TU-1135

#  Constraints on small-scale primordial density fluctuation
from cosmic microwave background through dark matter annihilation

Masahiro Kawasaki  ICRR, University of Tokyo, Kashiwa, 277-8582, Japan  Kavli
IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, 277-8583, Japan  Hiromasa
Nakatsuka  ICRR, University of Tokyo, Kashiwa, 277-8582, Japan  Kazunori
Nakayama  Department of Physics, Tohoku University, Sendai, Miyagi 980-8578,
Japan  Department of Physics, Faculty of Science, The University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan

###### Abstract

The cosmic microwave background (CMB) observation by the Planck satellite
precisely determines primordial curvature fluctuations on larger scales than
$\mathcal{O}(1)\,\mathrm{Mpc}$, while the small-scale curvature fluctuation is
still less constrained. The constraint on small-scale fluctuations is highly
improved if we assume the standard thermal relic dark matter scenario. When
small-scale fluctuations are large enough, dense regions collapse to form
small halos even in a redshift $z\gtrsim 10^{3}$, which is called
“ultracompact minihalos”. These minihalos enhance the annihilation of the dark
matter and it is constrained by observations such as extragalactic gamma rays
and the CMB. We revisit the effect of minihalos formed by the small-scale
density fluctuations and calculate the ionization history modified by the dark
matter annihilation. We perform the Markov Chain Monte Carlo method to
constrain the size of small-scale curvature fluctuations by the CMB power
spectrum. It is found that the constraint from the CMB power spectrum is
comparable to that from the extragalactic gamma rays. We confirm that our
constraint mainly comes from the energy injection in early time ($z\gtrsim
100$) and hence it is independent of the uncertainty of minihalo properties in
the late time.

###### Contents

  1. I Introduction
  2. II Abundance and property of halos
    1. II.1 NFW profile
    2. II.2 Moore profile
    3. II.3 Comparison
  3. III Modified thermal history due to dark matter annihilation
  4. IV Constraints on small-scale density fluctuation
  5. V Conclusion

##  I Introduction

The Planck satellite precisely measured the cosmic microwave background (CMB)
anisotropy and determined primordial curvature fluctuations on larger scales
than $\mathcal{O}(1)\mathrm{Mpc}$, which shows the nearly flat power spectrum
consistent with the slow-roll inflation. On small scales, on the other hand,
the sensitivity is limited due to the free streaming of photons (Silk
damping), and little is known about the power spectrum. However, the small-
scale fluctuations are expected to provide hints to new physics beyond the
slow-roll inflation. Some measurements have already put upper limits on the
small-scale power spectrum, e.g. the clustering of galaxies [1] the Lyman-
alpha forest [2] the gravitational waves [3, 4], the CMB spectral distortion
[5], and the primordial black holes [6]. These methods, however, only cover
limited scales or put much loose upper limits compared to the CMB anisotropy.

The constraints on the small-scale density fluctuation are highly improved
when the annihilation of dark matter particles is taken into account. Assuming
the standard thermal freeze-out scenario of the dark matter, the annihilation
cross section is estimated as $\braket{\sigma v}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}$ for the $s$-wave annihilation. When small-
scale fluctuations are large enough, dense regions collapse to form small
halos in the early universe, even in a redshift $z\gtrsim 10^{3}$, which is
often called “ultracompact minihalos” [7]. Such minihalos enhance the
annihilation of the dark matter, which is severely constrained by the
extragalactic gamma rays [8, 9, 10]. The constraint on the small-scale power
spectrum is obtained as $\mathcal{P}_{\zeta}\lesssim 10^{-6}$ over the broad
scales $k\in[10^{2},10^{6}]~{}\mathrm{Mpc}^{-1}$. This constraint, however,
has uncertainty related to the property of the dark matter, the formation of
minihalos, and the stability of minihalos on the late time.

The CMB anisotropy measurement also constrains the dark matter annihilation
signal. The dark matter annihilation injects the energetic decay products into
the thermal plasma, which ionizes the neutral hydrogen and helium. The CMB
power spectrum is sensitive to modification of the ionization history around
$z\sim 600$ [11], and it provides a reliable constraint on the exotic energy
injection in the early universe. Such constraints have been intensively
investigated for the case of (almost) spatially homogeneous dark matter
distribution [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], and the
Planck satellite puts the upper bound on the annihilation cross section as
$\braket{\sigma v}/m_{\chi}\lesssim
10^{-27}\mathrm{cm}^{3}/\mathrm{s}/\mathrm{GeV}$ for $s$-wave annihilation
with $\chi\chi\to e^{+}e^{-}$ channel [20], where $\chi$ denotes the dark
matter particle and $m_{\chi}$ is the dark matter mass. The annihilation
signal is enhanced if the dark matter distribution is (strongly)
inhomogeneous, but it is known that the enhancement is negligible for the
large halos formed under the nearly scale invariant primordial power spectrum
predicted by the standard slow-roll inflation [26], which we call the
“standard halos”, since the standard halos are formed in late time. On the
other hand, large fluctuations on small scales can enhance the production of
minihalos and the effect on the CMB anisotropy can also be significant. The
pioneering work [27] investigated the modified ionization history by the dark
matter annihilation from ultracompact minihalos. However, the author mainly
used the profile of minihalos unfavored from the view point of the recent
numerical calculation [28, 29] and derived the constraints just by comparing
the optical depth measured by the WMAP, and did not put the upper bound on the
primordial curvature power spectrum. Other works [10, 30, 31] also inherit
some incomplete treatments of Ref. [27]. Therefore, it is necessary to
consistently treat the CMB constraint on the small-scale fluctuations based on
the recent updates.

In this work, we revisit the minihalo formation due to the small-scale density
fluctuations and calculate the modified CMB power spectrum including the
effect of dark matter annihilation. In Sec. II, we describe the properties of
minihalos. Both the density profile of each minihalo and the abundance (or
mass distribution) of minihalos are uncertain and hence we adopt two methods:
(i) the Navarro-Frenk-White (NFW) profile with the Press-Schechter formalism
used in Ref. [32], and (ii) the Moore profile with the peak theory motivated
by the numerical calculation of minihalos [29]. We compare both methods and
estimate the enhancement of the dark matter annihilation, which is dubbed as
the “boost factor”. In Sec. III, the energy injection from the dark matter
annihilation in the minihalos is calculated assuming the standard thermal
freezeout scenario of dark matter. The modified ionization history of the
universe is calculated by the RECFAST [33, 34, 35] code with the exotic energy
injection. We use the CAMB [36] and CosmoMC [37] code to calculate the CMB
power spectrum and perform the Markov Chain Monte Carlo methods (MCMC) to put
the exclusion limit of the parameters, including the small-scale curvature
fluctuation. We conclude this paper in Sec. V.

##  II Abundance and property of halos

In this section, we summarize the abundance, profile, and mean squared density
of minihalos. Halos are formed when high-density regions collapse
gravitationally. The abundance and profile of the halos are estimated based on
the density power spectrum, while the estimation has some uncertainty
depending on methods. We discuss two methods to estimate halos and compare
their effects on the enhancement of the dark matter annihilation signal.

At first, let us introduce the small-scale density fluctuation. Since the
curvature fluctuations are constant on the superhorizon scale, it is useful to
characterize the small-scale density fluctuation in terms of the curvature
fluctuation. In this paper, we use the delta-function-type power spectrum of
the small-scale curvature perturbation defined by

$\displaystyle\mathcal{P}_{\mathcal{R}}(k)=A^{2}k_{*}\delta(k-k_{*}),$ (1)

where $k_{*}$ is a wave number of the fluctuation and $A^{2}$ is an amplitude,
which is assumed to be much larger than the naive value extrapolated from the
flat power spectrum at the CMB scales. We neglect the flat power spectrum to
estimate the halo abundance since structure formation from the flat power
spectrum is too late to affect the CMB constraints on the dark matter
annihilation [16, 38]. Our method is easily applied to other spiky power
spectra. The density fluctuations are related to the curvature fluctuations as

$\displaystyle\mathcal{P}_{\delta}(k,a)=\frac{4}{25}\left(D(a)\frac{k^{2}T(k)}{\Omega_{m0}H_{0}^{2}}\right)^{2}\mathcal{P}_{\mathcal{R}}(k),$
(2)

where $\Omega_{m0}\simeq 0.311$ is the current density parameter of the total
matter, $D(a)$ is a growth factor as a function of the cosmic scale factor $a$
and $T(k)$ is the transfer function. We follow Ref. [32] for a choice of them.
The growth factor is given by [39]

$\displaystyle D(a)$
$\displaystyle=a\cdot{}_{2}F_{1}\left[\frac{w-1}{2w},\frac{-1}{3w},1-\frac{5}{6w};1-X(a)\right],$
(3)

where $w$ is the equation of state parameter of dark energy (we will take
$w=-1$ in what follows), ${}_{2}F_{1}$ is the Hypergeometric function, and
$X(a)\equiv 1+a^{3}\Omega_{\Lambda}/\Omega_{m0}$ with $\Omega_{\Lambda}$ being
the density parameter of dark energy. The transfer function without baryonic
effects is given by [32, 40]

$\displaystyle T(q)$
$\displaystyle=\frac{\ln(2e+1.8q)}{\ln(2e+1.8q)+(14.2+\frac{731}{1+62.5q})q^{2}}\quad,\quad
q\equiv\frac{k}{\Omega_{m0}h^{2}\mathrm{Mpc}^{-1}},$ (4)

where $h$ is the present Hubble parameter in units of $100\,{\rm km/s/Mpc}$.
The above estimation ignores the kinetic decoupling and the free streaming of
the dark matter, which may washout density fluctuations on scales smaller than
the cut-off scale $k_{c}$. For the typical weakly interacting massive particle
(WIMP) dark matter, the cut off scale is given by $k_{c}\sim
10^{6-7}~{}\mathrm{Mpc}^{-1}$ [41, 42]. In this paper, we consider the spiky
power spectrum with peak scale
$k_{*}\in[1~{}\mathrm{Mpc}^{-1},10^{6}~{}\mathrm{Mpc}^{-1}]$.

In order to evaluate the energy flux from dark matter annihilation, we need
information about the number density and the density profile of minihalos. In
the context of ultracompact minihalos, previous works often used the steep
inner profile, $\rho(r)\propto r^{-9/4}$, which is analytically derived for
collisionless spherical infall on to an overdense region [43]. A recent N-body
numerical calculation [44, 28, 29], however, found that minihalos have the
shallower inner profiles with the inner slopes between $\rho(r)\propto r^{-1}$
and $r^{-3/2}$. In this paper, we investigate the halos with both NFW profile
with $\rho(r)\propto r^{-1}$ and Moore profile with $\rho(r)\propto r^{-3/2}$.

According to the numerical calculation in [29], the profile of minihalos
depends on their merger history. When we use the spiky power spectrum, the
isolated minihalos are formed with the Moore profile. The parameters of the
halo profile are determined by the environment at the formation, and the
parameters depend only on the formation time. In this case, the halo abundance
is estimated by the peak theory in which the isolated peak number is counted.
On the other hand, when the power spectrum has a step-like form on small
scales, the formed halos have a broad distribution on their mass and
experience many times of mergers. After mergers, the halo profile becomes
shallower, and the NFW profile well fits resultant halo profiles. In this
case, the Press-Schechter formalism is suitable for counting the halo number
since it is closely related to the coarse-grained density contrast and
includes the effect of mergers [45, 46, 47, 48].

Considering the above discussion, we adopt two methods to investigate
minihalos formed at high redshifts. The first method was investigated in Ref.
[32], where the authors used the NFW profile with the Press-Schechter
formalism to estimate the time-dependent concentration parameter of the halo
profile, which will be defined later. The second method was investigated in
Ref. [29], where the Moore profile was used with the peak theory to estimate
the collapsed time of halos. Although we use the delta-function type power
spectrum for simplicity, the primordial power spectrum inevitably has a peak
width. The evolution of minihalos follows either of two pictures depending on
the width, while a threshold between the above two pictures is not clear.
Thus, in this paper, we consider both pictures, which we simply call “NFW
profile” and “Moore profile”. The annihilation signal is evaluated on both
scenarios in the following discussion.

We compare both pictures to estimate the dark matter annihilation flux. Since
the annihilation rate is proportional to the squared number density of dark
matter particle, $n_{\chi}^{2}=\rho_{\chi}^{2}/m_{\chi}^{2}$, we calculate the
square average of the halo density profile,
${\braket{\rho_{\mathrm{halo}}(z)^{2}}}$, where the bracket represents the
average over halos and is defined in Eqs. (16) and (28).

###  II.1 NFW profile

The NFW profile is one of the standard halo profiles, which is defined as

$\displaystyle\rho_{\mathrm{NFW}}(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}},$
(5)

where $\rho_{s}$ and $r_{s}$ are parameters that depend on the halo mass and
virial radius. The virial radius of a halo with mass $M$ is calculated by the
spherical collapse model as

$\displaystyle
r_{h}(z,M)=\left(\frac{3M}{4\pi\Delta_{\mathrm{vir}}(z)\rho_{m}(z)}\right)^{1/3},$
(6)

where $\rho_{m}(z)$ is the matter density and $\Delta_{\mathrm{vir}}(z)$ is
given by the fitting function neglecting the radiation energy density [49]:

$\displaystyle\Delta_{\mathrm{vir}}(z)$
$\displaystyle\simeq\frac{18\pi^{2}+82x-39x^{2}}{1+x}\quad,\quad
x\equiv\frac{\Omega_{m}(z)}{\Omega_{m}(z)+\Omega_{\Lambda}}-1.$ (7)

To determine $\rho_{s}$, we require that the mass of the halo is given by the
integration of $\rho_{\mathrm{NFW}}(r)$ up to the virial radius,
$M=\int^{r_{h}}\mathrm{d}^{3}r\rho_{\mathrm{NFW}}(r)$. It is given by

$\displaystyle\rho_{s}=\frac{c}{\ln(1+c)-c/(1+c)}\frac{\Delta_{\mathrm{vir}}(z)\rho_{m}(z)}{3},$
(8)

with $c=r_{h}/r_{s}$ is called the concentration parameter and determined by
numerical simulations. There are various choice of $c$ [50, 51, 52, 53], and
in this paper, we use the fitting formula by [54]:

$\displaystyle
c(t)=4\times\left[1+\left(\frac{t}{3.75t_{0.04}}\right)^{8.4}\right]^{1/8},$
(9)

where $t_{0.04}$ is the time when a halo gained $4\%$ of its mass evaluated at
$t$. In summary, we can determine the halo profile by its mass $M$ and the
formation time $t_{0.04}$.

Next, we evaluate the halo abundance based on the Press-Schechter formalism
[55]. The halos are formed by density fluctuations growing on the subhorizon
regime. Although the Sheth-Tormen mass function [56] well fits the mass
spectrum of standard halos at low redshifts, it is unclear if this function is
applicable to minihalos formed at high redshifts. Thus we use the Press-
Schechter formalism in this paper for simplicity. In the Press-Schechter
formalism, the formation rate of halos is estimated by the coarse-grained
density fluctuation,

$\displaystyle\delta_{R}(t,\bm{x})=\int\mathrm{d}^{3}\bm{x}^{\prime}~{}W(|\bm{x}-\bm{x}^{\prime}|,R)\delta(t,\bm{x}^{\prime}),$
(10)

where $W(x,R)$ is the window function. We choose the Gaussian window function,

$\displaystyle W(x,R)$
$\displaystyle=(2\pi)^{-3/2}R^{-3}\exp\left(-\frac{r^{2}}{2R^{2}}\right)\quad,\quad\tilde{W}(kR)=\exp\left(-(kR)^{2}/2\right),$
(11)

where $\tilde{W}$ is the Fourier-transformation of $W$. The mass and radius of
the collapsed object are related as

$\displaystyle M(R)=W(0,R)^{-1}\rho_{m0}=(2\pi)^{3/2}R^{3}\rho_{m0},$ (12)

with the present matter density $\rho_{m0}$. The halo collapses when
$\delta_{R}$ overcomes the critical value $\delta_{c}=1.686$ in the linear
perturbation theory. The formation rate is estimated by assuming the Gaussian
distribution of fluctuations,

$\displaystyle\beta(M,z)=\frac{1}{\sqrt{2\pi}\sigma_{M}(z)}\int_{\delta_{c}}^{\infty}\mathrm{d}\delta\exp\left(-\frac{\delta^{2}}{2\sigma^{2}_{M}(z)}\right)=\frac{1}{2}\mathrm{erfc}(\nu_{M}/\sqrt{2}),$
(13)

where $\sigma_{M}(z)$ is the variance of coarse-grained density fluctuations
and $\nu_{M}\equiv\delta_{c}/\sigma_{M}$. The variance is estimated by

$\displaystyle\sigma_{M}^{2}(z)=\braket{\delta_{R}(t,\bm{x})\delta_{R}(t,\bm{x})}=\int\frac{\mathrm{d}k}{k}\mathcal{P}_{\delta}(k,z)\tilde{W}^{2}(kR).$
(14)

Based on the Press-Schechter formalism, the mass distribution of halos is
derived as

$\displaystyle\frac{\mathrm{d}n}{\mathrm{d}\ln
M}(M,z)=-2\frac{\rho_{m0}}{M}\frac{\mathrm{d}\beta}{\mathrm{d}\ln
M}=\sqrt{\frac{2}{\pi}}\frac{\rho_{m0}}{M}\frac{-\mathrm{d}\ln\sigma_{M}}{\mathrm{d}\ln
M}\nu_{M}\exp(-\nu_{M}^{2}/2),$ (15)

where $n$ represents the comoving number density of halos.

Using the mass distribution, we estimate the formation time $t_{0.04}$
following the method in Ref. [32]. The normalized mass distribution,
$f(M,z)\equiv({M}/{\rho_{m0}})\frac{\mathrm{d}n}{\mathrm{d}\ln M}$, has a peak
at mass $M_{c}(z)$ defined by $[\mathrm{d}f(M,z)/\mathrm{d}M]|_{M=M_{c}}=0$.
$M_{c}(z)$ represents the typical halo mass at redshift $z$. We need to
calculate the redshift $z_{0.04}$ when $M_{c}(z_{0.04})$ is 4% of $M_{c}(z)$.
We define the halo formation redshift $z_{0.04}(z)$ relative to the halo at
$z$ by $M_{c}(z_{0.04}(z))=0.04M_{c}(z)$. We show $z_{0.04}(z)$ for $z=1$ and
$100$ in Fig. 1 based on the power spectrum with $A^{2}=10^{-6}$ and
$k_{*}=10^{3}~{}\mathrm{Mpc}^{-1}$. The left panel of Fig. 1 shows the
normalized mass distribution at $z=1$ and $100$ (solid lines) . We also show
the mass distribution at $z\simeq 198$ and $548$ (dotted lines) where their
typical masses are $0.04$ times smaller than these at $z=1$ and $100$,
respectively. The right panel of the Fig. 1 shows the relation between
$z_{0.04}(z)$ and $z$. The formation redshift $z_{0.04}$ is much larger than
$z$ on low redshift due to the large density fluctuations. Using this
relation, we can calculate the concentration parameter (9).

Figure 1:  The mass distribution of minihalos calculated by the Press-
Schechter formalism (left panel) and the redshift of the halo formation (right
panel). In both case, we use the power spectrum (1) with $A^{2}=10^{-6}$ and
$k_{*}=10^{3}~{}\mathrm{Mpc}^{-1}$. In the left panel, the dotted lines
represent the mass distribution when the typical halo mass is $4\%$ times
smaller than that of solid lines. The red line of the right panel shows the
formation time $z_{0.04}$ relative to the observed redshift $z$. The blue and
green dotted lines correspond to the redshift in the left panel. The black
dotted line represents $z_{0.04}=z$ as a comparison.

We comment on the effect of baryons on the halo formation. In the above
analysis, we use the total matter density to calculate the halo abundance.
When the halo has a mass smaller than the Jeans mass, the pressure prevents
the baryons from collapsing, which decreases the density contrast by about
$0.05/0.3\sim 17\%$ compared to the above analysis. In this paper, we neglect
this effect and equally treat the dark matter and baryon in the halo
formation. We discuss this effect on the dark matter constraint in Sec. IV.

The energy flux from dark matter annihilation is proportional to the square of
the dark matter density averaged over the halo profile and the physical number
density of halos, which is given by

$\displaystyle\braket{\rho_{\mathrm{halo}}(z)^{2}}$
$\displaystyle\equiv\int\mathrm{d}M{(1+z)^{3}}\frac{\mathrm{d}n}{\mathrm{d}M}(M,z)\int^{r_{h}}\mathrm{d}^{3}r\rho_{\mathrm{halo}}(r;M,z)^{2},$
(16)

where $\rho_{\mathrm{halo}}(r;M,z)$ represents the dark matter distribution in
a halo with mass $M$ at redshift $z$. Here we simply assume that the dark
matter distribution is proportional to the matter distribution in the halo,
$\rho_{\mathrm{halo}}=(\Omega_{c0}/\Omega_{m0})\rho_{\mathrm{NFW}}$, with the
current dark matter density parameter $\Omega_{c0}$. The radial integration up
to the virial radius is calculated as

$\displaystyle\int^{r_{h}}\mathrm{d}^{3}r\rho_{\mathrm{halo}}(r,M,z)^{2}$
$\displaystyle=\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}\int^{r_{h}}\mathrm{d}^{3}r\rho_{\mathrm{NFW}}(r)^{2}$
(17)
$\displaystyle=\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}\Delta_{\mathrm{vir}}\rho_{m}Mf_{\mathrm{NFW}}(c),$
(18) $\displaystyle f_{\mathrm{NFW}}(c)$
$\displaystyle\equiv\frac{c^{3}}{9}\frac{1-1/(1+c^{3})}{[\ln(1+c)-c/(1+c)]^{2}},$
(19)

and the squared average of the dark matter density is given by

$\displaystyle\braket{\rho_{\mathrm{halo}}(z)^{2}}$
$\displaystyle=\int\mathrm{d}M~{}{(1+z)^{3}}\frac{\mathrm{d}n}{\mathrm{d}M}(M,z)~{}\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}\Delta_{\mathrm{vir}}\rho_{m}(z)Mf_{\mathrm{NFW}}(c)$
(20)
$\displaystyle=2\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}\Delta_{\mathrm{vir}}\rho_{m}(z)^{2}f_{\mathrm{NFW}}(c(z))\beta(M=0,z).$
(21)

###  II.2 Moore profile

In this section, we adopt the simulation result and analysis in Ref. [29]. The
authors suggest that the profile of minihalos follows the Moore profile: 111
Note that this profile is slightly different from the original profile
suggested by Moore et.al. [57],
$\displaystyle\rho(r)=\frac{\rho_{s}}{(r/r_{s})^{3/2}(1+(r/r_{s})^{3/2})}.$
but we call Eq. (22) “Moore profile” in this paper.

$\displaystyle\rho_{\mathrm{Moore}}(r)\equiv\frac{\rho_{s}}{(r/r_{s})^{3/2}(1+r/r_{s})^{3/2}}.$
(22)

The parameters of the profile are estimated by the numerical calculation using
a spiky power spectrum with a peak scale $k_{*}$ as

$\displaystyle r_{s}$
$\displaystyle=\left(\frac{17}{30}\right)^{2/3}k_{*}^{-1}a_{c}\quad,\quad\rho_{s}=30a_{c}^{-3}\rho_{m0},$
(23)

where $a_{c}$ is the scale factor at the halo formation time ($a_{0}=1)$.

Compared to the NFW profile in the previous subsection, the estimation of
$\braket{\rho_{\mathrm{halo}}(z)^{2}}$ is different in two points. First, we
use the peak theory instead of the Press-Schechter formalism. The simulation
in Ref. [29] suggests that the spiky spectrum leads to isolated halos and
their mergers are rare. In this case, the peak theory is favored since the
merger of halos is ignored in the peak theory [see Ref. [29] for more
details]. Second, the Moore profile is steeper than the NFW profile, and the
average of the dark matter density squared diverges at the center of the halos
if we naively use Eq. (22). In reality, the dark matter density has some upper
bound due to the effect of dark matter annihilation, which will be discussed
later.

Let us calculate the halo abundance based on the peak theory investigated by
Bardeen, Bond, Kaiser, and Szalay [58] (hereafter we call it BBKS). The peak
theory calculates the probability that random Gaussian fluctuations form a
peak. A halo is formed when a height of the peak overcomes the threshold value
$\delta_{c}=1.686$. Using the delta-function type power spectrum, the comoving
number density of halos at redshift $z$ is calculated by [29]

$\displaystyle n(z)$
$\displaystyle=\frac{k_{*}^{3}}{(2\pi)^{2}3^{3/2}}\int^{\infty}_{\delta_{c}/\sigma_{0}(z)}e^{-\nu^{2}/2}f(\nu)\mathrm{d}\nu,$
(24)

where $f(\nu)$ is the function defined by Eq. (A15) of BBKS, and
$\sigma_{0}(z)$ is the variance of the density power spectrum,

$\displaystyle\sigma_{0}(z)^{2}$
$\displaystyle=\int\frac{\mathrm{d}k}{k}\mathcal{P}_{\delta}(k,z).$ (25)

The differential number density at the formation time is given by

$\displaystyle\frac{\mathrm{d}n}{\mathrm{d}a_{c}}$
$\displaystyle=\frac{k_{s}^{3}}{a_{c}}h\left(\frac{\delta_{c}}{\sigma_{0}(z_{c})}\right),$
(26) $\displaystyle h(\nu)$
$\displaystyle=\frac{\nu}{(2\pi)^{2}3^{3/2}}e^{-\nu^{2}/2}f(\nu)\mathrm{d}\nu,$
(27)

where $z_{c}$ is the redshift of the halo formation $z_{c}=a_{c}^{-1}-1$ and
we assume the linear growth of the matter fluctuation, $\sigma_{0}\propto a$.
We show the typical distribution of the formation redshift in Fig. 2. As
$k_{*}$ and $A^{2}$ are smaller, the halo formation becomes later. Some halos
are formed even in $z>10^{3}$ for $k_{*}=10^{6}~{}\mathrm{Mpc}^{-1}$ and
$A^{2}=10^{-6}$.

Figure 2:  The number density of halos calculated by the peak theory in Eq.
(26). The horizontal axis represents a redshift of the halo formation. We use
three different parameters of the power spectrum.

We estimate the averaged dark matter density squared using the Moore profile,
which is given by

$\displaystyle\braket{\rho_{\mathrm{halo}}(z)^{2}}$
$\displaystyle\equiv\int_{a}^{0}\mathrm{d}a_{c}{(1+z)^{3}}\frac{\mathrm{d}n}{\mathrm{d}a_{c}}(a_{c})\int^{r_{h}}_{r_{\mathrm{min}}}\mathrm{d}^{3}r\rho_{\mathrm{halo}}(r;a_{c},z)^{2}.$
(28)

Compared to the NFW profile case given by Eq. (16), the average is performed
for the formation time $a_{c}$, and the $r$-integration has the lower bound
for the following reason. It is known that the annihilation of the dark matter
is saturated in the central cusp region of the halo. When the cusp region has
a number density $n$, the typical lifetime of the dark matter in the region is
$\Delta t\sim(\braket{\sigma v}n)^{-1}$. Then, the maximum density of the core
is given by [59]

$\displaystyle\rho_{\mathrm{max}}$
$\displaystyle=\frac{m_{\chi}}{\braket{\sigma v}(t-t_{i})},$ (29)

where $t_{i}$ is the time of the halo formation. We define $r_{\mathrm{min}}$
by
$(\Omega_{c0}/\Omega_{m0})\rho_{\mathrm{Moore}}(r_{\mathrm{min}})=\rho_{\mathrm{max}}$,
which is given by

$\displaystyle\epsilon$
$\displaystyle\equiv\frac{r_{\mathrm{min}}}{r_{s}}\simeq\left(\frac{\Omega_{m0}}{\Omega_{c0}}\frac{\rho_{\mathrm{max}}}{\rho_{s}}\right)^{-3/2}\simeq
6\times 10^{-11}a_{c}^{-2}\left(\frac{3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}/\mathrm{TeV}}{\braket{\sigma
v}/m_{\chi}}\frac{13.8~{}\mathrm{Gyr}}{t}\frac{30}{f_{2}}\frac{1.27\times
10^{-6}\mathrm{GeV}/\mathrm{cm}^{3}}{\rho_{\chi 0}}\right)^{-2/3},$ (30)

and we safely take $\epsilon\ll 1$ for our calculation.

We simply assume that the dark matter distribution in halos also follows the
Moore distribution as
$\rho_{\mathrm{halo}}=(\Omega_{c0}/\Omega_{m0})\rho_{\mathrm{Moore}}$.222There
are ambiguities in the baryon distribution and its feedback to the dark matter
distribution. If the halo mass is smaller the Jeans mass baryons cannot
collapse into dark matter halos. On the other hand, in large halos baryons are
cooled and take different distribution, which could affect dark matter
distribution. We neglect these ambiguities here The radial integration is
estimated as

$\displaystyle\int^{r_{h}}_{r_{\mathrm{min}}}\rho_{\mathrm{halo}}(r)^{2}$
$\displaystyle=\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}\int^{r_{h}}_{r_{\mathrm{min}}}\mathrm{d}^{3}r~{}\rho_{\mathrm{Moore}}(r;a_{c},z)^{2}$
(31)
$\displaystyle\simeq\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}2\pi\rho_{s}^{2}r_{s}^{3}\left[2\log\left(1+\frac{1}{\epsilon}\right)-\frac{3+2\epsilon}{(1+\epsilon)^{2}}\right]$
(32)
$\displaystyle\simeq\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}2\pi\rho_{s}^{2}r_{s}^{3}\left[2\log\left(\epsilon^{-1}\right)-3\right],$
(33)

where the choice of the upper bound of the integration is negligible as long
as $r_{h}\gg r_{s}$ and we use $r_{h}\to\infty$ in this calculation. Then, the
average of the dark matter density squared is given by

$\displaystyle\braket{\rho_{\mathrm{halo}}(z)^{2}}$
$\displaystyle=2\pi\left(\frac{\Omega_{c0}}{\Omega_{m0}}\right)^{2}\int_{0}^{a}\mathrm{d}a_{c}~{}\left(k_{s}^{-3}\frac{\mathrm{d}n}{\mathrm{d}a_{c}}(a_{c})\right)~{}\rho_{s}^{2}(r_{s}k_{s}/{a})^{3}\left[2\log\left(\epsilon^{-1}\right)-3\right].$
(34)

###  II.3 Comparison

Let us compare two pictures of halo formation: the NFW profile with the Press-
Schechter formalism in Sec. II.1 and the Moore profile with the peak theory in
Sec. II.2. The Moore profile has the denser core than the NFW profile, which
enhances the annihilation flux from minihalos. The total annihilation flux
also depends on the abundance of minihalos and it is different between the
Press-Schechter formalism and the peak theory. We quantitatively estimate
these points to specify the differences between the two pictures.

At first, we compare the annihilation flux from each minihalo for the NFW and
Moore profiles. Fixing the halo parameter $r_{s}$ and $\rho_{s}$, the ratio of
the density squared for the Moore profile to that for the NFW profile is given
by

$\displaystyle\frac{\int^{\infty}_{r_{\mathrm{min}}}\rho_{\mathrm{Moore}}(r)^{2}}{\int^{\infty}_{0}\rho_{\mathrm{NFW}}(r)^{2}}=\frac{2\pi\rho_{s}^{2}r_{s}^{3}\left[2\log\left(\epsilon^{-1}\right)-3\right]}{\frac{4\pi}{3}\rho_{s}^{2}r_{s}^{3}}=\frac{3}{2}\left[2\log\left(\epsilon^{-1}\right)-3\right]\sim
10^{2},$ (35)

where we estimate $\epsilon\equiv r_{\mathrm{min}}/r_{s}$ with $t=t(z=100)$,
$a_{c}=10^{3}$ and $\braket{\sigma v}/m_{\chi}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}/\mathrm{TeV}$. Thus, as for the
annihilation flux from a single halo, the Moore profile leads to the signal
about $10^{2}$ times larger than the NFW profile due to its dense core.

Second, we compare the formation rate of minihalos in the Press-Schechter
formalism and the peak theory. From Eq. (21), the annihilation signal is
proportional to $\beta(M=0,z)$ in the Press-Schechter formalism. We regard
$\beta(M=0,z)$ as the typical halo formation rate since it is written by the
mass-weighted number density per volume,

$\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}M}{M}~{}\left(\frac{M}{2\rho_{m0}}\right)\frac{\mathrm{d}n}{\mathrm{d}\ln
M}(M,z)=\left[-\beta(M,z)\right]_{0}^{\infty}=\beta(M=0,z).$ (36)

On the other hand, the formation rate in the peak theory is regarded as
$k_{*}^{-3}n(a)$ from Eq. (24). We show two formation rates in Fig. 3, which
is evaluated for the power spectrum with $A^{2}=10^{-6}$ and
$k_{*}=10^{3}~{}\mathrm{Mpc}^{-1}$. The Press-Schechter formalism predicts the
formation rate about $10^{2}$ larger than that of the peak theory at
$z\lesssim 300$. It reflects the different pictures of the halo formation. In
the peak theory, the halos are mainly formed with typical scale $k=k_{*}$ and
then stay separated. On the other hand, the Press-Schechter formalism includes
the effects of the merger and the growth of halos, which leads to the larger
halo formation rate in later times.

Figure 3:  The halo formation rate for the Press-Schechter formalism and the
peak theory. The solid line represents $\beta(M=0,z)$ in Eq (13) calculated by
the Press-Schechter formalism. The dotted line represents a normalized number
density calculated by the peak theory, $k_{*}^{-3}n(a)$ in Eq. (24).

In summary, the NFW profile with the Press-Schechter formalism predicts a
larger number of halos with smaller dark matter density than those of the
Moore profile with the peak theory. The two differences have opposite effects
on the total annihilation flux. We clarify this point by comparing the average
of the dark matter density squared in Fig. 4. The solid lines represent the
results of the NFW profile with the Press-Schechter formalism (21), and dotted
lines represent the Moore profile with the peak theory (34), respectively. We
show the results with three parameters of the power spectrum to check the
dependence on the scale and amplitude (red, blue, green lines). In both
pictures, the halo formation occurs faster when the amplitude $A^{2}$ is
larger and the peak scale $(k_{*})^{-1}$ is smaller. Interestingly, both
pictures predict similar annihilation fluxes. Therefore, we expect that these
two pictures result in a similar constraint on the small-scale curvature
fluctuation. We will confirm it in the next section.

Figure 4:  The squared average of the dark matter energy density normalized by
the homogeneous dark matter density. The solid lines represent the results of
the NFW profile with the Press-Schechter formalism (21), and dotted lines
represent the Moore profile with the peak theory (34), respectively. The
colors represent the different parametrization of the power spectrum.

##  III Modified thermal history due to dark matter annihilation

So far we have discussed the dark matter halo profile and abundance under the
small-scale density fluctuation. Now let us see the effect of dark matter
annihilation on the thermal history of the universe taking account of the halo
profile and abundance. The injected energy due to dark matter annihilation per
physical time and volume consists of two parts,

$\displaystyle\left[\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\right]=\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\bigg{|}_{\mathrm{BG}}+\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\bigg{|}_{\mathrm{halo}},$
(37)

where the former comes from the spatially homogeneous dark matter component
and the latter comes from the inhomogeneous component existing in minihalos.
The homogeneous part is given by

$\displaystyle\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\bigg{|}_{\mathrm{BG}}=m_{\chi}(n_{\chi}(z))^{2}\braket{\sigma
v}=\frac{(\rho_{\chi}(z))^{2}}{m_{\chi}}\braket{\sigma v},$ (38)

with the number density of the homogeneous component of dark matter
$n_{\chi}(z)$. The energy injection from the dark matter halos is given by

$\displaystyle\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\bigg{|}_{\mathrm{halo}}$
$\displaystyle=\int\mathrm{d}\alpha~{}\frac{\mathrm{d}n}{\mathrm{d}\alpha}(\alpha,z)F_{1}(\alpha,z),$
(39) $\displaystyle F_{1}(\alpha,z)$ $\displaystyle=m_{\chi}\braket{\sigma
v}\int^{R}\mathrm{d}^{3}r\left(\frac{\rho_{\mathrm{halo}}(r,\alpha,z)}{m_{\chi}}\right)^{2},$
(40)

where ${\mathrm{d}n}/{\mathrm{d}\alpha}$ is the number density of halos
differentiated with respect to the halo parameter $\alpha$ and
$F_{1}(\alpha,z)$ is the annihilation energy flux from halos. The halo
parameter $\alpha$ is the halo mass for the NFW profile and the formation time
for the Moore profile used in Sec. II.

We define the boost factor of the annihilation energy flux as

$\displaystyle B(z)$
$\displaystyle=\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\bigg{|}_{\mathrm{halo}}\bigg{/}\frac{\mathrm{d}E}{\mathrm{d}V\mathrm{d}t}\bigg{|}_{\mathrm{BG}}$
(41)
$\displaystyle=\int\mathrm{d}M~{}\frac{\mathrm{d}n}{\mathrm{d}\alpha}(\alpha,z)\int^{R}\mathrm{d}^{3}r\left(\frac{\rho_{\mathrm{halo}}(r,\alpha,z)}{\rho_{\chi}(z)}\right)^{2}$
(42)
$\displaystyle=\frac{\braket{\rho_{\mathrm{halo}}(z)^{2}}}{\rho_{\chi}(z)^{2}}.$
(43)

The boost factor is shown in Fig. 4 for the NFW profile (21) and the Moore
profile (34). The inhomogeneous part dominates the energy injection even when
the redshift is larger than $100$ if the small-scale perturbations are large
enough. Using this boost factor, we can estimate the effect of minihalos on
the energy injection into the thermal plasma.

The injected particles experience various scattering processes in the
background thermal plasma and lose their energy. A part of the annihilation
energy is finally used for heating of the thermal gas and ionizing hydrogen
and helium. Roughly speaking, about one-third of the injected energy is used
for the ionization while its ratio depends on the energy of the injected
particle, species, and the injected redshift. We define the consumption ratio
of the injected photon or electron/positron energy as
$Q^{(e,\gamma)}_{\alpha}(E,z_{\mathrm{in}},z_{\mathrm{abs}})/E$, where $E$ is
kinetic energy of the injected particle, $z_{\mathrm{in}}$ is the injected
redshift, $z_{\mathrm{abs}}$ is the redshift when the energy is used, and
$Q^{(e,\gamma)}_{\alpha}(E,z_{\mathrm{in}},z_{\mathrm{abs}})$ is a consumed
energy at $z_{\mathrm{abs}}$. The subscript $\alpha$ represents the
consumption channels, $\alpha=$heat, ionH, and ionHe. We use
$Q^{(e,\gamma)}_{\alpha}(E,z_{\mathrm{in}},z_{\mathrm{abs}})/E$ calculated in
our previous work [25], which includes various scattering processes and is
consistent with other work [21]. When the dark matter has various annihilation
channels, we average the injected energy by their branching ratio,

$\displaystyle Q_{\alpha}(z,z^{\prime})$
$\displaystyle=\int\mathrm{d}E^{\prime}\bigg{[}\left(\frac{\mathrm{d}N_{F}^{(e^{+})}}{\mathrm{d}E^{\prime}}+\frac{\mathrm{d}N_{F}^{(e^{-})}}{\mathrm{d}E^{\prime}}\right)Q^{(e)}_{\alpha}(E^{\prime},z,z^{\prime})+\frac{\mathrm{d}N_{F}^{(\gamma)}}{\mathrm{d}E^{\prime}}Q^{(\gamma)}_{\alpha}(E^{\prime},z,z^{\prime})\bigg{]},$
(44)

where ${\mathrm{d}N_{F}^{(a)}(E^{\prime})}/{\mathrm{d}E^{\prime}}$ is the
differential number density of the final state particles with the energy
$E^{\prime}$ for $a=e^{-},~{}e^{+},$ and $\gamma$. In this paper, we only
consider the $\chi\chi\to e^{+}e^{-}$ channel for simplicity. Then, the dark
matter annihilation modifies the evolution of the ionization fraction of
hydrogen $x_{\text{H}}$, that of helium $x_{\text{He}}$ and the gas
temperature $T_{M}$ as

$\displaystyle-\left[\frac{\mathrm{d}x_{\mathrm{H}}}{\mathrm{d}z}\right]_{\mathrm{DM}}$
$\displaystyle=\int_{z}\frac{\mathrm{d}z^{\prime}}{H(z^{\prime})(1+z^{\prime})}\frac{1}{n_{\mathrm{H}}(z^{\prime})B_{\mathrm{H}}}\left[\frac{\mathrm{d}E(z^{\prime})}{\mathrm{d}V\mathrm{d}t}\right]\frac{Q_{\mathrm{ion}_{\mathrm{H}}}(z^{\prime},z)}{m_{\chi}},$
(45)
$\displaystyle-\left[\frac{\mathrm{d}x_{\mathrm{He}}}{\mathrm{d}z}\right]_{\mathrm{DM}}$
$\displaystyle=\int_{z}\frac{\mathrm{d}z^{\prime}}{H(z^{\prime})(1+z^{\prime})}\frac{1}{n_{\mathrm{He}}(z^{\prime})B_{\mathrm{He}}}\left[\frac{\mathrm{d}E(z^{\prime})}{\mathrm{d}V\mathrm{d}t}\right]\frac{Q_{\mathrm{ion}_{\mathrm{He}}}(z^{\prime},z)}{m_{\chi}},$
(46)
$\displaystyle-\left[\frac{\mathrm{d}T_{M}}{\mathrm{d}z}\right]_{\mathrm{DM}}$
$\displaystyle=\int_{z}\frac{\mathrm{d}z^{\prime}}{H(z^{\prime})(1+z^{\prime})}\frac{2}{3}\frac{1}{(1+f_{\mathrm{He}}+x_{e})n_{H}(z^{\prime})}\left[\frac{\mathrm{d}E(z^{\prime})}{\mathrm{d}V\mathrm{d}t}\right]\frac{Q_{\mathrm{heat}}(z^{\prime},z)}{m_{\chi}}.$
(47)

We include these modifications in the RECFAST code [33, 34, 35] and calculate
the thermal history of the universe. The modified ionization history is shown
in Fig. 5, where we treat the energy injection as a perturbation and use the
cosmological parameters based on the Planck 2018 results [60]. The vertical
axis, $x_{e}$, is a number fraction of the free electron to the hydrogen atom.
The ionization ratio decreases at $z\sim 10^{3}$ due to the recombination of
the hydrogen. Without the energy injection from dark matter annihilation, the
ionization fraction monotonically decreases (blue line) until the reionization
epoch at $z_{\mathrm{re}}\sim 7$. When the dark matter annihilation is taken
into account, the injected energy ionizes the hydrogen and helium, and the
ionization fraction increases. The orange line shows the result for
$\braket{\sigma v}=3\times 10^{-26}~{}\mathrm{cm}^{3}/\mathrm{s}$,
$m_{\chi}=1~{}\mathrm{TeV}$, and the same boost factor as the red solid line
in Fig. 4.

Figure 5:  The ionization history of the universe with and without the energy
injection by the annihilation of the inhomogeneous dark matter. $x_{e}$ is a
ratio of the free electron number density to the hydrogen number density. The
orange line is calculated including the dark matter annihilation with
$m_{\chi}=1~{}\mathrm{TeV}$, $\braket{\sigma v}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}$, the $\chi\chi\to e^{+}e^{-}$ channel
decay, and the inhomogeneous dark matter distribution with the same boost
factor to the red solid line in Fig. 4. The black dotted line represents
reionization of the universe with the tanh model used in Planck 2018 [61, 60]
with $z_{\mathrm{re}}\simeq 7.7$.

We comment on the reionization of the universe. We use the tanh model of
reionization used in Planck 2018 [61, 60] where the redshift of the
reionization $z_{\mathrm{re}}$ is a free parameter. The Planck 2018 estimates
$z_{\mathrm{re}}\simeq 7.7$ [60] without the exotic energy injection. One
might mind whether there is a parameter degeneracy between $z_{\mathrm{re}}$
and the dark matter annihilation rate. We find that the degeneracy is solved
by the scale dependence of the CMB power spectrum, which will be discussed in
the next section. The reionization is also investigated by observations of
high redshift objects like quasars, which suggest that the neutral fraction of
the universe is $\mathcal{O}(10)\%$ around $z\sim 7.5$ [62, 63, 64]. The
energy injection modifies the thermal history by less than $\mathcal{O}(1)\%$,
and thus our calculations are consistent with the observations of the neutral
hydrogen.

The modified recombination history as discussed above affects the power
spectrum of the CMB anisotropy. The CMB photons are scattered by free
electrons through the Thomson scattering process. The optical depth due to the
Thomson scattering is given by [60]

$\displaystyle\tau=n_{H}(0)\sigma_{\mathrm{T}}\int\mathrm{d}z~{}x_{e}(z)\frac{(1+z)^{2}}{H(z)},$
(48)

where $n_{H}(0)$ is the present number density of the hydrogen and
$\sigma_{\mathrm{T}}$ is the Thomson scattering cross section. At low
redshifts, the number density of electrons decreases, and hence $\tau$ is less
dependent on $x_{e}(z)$. Thus, the optical depth is sensitive to the
modification of $x_{e}(z)$ at high redshifts. The Thomson scatterings decrease
temperature fluctuations and increase $E$-mode polarization. We calculate the
modified CMB power spectrum by using the CAMB [36] code with the modified
RECFAST code. The Fig. 6 show the modification of the CMB power spectrum by
the energy injection, and the vertical axes represent

$\displaystyle\Delta_{C_{l}}=\frac{C_{l}-C^{(0)}_{l}}{C^{(0)}_{l}},$ (49)

where $C_{l}$ and $C^{(0)}_{l}$ are the CMB power spectra with and without the
energy injection. We plot the temperature mode (top panel) and the $E$-mode
polarization (bottom panel) modified by the annihilation of the dark matter
with $m_{\chi}=1~{}\mathrm{TeV}$. The black solid line represents the modified
power spectrum by the inhomogeneous dark matter with $\braket{\sigma
v}=3\times 10^{-26}\mathrm{cm}^{3}/\mathrm{s}$ and the same boost factor as
the red solid line in Fig. 4. For comparison, we also plot the modified power
spectrum by the homogeneous dark matter with $\braket{\sigma v}=1\times
10^{-24}\mathrm{cm}^{3}/\mathrm{s}$ (black dotted line). Although we use the
$\mathcal{O}(10^{-2})$ times smaller cross section for the inhomogeneous case
than the homogeneous case, both cases have comparable results because of the
boost factor Eq. (43). The power spectra for both cases have different scale
dependence since the redshift dependence of the energy injection is different
for the homogeneous and inhomogeneous dark matter, where the latter injects
more energy in a later time as shown in Eq. (4).

Next, we investigate when the inhomogeneous energy absorption mainly
contributes to the CMB power spectrum. Note that the boost factor has an
uncertainty in the low redshift since we neglect the effect of the standard
structure formation. We evaluate the power spectrum by artificially limiting
the energy consumption epoch: early time ($z>100$, red line), middle time
($100>z>10$, green line), and late time ($10>z$, blue line). We find that the
early time contribution dominates the modification of the power spectrum for
both the temperature and the polarization modes in the case of inhomogeneous
energy injection, which is consistent with the fact that the CMB power
spectrum is most sensitive to the energy injection at $z\sim 600$ [11]. Thus
we conclude that the CMB power spectrum puts reliable constraints on the
energy injection by the inhomogeneous dark matter independent of the late time
cosmology.

Figure 6:  The modification of the CMB power spectrum induced by the energy
injection from the annihilation of the dark matter with
$m_{\chi}=1~{}\mathrm{TeV}$ and the $\chi\chi\to e^{+}e^{-}$ channel decay.
The top and bottom panels represent the power spectrum of the temperature and
the $E$-mode polarization. The energy injection is calculated for the
homogeneous dark matter with $\braket{\sigma v}=1\times
10^{-24}\mathrm{cm}^{3}/\mathrm{s}$ (black dotted line) and for the
inhomogeneous dark matter with $\braket{\sigma v}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}$ and the same boost factor to the red solid
line in Fig. 4. We also show the power spectrum by artificially limiting the
energy consumption epoch: early time ($z>100$, red line), middle time
($100>z>10$, green line), and late time ($10>z$, blue line), respectively.

##  IV Constraints on small-scale density fluctuation

When the dark matter abundance is determined by the thermal freeze out
scenario with the $s$-wave annihilation, the cross section is estimated as
$\braket{\sigma v}=3\times 10^{-26}~{}\mathrm{cm}^{3}/\mathrm{s}$. The current
upper bound on the dark matter annihilation cross section from the CMB
observation is $\braket{\sigma v}/m_{\chi}\lesssim
10^{-27}\mathrm{cm}^{3}/\mathrm{s}/\mathrm{GeV}$ for the $\chi\chi\to
e^{+}e^{-}$ channel annihilation [20] assuming the spatially homogeneous
distribution of dark matter. Therefore, the thermal freezeout scenario is
consistent with the CMB observation for the dark matter mass larger than $\sim
30$ GeV. If the small-scale curvature perturbation is large enough, on the
other hand, minihalos are formed and the annihilation signal may be
significantly enhanced. Thus we can put the upper bound on the small-scale
curvature perturbation by the CMB observation. In the following, we take
$m_{\chi}=1~{}\mathrm{TeV}$ and $\braket{\sigma v}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}$ as reference values.

We perform the Markov Chain Monte Carlo method (MCMC) with the CosmoMC code
[37] with the modified CAMB code to estimate the upper limit on small-scale
curvature fluctuations. We assume the base-$\Lambda$CDM model including
neutrinos with normal mass hierarchy and $\sum m_{\nu}=0.06$ eV used in the
Planck collaboration [60], and the energy injection by the dark matter
annihilation is added to the model.

In the MCMC, the eight cosmological parameters are varied
$\\{\Omega_{b}h^{2},~{}\Omega_{c}h^{2},~{}100\theta_{\rm
MC},~{}\tau,~{}n_{s}$, ${\rm{ln}}(10^{10}A_{s})$, $\log_{10}(k_{*})$,
$\log_{10}(A^{2})\\}$, where the latter two parameters determine the power
spectrum of the small-scale curvature fluctuations (1). We treat the energy
injection as a perturbation on the standard $\Lambda$CDM model, and the energy
consumption ratio on the thermal plasma and the boost factor are calculated
based on the Planck best-fit values. We use the top-hat prior on
$\log_{10}(k_{*}/\mathrm{Mpc}^{-1})$ and $\log_{10}(A^{2})$, and choose an
appropriate parameter range for each parameter. We use the following data
sets: the Planck 2018 data set including low-l ($l\leq 29$) TT- and EE-mode,
high-l ($l\geq 30$) TT-, TE- and EE-mode, and the CMB lensing power spectrum
[65], and the other cosmological data on galaxy correlation functions
including the baryon acoustic oscillation (BAO) [66, 67, 68, 69] and Dark
Energy Survey (DES) [70].

We show the results of the MCMC analysis in Fig. 7, where each line represents
the 95% exclusion contours. The halo formation is estimated by the methods of
the NFW profile with the Press-Schechter formalism (red solid line) and the
Moore profile with the peak theory (blue line). Both methods predict a similar
upper limit on the small-scale curvature fluctuation as we confirmed in Fig.
4. Our upper bound is comparable to or slightly weaker than that by the
extragalactic gamma rays [32, 29].

The red dotted line represents the exclusion contour for the dark matter with
mass $m_{\chi}=100~{}\mathrm{GeV}$. Since the annihilation signal is inversely
proportional to the dark matter mass, the constraint for
$m_{\chi}=100~{}\mathrm{GeV}$ is stronger than that of
$m_{\chi}=1~{}\mathrm{TeV}$. However, the constraint on the power spectrum is
less dependent on the property of minihalos and the mass of the dark matter
since the halo abundance exponentially depends on the power spectrum. The
boost factor for $A^{2}\gtrsim 10^{-6}$ increases steeply at $z\sim 500$, and
it mainly determines the upper bound on the small-scale density perturbation.
The constraint becomes weaker for smaller $k$ as we confirmed in Fig. 4. The
wiggling behavior for a small $k$ may be statistical fluctuations of the MCMC
analysis while the power spectrum on $\mathcal{O}(\mathrm{Mpc}^{-1})$ are
severely constrained by observations of large-scale structure.

Figure 7:  The constraints on the small-scale curvature fluctuation from the
CMB observation. We assume the dark matter with $\braket{\sigma v}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}$ and $m_{\chi}=1~{}\mathrm{TeV}$ for solid
lines, and $m_{\chi}=100~{}\mathrm{GeV}$ for a dotted line. The red and blue
lines represent the constraints by the halo formation estimated for the NFW
profile with the Press-Schechter formalism as discussed in Sec. II.1 and the
Moore profile with the peak theory as discussed in Sec. II.2. In the right
panel, we also show other constraints from the abundance of primordial black
holes [6], CMB spectral distortions [5] by the COBE/FIRAS [71], and the CMB
power spectrum by the Planck satellite [72].

We comment on the baryonic effect on the constraint. Our estimation of
minihalos neglects the baryonic pressure in the halo formation, which may lead
to an overestimation of density fluctuations by a factor
$\Omega_{m0}/\Omega_{c0}$. Thus, our constraint has uncertainty on the power
spectrum by $(\Omega_{c0}/\Omega_{m0})^{2}$, which is showed by a red band in
the right panel of Fig. 7, In the right panel of Fig. 7, we also show other
constraints on the curvature power spectrum from the CMB power spectrum by
Planck [72], CMB energy distortions [5] by the COBE/FIRAS [71], and the
abundance of primordial black halos [6].

We check the degeneracy between the energy injection and the reionization.
Since both effects increase the optical depth, they may degenerate in the MCMC
analysis. The Fig. 8 shows the two-dimensional contour plot on the redshift of
reionization, $z_{\mathrm{re}}$, and the amplitude of the power spectrum,
$A^{2}$, from the MCMC analysis for the NFW profile with the Press-Schechter
formalism (red line of the Fig. 7). $z_{\mathrm{re}}$ is derived by other
cosmological parameters assuming the tanh model of reionization. The
degeneracy of the parameters is solved in Fig. 8 since the exotic energy
injection and the reionization effect have different redshift dependence. A
similar result is confirmed in the case of the exotic energy injection by the
mixed WIMP and the primordial black hole scenarios [73].

Figure 8:  The two-dimensional contour plot of the redshift of reionization
and the amplitude of the small-scale power spectrum. The two contours
represent the $68\%$ and $95\%$ exclusion limits. We use the same parameter to
the red solid line of Fig. 7.

##  V Conclusion

Although the CMB observations have revealed the primordial fluctuations on
scales larger than $\mathcal{O}(1)\mathrm{Mpc}$, small-scale fluctuations are
still less constrained. When the small-scale density fluctuation is much
larger than the naive extrapolation from the CMB scales with the assumption of
scale-invariant power spectrum, minihalos are formed in the early stage of the
universe. The minihalos amplify the inhomogeneity of the dark matter
distribution, which increases the annihilation signal of the dark matter. In
the standard thermal freeze out scenario of dark matter production, the cross
section is expected to be $\braket{\sigma v}=3\times
10^{-26}\mathrm{cm}^{3}/\mathrm{s}$ for the $s$-wave annihilation. The dark
matter annihilation injects high-energy particles into the thermal plasma, and
they act as extra ionization sources of the neutral hydrogen and helium. The
modified ionization history is constrained by the observation of the CMB
anisotropy. Such annihilation signals from minihalos are intensively studied
in the context of the extragalactic gamma rays [32, 29], while there are fewer
works that focused on a constraint from the CMB anisotropy.

In this work, we have investigated the constraints on small-scale curvature
fluctuations from the CMB anisotropy through the dark matter annihilation. We
estimated the abundance and the profile of minihalos taking into account the
recent numerical simulation [29]. We have adopted two different methods to
estimate the properties of the minihalos. First, the halo abundance was
estimated by the Press-Schechter formalism assuming the NFW profile [32],
which is similar to the method often used for the standard halos. Second,
following the numerical simulation [29] we took the picture that minihalos
have the Moore profile and their formation time determines their property
[29]. It was found that both models highly amplify dark matter annihilation
rate as shown in Fig. 4. We estimated the energy injection from the dark
matter annihilation assuming the standard thermal freezeout scenario. The
modified ionization history was calculated by the RECFAST code, where we
precisely included the fraction of the injected energy used for ionization
taking into account all relevant scattering processes. Using the modified
RECFAST code, we calculated the CMB power spectrum by the CAMB code and
performed the MCMC analysis by the CosmoMC code to put the exclusion limit of
the parameters. The resultant constraints on the small scale power spectrum of
the curvature fluctuation are shown in Fig. 7 and comparable to that from the
extragalactic gamma rays [32, 29]. We have confirmed that our constraints
mainly come from the ionization process at the high redshifts, $z>100$.
Therefore, our constraint is independent of some uncertainty of the halo
properties in the late times, e.g. the destruction of the minihalos by the
standard halos [32]. Thus, the CMB constraint on the small scale curvature
fluctuations derived in this paper is reliable and complementary to the
constraint from the extragalactic gamma rays.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Nos. 17H01131 (M.K.), 17K05434
(M.K.), 20H05851(M.K.), 21K03567(M.K.), JP19J21974 (H.N.), 18K03609 (K.N.),
17H06359 (K.N.), Advanced Leading Graduate Course for Photon Science (H.N.),
and World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan (M.K.).

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