our starting point is the bouncing cosmologyinvestigated in Falciano et al. (2008), which we complete by a detailed study of thetransfer of cosmological perturbations through the bounce and a discussion of possi-ble observational effects of bouncing cosmologies. We focus on a symmetric bounceand compute the evolution of cosmological perturbations during the contracting,bouncing and inflationary phases. We derive an expression for the Mukhanov-Sasakiperturbation variable at the onset of the inflationary phase that follows the bounce.Rather than being in the Bunch-Davies vacuum, it is found to be in an excited statethat depends on the time scale of the bounce. We then show that this induces os-cillations superimposed on the nearly scale-invariant primordial spectra for scalarand tensor perturbations. We discuss the effects of these oscillations in the cosmicmicrowave background and in the matter power spectrum. We propose a new way toindirectly measure the spatial curvature energy density parameter Ω K in the contextof this model. Keywords: cosmology, bounce, inflation, CMB. ontents
1. Introduction 22. Background cosmology 4
3. Scalar perturbations 9 u u and v v
4. Tensor perturbations 19
5. The characteristic time scale ∆ η η
6. CMB and matter power spectra 26
7. Conclusions 30A. Time evolution of curvature perturbations 37
A.1 Numerical examples 37A.2 Growth of perturbations in the contracting phase 39– 1 – . Introduction
The flatness, homogeneity, isotropy and monopole problems of standard Big Bangcosmology are easily resolved by invoking a primordial epoch of inflation [1, 2, 3, 4].This framework also explains the origin of today’s cosmic microwave background(CMB) anisotropies and large scale structure with compelling naturalness and ease.It traces these anisotropies back to the quantum fluctuations of the scalar field re-sponsible for inflation. As the universe grows exponentially, these perturbations arestretched outside the Hubble radius at early times, become classical, source the tem-perature anisotropies of the CMB and provide the seeds for structure formation. Inaddition, all pre-existing super-Hubble fluctuations are conveniently stretched out-side the Hubble radius today and can be safely ignored. The statistical properties ofthe CMB and of the large scale structure of the universe in the observable range arethen directly related to the initial quantum state of sub-Hubble primordial fluctua-tions at the onset of inflation. The initial quantum state is unambiguously definedsince the wavelength of all perturbations observable today was much shorter thanthe Hubble scale at the onset of inflation so that perturbations start their evolutionin the adiabatic vacuum. With this initial state prescription, inflation yields a pri-mordial spectrum of density fluctuations, which, when evolved through the radiationand matter dominated epochs of standard cosmology, yields a CMB temperatureanisotropy angular power spectrum and a matter power spectrum compatible withobservations [5], lending strong support to the inflationary hypothesis, and any modelof the primordial universe lacking this unambiguous initial state prescription gener-ally suffers from a lack of predictability.Inflation however raises two new issues and leaves one unsolved. Firstly, theidentity and origin of the scalar field that drives inflation remain, for lack of candi-dates within the Standard Model (SM) of particle physics, entirely unknown despitethe fact that, in recent years, significant progress has been made by implementinginflation into a high energy framework beyond the SM. For example, in string theory,although scalar degrees of freedom abound, the derivation of their potentials remainschallenging and few candidate fields are suitable for inflation without (a more or lesssignificant amount of) finetuning.Secondly, the inflationary mechanism used to generate classical perturbationsfrom the stretching and amplification of quantum fluctuations leads to a conceptualissue of self-consistency. As spacetime expands during inflation, the physical wave-length of any given perturbation increases proportionally to the scale factor of theuniverse. Conversely, if we follow a perturbation backwards in time, its wavelengthbecomes shorter, possibly decreasing below the Planck length (cid:96) Pl if the duration ofinflation exceeds some 60 to 70 e -folds. However, close to the Planck scale, quantumgravity corrections become relevant. This is the well-known trans-Planckian problem of inflation. It has been studied both theoretically and numerically [6, 7, 8, 9, 10, 11]– 2 –y introducing modified dispersion relations or imposing non-vacuum initial con-ditions on the perturbations or by introducing a new cut-off scale of order of thePlanck mass in the inflationary mechanism. Although they remain phenomenolog-ical, these models generically predict oscillations superimposed onto the standard(approximately scale invariant) spectrum of primordial fluctuations. In Refs. [7, 8]it was shown that these oscillations then carry through to the CMB temperatureanisotropy and the matter density spectra.Thirdly, the usual models of inflation, even in the context of eternal infla-tion [12, 13], are not past complete, and inevitably contain a time-like singularityin the past at which the universe is of vanishing size. On the contrary, higher or-der curvature terms in the Einstein-Hilbert action [14, 15, 16, 17, 18, 19, 20, 21],non-minimal coupling of matter fields to gravity [22, 23, 24, 21] and the low energyeffective actions of some string theories [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] doallow non-singular cosmological background solutions. In these, instead of shrink-ing to zero size, the universe experiences a bounce: starting out large, the universeundergoes a contracting phase until it reaches a finite minimal size, after whichan expanding phase occurs. The existence of bouncing solutions in these high en-ergy effective theories warrants the study of simpler classical non-singular cosmo-logical models, in which background and perturbations are completely understoodand tractable. This simpler class includes models with minimally coupled scalarfields [36, 37, 38, 39], scalar fields with non-standard kinetic terms [40] or in geome-tries that depart from the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) geometry, i.e. Bianchi or Kantowski-Sachs spacetimes [41]. Another possibility is to considerunconventional forms of matter [42, 43, 44, 45], or a combination of radiation andscalar field matter [46]. For an extensive review, see e.g.
Refs. [47, 48].Although bouncing cosmologies have most often been discussed as alternativesto inflation, see e.g.
Refs. [49, 50, 32, 51], here we shall see both the contracting andthe non-singular bouncing phase as cosmological epochs that connect to inflation. Inthis paper, the cosmological singularity is replaced by a classical bounce which canentirely be described within General Relativity (GR), and after which inflation takesplace as usual. Moreover, we will mostly focus, for computational simplicity, on a symmetric bounce, meaning that the rate of contraction (up to a minus sign) andits duration are the same as in the inflationary phase that follows. The transition tostandard Big Bang cosmology after inflation through a phase of reheating remainsunchanged from the standard case without bounce.To determine the viability of a non-singular cosmology, it is important to inves-tigate the stability and attractor properties of the background cosmology and thenecessary level of fine-tuning required in order to obtain a bounce [36, 37, 52]. At theperturbative level, it is important to understand the transfer of cosmological pertur-bations through the bouncing phase, whether there can exist a (robust) prescriptionto choose the initial conditions for cosmological perturbations at some stage in their– 3 –volution, and finally, whether such bouncing models have distinctive observationalsignatures.In this paper, we focus on the latter three issues, using the background cosmologyof Ref. [36]. This model relies on a dynamical scalar field with a symmetric potentialof the small field type and a positive spatial curvature term. This is the minimalpossible setup in order to obtain a bounce but violate the least number of energyconditions [36, 53].This paper is organized as follows. We first give a detailed overview of thebackground cosmology in Sec. 2, and an introductory discussion of the evolution ofcosmological perturbations through the bouncing phase in Sec. 3.1. We then derive,in Sec. 3.2, the expression of the Bardeen variable at the onset of inflation at firstorder in slow roll by solving its mode equation prior to and after the bounce andby then applying the standard matching procedure at the bounce. We compute thecorresponding form of the initial state of the mode functions of the Mukhanov-Sasakivariable describing scalar perturbations at the onset of inflation (Secs 3.3 and 3.4).This initial state is an excited state rather than the vacuum state, and it is shown thatit can result in the presence of oscillations at either leading or subleading order in theprimordial spectrum of scalar perturbations, see Sec. 3.5. Similar results are obtainedfor tensor perturbations in Sec. 4. While the amplitude of the oscillations dependsonly on the initial state of the cosmological perturbations prior to the bounce, thefrequency of oscillation is given by a new time scale, which is discussed in Sec. 5. InSec. 5, we also discuss a new possibility to indirectly measure the spatial curvatureenergy density parameter Ω K in the context of the model proposed. Finally, in Sec. 6,we write down the form of the angular spectrum of CMB temperature anisotropies atlow multipoles, commenting on the appearance of slow oscillatory features on largeangular scales, and also provide numerical examples, using the CAMB code [54], of theangular power spectra of CMB temperature anisotropies and the spectrum of matterdensity fluctuations. In Appendix A, the evolution of the curvature perturbation inthe contracting phase is discussed in some more detail, and some numerical examplesof symmetric and asymmetric bouncing cosmologies are provided.
2. Background cosmology
We consider a homogeneous and isotropic FLRW universe with line elementd s = a ( η ) (cid:20) − d η + d r − K r + r (cid:0) d θ + sin θ d φ (cid:1)(cid:21) , (2.1)where a ( η ) is the scale factor as a function of conformal time η (where d η = d t/a ,with t the cosmic time). The constant parameter K describes the curvature of spatial– 4 –ections and can always be rescaled such that K = 0, ±
1. In standard inflation, oneusually sets K = 0. For the non-singular model studied in this paper, however,we shall require K = +1 (see below). For a perfect fluid, with equation of state P = wρ , where ρ and P and w are the energy density, pressure and equation ofstate parameter, respectively, the Friedmann equations derived from the Einsteinequations and the conservation of the stress energy tensor read κ ( ρ + P ) = 2 a (cid:0) H − H (cid:48) + K (cid:1) , (2.2) κ ( ρ + 3 P ) = − a H (cid:48) , (2.3) ρ (cid:48) + 3 H ( ρ + P ) = 0 , (2.4)where κ ≡ π/m Pl with m Pl the Planck mass, and where H ≡ a (cid:48) /a is the (conformal)Hubble parameter, with a prime denoting a derivative with respect to η .Consider a bounce occuring at conformal time η = 0 (or equivalently t = 0).At the bounce, the scale factor a reaches its (finite) minimum, a b , while the Hubbleparameter evaluated at the bounce, H b , is equal to zero and its derivative withrespect to conformal time, H (cid:48) b , is positive. For a perfect fluid, the null and strongenergy conditions derived from the Penrose-Hawking singularity theorems [55] read ρ + P ≥ , ρ + 3 P ≥ . (2.5)It can be seen from Eq. (2.2) that the null energy condition is preserved at thebounce only if K > H (cid:48) b . Recast in terms of cosmic time, this inequality reads K /a > ˙ H b where ˙ H b is the derivative of the (cosmic) Hubble parameter with respectto t evaluated at the bounce. Hereon, we set K = +1 and this condition turns into acondition on the combination a ˙ H b [see Eq. (2.24)]. From Eq. (2.3), one finds thatthe strong energy condition must necessarily be violated in order to obtain a bounce.Finally, note that the spatial curvature term K /a dominates the dynamics at thetime of the bounce. An inflationary phase is therefore required afterwards in order todrive the spatial curvature contribution to the energy content of the universe downto a level compatible with observations [56]. Let us first consider the de Sitter solution. If w = −
1, the energy content of theuniverse is in the form of a cosmological constant. For K = 0, one has a ( t ) ∝ e Ht , (2.6)where the (cosmic) Hubble parameter H ≡ ˙ a/a is a positive constant so that thescale factor of the universe increases monotonically. Re-expressing Eq. (2.6) usingthe conformal time coordinate, one has a ( η ) = − Hη and H = aH = − η . (2.7)– 5 –or t → ∞ , a ( t ) → ∞ so that η → η → −∞ in the past. In this case, onethus has η ∈ ] − ∞ , K = +1 de Sitter universe, one has a ( t ) = a b cosh (cid:18) ta b (cid:19) and H ( t ) = 1 a b tanh (cid:18) ta b (cid:19) , (2.8)For t <
0, one then has
H < t = 0,the dimensionful scale factor a ( t ) reaches its minimum a b , and the Hubble parametervanishes. For t >
0, on the other hand,
H > a ( η ) = a b sec ( η ) and H ( η ) = tan ( η ) , (2.9)so that the range of η for a de Sitter bounce is η ∈ ] − π/ , π/ If instead of a simple cosmological constant with w = −
1, the energy content of theuniverse is dominated by a scalar field ϕ with energy density and pressure ρ = ϕ (cid:48) a + V ( ϕ ) and P = ϕ (cid:48) a − V ( ϕ ) , (2.10)it follows from Eqs (2.2) and (2.3) that κ ( ρ + P ) = κ ϕ (cid:48) a = 2 a (cid:0) H − H (cid:48) + K (cid:1) , (2.11)and κ ( ρ + 3 P ) = − a H (cid:48) = − κV ( ϕ ) (cid:20) − ϕ (cid:48) a V ( ϕ ) (cid:21) , (2.12)while from Eq. (2.4) one has ϕ (cid:48)(cid:48) + 2 H ϕ (cid:48) + a d V d ϕ = 0 . (2.13)As stated earlier, at the bounce, H b = 0 and H (cid:48) b ≥ K > H (cid:48) b , the null energycondition is verified. This also implies ϕ (cid:48) ∈ R so that ghost fields are avoided. Again,the strong energy condition is not preserved at the bounce and implies a V ( ϕ b ) > ϕ (cid:48) in Eq. (2.12). If, as required in order to satisfy the observed flatness of spatial sectionstoday, an inflationary phase lasting N inf = 60 to 80 e -folds is to take place after thebouncing phase, one requires the usual slow roll conditions, a V ( ϕ ) (cid:29) ϕ (cid:48) , H ϕ (cid:48) (cid:29) ϕ (cid:48)(cid:48) , (2.14)to be verified for N inf e -folds. It is therefore seen that the strong energy conditionis consistently violated throughout both the bouncing and the inflationary phase. If– 6 –n addition, the cosmology is taken to be symmetric or close to symmetric, then adeflationary phase prior to the bounce is required. The strong energy condition isthen similarly violated in part of the contracting phase as well.The cosmological evolution of the early universe is conveniently described interms of a set of horizon flow functions defined by [57] (cid:15) = 1 − H (cid:48) H , (cid:15) i +1 = d ln | (cid:15) i | d N ( i ≥ , (2.15)which during inflation are slowly varying functions of time with (cid:15) i (cid:28) i . Thefirst three horizon flow functions are related to the usual slow roll parameters (cid:15) , δ and ξ by (cid:15) = (cid:15) , δ = (cid:15) − (cid:15) , ξ = 12 (cid:15) (cid:15) , (2.16)so that the functions (cid:15) and (cid:15) correspond to the inequalities in Eqs. (2.14) while thethird is related to the time derivative of (cid:15) and δ through its definition in terms of ξ .The field ϕ is in the slow-roll regime as long as (cid:15) i (cid:28) i and inflation endswhen (cid:15) (cid:39) (cid:15) i parameters themselves are considered constant and terms of O ( (cid:15) i ) orhigher are dropped. In the context of a symmetric bounce, the slow-roll contracting(deflationary) phase is obtained by imposing the same conditions on the (cid:15) i ’s but with H < both the deflationary and inflationary background evolutionsaway from the bouncing phase itself can be fully described in terms of the (cid:15) i ’s. In thebouncing phase itself, where spatial curvature is important, the horizon flow func-tions are ill-defined , but it is also possible to write down a generalized form of thescale factor describing deviations from a purely de Sitter bounce [38]. To describethe scale factor of a quasi de Sitter bouncing universe, we simply rewrite Eq. (2.9)in the generalized form a = a b cosh ( ωt ) , (2.17)where ω is some dimensionful scale characterizing the deviation from a de Sitterbounce for which ω = 1 /a b . In conformal time, one has a ( η ) = a b sec (cid:18) ηη c (cid:19) , (2.18) In fact, regardless of spatial curvature, in a bouncing cosmology with a bounce occuring atsay t = 0, the horizon flow functions are well-defined only in the separate contracting ( t <
0) andexpanding ( t >
0) domains but not globally since H ( N ) is not bijective throughout and since N diverges at the bounce. – 7 –here η c = 1 / ( ωa b ) is a dimensionless conformal time scale. For a symmetric bounce(where contracting and expanding phases last for the same number of e -foldings N )a Taylor expansion of Eqs. (2.11–2.13) and (2.18) around η = 0 yields [38, 36] ω = κV ( ϕ b ) − K a , (2.19)while the first Friedmann equation gives ϕ (cid:48) = 6 K κ − a V ( ϕ b ) , (2.20)where everywhere in these expressions the subscript “b” denotes quantities evaluatedat the bounce. Eqs. (2.19) and (2.20) imply that ω = K a − κ ϕ (cid:48) a = H (cid:48) b a . (2.21)If ϕ (cid:48) b = 0, ω = 1 /a b and the de Sitter solution is recovered. The null energy condition(2.5) is preserved provided ϕ (cid:48) b >
0, so that ω < a − .Let us discard the possibility that a bounce occurs when ϕ (cid:48) b = 0, in which case,for any bounce, the potential for the scalar perturbations diverges at the bounce,see Eq. (3.4) and Eq. (3.7). Then, once again under the assumption of a symmetricbounce, a bouncing solution is obtained only if [38, 36] ϕ b = 0 , ϕ (cid:48)(cid:48) b = 0 , V ( ϕ b ) = V , d V d ϕ (cid:12)(cid:12)(cid:12)(cid:12) b = 0 , d V d ϕ (cid:12)(cid:12)(cid:12)(cid:12) b ≤ . (2.22)Note that these conditions can be expected to hold approximately for a (slightly) asymmetric bounce as well. The simplest renormalizable potential that satisfies theconditions (2.22) and is bounded from below reads V ( ϕ ) = M (cid:34) − (cid:18) ϕµ (cid:19) (cid:35) , (2.23)with M and µ a priori free parameters. Given this form of potential, the phase ofinflation that follows the bounce falls into the class of small field models, for whichthe accelerated expansion sets in near the origin ϕ = 0. Requiring that both ϕ (cid:48) b and ω be real further yields the constraint a = (Υ /κ ) M − , < Υ < . (2.24) Having established the background cosmology, let us now briefly recall the relevantobservable range of scales and orders of magnitude of the parameters of the potential– 8 –hat will be used in the numerical work and observational predictions in Secs 5 and6, and in Appendix A. The Hubble scale and curvature radius today are given by (cid:96) H = 1 H and (cid:96) c = a √K . (2.25)Given that the Hubble scale today is H = 100 h km s − Mpc − with the reducedHubble parameter h (cid:39) .
7, one has (cid:96) H = 3 h − Gpc. Furthermore, for K = +1, giventhat | Ω K | ≤ − [5] and using | Ω K | = K / ( a H ) , one finds (cid:96) H a ≤ − , (2.26)so that a ≥ h − Gpc.Although part of the motivation for studying non-singular cosmologies is thepresence of bounce-inducing corrections to GR (or other bounce-inducing mecha-nisms) arising in some high energy theories, in this work, we shall remain in the do-main of applicability of GR and quantum field theory. We therefore set a b ∼ (cid:96) Pl .From the first Friedmann equation, one can see that this corresponds to an order ofmagnitude M (cid:39) O (10 − m Pl ) for the mass parameter of V ( ϕ ), consistent with smallfield inflationary model values for M [58], see Eq. (2.28) but also Eq. (2.24) where therange of allowed M parameter values is given. Using these values for a and a b , thenumber of e -foldings of expansion since the bounce then is N = ln( a /a b ) (cid:38) k phys , is 10 − h Mpc (cid:46) k phys (cid:46) h Mpc . (2.27)For a = 30 h − Gpc, the range of comoving wavenumbers is then 10 (cid:46) k (cid:46) .Finally, in order to obtain 60 to 80 e -foldings of inflation, the parameters of thepotential (2.23) must be of the order of the following small field inflation fiducialvalues [58] M (cid:39) O (cid:0) − m Pl (cid:1) , µ (cid:39) O ( m Pl ) . (2.28)
3. Scalar perturbations
We shall now study the evolution of cosmological perturbations around the homo-geneous and isotropic background cosmology discussed in the previous section. Wemainly consider the scalar part of the metric perturbations (given by the gauge invari-ant gravitational Bardeen potential Φ) in the presence of scalar field perturbations δϕ , but also give, although in less detail, the result obtained for tensor perturbations.In longitudinal gauge, the scalar part of the perturbed metric reads [59]d s = a ( η ) (cid:104) − (1 + 2Φ) d η + (1 − γ (3) ij d x i d x j (cid:105) , (3.1)– 9 –here γ (3) ij is the background metric of the spatial sections. Expressing the scalarparts of the perturbed Einstein equations in terms of Φ and the density perturba-tion variable δϕ and combining them in the appropriate way yields a second orderdifferential equation for the modes Φ k [59]Φ (cid:48)(cid:48) k + 2 (cid:18) H − ϕ (cid:48)(cid:48) ϕ (cid:48) (cid:19) Φ (cid:48) k + (cid:20) k − K + 2 (cid:18) H (cid:48) − H ϕ (cid:48)(cid:48) ϕ (cid:48) (cid:19)(cid:21) Φ k = 0 . (3.2)The wavenumber k is the eigenvalue of the Laplace-Beltrami operator on positivelycurved spatial sections. It is therefore a function of an integer n , and is given by k = (cid:112) n ( n + 2) with n = 0 and n = 1 corresponding to gauge modes [60]. As iswell-known, it is possible to define the variable u related to Φ through the relation [59]Φ = κ ρ + P ) / u = κ ϕ (cid:48) a u, (3.3)in terms of which the mode equation (3.2) reduces to u (cid:48)(cid:48) k + (cid:2) k − V u ( η ) (cid:3) u k = 0 , (3.4)where V u ( η ) = θ (cid:48)(cid:48) θ + 3 K (cid:0) − c s (cid:1) , θ = (cid:18) κ (cid:19) / H aϕ (cid:48) , (3.5)and c = δPδρ = − (cid:18) ϕ (cid:48)(cid:48) H ϕ (cid:48) (cid:19) . (3.6)The quantity c s can in some regimes be interpreted as the velocity of sound. Theexplicit form of V u ( η ) reads V u ( η ) = H + 2 (cid:18) ϕ (cid:48)(cid:48) ϕ (cid:48) (cid:19) − ϕ (cid:48)(cid:48)(cid:48) ϕ (cid:48) − H (cid:48) + 4 K , (3.7)which is manifestly regular when H = 0. Another possible choice of variable is givenby the generalized form of the Mukhanov-Sasaki variable, valid for arbitrary K andwhich we shall denote ˜ v . The modes of the variable ˜ v are related to Φ k , in Newtoniangauge, by the relation [38, 61, 62]˜ v k = − aχ k (cid:18) δϕ k + ϕ (cid:48) H Φ k − K κ H ϕ (cid:48) Φ k (cid:19) , (3.8)where χ k = 1 − K − c k . (3.9)The gauge-invariant combination of the modes Φ k and δϕ k defining the generalizedMukhanov-Sasaki variable in Eq. (3.8) yields an equation of motion for ˜ v k which isreminiscent of that for u k , ˜ v (cid:48)(cid:48) k + (cid:2) k − V ˜ v ( η ) (cid:3) ˜ v k = 0 , (3.10)– 10 –here V ˜ v ( η ) = ˜ z (cid:48)(cid:48) k ˜ z k + 3 K (1 − c ) and ˜ z k = a ϕ (cid:48) H χ k . (3.11)The variable ˜ v has the additional property that it appears naturally as the canonicalvariable in the total action for cosmological perturbations from which Eq. (3.10) isderived and, as such, should be the starting point for quantization and for the choiceof the quantum initial conditions seeding the growth of cosmological structures. Itcan however be seen from Eqs. (3.8-3.10) that using the combination defining ˜ v inEq. (3.8) to construct the relevant variable for the study of the evolution of cosmo-logical perturbations is inappropriate in the present context. The use of ˜ v acrossthe bounce is forbidden because the variable itself, c and the term ˜ z (cid:48)(cid:48) k / ˜ z k are eitherill-defined or divergent when H = 0 [38, 63, 64, 45]. Furthermore, the potential V ˜ v ( η )is k -dependent so that defining an asymptotically flat spacetime for high frequencymodes is ambiguous. From the expression for χ k , and setting K = +1, there wouldalso appear to be divergences whenever c = 1 − k / − k − ϕ (cid:48)(cid:48) H ϕ (cid:48) = δ, (3.12)where δ is the second slow roll parameter defined in terms of the horizon flow func-tions in Eq. (2.16). In the context of small field models and away from H = 0,the slow roll parameter δ verifies δ < | δ | (cid:28)
1, so that the condition (3.12)is clearly not a problem. It is therefore possible to retain the generalized forms ˜ v k and ˜ z k far from H = 0. In such regions, the modes ˜ v k are everywhere well-defined.Furthermore, in the regions where both K /a (cid:28) cosmic Hubble parameter H is slowly varying and different from zero. One maytherefore neglect the terms proportional to K , and use the approximation χ k (cid:39) v k and for the k -dependent function ˜ z k then reduceto their standard ( K = 0) forms v k and z , with z now k -independent. A quantuminterpretation of v is then possible and the standard quantization procedure on v k can in principle be achieved in the usual way [59]. Furthermore, in these regions, themodes of the variables u and v may be related to one another in the standard way. v k = (cid:18) κ (cid:19) / θ (cid:16) u k θ (cid:17) (cid:48) and k u k = − z (cid:16) v k z (cid:17) (cid:48) . (3.13)Using these relations, it is therefore possible to prepare an initial quantum state bychoosing the mode functions v k of the variable v in the regions η < u . The resulting formof u after the bounce can then be used to determine the form of the modes v k at theonset of inflation and then compute the primordial spectrum of scalar perturbations.– 11 –n the following sections, we study, analytically, the evolution of the perturba-tions using u after setting the initial state of perturbations using v . In Appendix A,the time evolution of perturbations and the issue of the growth of perturbations incontracting spacetimes is also addressed [65, 66, 67]. u Hereon, we restrict the analysis to the case of a symmetric bounce and in this section,we consider the evolution of the variable u in the contracting phase, at the bounceand in the inflationary phase. As shown in Ref. [36], the potential (3.7) is very wellapproximated by neglecting the curvature term K / H , neglecting the time depen-dence of the horizon flow functions, and joining a slow-roll exponentially contractingphase directly to a slow-roll inflationary phase. This is true because in the rangeof wavenumbers of observational interest, the spatial curvature term and the smallamplitude features in V u ( η ) near η = 0 are always negligible compared to k . Notethat, as discussed in the previous section, this approximation does not hold true inthe case of the variable ˜ v . Let us also point out that while it can be neglected whenstudying the perturbation u , the contribution of K , at the level of the backgroundcosmology, is crucial to obtain the bounce.We now proceed as in Ref. [36]. We neglect the spatial curvature term, in whichcase, at first order in slow roll, Eq. (3.7) simplifies to V u ( η ) = H (cid:16) (cid:15) + (cid:15) (cid:17) = 1 x ± (cid:16) (cid:15) + (cid:15) (cid:17) , (3.14)where we have used the expression H ± ( η ) = − (cid:15) η − η ± = ± (cid:15) x ± . (3.15)In the last step, we defined x ± = | η − η ± | , where the “ − ” or the “+” sign is chosenaccording to whether one considers times η before or after the bounce at η = 0. Upto first order in slow roll, the equation of motion for u therefore simply reads u (cid:48)(cid:48) + (cid:20) k − x ± (cid:16) (cid:15) + (cid:15) (cid:17)(cid:21) u = 0 , (3.16)where the subscript k on the mode function u is implicit. Note that Eq. (3.16) appliesfor symmetric bounces only because for an asymmetric bounce, the values of (cid:15) and (cid:15) cannot be assumed to be equal before and after the bounce. In Appendix A.1, wegive numerical examples showing that in cosmologies with strong departures fromsymmetry, additional features appear in the potential for u .We now briefly discuss the meaning of the conformal time parameters η − and η + . The expressions for H ± used in (3.15) are the analogs of the familiar expression– 12 –sed in standard inflation where one has, to first order in slow roll, H ( η ) = − (cid:15) η , (3.17)with η ∈ ] − ∞ , η − , , η + [ respectively, so that η − and η + are the asymptotic values of the conformal time for early and late cosmic times,which we shall label t − and t + , respectively. For instance, in the de Sitter bouncediscussed in Sec. 2.2, one has η ± = ± π/ u take the form of Hankel functions of the first andsecond kind H (1) ν and H (2) ν . We can write the solutions explicitly as u − ( η ) = (cid:112) kx − (cid:2) U − ( k ) H (1) ν ( kx − ) + U − ( k ) H (2) ν ( kx − ) (cid:3) , (3.18) u + ( η ) = (cid:112) kx + (cid:2) U +1 ( k ) H (1) ν ( kx + ) + U +2 ( k ) H (2) ν ( kx + ) (cid:3) , (3.19)where the order ν of the Hankel functions (equal before and after the bounce) isgiven by ν = 12 + (cid:15) + (cid:15) , (3.20)and where as before x − = | η − η − | = η − η − , x + = | η − η + | = η + − η . (3.21)It is important to realize that the arguments of the Hankel functions are different sothat the expressions given in Eqs. (3.18) and (3.19) are distinct. These two solutions(and their derivatives) can be matched at the time of the bounce η = 0 using thestandard procedure. Doing so and expanding the Hankel functions for large valuesof | kx ± | (that is, for η (cid:39) U +1 = U − ( σ k + i ) e − i ( k ∆ η − πν ) , (3.22) U +2 = U − ( σ k − i ) e i ( k ∆ η − πν ) , (3.23)where we retain only the leading order terms and where we have defined the param-eter σ k = 2 (cid:15) + (cid:15) k ∆ η (3.24)and rewritten the conformal time difference as∆ η = η + − η − . (3.25)Note that the expression for σ k is first order in slow-roll. This quantity will be seenin Sec. 3.5 to correspond to a slight tilt and amplitude modification of the standardform of primordial spectrum of scalar fluctuations. The quantity ∆ η is the conformal cosmological bouncing scale which we shall discuss in some more detail in Sec. 5. Weshall see that it is interpretable as the angular frequency of oscillations induced by thebouncing phase in the primordial power spectrum of scalar and tensor perturbations.In the case of a pure de Sitter bounce, ∆ η = π .– 13 – .3 Relating u and v In this section, we compute the form of the Mukhanov-Sasaki variable at the onsetof inflation in terms of its form prior to the bounce. Once determined, the expressionfor the Mukhanov-Sasaki variable in the expanding phase can be used in the standardway to compute the primordial spectrum of scalar perturbations.Let us recall that when expressed in terms of u , the time evolution of scalarperturbations is continuous across the bounce. The coefficients of the two linearlyindependent solutions for u + can then be expressed in terms of those for u − . Thiscalculation was the subject of Sec. 3.2. On the other hand, the time evolution interms of the Mukhanov-Sasaki variable ˜ v is singular at the bounce and in fact ˜ v itselfis ill-defined at the bounce (see Sec. 3.1). As a result, a calculation similar to the oneperformed in Sec. 3.2 is not possible in the case of ˜ v . One may nevertheless easilydetermine the functional form of the Mukhanov-Sasaki variable in the deflationaryand inflationary phases on either side of the bouncing phase because these are regionswhere K /a (cid:28) v (cid:39) v and V ˜ v ( η ) (cid:39) V v ( η ). Inthese regions, the equation of motion for the mode functions v is then given, at firstorder in slow roll, by the standard expression v (cid:48)(cid:48) + (cid:20) k − x ± (cid:18) (cid:15) + 3 (cid:15) (cid:19)(cid:21) v = 0 , (3.26)where the subscript k on v is implicit. The solutions of Eq. (3.26) are given by v − ( η ) = (cid:112) kx − (cid:2) V − H (1) (cid:37) ( kx − ) + V − H (2) (cid:37) ( kx − ) (cid:3) , (3.27) v + ( η ) = (cid:112) kx + (cid:2) V +1 H (1) (cid:37) ( kx + ) + V +2 H (2) (cid:37) ( kx + ) (cid:3) , (3.28)where (cid:37) = ν + 1 with ν defined in Eq. (3.20). From the relations in Eqs. (3.13)between u and v , and using Eqs. (3.18), (3.19), (3.27) and (3.28), one finds U ± i = ± (cid:16) κ (cid:17) / k − V ± i , (3.29)so that the coefficients of v − and v + are simply related by V +1 ( k ) = V − ( k ) ( σ k + i ) e − i ( k ∆ η − π(cid:37) ) , (3.30) V +2 ( k ) = V − ( k ) ( σ k − i ) e i ( k ∆ η − π(cid:37) ) . (3.31)We stress once more that although analogous to Eqs. (3.22) and (3.23), the expres-sions (3.30) and (3.31) cannot be obtained using the procedure followed to relate u + and u − because, as discussed above, the potential V ˜ v ( η ) in Eq. (3.11) is singular atthe bounce, and because ˜ v is ill-defined at the bounce.We shall now require that the canonical variable v be quantized. In regions wheremodes are sub-Hubble and spatial curvature is negligible, the standard quantization– 14 –rocedure of the canonical variable v can be applied. The consistency of the temporalmode functions v with the commutation relations required for quantization is ensuredprovided v satisfies the Wronskian condition [59]( v ) (cid:48) v ∗ − ( v ∗ ) (cid:48) v = i . (3.32)This imposes | V ± | − | V ± | = ∓ π k − (3.33)on the coefficients of the Hankel functions in (3.27) and (3.28). v Unlike for inflation, there exists no unique prescription to fix the initial conditionsfor the cosmological perturbations at the onset of the deflationary phase in a bounc-ing cosmology. This is because, rather than starting sub-Hubble in their vacuumstate, modes that enter the contracting phase have a prior history. For instance, andalthough this is by no means the only possibility, they may have started sub-Hubbleduring a radiation- or matter-dominated contracting era connected to the deflation-ary phase that preceeds the bounce. At the onset of inflation, therefore, modescannot be expected to be in their vacuum state. In this paper, we shall neverthelessassume, for definiteness, that V − and V − can be parameterized as V − = √ π ς k − α/ e iθ and V − = √ π ς k − β/ e iθ , (3.34)where ς , ς ∈ R and positive, α and β are numbers while θ and θ are phase angles.Given Eq. (3.34), one finds from Eq. (3.33) that ς k − β = ς k − α − k − . (3.35)In order to satisfy Eq. (3.35), one has to choose α = β = 1 and hence ς = ς − ς so that ς → ς , one finally has V − = √ π | ς | k − / e iθ and V − = √ π (cid:12)(cid:12) − ς (cid:12)(cid:12) / k − / e iθ , (3.36)where we have absorbed the choice of sign in going from | V − | and | V − | to V − and V − into the phases θ and θ . Additionally, it can be checked using Eqs. (3.30) and(3.31) that for η > | V +1 | − | V +2 | = − π k − (cid:0) σ k (cid:1) . (3.37)where σ k is second order in slow roll, as can be checked from (3.24).The initial conditions for v − at times η < θ and θ and through the amplitude parameter ς , a– 15 –eviation from the Bunch-Davies vacuum, which can be recovered provided ς = 1and θ = ( π/ (cid:37) + π/
4. Furthermore, provided one does choose the adiabatic vacuumprescription, it can be seen from Eqs (3.30) and (3.31) that v + is given by a singlemode function. Consequently, in this case and at first order in slow roll, we canexpect to recover the standard form of the primordial spectrum of scalar fluctuations(see Sec. 3.5).A comment on the order of magnitude of ς is also in order. It is well known thatcosmological perturbations on super-Hubble scales generically grow in contractingspacetimes [65, 66, 67]. This is because while in an expanding phase, a perturbationcan be split into a constant and a decaying mode, in a contracting phase, a pertur-bation is split into a constant and a growing mode. The amplitude of a perturbationwith wavenumber k , which we parameterize by ς , is therefore dependent on the rateand the duration of the contracting phase. As discussed in Appendix A.2, the am-plitude of perturbations can be made to deviate only slightly from the amplitude ofvacuum fluctuations provided the contracting phase is either slow, short, or both slowand short. In the case at hand, the contracting phase is a fast and short deflationaryphase ( ∼ e -folds in a short conformal time interval) during which super-Hubblemodes grow only very little. Furthermore, fluctuations can be assumed to remainsmall in any contracting phase preceding deflation, provided the rate of contractionis small. Finally, since ς turns up in the expression for the amplitude of the pri-mordial, CMB and matter power spectra, that it remains close to 1 is evidently anobservational necessity for the viability of the model. We shall therefore assume that ς (cid:39) We shall now use the late time (super-Hubble) behaviour of v + to compute theprimordial spectrum of scalar perturbations in the expanding phase, P ζ , where, inthe subscript, the variable ζ = v/z is the curvature perturbation in the comovinggauge discussed in the Appendix. The spectrum P ζ is given by the usual expression P ζ = k π | ζ | = k π (cid:12)(cid:12)(cid:12)(cid:12) v + z (cid:12)(cid:12)(cid:12)(cid:12) , (3.38)and on super-Hubble scales, v + is given in terms of the asymptotic forms of theHankel functions for kx + (cid:28) v + (cid:39) (cid:112) kx + (cid:40) V +1 ( k ) (cid:34) (cid:37) ) (cid:18) kx + (cid:19) (cid:37) + i Γ(1 − (cid:37) ) sin π(cid:37) (cid:18) kx + (cid:19) − (cid:37) (cid:35) + V +2 ( k ) (cid:34) (cid:37) ) (cid:18) kx + (cid:19) (cid:37) − i Γ(1 − (cid:37) ) sin π(cid:37) (cid:18) kx + (cid:19) − (cid:37) (cid:35)(cid:41) . (3.39)– 16 –iven that, as long as slow roll is not violated, (cid:37) >
0, we may keep only the secondterm in each line of (3.39). From the growing modes of (3.39) and from z − = (cid:16) κ (cid:17) / a √ (cid:15) , (3.40)where, since K /a (cid:28) χ (cid:39)
1, and where we have used the definitionof the first horizon flow function, we find P ζ = 2 m Pl k π a (cid:15) (cid:37) [Γ(1 − (cid:37) ) sin( π(cid:37) )] ( kx + ) − (cid:37) | V +1 − V +2 | . (3.41)It is convenient to split the explicit expression for P ζ into a first standard (tilt) part P std ζ and a second part P osc ζ , which will be seen to be oscillatory in k . Let us thuswrite P ζ = P std ζ × P osc ζ . (3.42)The standard part of the spectrum is given by P std ζ = k m Pl a (cid:15) (cid:37) [Γ(1 − (cid:37) ) sin( π(cid:37) )] ( kx + ) − (cid:37) which, up to first order in slow roll, reads P std ζ (cid:39) H m Pl π(cid:15) { − [2(1 + C ) (cid:15) + C(cid:15) + (2 (cid:15) + (cid:15) ) ln( kx + )] } , (3.43)where C = γ E + ln 2 − γ E is Euler’s constant. Eq. (3.43) is the standardinflationary result for P ζ . The oscillatory part of the spectrum reads P osc ζ ≡ π k | V +1 − V +2 | . (3.44)From (3.30) and (3.31), one has V +1 − V +2 = A (cid:0) | V − | e i ( θ − φ ) − | V − | e i ( θ + φ ) (cid:1) , (3.45)where A = (cid:0) σ k (cid:1) / , φ = (cid:2) (2 (cid:15) + (cid:15) ) σ − k − π(cid:37) (cid:3) − ϕ, ϕ = arctan σ − k , (3.46)with σ k defined by Eq.(3.24) and ∆ η by Eq. (3.25). The expression in Eq. (3.45) canalso be rewritten as V +1 − V +2 = A (cid:2) | V − | + | V − | − | V − || V − | cos ( θ − θ − φ ) (cid:3) / e i Ψ , (3.47)where Ψ = arctan (cid:20) | V − | sin ( θ − φ ) − | V − | sin ( θ + φ ) | V − | cos ( θ − φ ) − | V − | cos ( θ + φ ) (cid:21) . (3.48)– 17 –sing these expressions, the oscillatory contribution to the spectrum P osc ζ reads P osc ζ = 4 π A k (cid:2) | V − | + | V − | − | V − || V − | cos ( θ − θ − φ ) (cid:3) . (3.49)Up to this point, our calculations were exact to first order in slow-roll. We now makean approximation to simplify the cosine term in Eq. (3.49). We first rewrite it ascos ( θ − θ − φ ) = cos ( θ − θ − k ∆ η + 2 πν ) cos (cid:0) σ − k (cid:1) +sin ( θ − θ − k ∆ η + 2 πν ) sin (cid:0) σ − k (cid:1) . (3.50)Because σ − k (cid:29)
1, and as k grows, the cosine and sine terms that have the arctangentas their argument rapidly go to − θ − θ − k ∆ η + 2 πν ) in the first line of Eq. (3.50). As for A ,one can easily see from Eq. (3.46) that it is equal to one at first order in slow-roll.Furthermore, expanding P osc ζ at that order and using Eq. (3.36) one obtains P osc ζ (cid:39) ς + | − ς | − ς | − ς | / [cos(2 k ∆ η + θ − θ ) − π (2 (cid:15) + (cid:15) ) sin(2 k ∆ η + θ − θ )] . (3.51)The P osc ζ contribution to P ζ thus contains both a constant part and an oscillatorypart with frequency 2∆ η . The spectrum is then given by the product of P std ζ inEq. (3.43) and P osc ζ in Eq. (3.51).Let us finally expand H , (cid:15) and (cid:15) around the pivot scale k − = a ( x +p ) H ( x +p ),taken to be the logarithmic mean of the range of observable scales. At first order inthe expansion, P std ζ then reads P std ζ (cid:39) H p m Pl π(cid:15) (cid:26) − (cid:20) C ) (cid:15) + C(cid:15) + (2 (cid:15) + (cid:15) ) ln (cid:18) kk p (cid:19)(cid:21)(cid:27) , (3.52)where the (cid:15) i p and H p are evaluated at k p . Assuming for simplicity that θ and θ are k -independent, the expansion of P osc ζ around k p reads P osc ζ (cid:39) ς + (cid:12)(cid:12) − ς (cid:12)(cid:12) − ς (cid:12)(cid:12) − ς (cid:12)(cid:12) / × (3.53)[cos (2 k ∆ η + θ − θ ) + π (2 (cid:15) + (cid:15) ) sin (2 k ∆ η + θ − θ )] . At first order in slow roll, and in the limit ς →
1, the modes v − and v + are inthe Bunch-Davies vacuum, and any modifications to P ζ induced by the bouncingphase disappear. This conclusion is also true for ς → v − and v + (see Sec. 3.4). For ς very differentfrom 0 or 1, oscillations can appear at leading order in the spectrum, rather thanas small corrections. Note, finally that the form of P ζ obtained is reminiscent ofthe primordial spectrum obtained in trans-Planckian inflationary models but withsome important differences. In trans-Planckian models [6, 7], the oscillatory term– 18 –as a ln( k ) dependence rather than a dependence linear in k . In addition, theamplitude of oscillations in trans-Planckian models is suppressed by the productof the slow roll parameters with the ratio H inf /M c (cid:28) H inf is the Hubbleparameter during inflation and M c is a new mass scale of order m Pl . Here instead,the amplitude depends only on the deviation in amplitude from the vacuum stateand is not suppressed by factors of order (cid:15) i or by some new mass scale.
4. Tensor perturbations
In this section and the next, we compute the transfer of tensor perturbations throughthe bouncing phase and the primordial spectrum of gravitational waves. The evo-lution equation for the tensor modes h (where here h is not to be confused withthe reduced Hubble parameter h (cid:39) . h (cid:48)(cid:48) + (cid:2) k − V h ( η ) (cid:3) h = 0 with V h ( η ) = H (cid:48) + H − K , (4.1)where the subscript k on the mode functions h is implicit. Because tensor perturba-tions are not subject to gauge issues, h is everywhere well-defined and its evolutionequation is regular. As for V u ( η ) in Sec. 3.2, we neglect the spatial curvature termand use the approximate form V h ( η ) (cid:39) H (2 − (cid:15) ) . (4.2)Using Eq. (3.15) the equation for h becomes h (cid:48)(cid:48) + (cid:18) k − (cid:15) x ± (cid:19) h = 0 . (4.3)The solutions of this this equation before and after the bounce read h − = (cid:112) kx − (cid:104) T − ( k ) H (1) ν T ( kx − ) + T − ( k ) H (2) ν T ( kx − ) (cid:105) , (4.4) h + = (cid:112) kx + (cid:104) T +1 ( k ) H (1) ν T ( kx + ) + T +2 ( k ) H (2) ν T ( kx + ) (cid:105) , (4.5)where, keeping terms up to first order in slow roll, ν T = 32 + (cid:15) . (4.6)and where the coefficients T ± and T ± are unknown functions of the wavenumber k .Matching the coefficients of h + and h − at η = 0, expanding the Hankel functions for– 19 – kx ± | (cid:28)
1, and retaining terms in the expansion suppressed by negative powers of k ∆ η up to linear order, one obtains T +1 = T − ( τ k + i ) e − i ( k ∆ η − πν T ) , (4.7) T +2 = T − ( τ k − i ) e i ( k ∆ η − πν T ) , (4.8)where τ k = 2 + 3 (cid:15) k ∆ η . (4.9)Contrary to the expressions for U +1 and U +2 in Eqs. (3.22) and (3.23) which werefound to be exact at first order in slow roll, the truncated expressions for T +1 and T +2 in Eqs. (4.7) and (4.8) are not exact at first order in slow roll since, in principle,higher powers τ pk , with p ∈ N , contribute constant terms and terms of order (cid:15) to theexpansion. At first order in slow roll, these terms read τ pk = (1 + 3 p (cid:15) / / ( k ∆ η ) p .These higher order terms τ pk ( p >
1) can be safely neglected only if the constraint τ pk (cid:28) O ( (cid:15) ) is satisfied. It can be verified that for ∆ η ≥ π (∆ η = π correspondingto a de Sitter bounce) and for any (cid:15) < n (cid:38) k = n ( n + 2) for K > (cid:12)(cid:12) T ± (cid:12)(cid:12) − (cid:12)(cid:12) T ± (cid:12)(cid:12) = ∓ π k − . (4.10)As for v , we now assume that T − and T − are parameterized as T − = √ π ϑ k − α T / e iψ and T − = √ π ϑ k − β T / e iψ . (4.11)Eq. (4.10) then imposes α T = β T = 1 and ϑ = ϑ −
1. Again we drop the subscripton ϑ so that ϑ → ϑ , which gives T − = √ π | ϑ | k − / e iψ and T − = √ π (cid:12)(cid:12) − ϑ (cid:12)(cid:12) / k − / e iψ . (4.12)From Eqs. (4.7) and (4.8), the Wronskian condition on h + reads (cid:12)(cid:12) T +1 (cid:12)(cid:12) − (cid:12)(cid:12) T +2 (cid:12)(cid:12) = − π k − (cid:0) τ k (cid:1) . (4.13)Unlike for scalar modes, the Wronskian condition for h + only holds if τ pk (cid:28) O ( (cid:15) ). For tensor modes, the primordial power spectrum P T reads P T = k π πm Pl (cid:12)(cid:12)(cid:12)(cid:12) h + a (cid:12)(cid:12)(cid:12)(cid:12) . (4.14)– 20 –sing the late-time expansion of the h + solution in Eq. (4.5), this becomes P T = k π πm Pl a ν T [Γ (1 − ν T ) sin ( πν T )] ( kx + ) − ν T | T +1 − T +2 | . (4.15)Once again splitting Eq. (4.15) into a standard part and a bounce-induced oscillatorypart, P T = P std T × P osc T , we have P std T = k m Pl a ν T [Γ(1 − ν T ) sin( πν T )] ( kx + ) − ν T , (4.16) P osc T = 4 π k | T +1 − T +2 | . (4.17)Expanding at first order in (cid:15) , one recovers, from P std T , the standard result P std T = 16 π H m Pl [1 − C + 1) (cid:15) − (cid:15) ln ( kx + )] , (4.18)while P osc T is analogous to the form of P osc ζ with the substitution V + i → T + i ( i = 1 , P osc T (cid:39) | ϑ | + | − ϑ | − | ϑ | | − ϑ | / × [cos(2 k ∆ η + ψ − ψ ) − π(cid:15) sin(2 k ∆ η + ψ − ψ )] . (4.19)As for the scalar perturbations, Eqs. (4.18) and (4.19) can then easily be expandedaround the pivot scale k p . The tensor-to-scalar-ratio r also splits into a standard and a non-standard part, r = r std × r osc . The ratio of the standard parts of the tensor and scalar spectra yieldsthe usual result at leading order, r std = P std T / P std ζ = 16 (cid:15) , while the non-standardparts of the spectra yield the additional ratio r osc = P osc T P osc ζ = | T +1 − T +2 | | V +1 − V +2 | . (4.20)Given the choice in Eqs. (3.34) and (4.11), this ratio depends on the values of ς and ϑ , on the phases θ , θ , ψ and ψ , and it is scale-dependent through the 2 k ∆ η term in the sine and cosine arguments of P ζ and P T . The tensor-to-scalar ratio (4.20)therefore does not satisfy the standard consistency relation of single field inflationarymodels, for which one has r = − n T with n T = − (cid:15) the tensor spectral index.– 21 – . The characteristic time scale ∆ η In previous sections we saw that the term 2 k ∆ η plays a crucial role in the modifiedspectra after the bounce, appearing in the arguments of the sine and cosine functionsin both P osc ζ and P osc T . We shall therefore now discuss the significance of the timescale ∆ η defined in Eq. (3.25). We first review the relevant field space dynamics as afunction of the number of e -folds in slow roll inflation and then turn to the discussionon ∆ η . In inflation, the evolution of ϕ as a function of the number of e -folds is a knownfunction, depending only on the parameters of the potential and on the initial con-dition for ϕ at the onset of inflation. Indeed, as long as the slow-roll conditions aresatisfied, i.e. when (cid:15) i (cid:28)
1, the horizon flow functions introduced in Sec. 2.3 can bewritten in terms of field derivatives of the potential (2.23), (cid:15) (cid:39) κ (cid:18) V ,ϕ V (cid:19) , (cid:15) (cid:39) κ (cid:34)(cid:18) V ,ϕ V (cid:19) − (cid:18) V ,ϕϕ V (cid:19)(cid:35) , (5.1)which using the form of V ( ϕ ) in Eq. (2.23) gives (cid:15) (cid:39) ϕ κ ( µ − ϕ ) , (cid:15) (cid:39) µ + ϕ ) κ ( µ − ϕ ) . (5.2)Thus, if the primordial spectrum is to be evaluated at some pivot scale k p , the valuesof (cid:15) and (cid:15) (and therefore that of the scalar spectral index n s ) depend only on µ andon the value of the field, ϕ p , when modes of wavenumber k p exit the Hubble radius.The end of the inflationary expansion is defined by (cid:15) = 1 at ϕ = ϕ e . This yields ϕ e = (cid:114) κ (cid:20)(cid:16) κ µ (cid:17) / − (cid:21) . (5.3)The slow-roll trajectory is then found from N ( ϕ ) = − (cid:90) ϕϕ i d ϕ κVV (cid:48) = κ µ (cid:34)(cid:18) ϕ i µ (cid:19) − (cid:18) ϕµ (cid:19) + 2 ln (cid:18) ϕϕ i (cid:19)(cid:35) , (5.4)where ϕ i is the value of ϕ at some initial time. Inverting this expression, one canobtain the field value at Hubble exit of modes corresponding to the largest observablescales, ϕ (cid:63) µ = (cid:118)(cid:117)(cid:117)(cid:116) − W (cid:40) − (cid:18) ϕ e µ (cid:19) exp (cid:34) − (cid:18) ϕ e µ (cid:19) − N (cid:63) κµ (cid:35)(cid:41) , (5.5)– 22 –here W ( x ) is the principal branch of the Lambert function and ϕ (cid:63) is the fieldvalue evaluated at N (cid:63) , the number of e -folds between the Hubble exit of the largestobservable modes and the end of inflation. Given a value of µ and a value of N (cid:63) (typically 40 < N (cid:63) < ϕ (cid:63) is obtained from Eq. (5.5). This can then be usedto determine the value of M by adjusting the amplitude of the spectrum of scalarperturbations to the COBE normalization. For simplicity, we neglect the minormodifications to the expression for the COBE normalization that are introduced bythe new form of P ζ and by Ω K (cid:54) = 0 and simply write the normalization in the usualway, see e.g. ref. [58], M = 487 (cid:15) (cid:34) − (cid:18) ϕ (cid:63) µ (cid:19) (cid:35) − × − m Pl . (5.6)Let us take ϕ i to be the field value at the onset of inflation (in standard inflation, ϕ i is given as an initial condition). The total number of e -folds of inflation N inf is obtained when ϕ e of Eq. (5.3) is inserted into Eq. (5.4). It is useful, as a firstapproximation, to consider the case ϕ (cid:28) µ for which N inf can be approximated by N inf = N ( ϕ e ) (cid:39) κ µ ln (cid:18) ϕ e ϕ i (cid:19) . (5.7)Inflation should last at least
50 to 60 e -folds in order to solve the flatness, isotropyand horizon problems of Big Bang cosmology. Together with the value of ϕ e inEq. (5.3), one may then use Eq. (5.7) to determine an upper bound for ϕ i (in a smallfield model, the value of ϕ during inflation increases), which yields ϕ i ≤ ϕ e exp (cid:18) − N inf κµ (cid:19) . (5.8)This restricts ϕ i to very small values. Recall once again that in standard inflation, ϕ i is given as an initial condition. We see from Eqs. (5.5) and (5.7) that the fieldevolution between ϕ (cid:63) and ϕ e is independent of ϕ i but the total number of e -foldingsof inflation is a function of ϕ i . ∆ η Let us now turn to the case of the bouncing cosmology and the calculation of ∆ η .The conformal time interval ∆ η is obtained by integrating∆ η = η + − η − = (cid:90) η + η − d η. (5.9)Because the slow-roll contracting and expanding phases (sr) and the non-singularphase close to the bounce (b) cannot be approximated by a unique expression, itfollows that for a symmetric bounce∆ η = ∆ η − sr + ∆ η b + ∆ η +sr = ∆ η b + 2∆ η sr = (cid:90) η +i η − e d η b + 2 (cid:90) η + η +i d η sr . (5.10)– 23 –ere, η − e and η +i are the values of η at which the cosmology respectively exits andenters the slow-roll regime in the contracting and expanding phases. The slow-rollparts of the integral (5.9), evaluated in the interval from η +i to η + is given by∆ η sr = (cid:90) N e N i d N d H ( N ) , (5.11)where N i = ln( a i /a b ) and N e = ln( a e /a b ). In these expressions, a i is the scale factorat the onset of inflation and a e is the scale factor at the end of inflation. Expandingln H ( N ) around N p where N p = ln( a p /a b ) is the number of e -folds from the bounceup to the pivot scale, with N i < N p < N e , givesln H ( N ) = ln H ( N p ) + d ln H d N (cid:12)(cid:12)(cid:12)(cid:12) N p ( N − N p ) + 12 d ln H d N (cid:12)(cid:12)(cid:12)(cid:12) N p ( N − N p ) + . . . (cid:39) ln H p + (1 − (cid:15) ) ( N − N p ) − (cid:15) (cid:15) ( N − N p ) (5.12)At linear order in (cid:15) i , one thus has H ( N ) = H p e (1 − (cid:15) )( N − N p ) , (5.13)so that ∆ η +sr = e (1 − (cid:15) ) N p H p ( (cid:15) − (cid:2) e ( (cid:15) − N (cid:3) N e N i . (5.14)Since N e − N p = N (cid:63) , with 40 < N (cid:63) <
60, we finally have∆ η +sr (cid:39) H p [1 − (cid:15) ] e (1 − (cid:15) )( N p − N i ) , (5.15)while from Eq. (5.4) one also has N p − N i = κ µ (cid:34)(cid:18) ϕ i µ (cid:19) − (cid:18) ϕ p µ (cid:19) + 2 ln (cid:18) ϕ p ϕ i (cid:19)(cid:35) . (5.16)Here, the crucial point is that the field value at the onset of inflation ϕ i , althoughit is constrained through Eq. (5.8), is this time not given as an initial conditionbut depends on the detail of the bounce dynamics that preceed inflation and isan unknown function of a b and µ . This is directly related to the fact that theexact dynamics from the bouncing phase intermediate between the deflationary andinflationary phases is not a known (or easily approximated) function of a b and µ . Asa result, ∆ η sr but also ∆ η b and the initial field value ϕ i , and finally ∆ η are quantitiesthat cannot be obtained analytically.The value of ∆ η can however easily be obtained by integrating the backgroundcosmology numerically for any value of a b and µ . We do so in Fig. 1 where we plot,in the full lines, the quantity a ∆ η , which is the theoretical frequency of oscillations– 24 –or physical wave numbers, for the renormalized potential parameter µ in the range2 . ≤ µ ≤ .
5, over the allowed range for a b [see Eq. (2.24)] and in the range10 − ≤ | Ω K | ≤ .Furthermore, remembering that in the case of a K = +1 universe, the wavenum-ber k is discrete with k = (cid:112) n ( n + 2) for integer n , the arguments of the sine andcosine functions in Eq. (3.53), which are relevant for the oscillatory features in theprimordial spectra, read2 k ∆ η (cid:39) (cid:18) n + 1 − n (cid:19) ( mπ + δη ) ( n (cid:29) . (5.17)Here, m is an integer and hence 0 < δη < π , such that, in Eq. (3.53), one can makethe substitution 2 k ∆ η → (cid:18) n + 1 a (cid:19) a δη = k phys (2 a δη ) (5.18)for large n , where k phys , as previously, is the physical wavenumber and a is the scalefactor today. The effective frequency is therefore 2 a δη rather than 2 a ∆ η . Thequantity a δη is shown in the circles of Fig. 1. The quantity δη itself is shown ina two-dimensional plot in Fig. 2 in the ranges 3 . ≤ µ ≤ . . ≤ Υ ≤ . − ≤ | Ω K | ≤ . This reduced range for µ corresponds to n s (cid:39) .
95, inagreement with the observationally constrained spectral tilt of the primordial powerspectrum of scalar perturbations, while the range for Υ was chosen in order to satisfyobservational bounds on Ω K .Fig. 1 demonstrates that for µ = 3 . µ = 3 . effective frequency dropssignificantly in some narrow bands of Υ [or equivalently bands of a b , see Eq. (2.24)]while Fig. 2 shows the existence of this band for the case 3 . ≤ µ ≤ . η (cid:39) π with δη (cid:28)
1. The existence ofthese narrow bands is crucial. This is because, given that a is at the very least equalto 20 h − Gpc, the effective frequency of oscillations, a δη , can only be small enoughfor such features to be observable in the CMB or in the matter power spectrum if δη (cid:28) K should oscillations linear in k and attributableto a bouncing cosmology of the type described in this paper be measured in thetemperature anisotropy map and/or in the matter density power spectrum. Givenaccurate measurements of n s and a δη , a good estimate of µ can be obtained throughEqs. (5.1) and (5.2). Given an accurate measurement of a δη , the value of Υ is knownso that the value of the scale factor at the bounce is recovered. Furthermore, given V ( ϕ ), the value of µ , say of order 3 .
0, and the value of a b are enough to obtain ∆ η numerically. Finally, given the knowledge of ∆ η and that of a δη , the value of a iseasily obtained. Combined with H , it yields Ω K .– 25 –t last, one can estimate the number of oscillations per decade D in the wavenum-ber k from N osc ( D ) = δηπ i ( D ) , (5.19)where i ( D ) is an integer. For example, for µ = 3 m Pl [see Eq. (2.23)], a b (cid:39) . × (cid:96) Pl and setting δη = 0 . − h Mpc − < k phys < h Mpc − is10 (cid:46) k (cid:46) and | Ω K | (cid:46) − . This gives a total of four decades, D = 1 , , , i ( D ) = 3 , , ,
6. In this case, N osc ( D ) ranges from 0 . D = 1) to 3 × (for D = 4). If instead, δη = 0 .
01, then N osc ( D ) ranges from 3 (for D = 1) to 3 × (for D = 4). To see that these estimates are correct, see Sec. 6.2. Υ a ∆ η a nd a δ η ( h - M p c ) µ = 4 n s = 0.956 Υµ = 3.5 n s = 0.952 Υµ = 3 n s = 0.951 Υµ = 2.5 n s = 0.941 Υ µ = 2 n s = 0.918 Figure 1:
Numerical integration of a ∆ η and a δη . The full lines are a ∆ η , while the redcircles are a δη . For µ ≥ n s > .
95 and ∆ η (cid:39) π so that there exist regions in which a δη drops to significantly lower values, see Fig. 2.
6. CMB and matter power spectra
In previous sections, we derived the scalar and tensor primordial power spectra andshowed that oscillations induced by the combined effect of the deviation of incomingperturbations from the vacuum state and by the characteristic time scale ∆ η appearin P ζ and P h . In addition, we found that measuring the scalar spectral index n s and the frequency of oscillations makes possible, within the scope of this model, anindirect determination of the values of a b and of Ω K today.The oscillations can however only be measured if they are transferred to theangular power spectrum of the CMB (the C (cid:96) ’s) and/or to the matter power spectrum– 26 – igure 2: Numerical integration of δη . The thin lines represent | Ω K | isolines on a loga-rithmic scale, the red one denoting | Ω K | = 2 × − . In the range 3 ≤ µ ≤ n s (cid:39) . P δ . The transfer function relating P δ to P ζ is simply a k -dependent multiplicativefunction and oscillations can be expected to be transfered from P ζ to P δ unchanged,see Sec. 6.2. The C (cid:96) ’s, on the other hand, are related to P ζ through a complicatedexpression so that the transfer of oscillations to the angular power spectrum of theCMB is by no means obvious. This question is investigated in this section. Wefirst write down the analytic expressions for the angular power spectra in the range1 ≤ (cid:96) ≤
10. We then give numerical examples of C s (cid:96) over the entire (cid:96) range using the CAMB code [54].
For | Ω K | (cid:28) (cid:96) (cid:46)
10 but for K = +1, the scalar multipole moments C s (cid:96) can berelated to the primordial scalar curvature perturbation spectrum P ζ by [68, 69, 70] C (cid:96) = 2 π ∞ (cid:88) n =2 P ζ M n (cid:12)(cid:12)(cid:12) P − / − (cid:96) − / n [cos ( χ )] (cid:12)(cid:12)(cid:12) n ( n + 1) ( n + 2) sin ( χ ) , (6.1)– 27 –ith M n = (cid:96) (cid:89) i =0 ( i + 1) − n for (cid:96) ≤ n (cid:96) > n and where P − / − (cid:96) − / n are the associated Legendre polynomials while χ = η − η lss (cid:39) (cid:96) H /a = (cid:112) | Ω K | is the (conformal) radial distance to the last scattering surface, with η and η lss , the conformal times today and at the time of last scattering, respectively.With the exception that the integral over k should be replaced by a sum over n , thetextbook expression used in the case K = 0, see [70, 71], is a very good approximationto Eq. (6.1) for | Ω K | (cid:28)
1. The expression for the scalar C (cid:96) ’s can be carried over tothe case of the C (cid:96) ’s for tensor perturbations.The presence of the slowly oscillating sinusoidal functions in P ζ introduces anoscillation in the kernel of the sum in Eq. (6.1). In particular, one can in general ex-pect constructive or destructive interference between the oscillations of the P − / − (cid:96) − / n ’shaving as a characteristic scale χ with the oscillations of the cosine and sine termsin P ζ of characteristic frequency 2∆ η . Summed over n , this induces slow oscillationsin the C (cid:96) ’s at small (cid:96) values. The precise shape and locations of the deviations fromthe standard C (cid:96) ’s induced by the oscillations depend on Ω K via χ and on δη whiletheir amplitude depends on ς .The COBE normalization can be expected to be modified in two ways. First,note that for ς > P osc ζ is equal to 2 ς −
1. The amplitudeof the spectrum is therefore modified by this first factor. Secondly, the amplitudeof the spectrum will also be modified by a contribution coming from the oscillatorypart. This contribution depends, once again, on ς , on χ and on ∆ η .In Figure 3, we present the results of a numerical simulation using the CAMB code [54]. The quantities shown in the dashed red and full blue lines are defined by∆ C (cid:96) = C THEORY (cid:96) − C
WMAP (cid:96) (6.2)where the theoretical C (cid:96) ’s are the C (cid:96) ’s obtained, respectively, either from an inflation-ary cosmology or from the bouncing cosmology described in this paper. The errorbars are the WMAP error bars for the measured C (cid:96) ’s. The ∆ C (cid:96) ’s were computed for ς = 1 . , . . δη = 0 .
001 and 0 .
01. These results were obtained for µ = 3, N (cid:63) = 50, Υ (cid:39) . (cid:15) = 0 . (cid:15) = 0 . N inf = 65, and for a period of radiation and matter domination (neglecting reheating)lasting 65 e -folds [72], such that Ω K = − . δη = 0 .
01, and for ς = 1 .
01, the oscillatory features in the primordialspectrum appear to improve the fit to the WMAP data in comparison with whatis obtained in the absence of oscillations in P ζ . For larger values of ς , such as 1.1and 1.2, the effect of the oscillations worsens the fit near the first accoustic peak at (cid:96) (cid:39)
200 over a widening range of (cid:96) values as ς is increased. At higher frequency, for– 28 – η = 0 .
01, the resulting C (cid:96) ’s have large fluctuations that appear to be phase-shiftedwith respect to the result obtained in the inflationary case. At low (cid:96) , oscillationsare not visible but there remains a change in amplitude with respect to the ∆ C (cid:96) ’sobtained from inflation. This could be expected from Eq. (6.1) for the values of ς and δη chosen.
10 20 50 100 200 500 1000 (cid:45) (cid:45) l (cid:72) Π (cid:76) (cid:45) l (cid:72) l (cid:43) (cid:76) (cid:68) C l
10 20 50 100 200 500 1000 (cid:45) (cid:45) l (cid:72) Π (cid:76) (cid:45) l (cid:72) l (cid:43) (cid:76) (cid:68) C l
10 20 50 100 200 500 1000 (cid:45) (cid:45) l (cid:72) Π (cid:76) (cid:45) l (cid:72) l (cid:43) (cid:76) (cid:68) C l
10 20 50 100 200 500 1000 (cid:45) (cid:45) l (cid:72) Π (cid:76) (cid:45) l (cid:72) l (cid:43) (cid:76) (cid:68) C l
10 20 50 100 200 500 1000 (cid:45) (cid:45) l (cid:72) Π (cid:76) (cid:45) l (cid:72) l (cid:43) (cid:76) (cid:68) C l
10 20 50 100 200 500 1000 (cid:45) (cid:45) l (cid:72) Π (cid:76) (cid:45) l (cid:72) l (cid:43) (cid:76) (cid:68) C l Figure 3: ∆ C (cid:96) spectra for ς = 1 .
01, 1 . .
2, from top to bottom respectively and for δη = 0 .
001 and δη = 0 .
01 from left to right, generated using
CAMB . The error bars are theWMAP error bars on the measured values of the C (cid:96) ’s, the red dashed lines are the ∆ C (cid:96) ’sfor an inflationary cosmology and the blue line in full represents the ∆ C (cid:96) ’s for the bouncingcosmology presented in this work. The matter power spectrum P δ is related to the primordial power spectrum P ζ ( k )by a k -dependent transfer function T ( k ), i.e. one can write P δ ( k ) = T ( k ) P ζ ( k ).– 29 –oughly speaking, T ( k ) (cid:39) k , and T ( k ) ∝ k − at large wavenumbers. Theoscillating behaviour observed in the primordial perturbation spectrum is thereforeexpected to be transmitted to the matter power spectrum. Again, using the CAMB code [54] with standard cosmological parameter values, we have computed the theo-retical matter power spectra for the values of the parameters specified in Section 6.1.The results are shown in blue in Fig. 4. The theoretical spectra were then convolvedwith the observational window functions of the Sloan Digital Sky Survey (SDSS) [73]and are shown by the red dots in Fig. 4. These theoretical results are compared withthe SDSS data, shown in black in the figure. The convolution evidently smoothesout the oscillations, and it is clear from Fig. 4 that there exist values of ς and ∆ η such that the resulting matter spectra are fully degenerate with those obtained froma standard slow roll power spectrum. The range of scales displayed is centered onthe second decade ( D = 2) of observable scales. The number of oscillations in thisdecade is about 3 or 4 for δη = 0 .
001 and about 20 to 30 for δη = 0 .
01. These arethe estimates that were found in Sec. 5.
7. Conclusions
This paper refines and completes the analysis begun in Ref. [38] and continued inRef. [36]. In Ref. [38], the authors focused on the immediate vicinity of the bouncingphase and conducted a detailed analysis of the bounce-inducing background cosmol-ogy and on the transfer of fluctuations through the bounce by modifying the exactsolution obtained when considering a de Sitter universe with closed spatial sections.In Ref. [36], the framework used in Ref. [38] was used to show that for symmetric orquasi-symmetric general relativistic bouncing cosmologies with K = +1 the peak inthe potential for the variable u could never be large so that the metric perturbation isalways sub-Hubble in the vicinity of the bounce. A cosmology smoothly connecting acontracting phase and a bouncing phase to an inflationary phase was then proposedand analyzed at both the background and linear perturbations level. In the presentwork, we have provided a detailed summary of the background cosmology exploitedin Ref. [36] and have performed a much more detailed calculation of the transferof perturbations through the contracting, bouncing and inflationary phases of thecosmological background discussed in Ref. [36]. We have parameterized the initialstate of perturbations prior to the bounce in terms of the Bunch Davies vacuum stateand used the solution for the Mukhanov-Sasaki variable (far from the high spatialcurvature region) in terms of Hankel functions, to reduce the number of unknownparameters needed to define the initial state of first order perturbations, see Sec. 3.4.We then computed the effects of the bounce and the choice of initial conditions on thescalar and tensor mode primordial spectra P ζ and P h . We discussed the modified lowmultipoles of the CMB angular power spectrum, the modified COBE normalizationand also provided numerical evidence, using CAMB , that the C (cid:96) ’s and matter power– 30 – .200.100.050.021 (cid:180) (cid:180) (cid:180) (cid:180) (cid:64) h Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) h (cid:45) M p c (cid:68) (cid:180) (cid:180) (cid:180) (cid:180) (cid:64) h Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) h (cid:45) M p c (cid:68) (cid:180) (cid:180) (cid:180) (cid:180) (cid:64) h Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) h (cid:45) M p c (cid:68) (cid:180) (cid:180) (cid:180) (cid:180) (cid:64) h Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) h (cid:45) M p c (cid:68) (cid:180) (cid:180) (cid:180) (cid:180) (cid:64) h Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) h (cid:45) M p c (cid:68) (cid:180) (cid:180) (cid:180) (cid:180) (cid:64) h Mpc (cid:45) (cid:68) P (cid:72) k (cid:76) (cid:64) h (cid:45) M p c (cid:68) Figure 4:
Matter power spectrum for ς = 1 .
01, 1 . .
2, from top to bottom respec-tively and for δη = 0 .
001 and δη = 0 .
01 from left to right. The black dots with error barsare the data points from the Sloan Digital Sky Survey (SDSS) [73], the blue line is thetheoretical prediction of the bouncing cosmological model obtained using
CAMB , and thered dots are the simulated data points obtained by convolving the blue line with the SDSSwindow functions. spectrum P δ are affected by the combined effect of the choice of initial conditionsand the bouncing cosmology.The crucial point is the appearance of oscillations in the power spectra. Theseoscillations mainly depend on the initial state of the perturbations through the freeparameter ς and on a new cosmological scale ∆ η . The former sets the amplitude ofthe oscillations while the latter sets the oscillatory frequency. As shown in Sec. 6,– 31 –here exists values of ς and ∆ η for which the oscillations induced by the bouncingphase and by the choice of initial conditions do not conflict with the WMAP andSDSS data. In fact, there appears to be parameter ranges in which the C (cid:96) ’s andthe P δ derived from P ζ are fully degenerate with those obtained from a standardslow-roll primordial spectrum. The tensor-to-scalar ratio is also modified. It departsfrom the standard result in two ways, first by a modification of the overall amplitudeand secondly by the existence of a scale-depedent oscillation, see Eq. (4.20).We also identified a new way of indirectly measuring the spatial curvature ofthe universe, in Sec. 5, assuming the oscillations can be attributed to a bouncingcosmology of the kind described in this paper and can indeed be measured. Thescale factor at the bounce as well as the model parameter µ determine both Ω K K and∆ η , thereby establishing a one-to-one relationship between the two. If the frequencyof oscillations could be measured together with the spectral index of P ζ , then Ω K isdetermined, as well as a b .In Appendix A, we discussed the issue of the growth of perturbations in thecontracting phase and pointed out the possible necessity of studying asymmetricbouncing cosmologies. This will be the subject of future work. Finally, a well-knownproblem of contracting spacetimes with spatial curvature is the instability of thebackground evolution [36, 37, 52]. Again, this will be the subject of future work [74]. Acknowledgments
The authors would like to thank J. Martin, P. Peter, C. Ringeval and D. Steer forfruitful discussions, useful suggestions, and careful proof-reading of the manuscript.M. L. also wishes to thank L. Sriramkumar for valuable discussions as well asLouvain University and the Harish Chandra Research Institute for their hospitalityin the final stages of this work. M. L. is supported by ANR grant SCIENCE DELISA, ANR–07-BLANC-0339. L. L. is partially supported by the Belgian Federaloffice for Science, Technical and Cultural Affairs, under the Inter-universityAttraction Pole Grant No. P6/11. S. C. is supported by the Belgian Fund forresearch (F.R.I.A.) and the Belgian Science Policy (IAP VI-11).
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In this Appendix, we discuss the time evolution of curvature perturbations. Wefirst provide some numerical examples for symmetric as well as for non-symmetricbounces and then discuss the growth of perturbations in the contracting phase.
A.1 Numerical examples
Before gauge fixing, the scalar part of the perturbed metric is given by [59]d s = a ( η ) (cid:2) − (1 + 2 A ) d η + (1 + 2 C ) γ ij d x i d x j (cid:3) , (A.1)– 37 –here A and C are small but gauge-dependent scalar perturbation variables. Thegauge-invariant scalar degrees of freedom are [59]Φ = A + H ( B − E (cid:48) ) + ( B − E (cid:48) ) (cid:48) , (A.2)Ψ = − C − H ( B − E (cid:48) ) (A.3)where B and E are the scalar parts of the g i and g ij components of the perturbedmetric and where Φ and Ψ are equal in the absence of anisotropic stress in the stress-energy tensor. The curvature perturbation on comoving and on constant energydensity hypersurfaces, denoted by R and ζ , respectively, are given by [75, 76] R = C − H qρ + P and ζ = C − H δρρ (cid:48) , (A.4)where q is the scalar potential of the momentum density q i of the perturbed energy-momentum tensor, defined as q i = ∇ i q + q i with ∇ i q i = 0. Both R and ζ can beexpressed in terms of gauge-invariant quantities as R = Ψ + 2 H H + K ) Ψ (cid:48) + H Φ1 + w (A.5)and ζ = Φ − H w )( H + K ) (cid:40) Φ (cid:48) + (cid:34) − KH + 13 (cid:18) k H (cid:19) (cid:35) H Φ (cid:41) . (A.6)On super-Hubble scales and in the absence of isocurvature perturbations, ζ (cid:48) = 0 ifΦ (cid:48) + H Φ does not diverge (this can be seen from the equation of motion for Φ), and
R (cid:39) − ζ provided K (cid:39)
0. Hence, both ζ and R are conserved after Hubble horizoncrossing. In the absence of isocurvature perturbations, this holds in very generalcircumstances. These two quantities are therefore of primary cosmological interest.For scalar field matter, R can also be written in Newtonian gauge as R = − Φ − H δϕϕ (cid:48) (A.7)and it can be seen from (3.8), taking K = 0, that R is related to v by v = − z R .If K /a (cid:28) k , we may work with v and z rather than with ˜ v and ˜ z . In this case, ζ (cid:39) −R = v/z , and knowledge of v is directly usable to obtain ζ .In Figure 5, we show the results of a numerical integration of the backgroundcosmology, see (2.11) to (2.12), and curvature perturbation on equal density hyper-surfaces ζ , by integrating (3.4) and then using (3.3) and (A.6), for three sets ofinitial conditions at the bounce with which the bouncing phase gets progressivelymore asymmetric. – 38 – .2 Growth of perturbations in the contracting phase We now discuss the issue of the growth of perturbations in contracting spacetimes asdiscussed in Ref. [65], see also [67]. Let us consider the simple example of power-lawcontraction of the universe in which a ( η ) ∼ X p/ (1 − p ) where p = 2 / w ) is positiveand X = | η − η c | for some characteristic conformal time η c <
0. If p < η < η c andif p > η > η c so that for any p > η goes from −∞ to 0.In the power law case, it is easy to solve the evolution equations for either Φ or R .The solutions can be expressed, on super-Hubble scales, as the sum of two modes,one constant and one time-dependent,Φ k ∼ C Φ , X ( p +1) / ( p − + C Φ , (A.8)and R k ∼ C R , X (3 p − / ( p − + C R , . (A.9)When η → p = 2 / p = 1 /
2, the C Φ , mode diverges as X − and X − respec-tively, while the C R , mode diverges as X − and X − respectively. If p (cid:39) C Φ , mode grows as X − − p but the C R , decays as X − p . If w < − /
3, corresponding to ¨ a >
0, then p > X grows. For w (cid:39) − p (cid:29) C Φ , and C R , modes grow as X /p and X /p respectively. How-ever, in this case, X ∼ /a and a ∼ e Ht , so that the conformal time interval that amode remains super-Hubble in this regime is exponentially small so that both modesremain approximately constant.It is clear that the perturbation variables Φ k and R k do not evolve with the samepower in X . Furthermore, they are related to each other through their derivatives,so that, generally, the time-dependent mode of Φ does not correspond to the time-dependent mode R , and vice versa. The example given above also implies that ina given cosmological setup, perturbation theory may be preserved or the possibilitythat it be violated be alleviated provided the right choice of perturbation variableis made [65, 66]. In an explicit model where modes evolve in and out of the Hubbleradius, and where the equation of state parameter varies, as is the case for scalar fieldmatter, the issue of whether modes grow large or not evidently becomes more subtle.Let us ask which variables must remain small. In longitudinal gauge, the scalar partof the perturbed metric is diagonal, B = E = 0, so that the line element takes theform given in (3.1). It is legitimate to ask whether Ψ and Φ should remain small, orwhether it is A and C that should remain small. If the latter choice is made, then thegauge must be suitably chosen so that A and C do remain small in the cosmologicalevolution being studied. As in the example above, the longitudinal gauge may not beappropriate if Φ is a growing function of conformal time. In order to decide whetherthere exists a gauge in which perturbation theory remains valid, it is interesting to– 39 –ook at the intrinsic curvature of spatial sections and at the extrinsic curvature ofcomoving surfaces. The intrisic curvature of spatial sections in the comoving gaugeis given by (3) R = 6 K a + 4 a (∆ + 3 K ) R , (A.10)while the scalar part of the extrinsic curvature on comoving hypersurfaces is givenby K ij = H a (1 + 2 R ) δ ij + (cid:20) a H ∂ k ∂ j R − w )2 ∂ k ∂ j ∆ R (cid:48) (cid:21) δ ik , (A.11)where ∆ − is the inverse Laplacian. The evolution of each term in (3) R and K ij as a function of X is shown in Table 1. In all cases but the ekpyrotic scenario,perturbative terms to (3) R and K ij grow faster than background terms. In orderfor perturbation theory to be preserved, the contracting phase must therefore beshort – this points to the study of strongly asymmetric bouncing cosmologies [67]– or the contracting phase must be induced by a potential for ϕ giving rise to verylarge w ; see however [77]. In this work, see Sec. 3.4, we assume that the deflationaryphase is short enough that perturbations remain small in the rapid contracting phase.Figure 5 shows that this is in fact a realistic assumption. We further assume thatthe amplitude of fluctuations at the onset of the deflationary phase is small. Wethus choose initial conditions for the Mukhanov-Sasaki variable v with an amplitudedeviating only slightly from that of vacuum fluctuations, see Sec. 3.4.Type p X K /a H /a R /a HR /a R /a H R (cid:48)
Ekpyrotic (cid:28) (cid:38) X − p X − − p X − p X − p X − p X − p Radiation 1 / (cid:38) X − X − X − X − X − X − Matter 2 / (cid:38) X − X − X − X − X − X − Exponential (cid:29) (cid:37) X /p X /p X /p X /p X /p X /p Table 1:
Super-Hubble conformal time evolution for each term in the intrinsic and extrinsiccurvatures in the comoving gauge. – 40 – (cid:45)
10 0 10 200102030405060 t (cid:72) t Pl (cid:76) N (cid:45) (cid:45)
10 0 10 20 (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) H (cid:72) (cid:45) m P l (cid:76) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) V u (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45)
10 0 10 20020406080 t (cid:72) t Pl (cid:76) N (cid:45)
10 0 10 20 (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) H (cid:72) (cid:45) m P l (cid:76) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) V u (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45)
20 0 20 40 60050100150200 t (cid:72) t Pl (cid:76) N (cid:45)
20 0 20 40 60 (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) H (cid:72) (cid:45) m P l (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) V u (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i (cid:45) (cid:45) (cid:45) (cid:45) t (cid:72) t Pl (cid:76) Ζ (cid:144) Ζ i Figure 5:
From left to right, number of e -foldings N , Hubble parameter H , potential V u [true potential (blue), approximate potential (red dashed)] in rows 1, 3 and 5. Thenormalized curvature perturbation ζ/ζ i [real part (blue), imaginary part (red dashed)] for n = 5 , ,
15 in rows 2, 4 and 6, with ζ i = ζ ( t i ) the curvature perturbation at time t i .From top to bottom : Υ = 2 . ϕ b = 0 (top rows), Υ = 2 . ϕ b = − m Pl / . ϕ b = − m Pl /6 (bottom rows).