CMB Spectral µ-Distortion of Multiple Inflation Scenario

Gimin Bae,1 Sungjae Bae,1 Seungho Choe,1, ∗ Seo Hyun Lee,1 Jungwon Lim,1 and Heeseung Zoe1, †

1School of Undergraduate Studies, College of Transdisciplinary Studies,

Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988, Republic of Korea

(Dated: May 8, 2018)

Abstract

In multiple inflation scenario having two inflations with an intermediate matter-dominated phase, the

power spectrum is estimated to be enhanced on scales smaller than the horizon size at the beginning of

the second inflation, k > kb. We require kb > 10 Mpc−1 to make sure that the enhanced power spectrum

is consistent with large scale observation of cosmic microwave background (CMB). We consider the CMB

spectral distortions generated by the dissipation of acoustic waves to constrain the power spectrum. The

µ-distortion value can be 102 times larger than the expectation of the standard ΛCDM model (µΛCDM '

2 × 10−8) for kb . 103 Mpc−1, while the y-distortion is hardly affected by the enhancement of the power

spectrum.

∗ schoe@dgist.ac.kr† heezoe@dgist.ac.kr

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I. INTRODUCTION

Inflation provides the seeds of statistical fluctuations for the structure formation of the universe

[1–5] It fits with the large scale observations of the cosmic microwave background (CMB) and

large scale structure (LSS) such as the Wilkinson Microwave Anisotropy Probe (WMAP) [6], the

Planck [7] and Sloan Digital Sky Survey (SDSS) [8]. However, there are many inflation models

consistent with those large scale observations, and we should develop proper methods to specify

the primordial inflation. One of the possible ways should be probing inflationary power spectrum

on small scales by the observations such as ultracompact minihalos [9, 10], primordial black holes

[11, 12], the lensing dispersion of SNIa [13–15], the 21cm hydrogen line at or prior to the epoch of

reionization [16, 17] or CMB distortions [18–21].

Multiple inflation scenario, having more than one inflationary periods after the first inflation,

could leave characteristic signatures on small scales. Since the double inflation model, or infla-

tion with a break, was introduced to give decoupling the power spectrum on large (CMB) and

small (cluster-cluster/galaxy-galaxy) scales [22–24], many versions of multiple inflation have been

suggested as theoretical possibilities in supersymmetric particle physics models [25–36].

CMB spectral distortions are a useful technique to probe small scales at k . 104 Mpc−1 [18–

21]. COBE/FIRAS measurements indicate that the CMB photons are subject to the blackbody

spectrum of temperature Tγ = 2.726 ± 0.001K with the spectral distortions ∆I/I . 5 × 10−5

[37, 38]. However, we have many astrophyscial/cosmological sources inducing spectral distortions:

decaying or annihilating particles [39–43], reionization and structure formation [44–51], primordial

black holes [52–54], cosmic strings [55–58], small-scale magnetic fields [59–62], the adiabatic cooling

of matter [19, 63], cosmological recombination [64–69], gravitino decay [70], and the dissipation of

primordial density perturbations [20, 21, 71–81] which is the main concern of this paper. Types of

spectral distortions are characterized by the redshifts at which energy releases to CMB photons:

At z � 2 × 106, Compton and double Compton scatterings and Bremsstrahlung can effectively

thermalize the energy release maintaining a black body spectrum [45, 82, 83]. At 3×105 . z . 2×

106, only Compton scattering can efficiently redistribute the energy injected to the CMB. Compton

scattering keeps the electrons and photons in kinetic equilibrium forming a chemical potential µ.

We define µ-distortion as the spectral distortions associated with this chemical potential [82]. At

z . 104, Compton scattering becomes inefficient, and the photons diffuse only a little in energy.

We define y-distortion as the spectral distortion caused by the energy release at this epoch. It is

also connected with the Sunyaev-Zeldovich effect on galaxy clusters [84].

2

In this paper, we focus on multiple inflation with an intermediate matter-dominated period

(See Figure 1). We calculate the power spectrum, and predict the spectral distortion due to the

dissipation of the density perturbations. In [34, 85], the power spectrum with suppression on small

scales is discussed, and its implications on the spectral distortions are already mentioned. However,

the power spectrum of multiple inflation with a matter-dominated break turns out to be enhanced

on small scales, and its pattern of the spectral distortions should be different from them.

We discuss the evolution of the curvature perturbations generated by the multiple inflation

scenario with an intermediate matter domination in Section II. In Section III, we estimate the

CMB spectral distortions by Silk damping of acoustic waves, due to the shear viscosity in the

baryon-photon fluid. We find that the enhanced power spectrum of multiple inflation scenario

could be constrained by the spectral distortion measurements. In Section IV, we summarize our

results and discuss possible ways to constrain multiple inflation scenario.

II. THE EVOLUTION OF COSMIC PERTURBATIONS

A. Overview

We discuss the evolution of cosmological perturbations through two inflations with an interme-

diate phase as in Figure 1. The first inflation ends up with an intermediate phase, and then the

second inflation begins after the intermediate phase and lasts until the usual radiation domination

of Big-bang Nucleosynthesis (BBN) gets started.

There are many ways realizing these three stages. As an illustration, we consider a potential of

two scalar fields V (φ, ψ) = V1(φ) + V2(ψ) having the following form. We get the first inflation and

the matter domination by V1(φ) = V ∗1 − 12m

2φφ

2 + · · · . For φ ' 0, the first inflation is driven by the

vacuum energy V ∗1 , but for φ > 0, φ oscillates around its minimum with driving the matter phase.

Note that there is no reheating after the first inflation, if φ is not coupled with other fields. This

is how we move from the first inflation to the matter domination without introducing an reheating

phase between them. Then, we have the second inflation by V2(ψ) = V ∗2 +H2ψ2 − 12m

2ψψ

2 + · · · .

For H ' (V ∗1 )1/4 � mψ, ψ is safely held at the origin during the first inflation and the matter

domination, and the second inflation starts as the matter energy density of φ is redshifted.

The characteristic scales are defined as

kx ≡ axHx (1)

where ax and Hx are the scale factor and the Hubble parameter at the era boundary tx. kx

3

Hubbleradius

IntermediatePhase

SecondInflation

RadiationDomination

ln(aH/k)

ln a

kb

ka

ta tb tc

kc

t

FirstInflation

ln aa

Superhorizon

Subhorizon

ln ab ln ac

FIG. 1. Characteristic scales of multiple inflation with an intermediate phase between the primordial and

the second inflations. Three characteristic scales ka, kb and kc correspond to the comoving scale of the

horizon at each of the era boundaries.

is the comoving scale of the horizon at t = tx. The first inflation occurs at t < ta generating

the perturbations. While modes with k < kb remain outside the horizon before the radiation

domination and are not affected by the phases after the first inflation, modes with k > kb enter

and exit the horizon and should evolve differently from those with k . kb as the followings:

• Modes with kb < k < ka enter the horizon during the intermediate phase and then exit the

horizon during the second inflation.

• Modes with ka < k < kc exit the horizon for the first time during the second inflation.

• Modes with k > kc never exit the horizon and are not our interest.

The power spectrum of the multiple inflation would fit with the large scale observation of CMB and

4

LSS at k < kb, but should be clearly different from the expectations of a simple single primordial

inflation or ΛCMD model at k > kb.

Generic features of the power spectrum of multiple inflation with breaking intermediate periods

are extensively studied in [86]. It is shown that the power spectrum is being enhanced with

oscillations on small scales when the intermediate phase is matter dominated. Hence, we expect

that the enhanced power spectrum could be explored by the CMB spctral distortions, and this

is the motivation of this paper. In our paper, however, we adopt the scheme of multicomponent

perturbation calculation in [33] to refine the calculation on the curvature perturbation by making

the background smoothly changing from the matter domination to the second inflation. From now

on, we consider only matter domination as the intermediate phase, and discuss the power spectrum

and its spectral distortions.

B. During the first inflation

The scalar part of the perturbed metric [87] is

ds2

= (1 + 2A) dt2 − 2B,i a(t) dt dxi −[(1 + 2R) a2(t) δij + 2C,ij

]dxi dxj (2)

We define

Rδφ ≡ R−H

φδφ (3)

where R is the intrinsic curvature perturbation on comoving hypersurfaces of Eq. (2) and φ is an

inflaton field. Its mode functions are calculated by

Rδφ(k, t) = −1

2

√πei(ν+ 1

2)π2

(1

aH

) 12(H

aφ

)H(1)ν

(k

aH

)(4)

where ν = 1+δ+ε1−ε + 1

2 with the slow-roll parameters ε ≡ − HH2 and δ ≡ φ

Hφ(e.g. [88]), and describe

the evolution of the perturbation during the first inflation. From Eq. (4), we express the curvature

perturbation at t = ta as

Rδφ(k, ta) =

√π

2α(ta)Ha

(1

ka

) 32

H(1)ν

(k

ka

)(5)

where we assume H ' Ha throughout the first inflation and α(ta) = Ha

φ(ta)is a constant depending

on inflation models, ka = aaHa, and H(1)ν (x) is the Hankel function of the first kind.

For large scales, i.e. k � ka,

Rδφ(k, ta)→ − iΓ(ν) 2ν−1

√π

α(ta)

(1

ka

) 32(ka

k

)ν(6)

5

and the power spectrum is

P12Rδφ(k) =

2ν−52

π

Γ(ν)

Γ(

32

) α(ta)

(k

ka

) 32−ν

(7)

where α(ta) and ν will be fixed by the Planck normalization [7].

C. Matter-dominated intermediate phase

We assume that the primordial universe can shift quickly from the first inflation to the matter

domination. The evolution of the curvature perturbation on constant (matter) energy hypersurface

during the matter-dominated intermediate phase is calculated by using the scheme of multicom-

ponent perturbations introduced in [33]1. We assume that the second inflation vacuum energy

becomes dominating as the matter energy density is gradually redshifted. The energy density is

ρ = ρm + ρv (8)

where ρm is the matter energy density and ρv is the vacuum energy density driving the second

inflation. The characteristic scale kb is naturally identified by the comoving scale at which the

expansion rate of universe is changed

a(tb) = 0 , (9)

and hence

kb ≡ abHb (10)

where Hb is the Hubble parameter during the second inflation and assumed to be constant. Note

that Eqs. (8) and (9) give

kb

ka=Hb

Ha

(2

3

H2a

H2b

− 1

2

) 13

(11)

which produces the e-folds during the matter domination

Nab = logab

aa= log

(2

3

H2a

H2b

− 1

2

) 13

. (12)

The curvature perturbation on constant matter energy hypersurface is defined by

Rδρm ≡ R−H

ρmδρm . (13)

1 In [33], the authors discuss the evolution of curvature perturbation on constant energy hypersurface during the

(moduli) matter domination. However, their result is very different from ours because they should calculate it in

terms of radiation hypersurface which is relevant to describe the thermal inflation period.

6

whose mode functions should be matched with Rδφ(k, t) at t = ta by requiring

Rδρm(k, ta) = Rδφ(k, ta) (14)

˙Rδρm(k, ta) = ˙Rδφ(k, ta) (15)

From [33], the governing equation is given by

¨Rδρm +H

(2 +

ρm

ρm + 23q

2

)˙Rδρm −

(q2

3

)(ρm

ρm + 23q

2

)Rδρm = 0 (16)

whose solution is reduced to

Rδρm(k, tb) = Am(k, ta)

[1 + S(ta, tb)

(k

kb

)2]

+Bm(k, ta)

(Hb

Ha

)(17)

where

S(ta, tb) =1

3

(3

2

) 32∫ 1

xab

(2x

2 + x3

) 32

dx (18)

with xab ≡ aaab

and

Am(k, ta) =1

1 + 13

(HbHa

)(kakb

)(kkb

)2

[Rδφ(k, ta) +

(Hb

Ha

)(ka

kb

)3 ˙Rδφ(k, ta)

Ha

], (19)

Bm(k, ta) =

(HbHa

)(kakb

)1 + 1

3

(HbHa

)(kakb

)(kkb

)2

[1

3

(k

kb

)2

Rδφ(k, ta)−(ka

kb

)2 ˙Rδφ(k, ta)

Ha

]. (20)

D. The second inflation

As the vacuum energy density of the second inflation gets dominating, the curvature perturba-

tion

Rδψ ≡ R−H

ψδψ (21)

with ψ the inflaton for the second inflation would be matched with Rδρm at t = tb by

Rδψ(k, tb) = Rδρm(k, tb) (22)

˙Rδψ(k, tb) = ˙Rδρm(k, tb) (23)

We treat the evolution of Rδψ in the same manner of the first inflation in Eq. (4), but have to

consider both H(1)ν and H

(2)ν for the second inflation as in [86]. We express the full solution by

using Bessel functions, instead of Hankel functions, to have concise forms.

Rδψ(k, t) =

(k

aH

)ν′ [Cψ(k, tb) Jν′

(k

aH

)+Dψ(k, tb)Yν′

(k

aH

)]. (24)

7

100 101 102 103 104 105 106 107

k[Mpc 1]

10 4

10 3

10 2

10 1

100

101

102

103

(k)

highly oscillating

FIG. 2. Transfer Function T (k) for kb = 150 Mpc−1, ka = 105 Mpc−1 and kc = 107 Mpc−1.

where ν ′ = 1+δ+ε1−ε + 1

2 with the slow-roll parameters ε ≡ − HH2 and δ ≡ ψ

Hψand

Cψ(k, tb) =π

2

(kb

k

)ν′−1[Rδρm(k, tb)Yν′−1

(k

kb

)+

˙Rδρm(k, tb)

Hb

(kb

k

)Yν′

(k

kb

)](25)

Dψ(k, tb) = −π2

(kb

k

)ν′−1[Rδρm(k, tb) Jν′−1

(k

kb

)+

˙Rδρm(k, tb)

Hb

(kb

k

)Jν′

(k

kb

)](26)

Therefore, the curvature perturbation should be calculated at t = tc by

Rδψ(k, tc) =

(k

kc

)ν′ [Cψ(k, tb) Jν′

(k

kc

)+Dψ(k, tb)Yν′

(k

kc

)]. (27)

E. Transfer Function and Power Spectrum

The evolution of the curvature perturbation after the first inflation is summarized by a transfer

function

T (k) ≡∣∣∣∣Rδψ(k, tc)

Rδφ(k, ta)

∣∣∣∣ (28)

and the resultant power spectrum is expressed by

P12 (k) =

√k3

2π2|Rδψ(k, tc)| =

√k3

2π2|T (k)Rδφ(k, ta)| . (29)

8

100 101 102 103 104 105 106 107

k[Mpc 1]

10 13

10 11

10 9

10 7

10 5(k

)

highly oscillating

FIG. 3. Power Spectrum P(k) for kb = 150 Mpc−1, ka = 105 Mpc−1 and kc = 107 Mpc−1. The red-dotted

line represents the power spectrum of ΛCDM model, i.e. P(k) = A∗(k/k∗)n∗−1.

For k � kb, the transfer function goes to T → 1, and the power spectrum is determined mainly

by the first inflation, We should take ν ' 1.52 in Eq. (7) to follow the Planck normalization

A∗ = 2.21× 10−9 and n∗ = 0.96 at k∗ = 0.05 Mpc−1 [7], and ν ′ ' 1.5 in Eq. (27) to include generic

models for the second inflation.

Figure 2 and Figure 3 are illustrations of the transfer function in Eq. (28) and the power

spectrum in Eq. (29). We take Ha ' 1014 GeV to satisfy the constraint of the energy scale of

primordial inflation [7]. If ka ' 105 Mpc−1 and Hb ' 106 GeV are taken, we get Nab ' 13 and

kb ' 150 Mpc−1 through Eqs. (11) and (12). If we putHc ' 102 GeV, we have kc = kbeNbcHc/Hb '

107 Mpc−1 by taking Nbc ' 20. Note that the e-folds during the first inflation are large enough

to make the total e-folds through the inflationary phases more than 60. The dips and peaks are

related with zeros of Bessel functions. For example, the first dip is found at the first zero of Jν′(x)

in Eq. (26), i.e. k ' 3.14kb.

9

III. CMB SPECTRAL DISTORTIONS

Now we are ready to calculate the CMB spectral distortions generated from Eq. (29) by the full

thermalization Green’s function [89, 90]2. The dissipation of acoustic waves with adiabatic initial

conditions heats up the CMB photons generating the spectral distortions. One can calculate this

heating rate d(Q/ργ)/ dz over the redshift z, and then the spectral distortion at a given frequency

∆Iν is estimated from the heating rate by

∆Iν '∫Gth

(ν, z′

) d(Q/ργ)

dz′dz′ (30)

where Gth(ν, z′) includes the relevant thermalisation physics, which is independent of the energy

release scenario. The spectral distortions is expressed in terms of the temperature shift ∆T , y and

µ contributions

∆Iν ≈∆T

TG(ν) + y YSZ(ν) + µMSZ(ν) . (31)

where G(ν) = T ∂B(ν)∂T with B(ν) = 2hν3

c2(ex − 1)−1 for x = hν

kBT, YSZ(ν) ' T ∂B(ν)

∂T (x coth(x)− 4)

and MSZ(ν) ' T ∂B(ν)∂T

(0.4561− 1

x

).

We should impose kb & 10 Mpc−1 from large scale observations. The power spectrum is en-

hanced around k = kb, and it could conflict with CMB observations unless kb should be larger

than 10 Mpc−1 [91]. Primordial balck holes constrain the power spectrum as P . 10−1 for all

scales [12, 92]. We observe that the enhancement of P(k) tends to grow as k gets large. For

kb ∼ 10 Mpc−1, P(kc) . 10−1 can be attained by restricting kc . 108.

The abundance of ultracompact minihalos (UCMHs) for neutralino dark matter gives P . 10−6

at 5 Mpc−1 . k . 107 Mpc−1 [9]. (1) For ka . 107 Mpc−1, we can satisfy P(ka) < 10−6 only

if kb is not so smaller than ka. In this case, the power spectrum is not enhanced within the

spectral distortion window 10 Mpc−1 . k . 104 Mpc−1. Thus, the spectral distortions of this

case is not distinguished from those of ΛCDM case. (2) For ka > 107 Mpc−1, the power spectrum

may not be constrained by UCMHs, but Ha/Hb and Nab in Eq. (12) would be unrealistically

huge to enhance the spectral distortions. For example, we need Ha/Hb ∼ 1018 and Nab ∼ 30

for ka = 107 Mpc−1 and kb = 10 Mpc−1. Hence, UCMHs do not allow the proper ranges for the

characteristic scales to distinguish the spectral distortions of our scenario from those of ΛCDM

[9]. However, this constraint is valid when the dark matter is a weakly interacting massive particle

(WIMP). If the dark matter is made up of axions whose masses are within the typical range

2 We use the numerical code of Green function method which has been developed by Jens Chluba.

10

101 102 103 104

kb[Mpc 1]

4.5

5.0[x10

9 ]

ka = 103Mpc 1

ka = 104Mpc 1

ka = 5 × 104Mpc 1

ka = 105Mpc 1

ka = 5 × 105Mpc 1

ka = 106Mpc 1

FIG. 4. y-distortion for various values of ka and kc = 107 Mpc−1. We take 10 Mpc−1 . kb . 0.1ka.

10−6eV . ma . 10−3eV obtained by astrophysical and cosmological data (e.g. [93–97]), they

cannot annihilate to produce active gamma-rays as WIMPs do. Note that the interactions of

these low mass axions are substantially suppressed by the Peccei-Quinn symmetry breaking scale

109 GeV . fa . 1012 GeV, while WIMP interactions are suppressed by the weak interaction

scale mW ∼ mZ ∼ 102 GeV [95]. Hence, UCMHs for axion dark matter cannot constrain the

power spectrum, and we do not consider the constraints from [9] in our estimation of the spectral

distortions. We summarize the ranges of the characteristic scales as

10 Mpc−1 � kb < ka < kc . 108 Mpc−1 , (32)

and could expect that CMB spectral distortions is enhanced within these ranges.

y- and µ-distortions for a few examples are shown in Figure 4 and Figure 5. For µ-distortion,

we include a small correction ∆µ ' −0.334 × 10−8 due to the energy extraction from photons to

baryons as CMB photons heat up the non-relativistic plasma of baryons by Compton scattering

[19]. Both µ- and y-distortions are safely below than the limit of COBE/FIRAS for 10 Mpc−1 . kb

[38].

We see that y-value hardly changes over the range of kb in Figure 4, while µ-distortion clearly

depends on the value of kb in Figure 5. y-distortion is sensitive to the power spectrum at k .

11

101 102 103 104

kb[Mpc 1]

2

5

10

20

50

100

200

500[×

108 ]

ka = 103Mpc 1

ka = 104Mpc 1

ka = 5 × 104Mpc 1

ka = 105Mpc 1

ka = 5 × 105Mpc 1

ka = 106Mpc 1

FIG. 5. µ-distortions for various values of ka and kc = 107 Mpc−1. We take 10 Mpc−1 . kb . 0.1ka. The

red-dotted line represents the µ-value of ΛCDM model.

50 Mpc−1 [21], but the power spectrum is enhanced at k > kb > 10 Mpc−1. Then, y-distortion is

generated by the dissipation of the modes within a very small range kb . k . 50 Mpc−1. As ka gets

large, the power spectrum could be enhanced in the range, and y-distortion gets slightly large in

Figure 4. However, µ-distortion is sensitive to the power spectrum at 50 Mpc−1 . k . 104 Mpc−1

[21]. When ka is large, the power spectrum could be substantially enhanced in the range, and

µ-distortion can be 102 times larger than the value of ΛCDM µΛCDM ' 2 × 10−8 in Figure 5.

Therefore, the observation of µ-value is a key to test the enhancement of power spectrum on small

scales in our multiple inflation scenario.

IV. DISCUSSION

In this paper, we consider multiple inflation scenario with having an intermediate matter dom-

ination between inflations in Figure 1, and find that the power spectrum is enhanced at k > kb

where kb is defined as the comoving horizon scale at the beginning of the second inflation by

Eq. (9). If we require kb > 10 Mpc−1, the enhancement of power spectrum at k > kb has no

trouble with large scale observations such as CMB. For kb > 103 Mpc−1, the spectral distortions

12

of the multiple inflation scenario are expected to be similar to that of the standard ΛCDM model.

For kb . 103 Mpc−1, however, µ-distortion value would be allowed within the COBE/FIRAS, but

could be 102 times larger than the expectation of ΛCDM model which future distortion experiments

may be able to test.

We do not cover various topics on the astrophysical and cosmological applications of the multiple

inflation scenario in this paper. First, there could be constraints on the energy scale of the second

inflation from particle physics and cosmology, especially in the context of supersymmetric model

[98–100]. Two inflations at different energy scales may produce a special profile of gravitational

waves background [101–103]. Second, the enhanced power spectrum may impact on the halo

abundance and galaxy substructure as in [35]. This is entangled with the issues such as primordial

black holes, minihalos and 21cm observations, which may also give feedbacks to µ-distortions

[53, 54, 104–106]. If we consider those issues altogether, then multiple inflation scenario would be

more constrained. We leave these aspects as future work.

ACKNOWLEDGEMENTS

The authors thank to Ewan Stewart, Kihyun Cho, Sungwook Hong, Hassan Firouzjahi and

Jens Chluba for helpful discussions. HZ thanks Subodh Patil for useful discussions stimulating

this work. We use Jens Chluba’s Green function code to estimate CMB spectral distortions. This

work is supported by the DGIST UGRP grant.

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