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Journal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journal
The effects of the small-scale
behaviour of dark matter power
spectrum on CMB spectral distortion
Abir Sarkar,a,b Shiv. K. Sethia and Subinoy Dasc
aRaman Research Institute, CV Raman Ave Sadashivnagar,
Bengaluru, Karnataka 560080, India
bIndian Institute of Science, CV Raman Ave, Devasandra Layout,
Bengaluru, Karnataka 560012, India
cIndian Institute of Astrophysics, 100 Feet Rd, Madiwala, 2nd Block,
Koramangala, Bengaluru, Karnataka 560034, India
E-mail: abir@rri.res.in, sethi@rri.res.in, subinoy@iiap.res.in
Received January 30, 2017
Revised May 2, 2017
Accepted June 19, 2017
Published July 5, 2017
Abstract. After numerous astronomical and experimental searches, the precise particle
nature of dark matter is still unknown. The standard Weakly Interacting Massive Parti-
cle(WIMP) dark matter, despite successfully explaining the large-scale features of the uni-
verse, has long-standing small-scale issues. The spectral distortion in the Cosmic Microwave
Background(CMB) caused by Silk damping in the pre-recombination era allows one to access
information on a range of small scales 0.3 Mpc < k < 104 Mpc−1, whose dynamics can be
precisely described using linear theory. In this paper, we investigate the possibility of using
the Silk damping induced CMB spectral distortion as a probe of the small-scale power. We
consider four suggested alternative dark matter candidates — Warm Dark Matter (WDM),
Late Forming Dark Matter (LFDM), Ultra Light Axion (ULA) dark matter and Charged
Decaying Dark Matter (CHDM); the matter power in all these models deviate significantly
from the ΛCDM model at small scales. We compute the spectral distortion of CMB for these
alternative models and compare our results with the ΛCDM model. We show that the main
impact of alternative models is to alter the sub-horizon evolution of the Newtonian potential
which affects the late-time behaviour of spectral distortion of CMB. The y-parameter dimin-
ishes by a few percent as compared to the ΛCDM model for a range of parameters of these
models: LFDM for formation redshift z = 105f (7%); WDM for mass mwdm = 1 keV (2%);
CHDM for decay redshift z 5 −24decay = 10 (5%); ULA for mass ma = 10 eV (3%). This effect
from the pre-recombination era can be masked by orders of magnitude higher y-distortions
generated by late-time sources, e.g. the Epoch of Reionization and tSZ from the cluster of
©c 2017 IOP Publishing Ltd and Sissa Medialab https://doi.org/10.1088/1475-7516/2017/07/012
JCAP07(2017)012
galaxies. We also briefly discuss the detectability of this deviation in light of the upcoming
CMB experiment PIXIE, which might have the sensitivity to detect this signal from the
pre-recombination phase.
Keywords: CMBR theory, cosmological perturbation theory, Sunyaev-Zeldovich effect, dark
matter theory
ArXiv ePrint: 1701.07273
JCAP07(2017)012
Contents
1 Introduction 1
2 Models of dark matter 3
2.1 WDM 3
2.2 LFDM 4
2.3 ULA 5
2.4 Charged decaying dark matter 5
2.5 Evolution of the transfer functions of different dark matter candidates 7
3 Physics of spectral distortion 8
3.1 Energy released due to Silk damping 11
4 Spectral distortion and dark matter models 12
5 Results 16
6 Conclusion 19
1 Introduction
After intensive searches throughout the world over decades, the nature of dark matter is yet
to be confirmed. The existence of this component, whose presence is revealed only through
its gravitational interaction, is well established by many observations covering a broad range
of length scales and epochs of the universe. This list contains cosmic microwave background
(CMB) anisotropy experiments [1–3], large scale structure surveys [4–6], the study of the
galaxy rotation curves [7], cosmological weak gravitational lensing observations [8, 9], etc.
One of the leading candidates for dark matter, the Weakly Interacting Massive Particle
(WIMP) or the traditional cold dark matter (CDM), is inspired by the well-known WIMP
miracle [10]. The supersymmetric extension of the standard model of particle physics gives
rise to a particle with self-annihilation cross-section 〈σv〉 ' 3 × 10−26cm3s−1 and mass in
the range of 100 GeV, which coincidentally produces the correct present abundance of dark
matter. Inspired by this discovery, a lot of direct [11–13], indirect [14–17] and collider [18, 19]
searches have been performed worldwide but none of these experiments have yet succeeded
in providing consistent information about the particle nature of dark matter. In fact, the
results from many of these experiments are found to be in conflict [20] with each other.
Besides, there also exist some long-standing astrophysical problems with the WIMP.
One of them is the cusp-core problem [21], indicated by the discrepancy between increasing
dark matter halo profile (cusp) towards the centre of the galaxy from CDM N-body sim-
ulations [22], while observationally relatively flat density profiles are found [23]. Another
issue with WIMP is the missing satellite problem [24, 25]; N-body simulations of structure
formation with CDM produce much more satellite halos of a Milky-Way type galaxy than
observed. Another issue is the “too big to fail” [26, 27] problem, which underlines the fact,
based on N-body simulations, that a majority of the most massive subhalos of the Milky Way
are too dense to host any of its bright satellites. Some recent works claim that even when the
– 1 –
JCAP07(2017)012
effects of small scale baryonic physics are included, these issues may still persist [28–30]. All
these problems have inspired a drive to go beyond the standard picture of dark matter and
consider alternative candidates, which differ from CDM on galactic scales but must reproduce
its success on cosmological scales.
We consider four alternative dark matter models in this work: Warm Dark Matter
(WDM), Late Forming Dark Matter (LFDM), Ultra Light Axion Dark Matter (ULADM),
dark matter produced by the decay of a heavy charged particle in the early universe (CHDM).
The best studied alternative dark matter model is the WDM model (e.g. [31, 32]); we con-
sider only thermally produced WDM in this work. This model has been proposed to alleviate
some of the small-scale problems of the ΛCDM model [33]. WDM has been extensively tested
against observations at small scales: number of satellites based on N-body simulations [34–36];
the cusp-core problem [37, 38]; Lyman-α forest flux power spectrum [32]; structure forma-
tion constraints based on hydrodynamic simulations [39]; the “too big to fail” issue [36].
These detailed studies prefer WDM masses (or lower limit on the mass) to lie in the range:
0.1 keV < mwdm < 4 keV. These constraints do not reveal a unique picture with respect to
thermal WDM being a possible dark matter candidate and underline the need to investigate
alternative models. The dark matter produced by the decay of a heavy charged particle in
the early universe(CHDM) has also been proposed to address issues with small-scale power
of the ΛCDM model [40]. LFDM has its origin in extended neutrino physics [41] while the
ULA dark matter originates from the string axiverse scenario [42]. In these cases, the dark
matter starts behaving as CDM after the epoch of Big Bang Nucleosynthesis(BBN) and be-
fore the epoch of Matter-Radiation Equality (MRE) and have similar features (suppression
in small scale power followed by damped oscillations) in matter power spectra. For LFDM
and CHDM the suppression happens at scales that enter the horizon before the dark matter
is formed. For WDM and ULADM models, the scales that suffer suppression are determined
by either free-streaming of the particle (WDM) or variable sound speed of the scalar field
(ULADM). Both of these scales depend on the mass of the particle/scalar field. Recent
N-body simulations based on the LFDM and ULA models suggest that these models offer
acceptable solutions to the cusp-core issue while being consistent with large-scale clustering
data ([43–46]). Cosmological constraints favour the following lower limits on the mass of
ULA: 10−22 eV < m < 10−23a eV [47, 48].
All the models we consider deviate from the ΛCDM model at small scales. One possible
probe of this small scale deviation is the spectral distortion of the CMB caused by Silk damp-
ing in the pre-recombination era. This damping pumps entropy into the thermal plasma and
can distort the CMB spectra if the energy injection occurs after z ' 106 [49–52]. The acoustic
waves on scales 104 Mpc−1 < k < 0.3 Mpc−1 get dissipated before the recombination which
means that the CMB spectral distortion can be used to constrain the matter power at these
scales (for details e.g. [53–68]). The COBE-FIRAS results gave the current upper bounds on
the spectral distortion parameters: |µ| . 9 × 10−5 and |y| . 1.5 × 10−5 [69]. Silk damping
generally results in spectral distortion amplitudes many orders of magnitude smaller than
these limits. The upcoming experiment PIXIE [70] will be able to measure y ' a few × 10−9
and µ ' 10−8, assuming zero foreground contamination. The foregrounds depend on several
physical parameters and marginalisation over all of them will degrade the sensitivity [71].
In this work, we study CMB distortion parameters for alternative dark matter models
that differ from the ΛCDM model at small scales as possible probes of these models. For
these models, we compute the distortion parameters using the tight-coupling approximation
in the pre-recombination era.
– 2 –
JCAP07(2017)012
The paper is organised as follows. In section 2, we provide a brief description of all
the models considered in this work. In section 3, we review the physical picture of the
creation and evolution of spectral distortion in the CMB and describe the relation between
the dissipation of acoustic wave and CMB spectral distortion. In section 4, we discuss in
detail the physical basis of how a change in the dark matter model impacts Silk damping in
the tight-coupling approximation. Section 5 contains the main results of this work. Section 6
is reserved for the conclusion and prospects of this paper.
Throughout this paper, unless specified otherwise, we use the best-fit cosmological pa-
rameters given by Planck [1]: spatially flat universe with Ωb = 0.049, Ωdm = 0.254 (both at
the present epoch), and h = 0.67.
2 Models of dark matter
In this section, we describe the origins of the models considered in this work by motivating
their particle physics origins and briefly discuss their cosmological signatures.
2.1 WDM
Warm dark matter (WDM) particles with a common mass of ' 1 keV, inspired by particle
physics models of sterile neutrinos, have been advocated as a solution to the small-scale
anomalies of CDM. A WDM particle is mainly produced non-thermally before the epoch
of BBN by active to sterile neutrino oscillation [72] in the early universe. However, there
are models of thermal keV mass Sterile neutrinos which were in thermal equilibrium with
some hidden sector particle and decouple with correct dark matter relic density [73]. Sterile
neutrinos in this mass range cannot be detected in standard WIMP searches at least with
current experimental capabilities, but in galactic X-ray data, its imprint can be captured if a
DM sterile neutrino decays to a photon and relativistic active neutrino. This photon would
have keV energy which in principal can be detected as a galactic X-ray excess. Recent X-ray
anomalies from XMM-Newton and Chandra data can be explained by the decay of a 7 keV
WDM [74].
In this work, we have considered only thermally produced WDM. In the early universe,
WDM is highly relativistic, and its free-streaming scale is the horizon at that time. As time
proceeds, it cools down and passing through its semi-relativistic phase it finally becomes
non-relativistic CDM. The free-streaming scale during its semi-relativistic phase is given
by [33] ( )
0.3 0.15 (m )wdm 1.15
kfs ' Mpc−1 (2.1)
Ωwdm keV
Here Ωwdm ≡ ρwdm/ρc is the relic density of WDM and mwdm is the mass. In the matter
power spectrum produced by WDM, the power is cut at the WDM free-streaming scale as
compared to the standard ΛCDM model and the decrement is smooth in nature owing to the
fact that the thermal velocity of WDM decays as 1/a and is not small enough to be clustered
at small scales for a long time (figure 1). Clearly, lighter WDM will cut power at larger scales
(e.g. [33]). This free-streaming also suppresses the formation of low-mass halos or sub-halos
and its finite phase-space density prevents the development of density cusps [32, 75]. In this
work we consider WDM particles in the mass range: 0.3 keV < mwdm < 5 keV.
– 3 –
JCAP07(2017)012
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3 ΛCDM
10 5
LFDM zf = 1 x 10
5
CHDM zdecay = 1 x 10-4
10 WDM mwdm = 0.33 keV
-24
ULADM ma = 10  eV
-5
10
-3 -2 -1 0 1
10 10 10 10 10
-1
k (h Mpc )
Figure 1. Normalized matter power spectra, at z=0, of the four dark matter candidates considered
in this work along with ΛCDM are shown. For each of the non-standard dark matter candidate, the
parameters are chosen such that the power is cut at nearly the same scale k ' 0.3hMpc−1. The
specifications of the models are mentioned in the legend.
2.2 LFDM
In this scenario, the dark matter is formed as a consequence of phase transition of a scalar
field. The scalar field, which was initially trapped in a metastable minimum by thermal
effects or due to its interaction with other scalar fields resulting in a hybrid potential in the
early universe, makes a phase transition (which is generally between the redshift of BBN
and MRE in this model) to the true minima. After this, the scalar field starts oscillating
coherently and its equation of state changes from w = −1 to w = 0, making it clump exactly
like the CDM. The dynamics of the phase transition is very similar to the case of the hybrid
inflation [76], where the scalar field plays the role of the waterfall field and behaves like dark
matter after the phase transition. The hybrid potential for two scalar fields in the context of
LFDM was derived in [41, 77] and is given by
V (φ1, φ2) = (λφ
2
1 − µ2)2 + 4λ2φ2 2 2 21φ2 +M φ2 (2.2)
For a high value of φ2(T ) in the early universe, the scalar field is trapped in a metastable
minimum φ1 = 0 behaving like the cosmological constant. Later, at a critical redshift zf , φ1
becomes Tachyonic and rolls down to the true minima. After reaching the true minima, φ1
exhibits coherent oscillations and behaves exactly like the CDM. For details of this model,
we refer the reader to [41].
LFDM can also originate from a Fermionic field which has been proposed in [78]. Here,
a relativistic Fermionic fluid like the massless neutrino stops free-streaming at a certain
redshift zf due to the onset of a strong fifth force mediated by a scalar field. As the fermions
are massive, eventually the fifth force binds all the fermions within a Compton volume of
the mediating scalar, forming nuggets. These nuggets are much heavier than the original
fermions, and since their formation, they behave exactly like the CDM.
For both of the scenarios mentioned above, LFDM gets its initial conditions for evolution
from the massless neutrino. If LFDM is formed at z = zf , power is suppressed at a scale
k = klfdm that entered the horizon at that redshift. Unlike WDM, the suppression is sharp as
– 4 –
3 -3
P(k)(h  Mpc )
JCAP07(2017)012
the dark matter is assumed to form via an almost instantaneous phase transition (figure 1).
At k < klfdm, the matter power spectrum carries damped oscillations, a typical feature
exhibited by the massless neutrino. This feature is in contrast to WDM where the power
falls monotonically without any oscillation. Earlier the dark matter forms, smaller is the
scale where the decrement of power occurs. The cosmology of this model is studied in [79]
and it is found that dark matter should form deep inside the radiation dominated era and
before zf ' 0.98× 105. In the present work, we consider LFDM having formation redshift in
the range 5× 104 < z 5f < 5× 10 .
2.3 ULA
Another very well studied dark matter is the dark matter from ultra light axion (ULA) fields
in the context of string axiverse [42]. An axion-like particle as a dark matter candidate can
be described by [46, 80–8∫2] a two-p[arameter model, whose action is gi]ven by:
4 √ 1I = d x g F 2gµν∂µφa∂νφa − µ4(1− cosφa) (2.3)
2
where φa is a dimensionless and periodic scalar field, represented as φa → φa + 2π. F and
µ are the two parameters of the model. For sufficient small value of µ (which is the case
2
for dark matter), it can be shown that mass of the scalar is given by m µa = F . For a
cosmologically and astrophysically acceptable dark matter candidate, a reasonable value for
the mass is m ' 10−22a eV. All models of particle physics derived from string theory have
several periodic scalar fields such as φa and it has been argued that such a low mass is quite
reasonable from particle physics perspective.
ULA obtains its initial conditions after spontaneous symmetry breaking in the early
universe and behaves like a coherent scalar field. Early when H is high, the friction term
dominates and the field is stuck at some random initial value and behaves like the cosmological
constant. Later, when ma ∼ H(z) at a certain redshift, the field rolls off and start oscillating
coherently around the nearest minima of the periodic potential and starts behaving like CDM.
The adiabatic perturbations in the scalar field have a momentum-dependent and thus mass-
dependent effective sound speed. At scales below the effective sound horizon, perturbations
are washed out due to free-streaming, and the matter power spectrum features very similar
to LFDM (figure 1) are found in the matter power spectrum. The free-streaming scale is
given by [46] ( )1/3( )
m 100 kms−1
km ' − hMpc
−1. (2.4)
10 33eV c
This means that lighter axions will push the scale of suppression to a higher value. The
cosmologically relevant mass of ULA ranges from 10−18 and can be as tiny as 10−33 eV.
Some recent works have put a lower limit on the mass of ULA: ma . 2.6× 10−23 [48, 83]. In
this paper, we consider the mass range: 10−21 eV > m −25a > 10 eV.
2.4 Charged decaying dark matter
This model, its variants, and their cosmological implications have been investigated in de-
tail [40, 84–88]. We consider a model in which a heavy negatively charged particle of mass
Mch decays into a heavy neutral particle of mass Mneu and a relativistic electron (supersym-
metric models in which a selectron decays into an electron and a gravitino might achieve
– 5 –
JCAP07(2017)012
this scenario [40]). These two masses and the decay time τ parameterize the model. The
decay time and the mass difference between the two heavy particles ∆M = Mch −Mneu are
tightly constrained because the relativistic electron thermalizes with electron-photon coupled
system, thereby causing spectral distortion of CMB if the decay time corresponds to redshifts
z 6decay < 10 . The three-momentum of the relativistic electron p = ∆M and in the limit all
this relativistic energy is transferred to the(photon g)a(s, we ge)t t(he frac)tional energy increase:
−2 Ωdmh
2 105 ∆M
δργ/ργ ' 4.2× 10 (2.5)
0.11 1 + z Mch
Using the current bounds on µ and y-parameters, we get ∆M/M −2 −3ch . 10 –10 for decay
times in the redshift range 106 > zdecay > 10
5.
This constrains the energy density of the relativistic electrons to be dynamically unim-
portant and allows us to assume that the masses of the two heavy particles are the same.
This means that the main difference between models such as the LFDM model and the
decay charged particle model is that whereas the initial conditions (density and velocity
perturbations) in the former case arise from massless neutrino, they are inherited from the
baryon-photon fluid in the latter case.1 This results in a qualitative difference between the
two cases as seen in figure 1. While density and velocity perturbations of massless neutrinos
decay after horizon entry, these perturbations oscillate with nearly constant amplitude for
the photon-baryon fluid after horizon entry for η < ηdecay.
It is seen in figure 1 that the matter power spectrum in this case oscillates for scales
that are sub-horizon during the pre-decay phase but its value can exceed the matter power
spectrum for the ΛCDM model for the same cosmological parameters.
This can be understood as follows. In the ΛCDM model, the density perturbations of the
CDM component, in Synchronous gauge, δcdm = −h/2 or they are completely determined
by metric perturbations and are independent of velocity perturbations which are zero at
all times, θcdm = 0. In this case, the CDM density perturbations at sub-horizon scales
grow logarithmically during the radiation-dominated era. However, in the decaying charged
particle model, the initial velocity perturbations of the CDM component (the post-decay
neutral particle) are derived from the photon-baryon fluid and constitute an additional source
of density perturbations. For wavenumbers at which the velocity perturbations of the initial
conditions combine in phase with density perturbations of the CDM component, the density
perturbations can overshoot perturbations in the ΛCDM model. On the other hand, this
effect also serves to increase the decrement as compared to the ΛCDM model for wavenumbers
at which velocity perturbations act to suppress the growth of density perturbations.
All the models we consider here leave the matter-radiation equality unchanged and their
matter energy densities at the present epoch are normalized to Ωdm = 0.254. In figure 1,
1This assumption needs further explanation. For a massive charged particle to share the bulk velocity of
the baryon-photon fluid, it must be tightly coupled to this fluid which means the time scale of energy exchange
between this particle and the photon-baryons fluid should be far smaller than the expansion rate. This time
scale τ ' 100 (106K/T )3/2chelec (100Gev/Mch) sec which is much shorter than the expansion rate at redshifts of
interest. Another complication, in this case, arises from the possibility that the charged dark matter particle
can form an atom with a proton. This hydrogenic atom would have binding energy on the order of 24 keV
(' (mp/me)13.6 eV). These atoms, in turn, could be converted into an ion containing the charged dark
matter particle and doubly ionized helium through charge exchange reactions(for details e.g. [89]). Only a
tiny fraction of these atoms will form until the epoch there are not enough number of energetic CMB photons
on the tail of black body spectrum to ionize the atom; this epoch corresponds roughly to a temperature
' 200 eV. This could be an additional source of spectral distortion which we neglect in this paper.
– 6 –
JCAP07(2017)012
the power spectra of all the four models are displayed and compared to that of ΛCDM.
In the figure, we have plotted models which induce cut in power at nearly the same scale
k ' 0.3hMpc−1. The power spectra shown in the figure have been computed using the
publicly-available codes CMBFAST (WDM, CHDM, and LFDM by modifying this code) and
axionCAMB (ULA).
2.5 Evolution of the transfer functions of different dark matter candidates
In this subsection, we briefly describe how the dark matter transfer function evolves for
different dark matter candidates considered in this work. The transfer functions are given in
figure 2 at z = 104, 105, 2× 106 and 107. The model specifications are identical to that used
in figure 1. This allows us to motivate the discussion in section 4.
WDM. In this work, we have considered keV-mass thermally produced WDM candidates,
which become non-relativistic at z ∼ 106–107. So in the very early universe, when they
are still relativistic, the transfer function at sub-horizon scales will be similar to that of a
relativistic particle like the massless neutrino. As time progresses they become non-relativistic
but still cause suppression as compared to the ΛCDM model at small scales owing to their
free-streaming velocity that decays at a slower pace, as 1/a. We have shown the evolution
of transfer functions for WDM with of mass 0.33 keV in figure 2, which is relativistic until
z ' 1.4×106, so the sub-horizon features at z = 107 and z = 2×106 are essentially the same.
WDM with higher masses become non-relativistic earlier, before the onset of the µ-distortion
era (discussed in the next section).
ULADM. The ULADM forms due to spontaneous symmetry breaking in the early universe
and behaves like a coherent scalar field. Once the mass of the field drops below the Hubble
constant, the field starts oscillating coherently around the true minima of the potential
and behaves like cold dark matter. We have plotted the evolution of transfer function for
m −24a = 10 eV. ULADM with this mass decouples from the Hubble drag at z ' 2 × 104
and the suppression occurs at scales smaller than k ' 0.3 hMpc−1. So both at the onset of
µ-distortion (figure 3 in section 3) and at earlier times the sub-horizon transfer functions are
the same, as seen from the figure 2.
LFDM. As mentioned in subsection 2.2, LFDM is formed due to phase transition in the
neutrino sector and gets its initial conditions from the massless neutrino. In figure 2, we show
the evolution of the transfer function for z = 105f . We notice that the transfer functions of
LFDM and the WDM with mass 0.33 keV are identical at z = 106 and 107 because at high
redshifts both behave like massless neutrinos. At z = 105, the WDM has already become
non-relativistic but LFDM continues to be relativistic which explains greater suppression in
the LFDM power spectrum. After z = zf , the LFDM behaves as CDM, but WDM continues
to cause suppression of power at small scales owing to its slowly decaying but still significant
free-streaming velocities. At more recent time, i.e., z = 104, the LFDM transfer function is
identical to that of CDM at scales k & 0.3hMpc−1, the scale that enters the horizon at the
formation redshift, but smaller scales still carry the massless neutrino-like features.
CHDM. This model is qualitatively different from the other models as the dark matter
candidate, in this case, is charged and coupled to the photon-baryon plasma at early times.
The time evolution of CHDM transfer function is identical to that of baryons at high redshifts
with R = 3(ρb + ρdm)/(4ργ) as explained in subsection 2.4. At lower redshifts and at scales
– 7 –
JCAP07(2017)012
6
10 2
10
4
10
0
10
2
10
-2
10
0
10
-2 -4
10 10
-4
10 ΛCDM ΛCDM
5 -6 5
LFDM z  = 1 x 10 10f LFDM zf = 1 x 10
5 5
CHDM zdecay = 1 x 10 CHDM zdecay = 1 x 10
-6 WDM mwdm = 0.33 keV WDM mwdm = 0.33 keV10
-24 -24
ULADM ma = 10  eV ULADM ma = 10  eV
-8
10
-2 -1 0 1 -1 0 1
10 10 10 10 10 10 10
-1 -1
k (h Mpc ) k (h Mpc )
-2 -5
10 10
-3 -6
10 10
-4 -7
10 10
-5 -8
10 10
-6 -9
10 10
-7 -10
10 10
-8 -11
10 10
-9 CDM -1210 Λ 10 ΛCDM
5 5
LFDM zf = 1 x 10 LFDM zf = 1 x 10
5 5
-10 CHDM zdecay = 1 x 10 -13 CHDM zdecay = 1 x 1010 10
WDM mwdm = 0.33 keV WDM mwdm = 0.33 keV
-24 -24
ULADM ma = 10  eV ULADM ma = 10  eV
-11 -14
10 10
0 1 1 2
10 10 10 10
-1 -1
k (h Mpc ) k (h Mpc )
Figure 2. The evolution of transfer functions of four dark matter candidates considered in this
work along with ΛCDM. Clockwise from top-left, transfer functions are plotted at z = 104, 105, 2×
106 and 107. For each of the non-standard dark matter candidate, the parameters are chosen such
that the power is cut at nearly the same scale k ' 0.3hMpc−1. The specifications of the models are
mentioned in the legends.
larger than the scale that entered the horizon at zdecay = 10
5, its transfer function is identical
to CDM. At smaller scales, the transfer function can exceed the matter power of the ΛCDM
model, as seen in figure 2 and explained in subsection 2.4.
3 Physics of spectral distortion
In this section, the physics of spectral distortion in the CMB will be discussed in more detail.
In figure 3 different epochs related to the evolution of CMB distortion are shown. There are
four phases in this regard, and we provide a brief physical picture of each of these.
In the early universe, there are two classes of physical processes that can cause energy
exchange between electrons and photons and therefore act to equilibrate photon distribution
function to a black body. One is photon conserving process, dominated by inverse Compton
scattering and photon producing processes which are mediated by double Compton scattering
and free-free emission with double Compton scattering dominating the rate of production of
photons in the early universe. If both these processes act on time scales shorter than the
expansion time scale of the universe, any injection of energy in the universe is rapidly shared
between the photons and charged particles, and the CMB spectrum relaxes to a black body
(see e.g. [51, 90–92]).
– 8 –
T(k) T(k)
T(k) T(k)
JCAP07(2017)012
Figure 3. Important epochs of the evolution of spectral distortion in the Cosmic Microwave Back-
ground.
This condition is obtained in the first era z & 2 × 106. Any injection of energy during
this era, e.g. e+/e− annihilation and BBN, leaves no trace on the CMB spectrum.
For z . 2 × 106 the rate of photon production is not large enough to equilibriate the
photon distribution function to a black body except at low photon frequencies. This redshift
thus approximately marks the time after which the signature of energy injection can no
longer be erased. As Compton scattering remains an efficient process for energy exchange
between photons and electrons, the resultant distribution function relaxes to a Bose-Einstein
distribution with Tγ = Te but with a non-zero chemical potential, µ. This arises because
Compton scattering is a photon conserving process that cannot cause a transition between
two Black bodies at two different temperatures. During this phase, the evolution of the
photon distribution function, n(ν, t) with Compton scattering being the only energy-exchange
process, is given by the well-known Kompaneets eq[uation: ]
∂n neσTkBTe 1 ∂ 4 ∂n ( )= x 2
∂t m c x2 ∂x e
+ n+ n (3.1)
e e e ∂xe
Here Te is the electron temperature, ne is the number density of electrons, me is the mass
of the electron and xe = hν/kBTe, ν being the frequency of the photon, is the dimensionless
frequency. The general equilibrium solution of this equation is given by [49, 93]
1
n(ν) = (3.2)
exe+µ − 1
The chemical potential µ can be related to the fractional energy absorbed by the photon gas:
µ = 1.4∆ργ/ργ (e.g. [49, 51]).
For z . 2× 105, the Comptonization process is not efficient enough to bring the photon
distribution function into equilibrium, and the equilibrium solution of Kompaneets equation
(eq. (3.2)) is not valid anymore. This redshift marks the end of µ-distortion era. This era is
followed by “i-distortion” and “y-distortion” eras. We first discuss the physics of y-distortion.
The distribution function in eq. (3.2) satisfies the following identity: ∂n∂x = −(n + n
2)
where x = hν/kT . Assuming small departure from equilibrium, eq. (3.1) can be written as
– 9 –
JCAP07(2017)012
follows: [ ]
∂n neσTkB(Te − T ) 1 ∂ 4 ∂n= x (3.3)
∂t mec x2 ∂x ∂x
The time variable of eq. (3.3) can be modified as a new parameter which denotes the tem-
perature difference between electrons and photons:
neσTkB(Te − T )
dy = dt (3.4)
mec
Eq. (3.3) can be solved to give [93, 94]: [ ]
xyex x
n(x, y) = n(x, 0) + − 4 (3.5)
(ex − 1)2 tanh(x/2)
In this case the fractional change in photon energy can be related to the y-parameter as:
y = 1/4∆ργ/ργ [94].
Between µ- and y-distortion era there exists another era called the “intermediate (i-type)
distortion era”. In this time, the Comptonization time scale is not short enough to relax the
spectrum to equilibrium. Instead, the system settles into a state where the distortion is
given by the sum of µ-,y- and some residual distortion [65, 95]. The thermalization in this
era is approximated as a weighted combination of pure µ- and y-distortion [95]. The residual
distortion (r-type) is between 10-30% of the total distortion. This r-type distortion depends
sensitively on the time of energy injection, which is not the case for pure µ- or y-distortion.
A different approach was taken in [60] to quantitatively describe the i-distortion. The i-
distortion can be characterised by a modified∫ y-parameter defined as:zinj neσTkBT
yγ(zinj, zmax) = − dz (3.6)
z m cH(1 + z)max e
The Kompaneets equation is written in terms of this new time parameter and then expanded
about yγ . The solution shows that the distorted spectrum and thus the distortion is depen-
dent on yγ . According to eq. (3.6), yγ is sensitive to the time when the energy was injected
into the system through zinj. This makes the i-distortion able to estimate not only the
amount of energy injected but also the time of injection, something that can’t be estimated
by observing the µ or y-type distortion.
After z ∼ 1.5× 104, yγ becomes very small and y-distortion epoch commences and lasts
up to present time. Late time phenomena of the universe like reionization [60, 96], heating of
photons by warm-hot intergalactic medium [97–99] and SZ effects from groups and clusters
of galaxies ([55, 64] and references therein) also contribute to the y-distortion. As long as the
amount of distortion is small, the three different distortions (µ+y+r) can be linearly added
to give the final spectrum.
Various processes like decay of massive particles [65, 100, 101], annihilation of parti-
cles [58, 102, 103], dissipation of acoustic wave [50, 54, 56–58, 104], adiabatic cooling of
electrons [58, 59] and cosmological recombination radiation [62, 63, 105–107] can contribute
energy into or extract energy from the photon-baryon plasma leading to spectral distortion
in the early universe. In this work, we study the spectral distortion caused by dissipation
of acoustic wave (Silk Damping) for different dark matter models. A detailed description of
that process is given in next section.
– 10 –
JCAP07(2017)012
3.1 Energy released due to Silk damping
At z & 1100, the photon and baryons are tightly coupled to√each other via Compton scatteringand behave like a single fluid. Adiabatic perturbations in this fluid behave like standing
waves inside the sound horizon rs = csη, where cs = c/ 3(1 +R) is the speed of sound
in the plasma and R = 3ρb/4ργ . At scales much smaller than the sound horizon, photon
diffusion causes damping of density perturbation and bulk motion of this fluid [108]. This
process (Silk damping) can be modelled by expanding the evolution of perturbations in the
coupled photon-baryon fluid to second order in the mean free path of photons [109, 110]
or as the dissipation of energy of sounds waves owing to radiative viscosity and also on
thermal conduction at late times [111]. Silk damping causes the injection of entropy into the
thermal plasma.
As this is a continuous energy injection process, it can lead to µ-type, i-type or y-
type distortion depending on the era when the energy was injected. In the redshift range,
106 & z & 103, the structures corresponding to scales k ' 0.3–104 Mpc−1 are completely
wiped out due to this damping. Thus, conversely, one possible way to study and constrain
the initial power spectrum at these scales is by observing the spectral distortion that is
imprinted in the CMB spectrum due to this process. It should be noted that this is the only
known probe of linear structures for such a wide range of small scales. In comparison, the
observed CMB temperature anisotropies from Planck probe scales k . 0.1 Mpc−1 [1] and
the smallest scales probed by galaxy clustering data correspond to nearly linear scales at the
present: k . 0.1 Mpc−1 (e.g. [4]).
The damping of adiabatic perturbations at small scales has been well studied in the
literature. Recently a precise calculation of µ- and y-type distortion due to Silk damping
has been performed using the second order perturbation theory [57]. It has been shown that
tight-coupling approximation provides a good approximation for modelling the Silk damping
in the pre-recombination era. In this approximation, the source function for heating due to
Silk damping is given by [57]:2 [ ]
k2 R2 16
SSD(k, η) ' + |Θ (k, η)|22 1 YSZ (3.7)τ̇c 1 +R 15
Where Θ1(k, η) is the CMB dipole anisotropy, τ̇c = cneσTa is the derivative of Compton
scattering optical depth with respect to the conformal time and YSZ is the frequency depen-
dent function representing the y-distortion defined in eq. (3.5). The first term in the square
bracket comes from heat conduction and the second one is due to radiative viscosity. At
high redshift, the radiative viscosity dominates as R → 0. As any time, the average source
function can be obtained by integrating ove∫r all wavenumbers:
〈 〉 d
3k
SSD (η) = SSD(k, η) (3.8)
(2π)3
To obtain effective heating rate 〈SSD〉 needs to∫be multiplied by τ̇c and integrated over all
frequencies. The integration over frequencies, x3YSZdx, yields 4ργ . This result provides
2the source function given in eq. (3.7) can be understood as arising from radiative viscosity. In this case the
energy pumped into the thermal plasma is proportion to the square of the product of the photon mean free
path with the velocity shear field (e.g. [53]) which is proportional to k2Θ2/τ̇ 21 c . Alternatively, the dissipation
of the energy can be modelled in terms of photon monopole [56]. Both these approaches give results within
a factor of 3/4 of each other [57]. While numerically computing CMB spectral distortion, we use both the
methods and find reasonable agreement.
– 11 –
JCAP07(2017)012
the heating rate as a function of conformal time which is converted to a function of redshift
by dividing it by H(1 + z). Thus, the final expression of heating rate is given by
1 da4Q ≈ 4τ̇c〈SSD〉 (3.9)
a4ργ dz H(1 + z)
The µ and y-parameters are relate∫d to heating rates as follows [57, 92]∞ dz da4Q
µ = 1.4 exp(−[z/z 5/2µ] ) (3.10)4
z a ρµ γ dz
and ∫
1 zi dz da4Q
y = (3.11)
4 z a
4ρ dz
dec γ
zdec ' 1100 and zi ' 1.5 × 104 are the redshift of decoupling and the redshift denoting the
end of i-distortion era, respectively.
4 Spectral distortion and dark matter models
In this section, we discuss how altering the dark matter model impacts the generation of
entropy owing to Silk damping in the pre-recombination era. As noted in the previous
section, the dynamics of this energy pumping can be captured by the time evolution of the
photon dipole, Θ1(k, η).
We discuss here how the photon dipole is altered when the dark matter model is changed.
We use Newtonian conformal gauge for motivating our discussion as the underlying physics
is more transparent in this gauge. In the tight-coupling approximation, the photon dipole
can be expressed[as (e.g. [112]):
1
Θ1(k, η) = √ (Θ∫ 0(k, 0) + Φ(k, 0)) sin(kc( s
η)
3
k η ) ]− dη′ Φ(k, η′)−Ψ(k, η′) cos(kcsη − kc η′s ) exp(−k2/k2d) (4.1)3 0
Here Θ0(k, η) is the photon monopole and Φ(k, η) and Ψ(k, η) are the two gravitational
potentials in the Newtonian gauge. If the sma√ll contribution from neutrinos is neglected,
Ψ = −Φ.3 Θ0(k, 0) = 0.5Φ(k, 0) and cs = 1/ 3(1 +R). In Eq. (4.1) we have neglected
R = 3ρB/4ργ and its evolution in the pre-factors of the equation.
4 kd(η) corresponds to the
3Ψ = −(1 + 2/5Rν)Φ with Rν = ρν/(ρν + ργ); Rν = 0.41 for three massless neutrino degrees of freedom.
Two of the models we consider here can alter Rν . This means that the first term in eq. (4.1), which corresponds
to initial conditions, can also be used to distinguish between different dark matter models [57]. In WDM
models, the dark matter particle is initially relativistic and therefore would increase Rν . However, even
during this phase the contribution of this particle to neutrino relativistic degrees of freedom is negligible as
compared to the standard model neutrino; for mwdm = 1 keV, the particle contributes less than 1% of the
energy density of a standard model neutrino. In LFDM models, a tiny fraction of neutrino energy density
is used to create the dark matter particle during a phase transition (e.g. [79]). For both these models, the
effect of initial conditions is generally negligible. We take into account these effects in our detailed modelling
using CMBFAST but drop it in this sub-section to uncover the main determinants of the impact of dark matter
models on the photon dipole.
4For the charged decay model, R = 3(ρB +ρdm)/(4ργ) at early times as the dark matter particle is charged
and tightly coupled to the baryon-photon fluid. For this model, this is an additional effect that determines the
evolution of Θ1(k, η). In the results presented in this section, we take into account this effect using CMBFAST.
– 12 –
JCAP07(2017)012
 10  10
 1
 1
 0.1
 0.1
 0.01
 0.01
 0.001
 0.0001
 0.001
 1e-05
 0.0001
5 5
CHDM zdecay = 10  k=0.4 LFDM z 1e-06 f
 = 10  k=0.4
ΛCDM k=0.4 ΛCDM k=0.4
5 5
 1e-05 CHDM zdecay = 10  k=0.8 LFDM zf = 10  k=0.8
ΛCDM k=0.8  1e-07 ΛCDM k=0.8
5 5
CHDM zdecay = 10  k=6.4 LFDM zf = 10  k=6.4
ΛCDM k=6.4 ΛCDM k=6.4
 1e-06  1e-08
 0.01  0.1  1  10  100  0.01  0.1  1  10  100
-1 -1
η(Mpc ) η(Mpc )
Figure 4. The evolution of Φ(k, η), for k = 0.4, 0.8, 6.4Mpc−1, for CHDM (Left panel) and
LFDM(Right panel) models are compared to the ΛCDM model. In each subfigure the solid lines
stand for the non-standard model and the dotted lines for ΛCDM model.
scale that undergoes Silk damping∫ at any(time:η )2
k−2
1 R 4
d (η) ' dη + (4.2)
0 τ̇ 6(1 +R)
2 27(1 +R)
Here τ̇ = neσTa. Under these approximations, the evolution of Θ1(k, η) can be completely
determined by the Newtonian potential Φ(k, η) and the Silk damping scale kd(η). As dis-
cussed in the next section, the dissipation of energy at a given scale k occurs predominantly
at a time ηd such that k = kd(ηd). As we argue below, the main effect we seek depends on
the time evolution of Φ(k, η) in the time range, 1/k < η < 1/kd(ηd), i.e. from the horizon
entry of a scale to the time at which the mechanical energy at this scale dissipates.
All the models we consider in this paper are based on altering the nature of CDM
with an aim to suppress power at small scales. While these models impact the radiation
content of the universe in the early universe (e.g. WDM model is based on a particle of
mass mwdm ' 1 kev which is relativistic in the early universe) they all leave unchanged the
matter-radiation equality epoch. This implies that their impact is proportional to the ratio
of dark matter to the radiation energy density, ρdm/ρr, which is much smaller than unity at
early times. This means that the impact of changing the dark matter change is essentially
captured by the second term of eq. (4.1), which depends on the evolution of the Newtonian
potential Φ(k, η).
In figure 4 we show the evolution of Φ(k, η) for two of the models we consider here for a
range of wavenumbers k. The Newtonian potential Φ(k, η) is computed from CMBFAST code,
which is written in Synchronous gauge, using th(e following transfo)rmation:ȧ
Φ(k, η) = β(k, η)− ḣ(k, η) + 6β̇(k, η) (4.3)
ak2
Here h and β are the potentials in Synchronous Gauge (eq. 18 of [113]). Some notable
features of the comparison between the models we consider and the ΛCDM models are: (a)
the difference between the potentials is negligible of for large scales (small k). This is expected
as the models we consider agree with the ΛCDM model on large scales, (b) As the scale gets
smaller, the difference between alternative DM models and the ΛCDM model becomes more
significant at larger η. We discuss the nature of this deviation below.
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Φ(k, η)
Φ(k, η)
JCAP07(2017)012
LFDM model. In this model, the CDM forms at a redshift z = zf from a tiny fraction of
relativistic neutrinos due to a phase transition. It inherits the density and velocity pertur-
bations of the neutrino component, resulting in a sharp reduction in matter power for scales
k > klfdm, where klfdm is the scale that enters the horizon at z = zf .
To understand figure 4 we consider scales smaller and larger than klfdm: (a) k < klfdm:
these scales are outside the horizon when the dark matter forms. These scales evolve outside
the horizon in a purely radiation dominated universe. The perturbations at these scales
are not affected by LFDM physics except in determining the initial conditions which are
not necessary for the following reasons: the photon perturbations outside the horizon are
constant and are set by the potential. For scales outside the horizon, the potential changes
by a factor 9/10 in making a transition from radiation to the matter-dominated era (e.g. [112]
and references therein). If the era of LFDM lies deep inside RD era, the change is negligible.
This explains the large scale behaviour of potential in figure 4.
(b) k > klfdm: these scales enter the horizon before the formation of the cold dark
matter. As zlfdm  zeq these scales evolve sub-horizon in the radiation dominated era. In
radiation dominated era, the potential Φ(k, η) decays as (kη)−2 for kη  1 if perturbations in
radiation determine the evolution of the potential (e.g. [112]). At sub-horizon scales, neutrino
perturbation decay exponentially while photon density perturbations equal baryon pertur-
bations which oscillate with nearly constant amplitude for η < ηd. The CDM perturbations,
on the other hand, grow logarithmically during this phase. This means matter perturbations
can determine the evolution of the potential Φ(k, η) even in deep radiation dominated era.
This behaviour is seen the evolution of the potential for k = 0.8 in figure 4 as flattening of the
potential for larger η. For LFDM models, the potential decay is sharper because of the CDM
forms late. This behaviour along with other features for kη > 1 seen in figure 4 explains
how the evolution of Φ in LFDM models differs from the ΛCDM model. It is important to
note that this difference is most pronounced for scales that enter the horizon at z ' zf . For
even smaller scales, the effect of matter perturbations is less important as these scales enter
the horizon when ρdm/ρr is lower and therefore matter perturbations have a smaller impact
on their evolution. Equivalently, it is seen from eq. (4.1) that the photon dipole depends
on the integral of the potential and as the potential decays inside the horizon, most of the
contribution comes from epochs close to the horizon entry. It is also seen in the evolution
of the potential for k = 6.4 Mpc−1. The potential deviates insignificantly from the ΛCDM
model for kη & 1 but shows sharp deviation at later times. The evolution of perturbations
at these scales (from η  1 to η = ηd) correspond roughly to the case of no dark matter
at all time. The potential for this case falls exponentially for large η which occurs because
the only source of potential in the LFDM model is photon and neutrino perturbations which
both decay exponentially for η > ηd. It is of interest to note that the potential at k = kd is
dominated by matter perturbations for z . 105.
For the LFDM model shown in figure 4, the scales of interest for measuring the deviation
from the ΛCDM model lie in the range 0.6–4 Mpc−1. These scales enter the horizon before,
but not significantly before, the era of matter formation and the perturbations at these scales
suffer Silk damping after z ' 104.
For models such as WDM and ULA, the preceding discussion is directly applicable. We
briefly discuss the charged decaying particle case below.
Charged decaying particle. For charged decay model, the variation in the dynamics of
Φ(k, η) closely follows the previous discussion for LFDM model: Φ(k, η) nearly follows the
– 14 –
JCAP07(2017)012
 10  2.5  0.04
5 5 5
LFDM z  = 2 x 10 LFDM z  = 10 LFDM z  = 10f f f
ΛCDM ΛCDM ΛCDM
 2
 8
 0.03
 1.5
 6
 0.02
 1
 4
 0.5  0.01
 2
 0
 0
 0
-0.5
-0.01
-2
-1
-4 -1.5 -0.02
 0  50  100  150  200 η = η  250  0  50  100 η = η  150  200  250  10  20  30 η = ηD D 40  50  60  70D
-1 -1 -1
η(Mpc ) η(Mpc ) η(Mpc )
 10  2.5  0.04
5 5 5
CHDM zdecay = 2 x 10 CHDM zdecay = 10
CHDM zdecay = 10
ΛCDM ΛCDM ΛCDM
 2
 8
 0.03
 1.5
 6
 0.02
 1
 4
 0.5  0.01
 2
 0
 0
 0
-0.5
-0.01
-2
-1
-4 -1.5 -0.02
 0  50  100  150  200 η = η  250  0  50  100 η = ηD  150  200  250  10  20  30 η = ηD 40  50  60  70D
-1 -1 -1
η(Mpc ) η(Mpc ) η(Mpc )
Figure 5. The evolution of Θ1(k, η) for k = 0.4, 0.8, 6.4Mpc
−1 for CHDM (lower panel) and LFDM
(upper panel) for zf = z
5
decay = 10 . In each plot we also show the evolution of the corresponding
scale for ΛCDM model. In each plot the vertical dotted line marks the time when k = kd (eq. (4.2))
or that particular scale enters the Silk damping regime.
potential for ΛCDM model at large scales. At very small scales the effect is suppressed
because the matter density is small when they enter the horizon (figure 4). The main impact
is captured by intermediate scales that enter the horizon around but before the time of decay.
Some of the salient differences between the two cases can be seen in figure 4. For the
Charged decaying particle model, the potential can exceed the ΛCDM values for a range of
wavenumbers, e.g. k = 6.4 Mpc−1 in the figure.
It follows from our discussion that the impact of alternative DM models on Φ(k, η)
declines if the transition (decay) redshift moves deeper into the radiation dominated era.
We note that the tight-coupling approximation while capturing the essential physics, tends
to break down close to the recombination era; this is relevant for the computation of
y-distortion [57, 114]. However, this approximation allows us to compare our results in two
different gauges and identify the main determinants of altering CMB distortion for alternative
dark matter models.
In figure 5 we show the dynamics of Θ1(k, η) for a range of alternative DM models.
We compute Θ1(k, η) directly from CMBFAST code but also verify that eq. (4.1), which shows
that Θ1(k, η) depends on the history of the variation of Φ(k, η), explains the main difference
expected for alternative DM models. We establish it by running a smaller set of equations
for density and velocity perturbations in photon-baryon fluid and dark matter along with
the evolution of Φ(k, η) in conformal Newtonian gauge [113].
In figure 6, we show the values of Θ1(k, η) at z = 1000 of different models at the same
redshift. All the models shown correspond roughly to cases where the decrement in the power
spectrum occurs at k ' 0.3hMpc−1.
The preceding discussion allows us to establish that the main impact of alternative dark
matter models is to alter y-distortion and not µ- or i-distortion. The µ-distortion era occurs
in the redshift range 2× 105 . z . 2× 106. In this era the scales that dissipate their energy
and therefore cause the CMB distortion lie in the range 500 Mpc−1 . k . 14000 Mpc−1. In
– 15 –
Θ1 Θ1
Θ1 Θ1
Θ1 Θ1
JCAP07(2017)012
8
10
6
10
4
10
2
10
0
10
-2
10
-4
10
-6
10
-8 ΛCDM
10 5
LFDM zf = 1 x 10
5
CHDM zdecay = 1 x 10-10
10 WDM mwdm = 0.33 keV
-24
ULADM ma = 10  eV
-12
10
-3 -2 -1 0 1
10 10 10 10 10
-1
k (h Mpc )
Figure 6. CMB dipole transfer function of the four dark matter candidates considered in this work
along with ΛCDM. The specifications of the models are same as that of figure-1.
the range of redshifts of interest, the matter power for alternative models at these scales is
highly suppressed as compared to the ΛCDM model, as seen in figure 2. However, as argued
above, this does not have a significant impact on Θ1 because it depends on the history of
the evolution of the potential Φ(k, η) (eq. (4.1)). As the potential decays after horizon entry
(figure 4), the photon dipole is governed by the initial condition of the potential and its
evolution closer to the time of horizon entry of the scale (η = ηe) rather than its value at
η = ηd. The scale k ' 500 Mpc−1 (which decays at the end of the µ-distortion era) enters
the horizon at z ' 2×108. As the ratio of radiation to matter energy density at this redshift
ρdm/ρr ' 10−5, the potential is insensitive to even a substantial change in the matter power
at this scale at early times. The initial conditions for different models are not the same as
the ΛCDM model. However, as discussed above, requiring these models to agree with large
scale observations forces the impact of the change in initial conditions on the potential to
be negligible.
This also implies that the dynamics of scales that determine µ-distortion of CMB are
minimally affected by a change in the dark matter model. As we showed above, the main
effect in the photon dipole is caused by scales in the range 0.6–4 Mpc−1, this also means the
impact of alternative dark matter on the i-distortion era is also negligible. We test this in our
study by computing the photon dipole for scales in the range k < 500 Mpc−1 for alternative
models numerically and using an analytic approximation for even smaller scales to discern
the impact of initial conditions (the first term of eq. (4.1)). For all the models we consider,
the µ- and i-distortion is less than 0.1%.
5 Results
Using eqs. (4.1) and (3.7) one can analytically show that the dissipation at any given time is
dominated by scales such that k ' kd. Such an estimate is based on approximating Θ1(k, η)
by the first term of eq. (4.1) (e.g. [57]). This gives:
k2 ( )
S 2 2 2SD (k, η) ∝ 2Pφ (k, 0) sin (kcsη) exp −k /kd YSZ (5.1)τ̇c
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Θ1(k,η)
JCAP07(2017)012
Here PΦ(k, 0) is the power spectrum of the potential at the initial time. Using PΦ(k, 0) ∝
k−4+ns (e.g. [57, 112]) where ns '∫1 is the scalar spectral index and integrating over k gives:( )
〈SSD〉 ∝ dkkns sin2 (cskη) exp −k2/k2d (5.2)
This k integral in the equation is dominated by k ' kd which means the energy dissipation
at any given time is governed by this condition. This result holds even when both the terms
in eq. (4.1) are retained.5
This means that the dissipation is dominated by photon dipole at k ' kd at any time η,
Θ1D ≡ Θ1(kd, η). In figure 7 we show the redshift evolution of ∆Θ1D, the difference between
Θ1D for different dark matter models and that of ΛCDM. The set of model parameters we
have chosen is provided in table 1. The difference decreases as the redshift of formation
zf for LFDM, and the decay redshift zdecay for CHDM and masses of WDM and Axion
are increased. One further point to note is that the difference ∆Θ1D for alternative models
increases as the redshift decreases. These results are in agreement with our discussion in the
previous section which suggested that the alternative dark matter models yield substantial
difference only during the y-distortion era.
Using eqs. (3.7), (3.8) and (3.11), we calculate y-parameter for different models; Θ1(k, η)
is normalized using the condition on mass dispersion at scale R = 8 h−1Mpc at the present
epoch: σ8(η0) = 0.8225 [1]. The list of y-parameters for different models is provided in
table 1. We note that the y-parameter can vary by up to 10% for a range of models. This
behaviour is in line with the variation of ∆Θ1D. The y-parameter approaches its value for
ΛCDM model as zf for LFDM and zdecay for CHDM models are increased, and also when the
mass of WDM and axion is increased as expected from our discussion above. Furthermore,
we can use eq. (3.5) to determine the shape of distorted CMB spectrum.
Can the y-distortion be used to distinguish between different dark matter models? We
address this question by comparing models where the power is cut at nearly the same scale. In
figure 8, we show the evolution of ∆Θ1D and the spectral shape of the distorted spectrum for
models for which the power has been cut at k ' 0.3hMpc−1. Figure 8 also shows the distorted
spectrum from y-distortion for these cases; the spectra have the same shape (eq. (3.5)) but
differ owing to slightly different values of the y-parameter. A more detailed analysis based
on understanding degeneracies between parameters of different models would be needed to
quantify the results shown in figure 8. We also note that this effect can be masked by several
late-time phenomena like the Epoch of Reionization, thermal Sunyaev-Zel’dovich(tSZ) effects
in the galaxy groups and clusters that give rise to the spectral shape that arises from y-type
distortion with orders of magnitude higher y-parameters.
Many of the models we consider are already tightly constrained by cosmological obser-
vations. From galaxy clustering and CMB anisotropy data and observed neutral hydrogen
(HI) abundance at high redshifts, the LFDM models require zf & 105 [48, 79]. Lyman-α
data puts even stronger constraints on such models [79]. Recent studies of ULA models
suggest that m −22a & 10 eV is consistent with current observational data [47]. Other studies
based on the abundance of HI at high redshift constrain the axion mass m & 10−23a eV [48].
5While the first term of eq. (4.1) gives a reasonable analytic estimate, the second term generally dominates
the first term even for the ΛCDM model. The factor proportional to sin(cskη) in the second term dominates
over the first term and cancels it but the remainder has the form of the first term and has a similar order of
magnitude. The factor proportional to cos(cskη) in the second term is generally sub-dominant. These two
factors contain information of the evolution of potential which is germane to our work.
– 17 –
JCAP07(2017)012
 1  1
4
LFDM z  = 5 x 10 mwdm = 0.33 keVf
5
LFDM z  = 1 x 10 mwdm = 0.7 keVf
5
LFDM z  = 2 x 10 mwdm = 1 keVf
 0.1 5LFDM z m  = 2 keVf = 5 x 10  0.1 wdm
mwdm = 5 keV
 0.01
 0.01
 0.001
 0.001
 0.0001
 0.0001
 1e-05
 1e-05
 1e-06
 1e-07  1e-06
 100  1000  10000  100000  100  1000  10000  100000
z z
 1  1
-21 4
ma = 10  eV CHDM zdecay = 5 x 10
-23 5
ma = 2.8 x 10  eV CHDM zdecay = 1 x 10
-24 5
ma = 10  eV CHDM zdecay = 2 x 10 0.1
-25 eV 5
ma = 2.8 x 10 CHDM zdecay = 5 x 10
 0.1
 0.01
 0.001
 0.01
 0.0001
 0.001
 1e-05
 1e-06
 0.0001
 1e-07
 1e-08  1e-05
 100  1000  10000  100000  100  1000  10000  100000
z z
Figure 7. The difference between Θ1D for dark matter models considered in this work and ΛCDM
model: ∆Θ1D. Clockwise from top-left ∆Θ1D for LFDM, WDM, charged particle decay dark matter
and ULADM are plotted, respectively.
These constraints suggest that some of the models shown in table 1 are ruled out and in
particular the ones that show largest deviations from the ΛCDM model. However, there is
significant variation in the predictions of different models, and we might expect to see up
to 5% deviation (e.g. zf ' 105) within the framework of constraints that arise from galaxy
clustering [43] and CMB anisotropies [115].
The charged decaying particle model offers a more complicated scenario of spectral
distortion of the CMB. As noted in section 2.4, the relativistic electron released as the decay
product rapidly thermalizes its energy with the thermal plasma. Eq. (2.5) gives the fractional
increase in the photon energy density due to this process. This constrains the mass difference
between the two heavy particles ∆M to be tiny. This scenario therefore presents two different
ways of distinguishing this model from the ΛCDM model. For instance, for z 5decay ' 10 ,
the decay will cause i-type spectral distortion in the CMB (whose amplitude will depend
on ∆M) while Silk damping causes a difference in y-distortion. In principle, given their
distinct spectral signatures, these two phases of distortions are distinguishable from each
other [100, 101].
The upcoming experiment Primordial Inflation Explorer (PIXIE) [70] is likely to im-
prove the FIRAS bounds on CMB spectral distortion by many orders of magnitude. Its
projected sensitivity corresponds to: y ' 2 × 10−9 and µ ' 10−8. However, unlike the µ-
and i-distortion, the y-distortion created during the pre-recombination era can be masked
by CMB distortion in the post-recombination era. For instance, the epoch of reionization
– 18 –
∆Θ1D ∆Θ1D
∆Θ1D ∆Θ1D
JCAP07(2017)012
Model Parameter y × 109 % difference of y from ΛCDM
ΛCDM [1] 4.4180 0.0
zf = 5× 104 3.8561 14.57
zf = 1× 105 4.1001 7.75
LFDM zf = 2× 105 4.3037 2.65
zf = 5× 105 4.3959 0.50
mwdm = 0.33 keV 4.2178 4.74
mwdm = 0.70 keV 4.3105 2.49
WDM mwdm = 1.00 keV 4.3398 1.80
mwdm = 2.00 keV 4.3680 1.14
mwdm = 5.00 keV 4.3798 0.87
zdecay = 5× 104 3.8913 13.53
CHDM z 5decay = 1× 10 4.1884 5.48
z 5decay = 2× 10 4.2945 2.87
z 5decay = 5× 10 4.4002 0.4
m −25a = 2.8× 10 eV 3.8840 13.74
ma = 1.0× 10−24 eV 4.2812 3.19
ULA DM ma = 2.8× 10−23 eV 4.3990 0.43
ma = 1.0× 10−21 eV 4.4177 6.8× 10−3
Table 1. This table lists the values of y-parameter for alternative dark matter models and compared
with ΛCDM model.
is likely to produce global CMB distortion corresponding to y ' a few × 10−7, though the
most dominating source of post-recombination y-distortion is the tSZ effect in galaxy groups
and clusters corresponding to y ' 2 × 10−6 [64]. These sources would constitute a strong
foreground to the pre-recombination y-distortion [70]. In principle, it might be possible to
distinguish the pre-recombination y-distortion from reionization signal using spatial informa-
tion (e.g. [57, 104]) but it would be a challenge.
6 Conclusion
The large-scale behaviour of dark matter has been well established by a range of cosmological
observables such as CMB anisotropies and clustering of galaxies. However, there remain issues
with this model at small scales, and the nature of the dominant fraction of dark matter in
the universe remains a mystery.
The evolution of spectral distortion in the pre-recombination era allows us to study
scales in the range 104 Mpc−1 < k < 0.3 Mpc−1 as density perturbations at these scale decay
and leave observable signatures on CMB spectrum. These perturbations are in the linear
regime of their growth in the pre-recombination and therefore can be theoretically modelled
very accurately.
– 19 –
JCAP07(2017)012
 1  1e-25
5
LFDM zf = 1 x 10
ΛCDM
5 5
CHDM zdecay = 1 x 10
LFDM zf = 1 x 10
5
WDM mwdm = 0.33 keV  8e-26 CHDM zdecay = 1 x 10
-24
ULADM ma = 10  eV
WDM mwdm = 0.33 keV
-24
 0.1 ULADM ma = 10  eV
 6e-26
 4e-26
 0.01
 2e-26
 0.001
 0
-2e-26
 0.0001
-4e-26
 1e-05 -6e-26
 100  1000  10000  100000  1  10
z x
Figure 8. Comparison of ∆Θ1D(Left) and y-parameter(Right) for four different dark matter candi-
dates having small scale power suppressed at same value of k ∼ 0.3hMpc−1.
We consider four alternative dark matter models which give significant deviations of
small-scale power as compared to the ΛCDM model and study the spectral distortion of
CMB owing to Silk damping for these models. These models are motivated by different
aspects of physics in the early universe: phase transition (LFDM), free-streaming of massive
particles (WDM), the decay of massive charged particles(CHDM), and dynamics of a scalar
field with nonzero effective mass (ULA).
We show that main impact of the models we consider is to alter the late time spectral
distortion history by lowering the y-parameter by a few percent for an acceptable range of
parameters for these models (table 1).
In this work, we only consider models that leave invariant the matter-radiation equality.
This excludes models such as decaying dark matter particles that create relativistic decay
products thereby delaying the matter-radiation equality epoch (e.g. [116, 117]). We also
excluded models that are based on extra relativistic degrees of freedom (for details see [1]
and references therein) or are based on a change in initial conditions (e.g. [67, 68]).
Our analysis suggests that all class of models that give suppression of power as compared
to the ΛCDM model should result in late time spectral distortion. In section 4 we present
general arguments which show that the change in spectral distortion is dominated by the
evolution of potential after the scale enters the horizon. This effect scales as the ratio of
energy densities of dark matter and radiation ρdm/ρr and is negligible at early times. This
might also allow one to distinguish the impact of modifying the initial matter power spectrum
on CMB spectral distortion from a change in the dark matter model that affects the matter
power at small scales, as the former would cause all the three forms of CMB distortions while
the latter would not.
Future experiments such as PIXIE and CMB-S4 have the potential to establish the
nature of dark matter (e.g. [115]) by unprecedented improvement in mapping the CMB
spectral and spatial structures. Our work is one step in that direction and points out the
challenges involved in such an endeavour.
Acknowledgments
We thank the anonymous referee for sending useful comments and suggestions that can
make the paper better. The authors would like to acknowledge Rishi Khatri, Jens Chluba,
– 20 –
∆Θ1D
-2 -1 -1
∆I  (W m  ster  Hz )
ν
JCAP07(2017)012
David J E Marsh, Ayuki Kamada, Sunny Vagnozzi and Farinaldo Queiroz for their useful
comments and suggestions.
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