What initial condition of inflation would suppress the large-scale CMB spectrum?
aa r X i v : . [ g r- q c ] J a n What initial condition of inflation would suppress the large-scale CMB spectrum?
Pisin Chen
1, 2, 3, 4, ∗ and Yu-Hsiang Lin
1, 2, 3, † Department of Physics, National Taiwan University, Taipei 10617, Taiwan Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 10617, Taiwan Kavli Institute for Particle Astrophysics and Cosmology,SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94305, USA Graduate Institute of Astrophysics, National Taiwan University, Taipei 10617, Taiwan (Dated: August 23, 2018)There is an apparent power deficit relative to the ΛCDM prediction of the CMB spectrum at largescales, which, though not yet statistically significant, persists from WMAP to Planck data. Proposalsthat invoke some form of initial condition for the inflation have been made to address this apparentpower suppression, albeit with conflicting conclusions. By studying the curvature perturbationsof a scalar field in the FLRW universe parameterized by the equation of state parameter w , wefind that the large-scale spectrum at the end of inflation reflects the super-horizon spectrum ofthe initial state. The large-scale spectrum is suppressed if the universe begins with the adiabaticvacuum in a super-inflation ( w < −
1) or positive-pressure ( w >
0) era. In the latter case, thereis however no causal mechanism to establish the initial adiabatic vacuum. On the other hand, aslong as the universe begins with the adiabatic vacuum in an era with − < w <
0, even if thereexists an intermediate positive-pressure era, the large-scale spectrum would be enhanced ratherthan suppressed. We further calculate the spectrum of a two-stage inflation model with a two-fieldpotential and show that the result agrees with that obtained from the ad hoc single-field analysis.
I. INTRODUCTION
The ΛCDM model of cosmology with an early infla-tionary era is very successful in explaining the cosmicmicrowave background (CMB) power spectrum. How-ever, it has been observed in the
COBE data that thequadrupole power is lower than the model prediction[1, 2]. This observation is further confirmed by
WMAP ,reporting the quadrupole power lower than the theoret-ical expectation by more than 1 σ but less than 2 σ [3].Although this stand-alone low quadrupole mode may beexplained by the cosmic variance, the Planck observationanalyzed the low- ℓ ( ℓ <
30) and high- ℓ ( ℓ ≥
30) spectraseparately, and showed that the best-fit amplitude for thelow- ℓ spectrum is 10% lower than that for the high- ℓ oneat 2.5–3 σ significance [4, 5]. There have been attempts to explain the low- ℓ powersuppression of the CMB by introducing some pre-inflation era that breaks the slow-roll condition at about60 e-folds before the end of inflation [6–15]. The ba-sic argument about how an era that deviates from theslow-roll dynamics could suppress the power is that theamplitude of the curvature perturbation, R ∼
Hδφ/ ˙ φ ,would decrease as | ˙ φ | increases. This scenario is first re-alized in the single-field chaotic inflation with potential V = m φ /
2, where m is the mass of the inflaton. If ∗ [email protected] † [email protected] This low- ℓ /high- ℓ tension is present even when the particularlylow quadrupole mode is excluded from the analysis [4]. In [5]it is further pointed out that the low- ℓ power deficit is mainlycaused by the low multipoles between ℓ = 20 and 30. the inflaton φ starts with large speed ˙ φ ≫ m φ , the ki-netic energy dominates the pre-inflation universe, and thepower at the horizon scale is suppressed [6]. Other sce-narios of violating the slow-roll evolution include the pre-inflation era filled with some radiation [7], the primor-dial black hole remnants [8], or the frustrated networkof topological defects [9]. There are also pre-inflationmodels in which the universe is dominated by the spa-tial curvature as the emergent property from a numberof moduli fields in the models of solid inflation [10, 16].All of the models above report power suppression at thelarge scales.On the other hand, the existence of the pre-inflationdecelerating era in models that predict multi-stage in-flation does not always result in power suppression [11–14, 17–20]. In the presence of two fields with mass hi-erarchy, there are two inflationary eras connected by adecelerating era. With the second inflationary era iden-tified as the last 60 e-folds of the inflation, it is shownthat the power is enhanced, rather than suppressed, atlarge scales that cross the horizon during the first infla-tionary era or the decelerating era [17]. Similar evolutionalso occurs in the early times of the hybrid inflation [21],in which the heavy field is played by the “waterfall field”who acquires the mass through the coupling to the infla-ton field. In this case, it is however inferred that whenthe coupling term dominates at the early times, the infla-ton field rolls faster due to the coupling, and eventuallyleads to the power suppression [6]. Among other modelsof multi-stage inflation which commonly have a deceler-ating era before the last inflationary era, some predictpower suppression at large scales [11, 12, 14], while somepredict enhancement [13, 17–20]. One therefore naturallywonders: What initial condition of inflation generated bythe pre-inflation era would actually suppress the CMBpower spectrum at large scales?In this work, we address the question through the fol-lowing steps. We first find the spectrum of the adiabaticvacuum in the universe with a constant equation of statedriven by a scalar field. The conditions of having theblue-tilted, red-tilted, or scale-invariant spectra at thesuper-horizon scales are found. The spectra obtained arebased on the assumption that the mode solutions ap-proach the Minkowski limit at small scales. We pointout that in the decelerating universe the super-horizonmodes would enter the horizon and become sub-horizon,which means that these super-horizon modes are initiallycausally disconnected. Such assumption in an initiallydecelerating universe therefore relies on the final state ofthe mode evolution to govern its initial state, which re-verses the cause and effect. Later it was shown that thelarge-scale power suppression in models with pre-inflationdecelerating era is actually a consequence of this unnat-ural yet widely adopted assumption. In the next step, we demonstrate that for the universeexperiencing several eras with different equations of state,the large-scale spectrum is determined by the earliest erain which the universe begins. Starting with a single slow-roll era with the scale-invariant super-horizon spectrum,we find how the spectrum changes as one incrementallystacks a kinetic era, and yet another slow-roll era, intothe early times. If the universe begins with the initialadiabatic vacuum in the kinetic era, the spectrum is sup-pressed at large scales, and we find that the suppressionis a direct consequence of the blue-tilted super-horizonspectrum in the initial kinetic era. With another slow-rollera preceding the kinetic era, the large-scale spectrumis enhanced because the super-horizon spectrum in theinitial slow-roll era is scale-invariant with the amplitudehigher than that generated in the second slow-roll era. Inthis case, the intermediate kinetic era only serves to con-nect the two scale-invariant spectra of the two slow-rolleras. One sees that the power suppression stems from theinitial blue-tilted super-horizon spectrum, and once theinitial spectrum is different, the large-scale power maynot be suppressed even if there is a pre-inflation kineticphase.We also investigate the scenario that the universestarts with a super-inflation era before the slow-roll infla-tion. The super-inflation era can be induced in theoriesof quantum gravity [15, 27–29], or by a scalar field that Such a choice of the initial state for inflation, often referred toas the Bunch-Davies vacuum, even if there exists a non-slow-rollpre-inflation phase, has been commonly assumed in the literature(see, for example, [6, 7, 9, 10, 13, 15, 18]). There also existnumerous proposals of a non-Bunch-Davies vacuum as the basisof the initial condition for a universe that does not begin with aslow-roll phase (see, for example, [22–25]). The effect due to piecewise changes of the model parameter wasstudied in [26], where the time-dependent effective inflaton masswas considered. violates the dominant energy condition [30, 31]. Modelsin the latter case generally suffer from quantum insta-bilities and should only be regarded as effective theories(for related discussions, see, for example, [32, 33]). Theinterest here lies in the fact that, opposite to the case ofa single pre-inflation kinetic era, it is causal to assumethe initial adiabatic vacuum in the pre-inflation super-inflation era. We found that in this case the large-scalepower is also suppressed due to the blue-tilted super-horizon initial spectrum in the super-inflation era. Powersuppression due to an early super-inflation era has alsobeen inferred in the models of loop quantum gravity [27]or bouncing cosmology [15], while in this work a moresystematic treatment to the evolution of perturbations isgiven.After understanding the character of the spectrum inthe multi-stage inflation using the ad hoc single-fieldanalysis, we calculate the spectrum of the curvature per-turbations in a two-field model with the given potential.We consider the chaotic potential with a coupling termto the second scalar field, which is similar to the effec-tive potential in the early stage of the hybrid inflation.By numerically solving the equations of motion and us-ing the
CAMB code [34, 35], we show that the large-scalespectra of curvature perturbations and CMB are indeedenhanced due to the initial inflationary era. The paper is organized as following. The super-horizonspectrum in the universe with constant equation of stateis derived in Sec. II. We demonstrate the transformationof the power spectrum in three different background evo-lutions in Sec. III. The curvature perturbation and CMBspectra of the two-field inflation are found in Sec. IV. Weconclude in Sec. V.
II. SCALING RELATION
To demonstrate the differences between the accelerat-ing eras and the decelerating ones on the mechanism ofgenerating the curvature perturbation spectrum, we an-alyze the single-field models with ad hoc actions in thegiven background evolutions. In this section, we first findthe single-field model that gives the background evolu-tion with a given w , and the corresponding general so-lutions to the curvature perturbations. We then discussthe common assumption on the small-scale behavior ofthe solution and its causal character. At the end we findthe power spectrum in the background with given w andits power-law relation with respect to the wavenumber k . As regards the treatment of the two-stage inflation, our approachis closest to that of [18, 19], in which a more complicated string-motivated two-field model is considered. In [17] the two fieldshave no direct coupling, and certain approximations are used toobtain the analytical solutions in various regimes of the modelparameters. In [11, 12, 14, 20], the single-field models are used.In [13] the system is also modeled by a single fluid.
A. Perturbations with constant equation of motion
We consider a scalar field φ in an expand-ing Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) uni-verse with the action S = Z d x √− gP ( X, φ ) , (2.1)where the kinetic term X = − ∂ µ φ∂ µ φ. (2.2)The signature of the metric is ( − , + , + , +), and we adoptthe Planck units throughout this paper. The energy-momentum tensor can be put into the form of the perfectfluid, T µν = P g µν + ( ρ + P ) u µ u ν , (2.3)by identifying P as the pressure, and the energy densityand the velocity as ρ =2 X ∂P∂X − P, (2.4) u µ = ∂ µ φ √ X . (2.5)The background evolution with constant equation ofmotion, − < w ≤
1, can be modeled by the Lagrangian, P = X − V ( φ ) , (2.6)where, in the homogeneous and isotropic background, X = ˙ φ / ≥
0, and assuming the potential V ≥ Thedots denote the time derivatives. If one only requires ρ = X + V to be positive and allows V to be negative,then one can have w > − X < V < In the conformal Newtonian gauge, the perturbedFLRW metric is ds = − (1 + 2Φ) dt + a ( t )(1 + 2Ψ) d x , (2.7) With (2.6), the evolution of constant w > − φ i and ˙ φ i can be realized by the ad hoc potential V ( φ ) = ˙ φ i (1 − w )2(1 + w ) exp hp π (1 + w ) ( φ − φ i ) i . A negative potential with w > w > where a is the scale factor, and Φ and Ψ are the metricperturbations. The equation of motion of the curvatureperturbation, R , is given by the Mukhanov-Sasaki equa-tion in the Fourier space [38, 39], R ′′ + 2 A ′ A R ′ + k R = 0 , (2.8)where R =Ψ − H ˙ φ δφ, (2.9) A = a √ ρ + PH , (2.10) k is the wavenumber, H = ˙ a/a is the Hubble parame-ter, δφ is the field perturbation. The primes denote thederivative with respect to the conformal time η , definedby dt = adη . Introducing the new variable u = − A R , wecan get rid of the first-derivative term, turning (2.8) into u ′′ + (cid:18) k − A ′′ A (cid:19) u = 0 . (2.11)If the evolution of the universe is described by a con-stant w > −
1, there is a simple relation A ′′ /A = a ′′ /a since A = r w )8 π a. (2.12)The scale factor evolves as a ( η ) = a i (1 + αξ ) /α , (2.13)where ξ = a i H i ( η − η i ), a i and H i denote the initialvalues at η = η i , and α = 1 + 3 w . (2.14)Equation (2.11) then reads d udξ + (cid:20) κ − β (1 + αξ ) (cid:21) u = 0 , (2.15)with κ = k/a i H i and β = (1 − w ) /
2. The generalsolution is u ( ξ ) = C M ,µ (cid:20) iκ (cid:18) ξ + 1 α (cid:19)(cid:21) + C W ,µ (cid:20) iκ (cid:18) ξ + 1 α (cid:19)(cid:21) , (2.16)in which M ν,µ ( z ) and W ν,µ ( z ) are the Whittaker func-tions, and µ = 32 (cid:12)(cid:12)(cid:12)(cid:12) − w w (cid:12)(cid:12)(cid:12)(cid:12) . (2.17) TABLE I. Corresponding values of α and µ for some ref-erence equation-of-state parameter w . The parameter α =(1 + 3 w ) / µ = | − w ) / w ) | describes the general solution ofthe perturbation through (2.16). Note that for the acceler-ating universe with w < − /
3, one has α <
0, while for thedecelerating universe with w > − /
3, one has α > w −∞ − − −
13 0 13 23 1 + ∞ α −∞ − −
12 0 12 1 32 2 + ∞ µ
12 32 52 + ∞
32 12 16 0 12
Note that w = − / Corresponding α and µ for some reference values of w are listed in TableI. If we try to model the slow-roll evolution by assigning w = −
1, we will end up with A = 0 and cannot proceedin the way we did in the previous paragraph. The wayaround that is to use the attractor solution of the slow-roll era. By writing the density and pressure in terms offield, we have A = − φ ′ H , (2.18)assuming φ ′ < H as well as ˙ φ = φ ′ /a are approximatelyconstant, so we can write A = − ˙ φ i H a, (2.19)which is proportional to a as it is in (2.12). Also it can beverified by solving the Friedmann equation with constant H that (2.13) reproduces the scale factor in the slow-rollcase, so the equation of motion (2.15) still holds. We willrefer to the slow-roll limit as w ≃ − w < −
1, wemodel it by the Lagrangian, P = − X − V ( φ ) , (2.20)with the sign of the kinetic term reversed. With therequirement ρ = − X + V >
0, one generally has
V > The super-horizon spectrum is asymptotically divergent for w = − / µ = 0). To understand why, first note that in this casethe universe does not accelerate nor decelerate (¨ a = 0), so thecomoving scale of Hubble horizon is constant in time. If w isslightly smaller than − /
3, the universe accelerates but slowly.It takes a long time for the horizon to shrink a little. At themeantime the amplitudes of the fluctuations inside the horizonkeep decaying, therefore the amplitude of the power spectrumchanges much within a small range of k . X >
0, and the scale factor still evolves as (2.13). Bysubstituting the original definition of A with A = a √− ρ − PH , (2.21)it turns out that the equation of motion of the curva-ture perturbation can still be written in the form of theMukhanov-Sasaki equation (2.8), and the rest of the anal-ysis follows.
B. Assumption and character of the small-scalesolution
The common assumption on the initial condition isthat the mode solution approaches the Minkowski limitin the short-wavelength limit, u ( η ) = 1(2 π ) / √ k e − ikη (for kη → ∞ ) . (2.22)The Whittaker function that matches this form when z ≫ W ,µ (2 iz ) = e − iz (cid:20) O (cid:18) z (cid:19)(cid:21) . (2.23)Therefore, the requirement of matching the adiabaticvacuum picks out the solution, u ( ξ ) = e iκ/α (2 π ) / √ a i H i κ W ,µ (cid:20) iκ (cid:18) ξ + 1 α (cid:19)(cid:21) . (2.24)If at the early times of the inflationary history, theuniverse is initially accelerating and evolves with a con-stant equation of state, then by assuming the adiabaticvacuum at the sub-horizon limit, we obtain the initialcondition (2.24). This sub-horizon initial condition, com-bined with the quasi-de Sitter expansion, predicts the ob-served nearly scale invariant super-horizon spectrum inthe ΛCDM universe. One of the reasons that make thisscenario attractive is that the quantum fluctuations welearn well in the local Minkowski spacetime, after beingstretched to the cosmic scale by inflation, also form theseed of the cosmic structure.When applying the limit (2.22) to a decelerating uni-verse, this sub-horizon assumption becomes acausal. Inthe decelerating universe, the Hubble horizon growsfaster than the perturbations do, just like in the late-time universe dominated by matter or radiation. Thesub-horizon spectrum is therefore formed after the super-horizon modes enter the horizon. The estimation wemake to the super-horizon spectrum is then based onthat, after the modes enter the horizon, they must fall inthe vacuum state at the small-scale limit. Through theanalysis in the next section, we will see that the large-scale power suppression caused by the pre-inflation ki-netic era actually originates from the super-horizon spec-trum deduced from this picture. C. Power spectrum
The power spectrum of R is defined through the ex-pectation value of ˆ R ( x , t ), h ˆ R ( x , t ) i = Z dkk P ( k ) . (2.25)Expanding R in terms of the creation and annihilationoperators, R ( x , t ) = Z d k h a k R k ( t ) e i k · x + a † k R ∗ k ( t ) e − i k · x i , (2.26)and using the commutator[ a k , a † k ′ ] = δ ( k − k ′ ) , (2.27)one finds that P ( k ) = 4 πk |R k | . (2.28)Recalling that R = − u/A , we find that, for w = − α = − P = H i κ | αξ | − /α π | α | (cid:12)(cid:12)(cid:12)(cid:12) W ,µ (cid:20) iκ (cid:18) ξ + 1 α (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . (2.29)For the slow-roll case, we approximate A by the slow-rolllimit (2.19), and the mode function in that limit is thengiven by (2.24) with w ≃ −
1. In terms of the slow-rollparameter, ǫ ≡ π (cid:18) V ∂V∂φ (cid:19) = 4 π ˙ φ H , (2.30)the power spectrum at the slow-roll limit is given by P = H i κ | αξ | − /α πǫ (cid:12)(cid:12)(cid:12)(cid:12) W , (cid:20) iκ (cid:18) ξ + 1 α (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . (2.31)Using the small-argument expansion of the Whittakerfunction [40], the super-horizon limits of the power spec-trum are found for different ranges of w . For w < − / w > w = −
1, one has P κ ≪ = H i Γ (2 µ ) π Γ ( µ + ) | α | (cid:12)(cid:12)(cid:12) α (cid:12)(cid:12)(cid:12) µ − κ − µ +3 . (2.32)For w ≃ −
1, the slow-roll power spectrum (2.31) recoversthe familiar scale-invariant spectrum P κ ≪ = H i πǫ . (2.33)For − / < w <
1, the super-horizon power spectrumdecays with time, given by P κ ≪ = H i Γ (2 µ ) π Γ ( µ + ) | α | (cid:12)(cid:12)(cid:12) α (cid:12)(cid:12)(cid:12) µ − κ − µ +3 T ( ξ ) , (2.34) with the time-dependence T ( ξ ) = | αξ | − − w ) / (1+3 w ) . (2.35)For w = 1, the spectrum is P κ ≪ = H i π κ T ( ξ ) , (2.36)with T ( ξ ) = (cid:12)(cid:12)(cid:12)(cid:12) ln (cid:20) iκ (cid:18) ξ + 1 α (cid:19)(cid:21) + γ − (cid:12)(cid:12)(cid:12)(cid:12) , (2.37)where γ is the Euler-Mascheroni constant.We can summarize the power-law relations of thesuper-horizon power spectrum with respect to the nor-malized wavenumber κ by the scaling relation P ∝ κ − µ +3 , (2.38)where the correspondence between the case of w ≃ − w . For w < − / w > P ∝ κ w ) / (1+3 w ) . (2.39)For − / < w < P ∝ κ w/ (1+3 w ) . (2.40)The super-horizon behavior of the spectrum can bedivided into three types according to the scaling relation.Some typical cases are plotted in FIG. 1. For µ = 3 /
2, thespectrum is scale-invariant. This is the case for w ≃ − w = 0. When µ < /
2, the spectrum isblue-tilted and the power is lower than the scale-invariantspectrum at super-horizon scales. This is attainable fromthe positive-pressure ( w >
0) or super-inflation era ( w < − µ > /
2, or equivalentlyan era with − < w < w = − / w > − /
3) isacausal.
III. EVOLUTION OF THE POWER SPECTRUM
The character of the large-scale spectrum at the endof inflation reflects the nature of the initial state. If at This relation is also derived in [41] as the approximation to thesuper-horizon spectrum at the end of the multi-stage inflationaryevolution. The authors focus on the recursive matrix formalismof the multi-stage pre-inflationary era, with the assumption thatevery pre-inflation era is an accelerating expansion (or deceler-ating contraction in the bounce inflation scenario).
Sub - horizonSuper - horizonw = - > - = - (cid:144) = = (cid:144) = = - - - - Κ l og @ Π H + Α L P (cid:144) H i D FIG. 1. The power spectra (2.29) with sample parameters w = − . − /
3, 0, 1 /
3, 1, 5, and at the slow-roll limit w ≃−
1. They are plotted with normalizations such that they havethe same magnitude at the sub-horizon limit. For w ≃ − πǫP/H i . blue - tilted H suppression L red - tilted H enhancement L s up e r - i n f l a ti on acce l e r a ti ng d ece l e r a ti ng n e g a ti v e po t e n ti a l - - - - - - - e xpon e n t FIG. 2. The plot of − µ + 3, the exponent of the power-lawscaling relation (2.38) with respect to the equation-of-stateparameter, w . the early times before the onset of inflation, the universeis initially described by some adiabatic vacuum in thebackground with constant equation of state, the powerspectrum is given by (2.29) during that era. As the uni-verse evolves with time, the power spectrum continuouslytransforms accordingly. The shape of the spectrum at thelong-wavelength limit is nevertheless not affected by theevolution, and is preserved in the late-time spectrum.We demonstrate in this section that, as a consequenceof the spectrum evolution, the spectra can be radicallydifferent at large scales for distinct initial vacua. Par-ticularly, if the universe transits from a kinetic era intothe inflation era, the large-scale spectrum is suppressedbecause of the initial adiabatic vacuum assumed in thekinetic era. If the initial vacuum is different—for ex-ample, changed by an earlier accelerating era before thekinetic era, as discussed in this section—the large-scalespectrum may even become enhanced. We compare the evolution of the power spectrum inthree cases, modeled phenomenologically by the singlefield dynamics. The first one is a single slow-roll era(denoted as era C) with Hubble parameter H C . Thesecond one is the slow-roll era (era C) preceded by akinetic era (era B). In the third case we add one moreslow-roll era (era A), with Hubble parameter H A , beforethe kinetic era (era B), which is again followed by theslow-roll era (era C). When applicable, the quantities atthe transition from era A to B are denoted by subscript1 (so, for example, the scale factor at the transition isequal to a ), and those at the transition from B to C areby subscript 2.In view of the acausal character of the adiabatic vac-uum in the initially decelerating era (the second case withonly era B and C), we also analyze the case of having asuper-inflation era (era S) before the slow-roll era (eraC), which also implies power suppression at large scalesbut is free from the acausal property. Analogously, thequantities at the transition from era S to C are denotedby subscript 2. A. Slow-roll
In the slow-roll era (era C), the general solution to themode function at the slow-roll limit w ≃ − u C = C + (cid:18) − i ˜ a C ˜ k C (cid:19) e − i ˜ k C / ˜ a C + C − (cid:18) i ˜ a C ˜ k C (cid:19) e i ˜ k C / ˜ a C , (3.1)where ˜ k C = k/a H and ˜ a C = a/a = [1 − a H ( η − η )] − . Here a and H can be viewed as quantitiesat some reference time, η . The notations are chosenfor the convenience of later comparison and should notcause confusion. Note that in the slow-roll era, one canapproximate H C ≈ H as a constant. The normalizedpower spectrum is given by˜ P C = ˜ k C ˜ a C (cid:12)(cid:12)(cid:12)(cid:12) ˜ C + (cid:18) − i ˜ a C ˜ k C (cid:19) e − i ˜ k C / ˜ a C + ˜ C − (cid:18) i ˜ a C ˜ k C (cid:19) e i ˜ k C / ˜ a C (cid:12)(cid:12)(cid:12)(cid:12) , (3.2)where ˜ C ± = √ a H C ± , and˜ P C = ǫP π H C , (3.3) ǫ = 4 π ˙ φ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C . (3.4)are the normalized power spectrum and the slow-roll pa-rameter evaluated in era C, respectively. - - k Ž C l og P C Ž FIG. 3. Time evolution of the power spectrum in era C in thecase of single-stage evolution (only era C). The solid curvesare the spectra of R from early time to late time (from light todark). The vertical dashed lines denote the comoving horizonsize at the corresponding instants (also from light to dark). In the adiabatic vacuum (2.22), only the second termin (3.1) remains, and the mode function reduces to u C = 1(2 π ) / √ k (cid:18) i ˜ a C ˜ k C (cid:19) e i ˜ k C / ˜ a C . (3.5)The power spectrum is˜ P C = 116 π k C ˜ a C ! , (3.6)which recovers the well-known form in the super-horizonlimit, P = 1 π (cid:18) H C ǫ (cid:19) . ( ˜ k C ≪ a C = 1 ) (3.7)The comoving horizon size decreases with time, movingtoward the right to the small scales in FIG. 3. After themode exits the horizon, lying on the left-hand side of thehorizon scale, the power stays scale-invariant. B. Kinetic—slow-roll
In the second case, the energy density is dominated bythe kinetic energy at η < η , with H ≈ − r π φ ′ a , (3.8)assuming φ ′ ≤ w = 1 and the mode function can be written in termsof the Hankel functions, u B = B + ˜ a B H (1)0 (cid:18)
12 ˜ k B ˜ a B (cid:19) + B − ˜ a B H (2)0 (cid:18)
12 ˜ k B ˜ a B (cid:19) , (3.9) where we denote ˜ k B = k/a H and ˜ a B = a/a = p a H ( η − η ). Similarly, for later convenience, wechoose η < η to be some reference time for era B. Thepower spectrum in the kinetic era is given by˜ P B = ˜ k B (cid:12)(cid:12)(cid:12)(cid:12) ˜ B + H (1)0 (cid:18)
12 ˜ k B ˜ a B (cid:19) + ˜ B − H (2)0 (cid:18)
12 ˜ k B ˜ a B (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (3.10)where ˜ B ± = √ a H B ± , and˜ P B = 3 P π H . (3.11)At η > η , the universe shifts into the slow-roll stage,and the solution is given by (3.1). To match the bound-ary between the eras, we fix the coefficients C + and C − by the continuity of R and R ′ . One finds˜ C + = e i ˜ k C k C nh ˜ k C H (1)0 ,C − (1 − i ˜ k C ) H (1)1 ,C i ˜ B + + h ˜ k C H (2)0 ,C − (1 − i ˜ k C ) H (2)1 ,C i ˜ B − o , (3.12)˜ C − = e − i ˜ k C k C nh ˜ k C H (1)0 ,C − (1 + i ˜ k C ) H (1)1 ,C i ˜ B + + h ˜ k C H (2)0 ,C − (1 + i ˜ k C ) H (2)1 ,C i ˜ B − o , (3.13)where H (1)0 ,C is the shorthand of H (1)0 (˜ k C /
2) and so on.The power spectrum in era C is given by (3.2) with ˜ C + and ˜ C − substituted by (3.12) and (3.13), respectively.If the universe is in the adiabatic vacuum in era B, onlythe second term in (3.9) presents in the mode function, u B = 18 π ˜ a B H (2)0 (cid:18)
12 ˜ k B ˜ a B (cid:19) , (3.14)where the normalization is chosen to recover the small-scale limit (2.22). In the kinetic era, the comoving hori-zon size increases with time, moving toward the left tothe large scale (FIG. 4). The power drops after the modeenters the horizon, as in the slow-roll era, but it alsodecreases at the super-horizon scales before the horizonentry. This is actually a salient feature of the adiabaticvacuum in the decelerating era with − / < w < - - - - - - - -
50 log k Ž B l og P B Ž FIG. 4. Time evolution of the power spectrum in era B in thecase of two-stage evolution (era B and C). The solid curvesare the spectra of R from early time to late time (from light todark). The vertical dashed lines denote the comoving horizonsize at the corresponding instants (also from light to dark). - - k Ž C l og P C Ž FIG. 5. Time evolution of the power spectrum in era C in thecase of two-stage evolution (era B and C). The solid curvesare the spectra of R from early time to late time (from light todark). The vertical dashed lines denote the comoving horizonsize at the corresponding instants (also from light to dark). kinetic era (cf. [6]). The other modes with shorter wave-lengths enter the horizon in era B, and exit the horizon inera C. They are scale-invariant outside the horizon, as inthe case of the single slow-roll scenario. This is becausethe adiabatic vacua approach the same short-wavelengthlimit (2.22) in either the kinetic era or the slow-roll era.Therefore, although the spectrum has a scale-invariantsegment due to the slow-roll era, the largest scales of thespectrum reveal the initial vacuum stemming from thekinetic era.The results we obtain crucially depend on the assump-tion about the initial vacuum of the universe. The predic-tion of power suppression is challenged by the fact thatit originates from the blue-tilted super-horizon initialspectrum in the pre-inflation decelerating era, in whichthe super-horizon modes have not been in causal con-tact throughout the history. We demonstrate this pointby an illustration showing the evolution of the Fourier Kinetic Slow - roll L CDM Present
Hubble radius
Wavelengths
ScalescausallyconnectedIR suppressionKink at IR suppressionUV scale - invariant NPhysicallength H log scale L FIG. 6. Illustration of the evolution of the physical wave-lengths of the modes and the Hubble radius with respect tothe number of e-fold, N . The three parallel straight linesdenote the modes with three different wavelengths, long toshort from top to bottom (color online). The correspondingfeatures they generate in the power spectrum, FIG. 5, arelabeled in the legends (top to bottom corresponding to longto short wavelengths). The black piecewise-connected linesdenote the Hubble radius evolving from the kinetic era to theslow-roll era, and finally into the ΛCDM era. The shaded re-gion denotes the scales within which are causally connected. wavelengths and the Hubble horizon (FIG. 6). In thedecelerating universe, such as the initial kinetic era, theHubble horizon grows faster than the Fourier wavelengthsdo. At the end of the initial decelerating era and the be-ginning of the accelerating inflation, the modes that areabout to enter the horizon—those who are the origin ofthe suppressed large-scale modes today—will soon be ex-panded and kept outside the horizon by inflation. If theuniverse starts with the decelerating era before inflation,these mode are then insulated from any sub-horizon dy-namics throughout the history. Therefore, the spectrumof the perturbations beyond the horizon size at the endof the decelerating era is not the consequence of causalphysics, and the common approach of deducing the ini-tial conditions through requiring the spectrum recoverthe Minkowski limit at the sub-horizon scale is therefore a posteriori . This is of the same footing as the “hori-zon problem” the big bang cosmology faced before thepicture of inflation was introduced. C. Slow-roll—kinetic—slow-roll
With the additional slow-roll era (era A) prependingthe kinetic era (era B), we simply apply solution (3.1) at η < η , u A = A + (cid:18) − i ˜ a A ˜ k A (cid:19) e − i ˜ k A / ˜ a A + A − (cid:18) i ˜ a A ˜ k A (cid:19) e i ˜ k A / ˜ a A , (3.15)where ˜ k A = k/a H and ˜ a A = a/a = [1 − a H ( η − η )] − . The mode function of the adiabatic vacuum inera A is u A = 1(2 π ) / √ k (cid:18) i ˜ a A ˜ k A (cid:19) e i ˜ k A / ˜ a A . (3.16)Matching the boundary between era A and B, one findsthe adiabatic vacuum in era A excites both modes of (3.9)in era B with coefficients˜ B + = − e i ˜ k B p π ˜ k B h ˜ k B H (2)0 ,B − (1 − i ˜ k B ) H (2)1 ,B i , (3.17)˜ B − = e i ˜ k B p π ˜ k B h ˜ k B H (1)0 ,B − (1 − i ˜ k B ) H (1)1 ,B i . (3.18)These results can be fed into (3.12) and (3.13), identifying H = H A as the Hubble constant in era A, and obtainthe initial coefficients ˜ C ± in era C. The power spectrumin era C is again given by (3.2) with the ˜ C ± found.With the initial vacuum in the slow-roll era (era A), inwhich the spectrum evolves in the same way as it doesin FIG. 3, the super-horizon spectrum in era B is scale-invariant (FIG. 7), different from the blue-tilted spec-trum of the adiabatic vacuum in era B (FIG. 4). More-over, after the modes enter the horizon, the power de-crease and the spectrum becomes red-tilted (FIG. 7), op-posite to the blue-tilted spectrum in era B without eraA (FIG. 4).In era C, there are three different types of history in-herited by the super-horizon modes (FIG. 8). The modeswith the longest wavelengths do not enter the horizon inera B and are kept outside of the horizon in era C. Theypreserve the scale-invariant spectrum as the proof of theexistence of the initial vacuum in a slow-roll era (era A).For the modes with shorter wavelengths that enter thehorizon in era B and exit the horizon in era C, the red-tilted shape of the spectrum persists, denting more as thepower decreases inside the horizon. The super-horizonmodes with shortest wavelengths exit the horizon for thefirst time in era C. These modes behave like the onesin era A, leaving the power scale-invariant as they exitthe horizon. Note the magnitude of the scale-invariantspectrum generated in era C is lower than that generatedin era A, since according to (3.7) the mode exiting thehorizon from the adiabatic vacuum acquires the powerthat is proportional to the Hubble expansion rate, whichis lower in era C.With the accelerating era A before the kinetic era B, allthe modes are initially sub-horizon at the early times andare causal connected (FIG. 9). However, although this - - - - - -
505 log k Ž B l og P B Ž FIG. 7. Time evolution of the power spectrum in era B inthe case of three-stage evolution (era A, B, and C). The solidcurves are the spectra of R from early time to late time (fromlight to dark). The vertical dashed lines denote the comovinghorizon size at the corresponding instants (also from light todark). - - k Ž C l og P C Ž FIG. 8. Time evolution of the power spectrum in era C inthe case of three-stage evolution (era A, B, and C). The solidcurves are the spectra of R from early time to late time (fromlight to dark). The vertical dashed lines denote the comovinghorizon size at the corresponding instants (also from light todark). scenario is free from the acausal initial conditions, the in-termediate kinetic era no longer leads to power suppres-sion at the large scales. The accelerating era precedingthe kinetic era changes the initial conditions, generatingthe the scale-invariant spectrum at the large scales. Thepower spectrum now has two scale-invariant segments:one generated in era A with larger power at the largerscales, and the other generated in era C with lower powerat the smaller scales. The intermediate kinetic era in thiscase generates the spectrum that connects the two scale-invariant segments (FIG. 8).We obtain three different large-scale behaviors fromthree different initial vacua and intermediate evolutions.Matching them with the inflation scenarios, we have thefollowing interpretations. In the first case, with a single0 Firstslow - roll KineticSecondslow - roll L CDMPresent
Hubble radius Wavelengths
ScalescausallyconnectedIR scale - invariantKink near IR plateauJoining segmentKink near UV plateauUV scale - invariant NPhysicallength H log scale L FIG. 9. Illustration of the evolution of the physical wave-lengths of the modes and the Hubble radius with respectto the number of e-fold, N . The five parallel straight linesdenote the modes with different wavelengths, long to shortfrom top to bottom (color online). The corresponding fea-tures they generate in the power spectrum, FIG. 8, are la-beled in the legends (top to bottom corresponding to long toshort wavelengths). The black piecewise-connected lines de-note the Hubble radius evolving from the first slow-roll era tothe kinetic era, then to the second slow-roll era, finally intothe ΛCDM era. The shaded region denotes the scales withinwhich are causally connected. era C, the super-horizon spectrum is scale-invariant. Thiscorresponds to the picture of inflation with large e-foldingnumbers (much larger than 60 e-folds). In the secondcase, the spectrum is suppressed at the large scales ifthe universe is in the adiabatic vacuum of era B beforethe onset of era C. This is the scenario of having a fast-roll era before the “just enough” inflation (with about50–60 e-folds). In the third case, if before the fast-rollera (era B) the universe is in another slow-roll era (eraA) at the early times, the spectrum is enhanced at thelarge scales. This corresponds to many supersymmetryor string motivated models that manifest the fast-roll eraas a transient between two slow-roll eras [11, 12, 20]. D. Super-inflation—slow-roll
Consider the case when η < η the universe is in thesuper-inflation era (era S), which is modeled by the La- grangian (2.20). With w < −
1, the mode function is u S = S + M ,µ i ˜ k S α ˜ a αS ! + S − W ,µ i ˜ k S α ˜ a αS ! , (3.19)where ˜ k S = k/a H , ˜ a S = a/a = [1+ αa H ( η − η )] /α ,and α and µ are given by (2.14) and (2.17), respectively.Here η < η also denotes the reference time for era S.The normalized power spectrum is˜ P S = ˜ k S ˜ a S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ S + M ,µ i ˜ k S α ˜ a αS ! + ˜ S − W ,µ i ˜ k S α ˜ a αS !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.20)where ˜ S ± = √ a H S ± and˜ P S = − (1 + α ) P S π H . (3.21)The universe goes into the slow-roll era (era C) at η , with mode function given by (3.1). Using the samematching conditions, the coefficients ˜ C ± are found to be˜ C + = √ ǫe − αN S e i ˜ k C p − (1 + α )˜ k C × n h k C + 2 i ˜ k C − i ˜ S + M ,µ + 2 h k C + 2 i ˜ k C − i ˜ S − W ,µ + α (1 − i ˜ k C )(2 µ + 1) ˜ S + M ,µ − α (1 − i ˜ k C ) ˜ S − W ,µ o , (3.22)˜ C − = √ ǫe − αN S e − i ˜ k C p − (1 + α )˜ k C × n − S + M ,µ − S − W ,µ + α (2 µ + 1)(1 + i ˜ k C ) ˜ S + M ,µ − α (1 + i ˜ k C ) ˜ S − W ,µ o , (3.23)with all Whittaker functions evaluated at 2 i ˜ k C /α . Theslow-roll parameter in era C is still given by ǫ , and N S isthe number of e-folds from η to η .The adiabatic vacuum (2.24) in era S corresponds tothe coefficients˜ S + =0 , (3.24)˜ S − = exp (cid:16) i ˜ k C α e − αN S (cid:17) (2 π ) / p k C e − αN S / , (3.25)1 - - - - - - k Ž S l og P S Ž FIG. 10. Time evolution of the power spectrum in era S inthe case of two-stage evolution (era S and C), with w = − . R from early time to latetime (from light to dark). The vertical dashed lines denotethe comoving horizon size at the corresponding instants (alsofrom light to dark). - - - - - k Ž C l og P C Ž FIG. 11. Time evolution of the power spectrum in era C inthe case of two-stage evolution (era S and C), with w = − . N S = 6, and ǫ = 0 .
1. The solid curves are the spectra of R from early time to late time (from light to dark). Thevertical dashed lines denote the comoving horizon size at thecorresponding instants (also from light to dark). where we have used the relation ˜ k S = ˜ k C e − αN S . Thepower spectrum in era S is plotted in FIG. 10, taking w = − . IV. TWO-FIELD CASCADE INFLATION
After establishing the understanding of the spectrumevolution in multi-stage inflation using the ad hoc single-field analysis, in this section we calculate the spectrumgenerated by a two-stage inflation model from the givenpotential. The purposes are to show that the evolutionpattern we obtain in the single-field analysis is also re-flected in the two-field dynamics, and to check whetherthe large-scale power is suppressed due to the couplingbetween the two fields.We investigate a simple two-field cascade inflation,which is driven by a heavy scalar field ψ and a lightscalar field φ with the action S = Z d x √− g (cid:20) − ∂ α φ∂ α φ − ∂ α ψ∂ α ψ − V ( φ, ψ ) (cid:21) . (4.1)In large field inflation, the field operates at the super-Plankian scale when the coefficient of the kinetic termis normalized to 1 /
2. If the field value is of the orderof the Planck mass, M P = 1 / √ G , the energy scale ofinflation is determined by the mass of the field. The typ-ical evolution can therefore be divided into four stages.The first stage is the inflationary era driven by the heavyfield, with fields starting far away from the potential min-imum and rolling down alone the hillside of the potential.At the second stage the heavy field falls into the poten-tial minimum and oscillates with damping amplitude. Asthe energy drops to the scale of the light field mass, theuniverse enters the third stage in which the light fieldinitiates the other inflation era. Finally at the fourthstage the light field decays, ending the inflation, and thestandard ΛCDM evolution begins.Consider the cascade inflation realized by the potential V ( φ, ψ ) = λ ′ φ ψ + 12 m φ . (4.2)The heavy-field inflation is driven by the coupling term λ ′ φ ψ /
2, and the light-field inflation is driven by themass term m φ /
2. At the stage of heavy-field inflation,the fields follow the attractor solutions, which can beobtained through expressing the equations of motion interms of the number of e-folds, N ≡ ln( a/a i ), where a i is some initial scale factor. The Friedmann equation is H = π V − π (cid:20)(cid:16) dφdN (cid:17) + (cid:16) dψdN (cid:17) (cid:21) , (4.3)where H = ˙ a/a . The dots denote the derivatives withrespect to t . The equations of motion can be casted into2the form d φdN − π "(cid:18) dφdN (cid:19) + (cid:18) dψdN (cid:19) − π × (cid:18) dφdN + 18 πV ∂V∂φ (cid:19) = 0 , (4.4) d ψdN − π "(cid:18) dφdN (cid:19) + (cid:18) dψdN (cid:19) − π × (cid:18) dψdN + 18 πV ∂V∂ψ (cid:19) = 0 . (4.5)It can be shown that the attractor solutions satisfy dφdN = − πV ∂V∂φ , (4.6) dψdN = − πV ∂V∂ψ , (4.7)provided ( dφ/dN ) +( dψ/dN ) − / π < λ ′ φ ψ /
2, theattractor solutions are φ ( N ) = r φ i − π ( N − N i ) , (4.8) ψ ( N ) = r ψ i − π ( N − N i ) , (4.9)where the subscripts i denote the initial values.At the second stage, ψ exits the slow-roll regime andoscillates at the minimum of the potential with its ampli-tude damped with time. The potential is still dominatedby the coupling term before the next inflation begins.During the oscillatory stage, φ remains slow roll, while ψ acquires a large kinetic energy that is of the same order ofthe potential energy, ˙ ψ ∼ λ ′ φ ψ ∼ H , and the energydensity evolves effectively according to w = 0 (zero pres-sure). Ignoring the kinetic energy of φ in the Friedmannequation, we have (cid:18) dψdN (cid:19) + λ ′ φ H ψ = 34 π . (4.10)With A ≡ H/ √ λ ′ φ , we parametrize ψ and dψ/dN by theamplitude A and the phase θ , dψdN = r π cos θ, (4.11) ψ = r π A sin θ. (4.12)Combining (4.5) and (4.10), we obtain a set of differentialequations of A and θdAdN = − A cos θ (cid:18) φ dφdN (cid:19) , (4.13) dθdN = 1 A + cos θ sin θ (cid:18) φ dφdN (cid:19) . (4.14) Among the two terms contributing to the frequency dθ/dN , the first term 1 /A is of the order of 1 /ψ , since A = H/ √ λ ′ φ ≈ √ λ ′ φψ/ √ λ ′ φ = ψ . The second termis of order unity as φ slow rolls. After ψ drops belowthe Planck mass, 1 /A dominates and θ oscillates rapidly,so we can approximate cos θ in (4.13) by 1 /
2. With φ given by the slow-roll solution (4.8), A and θ can then beintegrated to yield A ( N ) = A i (cid:18) − N − N i πφ i (cid:19) − / e − ( N − N i ) , (4.15) θ ( N ) = θ i + N − N i A i − ( N − N i ) πφ i A i . (4.16)As the field ψ attenuates, gradually the energy densityfrom the coupling term is taken over by the m φ / φ .Here we consider the perturbation to the homogeneousbackground in the conformal Newtonian gauge. The fieldperturbations are φ ( x , t ) = ¯ φ ( t ) + δφ ( x , t ) , (4.17) ψ ( x , t ) = ¯ ψ ( t ) + δψ ( x , t ) , (4.18)where ¯ φ ( t ) and ¯ ψ ( t ) denote the background solutions.The equations of motion of the perturbations in theFourier space are δ ¨ φ + 3 Hδ ˙ φ + (cid:18) k a + ∂ V∂φ (cid:19) δφ = − ∂ V∂φ∂ψ δψ − ∂V∂φ Ψ + 4 ˙¯ φ ˙Ψ , (4.19) δ ¨ ψ + 3 Hδ ˙ ψ + (cid:18) k a + ∂ V∂ψ (cid:19) δψ = − ∂ V∂φ∂ψ δφ − ∂V∂ψ Ψ + 4 ˙¯ ψ ˙Ψ , (4.20)˙Ψ + H Ψ = 4 π ( ˙¯ φδφ + ˙¯ ψδψ ) . (4.21)Note that the energy-momentum tensor of action (4.1)has no velocity perturbations either anisotropic inertia,so the Einstein equation gives Φ = Ψ. The curvatureperturbation in two-field system is R = − Ψ − H ˙¯ φδφ + ˙¯ ψδψ ˙¯ φ + ˙¯ ψ . (4.22)The initial conditions for the sub-horizon modes are setin the era of the heavy-field inflation by the method of it-eration. We first neglect the metric perturbations Ψ andfind the solutions to δφ and δψ at the short-wavelengthlimit. The initial conditions for the sub-horizon field per-turbations are set as the normalized positive-frequencysolutions. We then feed the initial δφ and δψ back into3the Einstein equations and obtain the initial conditionsof Ψ. At the end we perform the consistency check to seewhether the initial Ψ obtained are indeed much smallerthan δφ and δψ at the short-wavelength limit.To solve the field perturbations, first note that theequations of motion (4.19) and (4.20) can be put intoa simpler form by substitute t by the conformal time η ,and introducing the new variables w = aδφ and q = aδψ .Neglecting the metric perturbation, the equations for w and q are w ′′ + (cid:18) k + ∂ V∂φ − a ′′ a (cid:19) w = − ∂ V∂φ∂ψ a q, (4.23) q ′′ + (cid:18) k + ∂ V∂ψ − a ′′ a (cid:19) q = − ∂ V∂φ∂ψ a w, (4.24)where the primes denote the derivatives with respect tothe conformal time η . For k ≫ aH the two equationsdecouple and reduce to the equations of harmonic oscil-lators w ′′ + k w = 0 , (4.25) q ′′ + k q = 0 . (4.26)After the quantization, the normalized positive-frequencymode functions are δφ ( η ) = e − ikη (2 π ) / √ k a , (4.27) δψ ( η ) = e − ikη (2 π ) / √ k a , (4.28)which are the initial conditions for the sub-horizon fieldperturbations. Feeding (4.27) and (4.28) into the Ein-stein equationΨ = 1˙¯ φ + ˙¯ ψ − k πa h ˙¯ φδ ˙ φ + ˙¯ ψδ ˙ ψ + (cid:18) H ˙¯ φ + ∂V∂φ (cid:19) δφ + (cid:18) H ˙¯ ψ + ∂V∂ψ (cid:19) δψ (cid:21) , (4.29)one obtains the initial conditions for the sub-horizon met-ric perturbations.To justify that the metric perturbations are negligiblewhen finding solutions to the field perturbations, in thisparagraph we are going to show that Ψ obtained by (4.29)is much smaller than δφ and δψ at the short-wavelengthlimit in the era of heavy-field inflation. The followingdiscussion in this paragraph assumes k ≫ aH . Firstnote that from (4.8) and (4.9) one has ˙¯ φ ∼ H/ ¯ φ and˙¯ ψ ∼ H/ ¯ ψ . The metric perturbation (4.29) therefore goeslikeΨ ∼ (cid:0) kaH (cid:1) " δ ˙ φH ¯ φ + δ ˙ ψH ¯ ψ + (cid:18) φ + λ ′ ¯ φ ¯ ψ H (cid:19) δφ + (cid:18) ψ + λ ′ ¯ φ ¯ ψH (cid:19) δψ (cid:21) , (4.30) where we have omitted all the constant coefficients andused the fact that the potential is dominated by λ ′ φ ψ / δ ˙ φ = − Hδφ (cid:18) ikaH (cid:19) ∼ ka δφ (4.31)and similarly for δ ˙ ψ . The first two terms in the bracketof (4.30) then go like ( k/aH ) · ( δφ/ ¯ φ ) and ( k/aH ) · ( δψ/ ¯ ψ ). Using the Friedmann equation we know that λ ′ ¯ φ ¯ ψ /H ∼ / ¯ φ and λ ′ ¯ φ ¯ ψ/H ∼ / ¯ ψ , therefore thelast two terms in the bracket of (4.30) go like δφ/ ¯ φ and δψ/ ¯ ψ . Since during the heavy-field inflation ¯ φ and ¯ ψ isof order O (1) in Planck units, one hasΨ ∼ (cid:0) kaH (cid:1) (cid:18) δφ ¯ φ + δψ ¯ ψ (cid:19) . (4.32)Therefore the initial Ψ is indeed much smaller than theinitial δφ and δψ at the limit of k ≫ aH in the era ofheavy-field inflation.The CMB spectrum is found by three steps of numer-ical calculations. We first obtain the background dy-namics by solving (4.4) and (4.5). The reheating energyscale is assumed to be 7 . × − M P ∼ GeV. Theevolution of perturbations is then solved by integrating(4.19), (4.20), and (4.21), with the initial conditions setby (4.27), (4.28), and (4.29). The spectrum of the cur-vature perturbations is found by evolving each Fouriermode until it reaches the steady value after the hori-zon exit. The resulting spectrum is then fed into
CAMB [34, 35], which is modified to accept arbitrary initial spec-trum represented by an interpolating function, to calcu-late the spectrum of the CMB temperature fluctuations.There are two parameters and four initial conditionsin our model. The two parameters are the coupling con-stant, λ ′ , and the mass of the light field, m . The fourinitial conditions are the initial values and derivatives ofthe fields φ and ψ . We assume that the light field drivesabout the last 60 e-folds of inflation, so the mass m isdetermined by the Hubble scale during inflation deducedby the observation. The initial derivatives of φ and ψ are set to zero, leaving the system released from rest andevolving into the attractor solutions.The initial value of the light field φ determines the be-havior of the system in two ways. First, it determinesthe energy scale of the heavy-field inflation as well asthat of the oscillatory period, since at the beginning ofthe oscillatory stage ψ ∼ M P and H ∼ √ λ ′ φ . Sec-ond, it determines the number of e-folds of the light-field inflation. With larger initial φ , the light-field in-flation begins at a higher energy scale and lasts longer.In this case, only those modes with larger wavelengthsthat are affected by the heavy-field inflation and the os-cillatory stage (FIG. 12). Therefore, less deviations aremanifested in the present-day CMB spectrum since thoselarge modes have not entered the Hubble horizon today(FIG. 13).4 Φ i = Φ i = Φ i = - - - - - - - - - - - - H k (cid:144) Mpc - L l og P FIG. 12. The spectra of the curvature perturbations with φ i = 3 . M P , 3 . M P , and 3 . M P . The other parametersare held fixed as λ ′ = 10 − , ψ i = 1 . M P , and m = 1 . × − M P . L CDM best fit Φ i = Φ i = Φ i = H { L { H { + L C { TT (cid:144) H Π L @ Μ K D FIG. 13. The CMB temperature-temperature correlation(TT) spectra with φ i = 3 . M P , 3 . M P , and 3 . M P . Thedots with error bars are the Planck 2013 data. The otherparameters are held fixed as λ ′ = 10 − , ψ i = 1 . M P , and m = 1 . × − M P . Raising the value of initial ψ affects the spectrum bymaking the heavy-field inflation longer and, while holdingthe initial φ fixed, the light-field inflation shorter, withoutchanging the duration of the transition era. The largerthe initial ψ is, more visible k modes and ℓ modes areaffected (FIG. 14).The coupling λ ′ determines the strength of the interac-tion between the two fields. With stronger interactions,it takes longer for the transient oscillation to settle, andtherefore more modes with short wavelengths are affectedwhile other conditions held fixed (FIG. 15).We see from both the spectra of the curvature pertur-bations (FIG. 12) and CMB (FIG. 13) that the large-scalespectrum is determined by the initial era with which theuniverse begins: The large-scale spectrum reflects thescale-invariant shape of the super-horizon spectrum inthe initial slow-roll (heavy-field inflation) era. Althoughthe oscillatory stage behaves like w = 0, the spectrum L CDM best fit Ψ i = Ψ i = Ψ i = H { L { H { + L C { TT (cid:144) H Π L @ Μ K D FIG. 14. The CMB TT spectra with ψ i = 1 . M P , 1 . M P ,and 1 . M P . The dots with error bars are the Planck 2013data. The other parameters are held fixed as λ ′ = 10 − , φ i = 3 . M P , and m = 1 . × − M P . L CDM best fit Λ ' = ´ - Λ ' = - Λ ' = -
10 100 10000100020003000400050006000 Multipole Moment H { L { H { + L C { TT (cid:144) H Π L @ Μ K D FIG. 15. The CMB TT spectra with λ ′ = 10 − , 10 − , and3 × − . The dots with error bars are the Planck 2013 data.The other parameters are held fixed as φ i = 3 . M P , ψ i =1 . M P , and m = 1 . × − M P . evolution is qualitatively similar to that shown in Sub-sec. III C. A quantitative analysis of the spectrum evolu-tion involving the zero-pressure ( w = 0) era is given inthe Appendix, showing that the ad hoc single-field anal-ysis does capture the pattern of the spectrum evolutionand is in agreement with the numerical results. V. CONCLUSIONS
We show that if the universe begins in the super-inflation era ( w < −
1) or that with positive pressure( w > w >
0) or super-inflation ( w < −
1) era, and5red-tilted for the era with − < w <
0, except the sin-gular case with w = − /
3. In the slow-roll ( w ≃ − w = 0) background, the super-horizonspectrum is scale-invariant. We also point out that theconclusions are drawn from assuming the mode func-tion approaches the Minkowski limit at small scales. Al-though being natural in the accelerating universe, thisassumption becomes a posteriori in the decelerating uni-verse since the sub-horizon modes are evolved from thesuper-horizon modes, which are initially across causallydisconnected regions before entering the horizon.By analyzing three scenarios: a single slow-roll era, aslow-roll era preceded by a kinetic era, and two successiveslow-roll eras connected by a kinetic era, we show the fol-lowing two facts. First, the large-scale power suppressionin the model with a single pre-inflation kinetic era stemsfrom the blue-tilted super-horizon spectrum of the ini-tial kinetic era. Second, the additional slow-roll era pre-ceding the kinetic era changes the super-horizon initialspectrum, so the large-scale power is enhanced, ratherthan suppressed. These results show that the large-scalespectrum depends sensitively on the initial vacuum. Inthe universe beginning with the positive-pressure era, aswe pointed out earlier, the super-horizon modes are ini-tially across causally disconnected regions, and the well-motivated assumption on the initial state is still lacking.Some investigations about the effect of the different ini-tial vacuum on the spectrum have been carried out in theliterature [22–25]. We also explore the case that is freefrom the acausal issue: a super-inflation era precedingthe slow-roll era, and show that the large-scale spectrumis suppressed due to the initially blue-tilted spectrum inthe super-inflation era.We calculate the curvature perturbation and CMBspectra of a two-stage inflation model from the given two-field potential. We show that the large-scale power is en-hanced due to the initial spectrum set in the first acceler-ating era, and the effect of the intermediate deceleratingera on the spectrum is connecting the two plateaus gen-erated in the two accelerating eras, which agrees with thepicture obtained through the ad hoc single-field analysis. ACKNOWLEDGMENTS
We are grateful for the discussions with F. Arroja,S. Downes, C. Gauthier, J. Gu, K. Izumi, L. Labun,A. Linde, A. Mazumdar, T. Qiu, M. Sasaki, L. Sen-atore, T. Suyama, and D. Yeom. Y. L. also wantto thank KIPAC, SLAC National Accelerator Labora-tory and SITP, Stanford University for the hospitality.P. C. and Y. L. are supported by Taiwan National Sci-ence Council under Project No. NSC 97-2112-M-002-026-MY3 and by Taiwan National Center for Theoret-ical Sciences (NCTS). P. C. is in addition supportedby U.S. Department of Energy under Contract No. DE-AC03-76SF00515. Y. L. is in addition supported by theGraduate Students Study Abroad Program No. 104-2917- I-002-008 sponsored by Taiwan Ministry of Science andTechnology.
Appendix: Spectrum evolution involving azero-pressure era
Consider the universe consists of three successive eras:a first slow-roll era (era A), an intermediate zero-pressure( w = 0) era (era B), and a second slow-roll era (era C).Much parallel to the discussion in Subsec. III C, we de-note the quantities at the transitions from era A to Band from era B to C by subscripts 1 and 2, respectively.In era A, the mode function of the initial adiabaticvacuum is given by (3.16). In era B, we first keep thediscussion general for w > −
1. The mode function u isgiven by (2.16), u B = B + M ,µ i ˜ k B α ˜ a αB ! + B − W ,µ i ˜ k B α ˜ a αB ! , (A.1)where ˜ k B = k/a H and ˜ a B = a/a = [1 + αa H ( η − η )] /α . Parameters α and µ are given by (2.14) and(2.17), respectively. The normalized power spectrum is˜ P B = ˜ k B ˜ a B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ B + M ,µ i ˜ k B α ˜ a αB ! + ˜ B − W ,µ i ˜ k B α ˜ a αB !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A.2)where ˜ B ± = √ a H B ± and˜ P B = (1 + α ) P B π H . (A.3)In era C, the mode function is (3.1), and the normalizedpower spectrum is (3.2). For simplicity, we assume thatthe slow-roll parameter ǫ in era A and C has the samevalue.By matching at the boundary η = η , we obtain˜ B + =∆ B h ( − i ˜ k B + 2˜ k B + i ) W ,µ + α (˜ k B + i ) W ,µ i , (A.4)˜ B − =∆ B h − ( − i ˜ k B + 2˜ k B + i ) M ,µ + 2 µ + 12 α (˜ k B + i ) M ,µ (cid:21) , (A.5)where∆ B = √ α + 1 e i ˜ k B π ˜ k B ) / α √ ǫ × µ + 1) M ,µ W ,µ + 2 M ,µ W ,µ , (A.6)6and the Whittaker functions are evaluated at 2 i ˜ k B /α .Matching at η = η gives˜ C + = √ ǫe − αN B e i ˜ k C √ α ˜ k C × n h k C + 2 i ˜ k C − i ˜ B + M ,µ + 2 h k C + 2 i ˜ k C − i ˜ B − W ,µ + α (1 − i ˜ k C )(2 µ + 1) ˜ B + M ,µ − α (1 − i ˜ k C ) ˜ B − W ,µ o , (A.7)˜ C − = √ ǫe − αN B e − i ˜ k C √ α ˜ k C × n − B + M ,µ − B − W ,µ + α (2 µ + 1)(1 + i ˜ k C ) ˜ B + M ,µ − α (1 + i ˜ k C ) ˜ B − W ,µ o , (A.8) with the Whittaker functions evaluated at 2 i ˜ k C /α . Onealso has the relation ˜ k B = ˜ k C e − αN B .Setting w = 0 in era B, we find the spectrum evolu-tion as FIG. 16. Similar to the case of the intermediatekinetic era in Subsec. III C, the super-horizon modes in-herits the scale-invariant spectrum in the initial slow-rollera (era A), and the modes that enter the horizon duringthe zero-pressure era (era B) have the red-tilted spec-trum. Entering into the second slow-roll era (era C), asshown in FIG. 17, the modes that enter the horizon in eraB are expelled out of the horizon again, leading to thesteep red-tilted spectrum connecting the two plateaus.The right plateau is formed by the modes that are sub-horizon during eras A and B and then exit the horizon inera C. Comparing to the spectrum of the two-field model(FIG. 12), which also has a zero-pressure era between twoinflationary eras, we see that the ad hoc single-field anal-ysis agrees with the numerical result and largely capturesthe physics of the spectrum formation. [1] G. F. Smoot et al. , Astrophys. J. , L1 (1992).[2] M. Tegmark, Astrophys. J. , L35 (1996).[3] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik,G. Hinshaw, N. Odegard, K. M. Smith, R. S. Hill,B. Gold, M. Halpern, E. Komatsu, M. R. Nolta, L. Page,D. N. Spergel, E. Wollack, J. Dunkley, A. Kogut,M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright,Astrophys. J. Suppl. Ser. , 20 (2013).[4] Planck Collaboration, Astron. Astrophys. , A15 (2014).[5] Planck Collaboration, arXiv:1507.02704 .[6] C. R. Contaldi, M. Peloso, L. Kofman, and A. Linde,J. Cosmol. Astropart. Phys. , 002 (2003).[7] I.-C. Wang and K.-W. Ng,Phys. Rev. D , 083501 (2008).[8] F. Scardigli, C. Gruber, and P. Chen,Phys. Rev. D , 063507 (2011).[9] M. Bouhmadi-L´opez, P. Chen, Y.-C. Huang, and Y.-H.Lin, Phys. Rev. D , 103513 (2013).[10] S. Kouwn, O.-K. Kwon, and P. Oh,Phys. Rev. D , 063521 (2015).[11] R. K. Jain, P. Chingangbam, J.-O. Gong,L. Sriramkumar, and T. Souradeep,J. Cosmol. Astropart. Phys. , 009 (2009).[12] E. Dudas, N. Kitazawa, S. Patil, and A. Sagnotti,Journal of Cosmology and Astroparticle Physics , 012 (2012).[13] M. H. Namjoo, H. Firouzjahi, and M. Sasaki,arXiv:1207.3638 (2012).[14] J. White, Y.-l. Zhang, and M. Sasaki,Phys. Rev. D , 083517 (2014).[15] T. Biswas and A. Mazumdar,Classical and Quantum Gravity , 025019 (2014).[16] S. Endlich, A. Nicolis, and J. Wang,J. Cosmol. Astropart. Phys. , 011 (2013).[17] D. Polarski and A. Starobinsky, Nucl. Phys. B , 623(1992). - - - - - - -
20 log k Ž B l og P B Ž FIG. 16. Time evolution of the power spectrum in era Bin the case of three-stage evolution (era A, B, and C), inwhich w = 0 in era B. The solid curves are the spectra of R from early time to late time (from light to dark). Thevertical dashed lines denote the comoving horizon size at thecorresponding instants (also from light to dark).[18] A. Ashoorioon and A. Krause, arXiv:hep-th/0607001 .[19] A. Ashoorioon, A. Krause, and K. Turzynski,J. Cosmol. Astropart. Phys. , 014 (2009).[20] D. Yamauchi, A. Linde, A. Naruko, M. Sasaki, andT. Tanaka, Phys. Rev. D , 043513 (2011).[21] A. Linde, Phys. Rev. D , 748 (1994).[22] L. Sriramkumar and T. Padmanabhan,Phys. Rev. D , 103512 (2005).[23] D. Boyanovsky, H. J. de Vega, and N. G. Sanchez,Phys. Rev. D , 123006 (2006).[24] R. Holman and A. J. Tolley,Journal of Cosmology and Astroparticle Physics , 001 (2008). - - k Ž C l og P C Ž FIG. 17. Time evolution of the power spectrum in era Cin the case of three-stage evolution (era A, B, and C), inwhich w = 0 in era B. The solid curves are the spectra of R from early time to late time (from light to dark). Thevertical dashed lines denote the comoving horizon size at thecorresponding instants (also from light to dark).[25] I. Agullo and L. Parker, Phys. Rev. D , 063526 (2011).[26] V. Mukhanov and M. Zelnihov,Physics Letters B , 169 (1991).[27] S. Tsujikawa, P. Singh, and R. Maartens, Classical and Quantum Gravity , 5767 (2004).[28] A. Ashtekar and D. Sloan,Physics Letters B , 108 (2010).[29] G. Dom`enech and M. Sasaki,Journal of Cosmology and Astroparticle Physics , 022 (2015).[30] Y.-S. Piao and Y.-Z. Zhang,Phys. Rev. D , 063513 (2004).[31] M. Baldi, F. Finelli, and S. Matarrese,Phys. Rev. D , 083504 (2005).[32] S. M. Carroll, M. Hoffman, and M. Trodden,Phys. Rev. D , 023509 (2003).[33] J. M. Cline, S. Jeon, and G. D. Moore,Phys. Rev. D , 043543 (2004).[34] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. , 473 (2000).[35] A. Lewis and A. Challinor, http://camb.info/ .[36] P. J. Steinhardt and N. Turok,Phys. Rev. D , 126003 (2002).[37] D. Battefeld and P. Peter,Physics Reports , 1 (2015).[38] V. Mukhanov, JETP Lett. , 493 (1985).[39] M. Sasaki, Progress of Theoretical Physics , 1036 (1986).[40] F. Olver, D. Lozier, R. Boisvert, and C. Clark, NISTHandbook of Mathematical Functions (National Instituteof Standards and Technology and Cambridge UniversityPress, Cambridge, England, 2010).[41] Y. Cai, Y.-T. Wang, and Y.-S. Piao,Phys. Rev. D92