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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.
∗ [email protected]† [email protected]
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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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