Analytical approximations for curved primordial power spectra
PPrepared for submission to Phys. Rev. D
Analytical approximations for curved primordial power spectra
Ayngaran Thavanesan,
1, 2, ∗ Denis Werth,
1, 2, 3, † and Will Handley
1, 2, 4, ‡ Astrophysics Group, Cavendish Laboratory, J.J.Thomson Avenue, Cambridge, CB3 0HE, UK Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK ´Ecole Normale Sup´erieure Paris-Saclay, Avenue des Sciences, Gif-sur-Yvette, France Gonville & Caius College, Trinity Street, Cambridge, CB2 1TA, UK (Dated: September 14, 2020)We extend the work of Contaldi et al. [1] and derive analytical approximations for primordialpower spectra arising from models of inflation which include primordial spatial curvature. Theseanalytical templates are independent of any specific inflationary potential and therefore illustrateand provide insight into the generic effects and predictions of primordial curvature, manifesting ascut-offs and oscillations at low multipoles and agreeing with numerical calculations. We identifythrough our analytical approximation that the effects of curvature can be mathematically attributedto shifts in the wavevectors participating dynamically.
I. INTRODUCTION
The inflationary scenario [2–4] was invoked to resolveseveral issues within the basic Hot Big Bang model, andit is upon this that the current concordance cosmology(ΛCDM) is built. Through a brief period of rapid expan-sion at early times, the inflationary framework success-fully predicts the minimal present-day curvature, as wellas the generation and growth of nearly scale-invariantadiabatic scalar perturbations. These perturbations thenmanifest themselves in the cosmic microwave background(CMB), as anisotropies, giving us the measured spectrumwe observe on the sky today [5, 6].If one is to study inflation in a theoretically completemanner, one cannot assume that the universe was flat atthe start of the expansion. Furthermore, the presence ofsmall discrepancies at low multipoles in the spectrum ofthe CMB [7] from those predicted by flat inflationary dy-namics, motivate a study of the effects of primordial cur-vature. Typically the spectra contain generic cut-offs andoscillations within the observable window for the level ofcurvature allowed by current CMB measurements. Previ-ous numerical calculations [8] of such models have shownthat the primordial power spectra generated for curvedinflating universes provide a better fit to current data.Additionally, the introduction of a small amount oflate-time curvature, creating a K ΛCDM cosmology [9,10], has been suggested as a potential resolution to thetensions observed between datasets probing the early uni-verse and those that measure late-time properties [11–18]. Planck 2018 data without the lensing likelihood [19]presents relatively strong evidence for a closed uni-verse [5]. Adding in lensing and Baryon acoustic oscilla-tion data [20–22] reduces this evidence considerably, butit remains an open question as to why the CMB alone ∗ [email protected] † [email protected] ‡ [email protected] prefers universes with positive spatial curvature. Whilstinterpretations of the level of experimental support fora moderately curved present-day universe differ [23–25],universe models with percent-level spatial curvature re-main compatible with CMB datasets. The appearanceof any present-day curvature is arguably incompatiblewith eternal inflation, and strongly constrains the totalamount of inflation, providing a powerful justification forjust-enough-inflation theories [26–31].In this paper, we generalise the approach of Contaldiet al. [1] to the curved case, obtaining analytical back-ground solutions and primordial power spectra for uni-verses including spatial curvature. The approximationmodels the background universe as beginning in a kinet-ically dominated regime, followed by an instantaneoustransition to a regime with no potential dependence,which we term ultra-slow-roll . Despite such an idealisedsituation, this simple approximation qualitatively repro-duces the exact spectrum obtained by numerical compu-tation, with the notable advantage of using this methodthat the results are independent of the scalar field po-tential. The analytic solutions yield better insight intothe physics and effects of curvature on the primordialuniverse which may potentially be overlooked through apurely numerical approach.This paper is organised as follows. In Sec. II the con-formal time equivalent of the background equations andgeneral Mukhanov-Sasaki equation for curved inflatinguniverses are presented. We solve the curved Mukhanov-Sasaki equation and plot the corresponding spectra forour potential-independent curved inflationary model inSec. III. This is proceeded by a discussion of our re-sults in Sec. IV, after which we present our conclusions inSec. V. Supplementary material, such as Python code forgenerating figures and Mathematica scripts for computeralgebra, is found at [32]. a r X i v : . [ a s t r o - ph . C O ] S e p II. BACKGROUND
In this section we establish notation and sketch aderivation of the background and first-order perturbationequations in conformal time. Further detail and expla-nation may be found in [33–35].The action for a single-component scalar field mini-mally coupled to a curved spacetime is S = (cid:90) d x (cid:112) | g | (cid:26) R + 12 ∇ µ φ ∇ µ φ − V ( φ ) (cid:27) . (1)Extremising this action generates the Einstein fieldequations and a conserved stress energy tensor.In accordance with the cosmological principle, the so-lutions to the Einstein field equations are assumed tobe homogeneous and isotropic at zeroth order. One thenperturbatively expands about the homogeneous solutionsto first order in the Newtonian gauge.In conformal time and spherical polar coordinates inthe Newtonian gauge, the metric may be written asd s = a ( η ) [(1 + 2Φ) d η − (1 − c ij + h ij ) d x i d x j ] ,c ij d x i d x j = d r − Kr + r (d θ + sin θ d φ ) , (2)where K ∈ { +1 , , − } denotes the sign of the curvatureof the universe: flat ( K = 0), open ( K = −
1) and closed( K = +1). The longitudinal metric perturbation Φ andcurvature metric perturbation Ψ along with the pertur-bation to the field δφ are scalar perturbations, whilst h ij is a divergenceless, traceless tensor perturbation withtwo independent polarisation degrees of freedom. Thecovariant spatial derivative on comoving spatial slices isdenoted with a Latin index as ∇ i .By taking the (00)-component of the Einstein fieldequations and the (0)-component of the conservationof the stress-energy tensor, one can show that thebackground equations for a homogeneous Friedmann-Robertson-Walker (FRW) spacetime with material con-tent defined by a scalar field are H + K = 13 m (cid:18) φ (cid:48) + a V ( φ ) (cid:19) , (3)0 = φ (cid:48)(cid:48) + 2 H φ (cid:48) + a dd φ V ( φ ) , (4)where H = a (cid:48) /a is the conformal Hubble parameter, m p is the Planck mass, φ is the homogeneous value of thescalar field, V ( φ ) is the scalar potential, a is the scalefactor and primes indicate derivatives with respect toconformal time η defined by d η = d t/a . A further usefulrelation to supplement Eqs. (3) and (4) is H (cid:48) = − m (cid:16) φ (cid:48) − a V ( φ ) (cid:17) , (5) note this is opposite to the curvature density parameter Ω K , K = +1 ⇒ Ω K < which is derived from the trace of the Einstein fieldequations. For the remainder of this paper we set thePlanck mass to unity ( m p = 1), but note that one mayreintroduce m p at any time by replacing φ → φ/m p , V → V /m .Another useful physical perturbation to consider is thegauge-invariant comoving curvature perturbation R = Ψ + H φ (cid:48) δφ. (6)The equation of motion for this quantity is termed theMukhanov-Sasaki equation. To derive this equation forcurved universes, one can take a direct perturbative ap-proach as that introduced by Mukhanov et al. [33]. Thiscomputation has been performed historically by [36–42].One can also arrive at Eq. (7) via the Mukhanov action,by following the notation of Baumann [35, AppendixB] and generalising the ADM formalism [43, 44] to thecurved case.Employing both approaches, a general version of theMukhanov-Sasaki equation for curved universes was com-puted by Handley [8] in cosmic time. Extending thesecalculations to conformal time, we now show that thecurved Mukhanov-Sasaki equation is given by (cid:0) D − K E (cid:1) R (cid:48)(cid:48) + (cid:32)(cid:32) φ (cid:48) H + 2 φ (cid:48)(cid:48) φ (cid:48) − K H (cid:33) D − K HE (cid:33) R (cid:48) + (cid:18) −D + K (cid:18) K H − E + 1 − φ (cid:48)(cid:48) φ (cid:48) H (cid:19) D + K E (cid:19) R = 0 , D = ∇ i ∇ i + 3 K, E = φ (cid:48) H , (7)where primes denote derivatives with respect to confor-mal time. Eq. (7) can be expressed in a more famil-iar form by Fourier decomposition and a redefinition ofvariables. In the flat case one normally redefines vari-ables in terms of the Mukhanov variable v = z R , where z = aφ (cid:48) / H . In the curved case, this is impossible, butone can define a wavevector-dependent Z and v via v = ZR , and Z = aφ (cid:48) H (cid:114) D D − K E . (8)Fourier decomposition acts to replace the D operatorin Eq. (7) with its associated scalar wavevector expres-sion [34] D ↔ −K ( k ) + 3 K, (9) K ( k ) = (cid:26) k , k ∈ R , k > , K = 0 , − ,k ( k + 2) , k ∈ Z , k > , K = +1 . (10)After some algebraic manipulation, the curvedMukhanov-Sasaki equation may be written as v (cid:48)(cid:48) k + (cid:20) K − (cid:18) Z (cid:48)(cid:48) Z + 2 K + 2 K Z (cid:48) HZ (cid:19)(cid:21) v k = 0 . (11) III. ANALYTICAL PRIMORDIAL POWERSPECTRA FOR CURVED UNIVERSES
To obtain spectra for curved inflating cosmologies, wewill now generalise to the curved case an approximate an-alytical approach first applied by Contaldi et al. [1]. Forour model we assume a pre-inflationary kinetically dom-inated regime defined by φ (cid:48) (cid:29) a V ( φ ). We then invokean instantaneous transition to a regime where the scalarfield motion has significantly slowed φ (cid:48) (cid:28) a V ( φ ), andthe standard slow-roll constraints to solve the horizonproblem are satisfied. This rather brutal approximationhas the advantoge that it does not depend on a specificpotential choice V ( φ ) and illustrates the effects of cur-vature on the primordial power spectrum. Furthermore,this model grants a framework, within which potentialdependence can be added via higher order terms in thesolutions for curved inflationary dynamics.In Sec. III A we provide analytic solutions and powerseries expansions for the background variables in the tworegimes. In Secs. III B and III C we derive analytic solu-tions for the mode equations in each regime, and matchthese together at the transition point. Sec. III D thenuses the freeze-out values of the ultra-slow-roll solutionto produce our analytic template in Eq. (32).To avoid confusion, note that we work in a conventionwhere the conformal time η = 0 at the singularity, i.e. a ( η = 0) = 0, which is different from Contaldi et al. [1],who places η = 0 at the transition time. A. Background dynamics
To solve for the background variables a , H and φ wecan rearrange Eqs. (3) and (5) into two useful forms H (cid:48) + 2 H + 2 K = a V ( φ ) , (12) H (cid:48) − H − K = − φ (cid:48) . (13)In the initial stages of kinetic dominance φ (cid:48) (cid:29) a V ( φ ), we can neglect the right-hand-side of Eq. (12),and similarly in the ultra-slow-roll stage φ (cid:48) (cid:28) a V ( φ )we can set the right hand side of Eq. (13) similarly tozero.If we define S K ( x ) = sin( x ) K = +1 x K = 0sinh( x ) K = − , (14)then solving Eq. (12) with the right-hand-side set to zeroyields a ∼ (cid:112) S K (2 η ) for the kinetically dominanted, andsolving Eq. (13) similarly yields a ∼ /S K ( η ) for ultra-slow-roll. In both cases these solutions have two freeintegration constants corresponding to an additive coor-dinate shift in η and a linear scaling of a . Matching a η t η t η a ( η ) kinetic dominance φ (cid:29) a V ( φ ) ultra-slow-roll φ (cid:28) a V ( φ ) K = +1 K = 0 K = − FIG. 1:
Evolution of the scale factor a over conformal timefrom the analytical calculation in Eq. (15), where the initialsingularity has been set at η = 0 and the scale factornormalised a = 1 at the transition time η t for the case of aflat universe ( K = 0). and a (cid:48) for these two solutions at some transition time η t gives a = (cid:26) (cid:112) S K (2 η ) : 0 ≤ η < η t [ S K (2 η t )] / /S K (3 η t − η ) : η t ≤ η < η t , (15)with the conformal coordinate freezing out into the in-flationary phase as η → η t . The evolution of the scalefactor a is plotted in Figure 1.Note that for the closed case ( K = +1), there is amaximum sensible value of η t = π/
4. At values of η t greater than this, the universe begins collapsing beforethe transition is reached and should be regarded as abreakdown of the approximation.The remaining background variables may also besolved in the kinetically dominated regime with curva-ture and conformal time [26], but for the purposes of thisanalysis we only need power series expansions, which upto the first curvature terms read N = N p + 12 log η − K η + O ( η ) , (16) φ = φ p ± (cid:114)
32 log η ± √ K η + O ( η ) , (17)where N = log a . Other derived series include φ (cid:48) = ± (cid:114)
32 1 η ± √ K η + O ( η ) , (18) H = N (cid:48) = 12 η − K η + O ( η ) , (19) a = e N = e N p η / − e N p K η / + O ( η / ) . (20)A complete derivation of these series requires a con-sideration of logolinear power series expansions [10, 45],which we detail further in Appendix A.The ultra-slow-roll regime is defined loosely as φ (cid:48) (cid:28) a V ( φ ), but can be more precisely thought of asthe limit where E → a found in Eq. (15). B. Mukhanov-Sasaki solutions underkinetically dominance
The evolution of the Mukhanov variable v k is definedby the Mukhanov-Sasaki Eq. (11). Combining the re-sults from Eqs. (18) to (20) show that for the kineticallydominated regime Z (cid:48)(cid:48) Z +2 K + 2 K H Z (cid:48) Z = − η + 32 K − K K ( k ) + O ( η ) . (21)Substituting (21) into (11) we can write theMukhanov-Sasaki for the kinetically dominated regimeas v (cid:48)(cid:48) k + (cid:20) k − + 14 η (cid:21) v k = 0 ,k − ( k ) = K ( k ) − K K K ( k ) . (22)From Eqs. (10) and (22) we can see that the first-ordereffects of curvature on the Mukhanov-Sasaki Eq. (11)in the kinetically dominated regime manifest themselvespurely as an effective shift in the wavevector participat-ing dynamically.By solving Eq. (22) we find that during the kineticdominance regime the Mukhanov variable v k evolves as v k ( η ) = (cid:114) π √ η (cid:104) A k H (1)0 ( k − η ) + B k H (2)0 ( k − η ) (cid:105) , (23)where H (1 , are zero-degree Hankel functions of the firstand second kind , and quantum mechanical normalisa-tion requires | B k | − | A k | = 1 [46]. Following Contaldi and should not be confused with the present day Hubble constant et al. [1] and Sahni [47] we choose initial conditions whichselect the right-handed mode A k = 0 , B k = 1 , (24)leaving a consideration of alternative quantum initialconditions to a future work. C. Mukhanov-Sasaki solutions under ultra-slow-roll
For the ultra-slow-roll regime ( η ≥ η t ), taking the limit E → Z (cid:48)(cid:48) Z + 2 K + 2 K Z (cid:48) HZ → a (cid:48)(cid:48) a + 4 K = 2( η − η t ) − K O [( η − η t ) ] . (25)Substituting this result from (25) into (11), allows usto express the Mukhanov-Sasaki equation for the subse-quent ultra-slow-roll regime as v (cid:48)(cid:48) k + (cid:20) k − η − η t ) (cid:21) v k = 0 ,k = K ( k ) − K . (26)Note that the shifted dynamical wavevector k + forultra-slow-roll ( η ≥ η t ) is distinct from that defined forthe kinetically dominated regime k − ( η ≤ η t ).By solving Eq. (26) we find that during the ultra-slow-roll stage, the Mukhanov variable v k evolves as v k ( η ) = (cid:114) π (cid:112) η t − η (cid:104) C k H (1)3 / ( k + (3 η t − η ))+ D k H (2)3 / ( k + (3 η t − η )) (cid:105) . (27)One can now invoke the condition of continuity of v k and v k (cid:48) at the transition time η t and match Eqs. (23),(24) and (27), to show the coefficients of the two modesof the Mukhanov variable v k in the ultra-slow-roll regimeare C k = iπη t √ (cid:104) k + H (2)0 ( k − η t ) H (2)1 / (2 k + η t ) − k − H (2)1 ( k − η t ) H (2)3 / (2 k + η t ) (cid:105) , (28) D k = iπη t √ (cid:104) k − H (2)1 ( k − η t ) H (1)3 / (2 k + η t ) − k + H (2)0 ( k − η t ) H (1)1 / (2 k + η t ) (cid:105) . (29)This recovers the results obtained by Contaldi et al. [1]in the limit of zero curvature ( K = 0), i.e. k − → K → k and k → K → k . D. The Primordial Power Spectrum
With these complete solutions of the Mukhanov vari-able, we have the means to compute a primordial powerspectrum.By extending upon the analysis of Contaldi et al. [1]and generalizing to the curved case, we derive the curvedprimordial power spectrum of the comoving curvatureperturbation R under our approximation to be P R ( k ) ≡ k π |R k | → lim η → η t a E π (3 η t − η ) k k | C k − D k | , = A s k k | C k − D k | , (30)where we have used that R k = v k / Z k , and Z → aφ (cid:48) / H = a √ E where the transition time parameter η t , slow-rollparameter E and formally diverging parameters can beabsorbed into the usual scalar power spectrum amplitude A s .At short wavelengths, where k − → k + → k (cid:29) /η t ,one recovers the standard result of a scale invariant spec-trum | C k | (cid:39) , | D k | (cid:28) | C k | , P R (cid:39) A s . (31)It should be noted that as we are working in the ultra-slow-roll regime, as in Contaldi et al. [1] there is no tilt n s to this power spectrum. Whilst there exist more sophis-ticated ways to incorporate higher order terms and hencerecover the tilt, in this work we insert this by replacing A s with the standard tilted power spectrum parameteri-sation.Our analytical form of the primordial power spectrumfor each curvature K ∈ { +1 , , − } therefore is parame-terised by an amplitude A s , spectral index n s and tran-sition time η t P R ( k ) = A s (cid:18) kk ∗ (cid:19) n s − k k | C k ( η t ) − D k ( η t ) | , (32)where C k and D k are defined by Eqs. (28) and (29), usingHankel functions and wavevectors k ± defined in Eqs. (23)and (26).The spectra of P R generated by our analytical calcula-tion are plotted in Figure 2. We note that they reproducethe spectra obtained by Contaldi et al. [1] in the case ofzero curvature ( K = 0). Contaldi et al. [1] considered the spectrum of a uniquely definedvariable Q in the limit where it becomes constant at late time. IV. DISCUSSION
Upon review of the calculations presented in III, we seethat when applying a purely analytical approach to solvecurved inflationary dynamics, the effects of curvature canbe mathematically attributed to shifts in the wavevectorsparticipating dynamically Eqs. (22) and (26). Furtherinspection of the curved Mukhanov-Sasaki equation inEq. (7) provides a sanity check of this mathematical re-sult, as we see that, within Fourier space, the differentialoperator is replaced by a scalar wavevector shifted bya curvature term. At a dynamic level we see that thisshifted wavevector manifests itself in the spectra of P R as phase-based ringing effects for large enough values ofthe transition time, η t . This gives us a physical intu-ition of the oscillations seen in the numerically generatedcurved primordial power spectra for closed inflating uni-verses [8].Furthermore, we have shown through our generallycurved approach that curvature also manifests itself asa shifted wavevector in the open case, and thus thesephase-based ringing effects are present in open inflatinguniverses. Unlike the open case, for the closed case wedo not obtain a sensible spectrum for all values of thetransition time, η t ; for large η t , we observe a naturalbreakdown of the approximation at low k at the limit of k = 3 for comoving k . This is in agreement with theconstraint k ∈ Z > z (cid:48)(cid:48) /z . The flat ( K = 0)spectra generated by our general curved approach alsodemonstrate these effects, but in the curved case there isa discontinuity in Z (cid:48)(cid:48) / Z + 2 K + 2 KZ (cid:48) / HZ , which in theflat case ( K = 0) reduces to a discontinuity of z (cid:48)(cid:48) /z . K ΛCDM is a commonly considered extension to stan-dard flat ΛCDM, where there is an additional degree offreedom of spatial curvature Ω K . In K ΛCDM an almostflat power spectrum is assumed P K ΛCDM R ( k ) = A s (cid:18) kk ∗ (cid:19) n s − , (33)where k ∗ corresponds to the pivot perturbation modeand by convention is set to have a length-scale todayof 0 . − .The Planck A s = 2 . ± . × − and n s =0 . ± . k , independent of initial conditions, and hence deviatesfrom the form of Eq. (33). It has been shown in previ-ous work that the ( k/k ∗ ) n s − tilt is a higher-order effect − − − − physical k [Mpc − ] comoving k . . . l og ( P R ) K = +1 η t = 0 . η max η t = 0 . η max η t = 0 . η max η t = η max P K ΛCDM R − physical k [Mpc − ] ‘ D TT ‘ [ µ K ] K = +1 P K Λ CDM R Planck 2018 − − − − physical k [Mpc − ] comoving k . . . l og ( P R ) K = − − physical k [Mpc − ] ‘ D TT ‘ [ µ K ] K = − FIG. 2:
Left: Primordial power spectra P R corresponding to the range of allowed values of the transition time η t for openand closed universes K ∈ {− , +1 } . Oscillations and a generic suppression of power are visible at low- k . For K = +1, onlyinteger values of comoving k with k ≥ k . For clarity, we include thecontinuous spectrum. Right: The corresponding low- (cid:96) effects on the CMB power spectrum. The power law K ΛCDMspectrum is highlighted in grey along with Planck data. There is no appreciable deviation from the traditional powerspectrum at higher k and (cid:96) values. Note that the spectra of P R and D TT(cid:96) qualitatively reproduce those found numericallyin [8]. Multipole (cid:96) and comoving & physical k are related by the conversions presented in Agocs et al. [48]. manifested from the nature of the scalar field potentialchosen for the slow-roll regime. To compute such effects,one can determine the higher order terms of the logolin-ear expansions listed in Appendix A.As a good phenomenological approximation for a gen-eral inflationary setting, we scale our normalised P R with the best-fit scalar power spectrum amplitude A s and manually add in the tilt, we present our analyticalprimordial power spectrum P R for varying values of thetransition time η t , which we then follow through to theCMB, in Figure 2 [49]. The CMB spectra, correspond-ing to these primordial power spectra, are generated us-ing parameters values set in accordance with the best fitdata for each curved scenario. For the closed case we usethe Planck anesthetic package, subject tothe constraint that Ω k > K = −
1) [50].The requirement that the horizon problem is solved i.e. that the amount of conformal time during inflation η i is greater than the amount of conformal time before η t and afterwards η ↑ , bounds the transition time η t fromabove. The condition of the amount of conformal timeduring inflation η i being greater than the conformal timein the kinetically dominated regime preceding inflation η t is naturally satisfied by the ultra-slow-roll solution of(15), since η i = 2 η t . The additional condition regardingthe conformal time after inflation η ↑ places the constraintthat ( η ↑ < η t ). The implications of these constraintson exact numerical integration methods for computingcurved primordial power spectra, are discussed in moredetail in Handley [8, Section IV].Figure 2 demonstrates how the computed spectra varyfor different values of the transition time η t . The loca-tion of the cutoff, suppression of power and oscillationsare changed by adjusting the transition time, and as ex-pected [10] the depth of the suppression in closed uni-verses ( K = +1) is greater for the case when primor-dial curvature has a larger magnitude (corresponding toa higher transition time). Interestingly, we find that forlarge enough values of the transition time, a suppressionof power is also seen in open universes ( K = − i.e. the case of zero curvature ( K = 0).With this work we have not only developed an analyti-cal framework to solve curved inflationary dynamics, buta means to study curvature in isolation, without com-plicating factors, such as the choice of the scalar fieldpotential. V. CONCLUSIONS
The inflationary scenario addresses the initial valueproblem of the Hot Big Bang, but provides us with noinsight into the universe’s state pre-inflation. Therefore,in order to truly understand the physics of inflation, wemust study it with no bias towards the conditions of theuniverse at inflation start; more specifically, we can notinfer the shape of the universe prior to inflation fromthe observed flatness seen at recombination through thecosmic microwave background (CMB).In [8], it was shown through exact numerical calcu-lations of curved inflating universes generated spectrawith generic cut-offs and oscillations within the observ-able window for the level of curvature allowed by currentCMB measurements and provide a better fit to currentdata. In this work we have used the formalism popu-larised by [33–35] and subsequent manipulation to writethe Mukhanov-Sasaki equation for curved universes inconformal time. This has allowed us to derive analyticalsolutions of the Mukhanov-Sasaki equation for a gener-ally curved universe scenario, which show that curvaturemathematically manifests itself as a shifted dynamicalwavevectors, and physically at low k as a suppression ofpower and oscillations in the primordial power spectra,which then follow through to the CMB.The main emphasis of our paper was related to mod-ifications of the simple model utilised by Contaldi et al.[1], which invokes an instantaneous transition betweenan initial kinetic stage (when the velocity of the scalarfield was not negligible) and an approximate de-Sitter in-flationary stage, to generate the significant suppressionof the large scale density perturbations. Through theapplication of logolinear series expansions and a newlydefined inflationary regime, we generalise this model tothe curved inflating case, to introduce oscillations and asuppression of power at low k , as well as generic cut-offs,which is in agreement with exact numerical calculations.Varying the remaining degree of freedom, specifically theamount of primordial curvature (provided through thetransition time), alters the oscillations and level of sup-pression in a non-monotonic manner, whilst there is aconsistent lowering in the position of the cut-off at low k with increasing transition time.The addition of an extra curvature parameter in thetheory to obtain a better fit with data comes with costs,but given the recent discrepancies that have arisen withthe standard ΛCDM model, this is something that nowrequires strong consideration. A natural extension is K ΛCDM. Our work has now shown, both analyticallyand numerically, that for all allowed values of initial pri-mordial curvature, incorporating the exact solutions forclosed universes results in observationally significant al-terations to the power spectrum. Furthermore, the dataare capable of distinguishing a preferred vacuum state,with the best fit preferring Renormalised-Stress-Energy-Tensor (RSET) initial conditions over the traditionalBunch–Davies vacuum. Future work will involve extend-ing our analytical approach to RSET and other initialconditions.
ACKNOWLEDGMENTS
D.W. thanks the ENS Paris-Saclay for its continuingsupport via the normalien civil servant grant. W.H.would like to thank Gonville & Caius College for their on-going support. The authors would like to thank FruzsinaAgocs, Julien Lesgourges, Oliver Philcox and ThomasGessey-Jones for their conversations on the nature ofcurved primordial power spectra.
Appendix A: Logolinear expansions in conformaltime
Logolinear series expansions [45] for a general function x ( η ) have the form x ( η ) = (cid:88) j,k [ x kj ] η j (log η ) k , (A1)where [ x kj ] are twice-indexed real constants defining theseries, with square brackets used to disambiguate powersfrom superscripts.We begin with Eqs. (4) and (5) N (cid:48)(cid:48) + N (cid:48) + 13 (cid:16) φ (cid:48) − a V ( φ ) (cid:17) = 0 , (A2) φ (cid:48)(cid:48) + 2 N (cid:48) φ (cid:48) + a dd φ V ( φ ) = 0 . (A3)Here N = log a has been used rather than H as itrestates Eqs. (4) and (5) in the form of second orderdifferential equations, which we can then in turn convertto a first order system of equations˙ N = h, ˙ φ = v, ˙ h = h − v + a η V ( φ ) , ˙ v = v − vh − a η dd φ V ( φ ) , (A4)where dots indicate derivatives with respect to logarith-mic conformal time log η , i.e. ˙ x = ddlog η x .To analytically determine approximate solutions forcurved cosmologies we will consider series expansions fora general function x ( η ) of the form x ( η ) = (cid:88) j x j ( η ) η j ⇒ ˙ x ( η ) = (cid:88) j ( ˙ x j + jx j ) η j . (A5) Note that this indexing convention differs from that adoptedin [10], which also utilised series expansions to solve cosmologi-cal evolution equations. For our purposes an expansion in η wasrequired, hence the unique convention used in our series defini-tions. Substituting in our series definition from Eq. (A5) andequating coefficients of η j , we find that Eq. (A4) becomes˙ N j + jN j = h j , ˙ φ j + jφ j = v j , ˙ h j + jh j = h j + 13 V ( φ ) e N p e (cid:80) q> N q ( η ) η q | j − − (cid:88) p + q = j v p v q , ˙ v j + jv j = v j − d V ( φ )d φ e N p e (cid:80) q> N q ( η ) η q | j − − (cid:88) p + q = j v p h q . (A6)One should also consider the equivalent of Eq. (3)13 V ( φ ) e N p e (cid:80) q> N q ( η ) η q | j − + (cid:88) p + q = j v p v q − h p h q = K | j − , (A7)where exponentiation of logolinear series was discussedin [45].We may solve for the j = 0 case of Eq. (A6) usingthe kinetically dominated solutions, as it is equivalent toEq. (A4) with V = 0 N = N p + 12 log η, h = 12 ,φ = φ p ± (cid:114)
32 log η, v = ± (cid:114) , (A8)where N p and φ p are constants of integration. As men-tioned previously we expect there to be four constantsof integration a priori . One of the missing constants isfixed by defining the singularity to be at η = 0, whilst theother is effectively set by the curvature. Hence Eq. (A8)represents a complete solution to j = 0 for only the flatcase ( K = 0). Nevertheless, we may still adopt Eq. (A8)as the base term for the logolinear series. The final con-stant of integration will then effectively emerges from aconsideration of higher order terms.For j (cid:54) = 0, we can rewrite Eq. (A6) in the form of a firstorder linear inhomogeneous vector differential equation˙ x j + A j x j = F j , (A9)where x = ( N, φ, h, v ), A j is a (constant) matrix A j = j − j −
10 0 j − v v j − h , = j − j −
10 0 j − ± (cid:113) ±√ j , (A10)and F j is a vector polynomial in log η depending only onearlier series x p
62 ( √ ∓√ (cid:17) , A j e b = ( j − · e b ,e n = (cid:0) (cid:1) , A j e n = j · e n ,e φ = (cid:0) (cid:1) , A j e φ = j · e φ . (A13)Parametrising initial conditions [ x j ] using the eigenba-sis in Eq. (A13) with parameters ˜ N , ˜ φ, ˜ b, ˜ β , yields x cf j = e − A j log η ( ˜ N e n + ˜ φe φ + ˜ be b + ˜ βe β )= (cid:16) ˜ N e n + ˜ φe φ + ˜ be b η + ˜ βe β η − (cid:17) η − j . (A14) We may absorb all ˜ N and ˜ φ into our definitions of N p and φ p . Choosing ˜ β = 0 amounts to setting the singu-larity to be at η = 0 as an initial condition without lossof generality, as it grows faster than our leading term as η →
0. The only remaining undetermined integrationconstant is ˜ b , which amounts to the integration constantthat was missing from Eq. (A8). The constant ˜ b is con-trolled by the curvature of the universe via Eq. (A7)˜ b = − K. (A15)Applying the standard definition of conformal timed η = d t/a , show a clear equivalence between Eq. (A15)and the cosmic time version found in the series solutionsderived in [45] [45]. We can now exchange K for ˜ b viathis relation, and for the proceeding analysis in the mainbody of the paper we shall drop the notation of ˜ b andexplicitly denote curvature terms with K in the seriessolutions. We also note from (A14) that the curvature ofthe universe depends on a term in η .All that remains to be determined is a particular in-tegral of Eq. (A9), given that one has the form of F j at each stage of recursion. The trial solution is x j ( η ) = (cid:80) N j k =0 [ x kj ](log η ) k . Defining F j = (cid:80) N j k =0 [ F kj ](log t ) k andequating coefficients of (log η ) k gives( k + 1)[ x k +1 j ] + A j [ x kj ] = [ F kj ] , (A16)giving a descending recursion relation in k [ x N j +1 k ] = 0 , [ x k − j ] = A − j ([ F k − j ] − k [ x kj ]) . (A17)The recursion relation in Eq. (A17) fails when A j isnon-invertible, which occurs when any of the eigenvaluesin Eq. (A13) are zero ( j = − , , N p , φ p , ˜ b , or an introduction of non-zero ˜ β , which wedisallow due to the consequent shift of the singularity toa non-zero conformal time η . [1] Carlo R. Contaldi, Marco Peloso, Lev Kofman, and An-drei Linde. Suppressing the lower multipoles in the CMBanisotropies. J. Cosmology Astropart. Phys., 2003(7):002, Jul 2003. doi:10.1088/1475-7516/2003/07/002.[2] A. A. Starobinskii. Spectrum of relict gravitational radi-ation and the early state of the universe. ZhETF PismaRedaktsiiu , 30:719–723, December 1979.[3] Alan H. Guth. Inflationary universe: A possi-ble solution to the horizon and flatness problems. Phys. Rev. D, 23(2):347–356, January 1981. doi:10.1103/PhysRevD.23.347.[4] A. D. Linde. A new inflationary universe scenario: Apossible solution of the horizon, flatness, homogeneity,isotropy and primordial monopole problems.
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