# Anisotropies in Cosmological 21 cm Background by Oscillons/I-balls of Ultra-
light Axion-like Particle

Masahiro Kawasaki  Kazuyoshi Miyazaki  Kai Murai  Hiromasa Nakatsuka  and
Eisuke Sonomoto

###### Abstract

Ultra-light axion-like particle (ULAP) with mass $m\sim
10^{-22}~{}\mathrm{eV}$ has recently been attracting attention as a possible
solution to the small-scale crisis. ULAP forms quasi-stable objects called
oscillons/I-balls, which can survive up to a redshift $z\sim 10$ and affect
the structure formation on a scale $\sim\mathcal{O}(0.1)~{}\mathrm{Mpc}$ by
amplifying the density fluctuations. We study the effect of oscillons on 21 cm
anisotropies caused by neutral hydrogen in minihalos. It is found that this
effect can be observed in a wide mass range by future observations such as
Square Kilometer Array (SKA) if the fraction of ULAP to the total dark matter
density is $\mathcal{O}(0.01\text{--}0.1)$.

##  1 Introduction

Dark matter (DM) accounts for about 25% of energy in the current universe,
while its origin remains unexplained [1, 2, 3]. The $\Lambda$CDM model
successfully explains observations of the cosmic microwave background
fluctuations and the large-scale structures of the universe [4, 5, 6].
However, when focusing on small scales, the numerical simulations based on the
$\Lambda$CDM model show results contrary to some observations, giving rise to
three cosmological problems; missing satellite problem [7, 8], core-cusp
problem [9, 10], and too-big-to-fail problem [11, 12]. These problems could be
solved by ultralight bosonic DM with a mass of $\sim 10^{-22}$ eV [13, 14],
which suppresses density fluctuations on small scales and behaves like CDM on
large scales.

Ultra-Light Axion-like Particle (ULAP), which is predicted from the string
theory [15], has been attracting attention as a possible solution to the
small-scale problems. In the early universe, the ULAP field starts oscillation
and the oscillating ULAP field has spatial instabilities which generate local
objects called oscillons/I-balls if the potential is shallower than quadratic
[16, 17, 18]. Oscillons are quasi-stable due to their approximate conservation
of the adiabatic invariant, and they could survive until the present time
depending on the ULAP mass and the shape of the potential [19, 20, 21]. Since
oscillons are randomly produced, they generate the Poisson fluctuations on the
ULAP density power spectrum. Thus, if oscillons exist, they can affect the
structure of the universe at scale $\sim\mathcal{O}(0.1)$ Mpc.

The 21 cm line, which is produced by the hyperfine structure of neutral
hydrogen, is useful to probe density fluctuation on small scales. Some works
investigated the 21 cm line to constrain the axions [22, 23, 24], warm dark
matter [25], primordial black holes [26, 27], and oscillons [28]. Ref. [28]
discussed the halo formation including the effects of oscillons and evaluated
the isotropic 21 cm line signal. However, the anisotropies on the 21 cm line
have not yet been investigated.

In this paper, we study anisotropies in the cosmological 21 cm background
induced by oscillons made of the ULAP field. We assume that the ULAP accounts
for a fraction of the total DM. We consider the coexistence of the oscillons
and homogeneous ULAP following the numerical simulation [29]. We found that
future experiments such as Square Kilometer Array (SKA) [30] can detect the
effect of oscillons on the 21 cm line.

The structure of this paper is as follows. In Sec. 2, we consider the matter
power spectrum when ULAP partially accounts for the DM. We analytically
calculate the matter power spectrum in the presence of oscillons and confirm
that the oscillons enhance the fluctuations on small scales. In Sec. 3, we
derive the 21 cm fluctuation. We will see that oscillons affect the 21 cm
fluctuations by changing the number density and formation time of minihalos.
In Sec. 4, we discuss the detectability of the 21 cm fluctuation with
oscillons by future observations. We conclude the results in Sec. 5. All
cosmological parameters in this paper are extracted from the results of Planck
2018 [6].

##  2 Matter Power Spectrum of ULAP

In this paper, we consider that the dark matter consists of two components,
unknown cold dark matter and ULAP. The ULAP is further decomposed into two
parts, (almost) homogeneous ULAP and ULAP oscillons.111We use the term
“homogeneous” to describe the ULAP component that does not form oscillons,
including perturbations around the homogeneous component.  We define the
density parameter of each component as

$\displaystyle\Omega_{\mathrm{DM}}$
$\displaystyle=\Omega_{\mathrm{CDM}}+\Omega_{\mathrm{ULAP}}$ (2.1)
$\displaystyle=\Omega_{\mathrm{CDM}}+(\Omega_{\mathrm{homo}}+\Omega_{\mathrm{osc}}),$
(2.2)

where $\Omega_{\mathrm{CDM}}$ is the density parameter of the CDM besides
ULAP, and $\Omega_{\mathrm{ULAP}}$ represents that of the total ULAP including
the homogeneous part $\Omega_{\mathrm{homo}}$ and oscillon part
$\Omega_{\mathrm{osc}}$. We also define the ratio of ULAP to DM and the ratio
of oscillons to the total ULAP as

$\displaystyle f_{\mathrm{ULAP}}$
$\displaystyle\equiv\frac{\Omega_{\mathrm{ULAP}}}{\Omega_{\mathrm{DM}}}\quad,\quad
r_{\mathrm{osc}}\equiv\frac{\Omega_{\mathrm{osc}}}{\Omega_{\mathrm{ULAP}}}.$
(2.3)

The oscillon energy fraction depends on the ULAP potential. Here, we adopt the
monodromy-type potential [31, 32, 33], which is written as

$V(\phi)=\frac{m_{a}^{2}F^{2}}{2p}\left[1-\left(1+\frac{\phi^{2}}{F^{2}}\right)^{-p}\right],$
(2.4)

where $m_{a}$ is the ULAP mass, and $F$ is the ULAP decay constant. Oscillons
are formed only when the potential is shallower than the quadratic term,
$p>-1$ [34]. Due to the approximate conservation of adiabatic invariant, they
can exist for a long time and can survive long enough until the epoch of our
interest ($z\sim 10$) [35, 36, 37]. Since the oscillon mass is related to the
ULAP mass by $M_{\mathrm{osc}}\propto F^{2}/m_{a}$, a tiny ULAP mass leads to
heavy oscillons. For example, when $m\sim 10^{-22}$ eV and $F\sim 10^{15}$
GeV, the oscillon mass is $M_{\text{osc}}\sim 10^{6}\,M_{\odot}$. In such a
case, the number density of the oscillons is so small that their Poisson
fluctuations cannot be neglected. For this reason, the oscillons can affect
the matter power spectrum. The formed oscillons contain at most about 70% of
the total energy of the ULAP [29], and the rest exists as the homogeneous
component of the ULAP.

The homogeneous ULAP and oscillons affect the matter power spectrum
differently. The homogeneous ULAP suppresses the fluctuations on scales
smaller than its de Broglie length. On the other hand, oscillons induce the
Poisson fluctuations. In the following section, we quantitatively estimate the
matter power spectrum taking these effects into account.

###  2.1 Matter Power Spectrum without Oscillons

When ULAP does not form oscillons or oscillons totally decay, we can decompose
the total matter power spectrum in the following form:

$P(k,t)=P_{\mathrm{CDM}}(k,t)+P_{\mathrm{ULAP}}(k,t),$ (2.5)

where $P_{\mathrm{ULAP}}(k,t)$ denotes the matter power spectrum of ULAP. We
calculate the power spectrum without oscillons using the AxionCAMB code [38].
The calculation results are shown in Fig. 1 as the dotted lines. Since the
ULAP without oscillons suppresses the small-scale fluctuations, the matter
power spectrum is also suppressed compared with the $\Lambda$CDM power
spectrum at $k\gtrsim\mathcal{O}(10)~{}\mathrm{Mpc}^{-1}$, which can be seen
in Fig. 1.

###  2.2 Matter Power Spectrum with Oscillons

Next, let us show the oscillon matter power spectrum $P_{\mathrm{osc}}(k,t)$
following Ref. [29]. Assuming that the positions of oscillons are not
correlated and neglecting the oscillon size ($k/a\ll m_{a}$), the matter power
spectrum of oscillons in the matter dominated era is written as

$P_{\mathrm{osc}}(k,t)=\left(\frac{3a}{2a_{\mathrm{eq}}}\right)^{2}\frac{\left(r_{\mathrm{osc}}f_{\mathrm{ULAP}}\right)^{2}}{n_{\mathrm{osc}}a^{3}}\left(\frac{\Omega_{\mathrm{DM}}}{\Omega_{m}}\right)^{2}\frac{\left\langle
M_{\mathrm{osc}}^{2}\right\rangle}{\left\langle
M_{\mathrm{osc}}\right\rangle^{2}}\left[1-\left(\frac{2}{kL_{s}}\right)^{2}\sin^{2}\frac{kL_{s}}{2}\right],$
(2.6)

where $a_{\mathrm{eq}}$ is the scale factor at the matter-radiation equality,
the bracket $\langle~{}\rangle$ represents the ensemble average over the mass
distribution of oscillons, $M_{\mathrm{osc}}$ is the mass of an oscillon,
$n_{\mathrm{osc}}$ is the physical number density of oscillons, and $L_{s}$ is
the cut-off scale due to energy conservation, which is determined by a
simulation. Because of the gravitational growth of the isocurvature
fluctuations and the evaporation of oscillons, the oscillon power spectrum
evolves over time. The prefactor proportional to $a^{2}$ reflects the linear
growth of the fluctuations after the matter-radiation equality.

In the presence of oscillons, the power spectrum is contributed by the Poisson
fluctuations of oscillons in addition to that in Eq. (2.5). When evaluating
the contribution of Eq. (2.5) by AxionCAMB, we set the homogeneous ULAP
fraction as $f_{\mathrm{ULAP}}(1-r_{\mathrm{osc}})$ and consider oscillons as
CDM.222Although the oscillon fraction $r_{\mathrm{osc}}$ gradually decreases,
$r_{\mathrm{osc}}$ does not change that much during the time we are interested
in. Considering this point, we used $r_{\mathrm{osc}}$ averaged over $z=10$ to
$20$ in AxionCAMB.  On the other hand, we evaluate the power spectrum of the
Poisson fluctuations of oscillons following Refs. [29, 28]. Since the oscillon
fraction $r_{\mathrm{osc}}$, number density $n_{\mathrm{osc}}$, and mean mass
$\langle M_{\mathrm{osc}}\rangle$ gradually change due to the evaporation of
oscillons, we follow the time evolution of oscillons to obtain such parameters
for a fixed $z$. The initial mass distribution of oscillons is determined by
the numerical lattice simulation of the oscillon formation. In this paper, we
use the result of the lattice simulation in Ref. [28]. (See appendix B of Ref.
[28] for the setup of the numerical calculation.) After the formation, the
subsequent time evolution of the mass distribution is calculated using the
analytical decay rate of an oscillon [36, 35],

$\Gamma\equiv\frac{1}{M_{\mathrm{osc}}}\left|\frac{\mathrm{d}M_{\mathrm{osc}}}{\mathrm{d}t}\right|.$
(2.7)

The RHS of Eq. (2.7) is evaluated by decomposing the ULAP field into the
oscillon profile and perturbation part
$\phi(t,\bm{x})=\phi_{\mathrm{osc}}(t,\bm{x})+\xi(t,\bm{x})$, and analytically
solving the equation of motion of the perturbation part. Then, the energy
emission from an oscillon is given by the time average of the Poynting vector
of the perturbation part

$\frac{\mathrm{d}M_{\mathrm{osc}}}{\mathrm{d}t}=4\pi
r^{2}\overline{\partial_{t}\xi(t,r)\partial_{t}\xi(t,r)},$ (2.8)

where we assume that $\xi$ is spherically symmetric and the overline
represents a time average.

The oscillon matter power spectrum for the ULAP mass
$m_{a}=10^{-22}~{}\mathrm{eV},\,10^{-21}~{}\mathrm{eV},$ and
$10^{-20}~{}\mathrm{eV}$ are plotted in Fig. 1 by the solid lines. In the
presence of oscillons, the small scale fluctuations are amplified and so is
the matter power spectrum as can be seen in Fig 1. We should note that, when
one considers a single oscillon, the expression of $P_{\mathrm{osc}}$ is
invalid because the fluctuations are non-linear on the scale containing a
single oscillon. So, we cut off the oscillon matter power spectrum on the
scale $k_{\mathrm{c}}$ where the number of oscillons equals to 10, or
equivalently $n_{\mathrm{osc}}(2\pi/k_{c})^{3}=10$. Corresponding regions with
$k>k_{\mathrm{c}}$ are plotted with the dot-dashed lines. We have confirmed
that changing this criterion from 10 to 2 does not largely change the final
result.

Figure 1: Matter power spectrum at $z=10$ for $f_{\mathrm{ULAP}}=0.5$. The
blue, green, and red solid lines show the oscillon matter power spectra for
$m_{a}=10^{-20}~{}\mathrm{eV},\,10^{-21}~{}\mathrm{eV},$ and
$10^{-22}~{}\mathrm{eV}$, respectively. The dotted lines are the matter power
spectra for the case of homogeneous ULAP, and the color difference is the same
as in the previous case. We cut off the oscillon matter power spectra on the
scale where the number of oscillons equals 10 because the fluctuations are
non-linear. Corresponding scales are plotted with the dot-dashed lines. The
matter power spectrum in the $\Lambda$CDM model is plotted with the black
solid line for comparison.

##  3 Minihalo and 21 cm Fluctuations

In this section, following Ref. [39, 23, 24, 40], we formulate the properties
of minihalos and the 21 cm fluctuations from minihalos in the presence of
ULAP, which are different from the $\Lambda$CDM case.

###  3.1 Minihalo Mass Function

Figure 2: Minihalo mass functions for $m=10^{-20}$ eV (red), $m=10^{-21}$ eV
(Green), $m=10^{-22}$ eV (Blue) are shown. The solid and dotted lines denote
the halo mass functions with and without oscillons, respectively. We also show
the case of the $\Lambda$CDM model with the black solid line for comparison.

First, we introduce the comoving number density of minihalos per mass interval
$M\sim M+\mathrm{d}M$, denoted by $\mathrm{d}n/\mathrm{d}M$. In calculating
the halo mass function, the Press-Schechter formalism is often used in the
literature. In this paper, we adopt a more accurate fitting formula, Sheth-
Tormen mass function [41]333Exactly speaking, the Sheth-Tormen mass function
cannot be used for the Poisson fluctuations of the oscillons because it
assumes the Gaussian distribution for density fluctuations. However, the
Poisson distribution approaches the Gaussian distribution in the large number
limit. Therefore, we can approximately use the Sheth-Tormen mass function for
the oscillon number $\gtrsim 10$.

$\frac{\mathrm{d}n}{\mathrm{d}M}(z,M)=\frac{\bar{\rho}_{m}}{M}\frac{\mathrm{d}\ln\nu}{\mathrm{d}M}\nu
f(\nu),\quad\left(\nu(z,M)\equiv\frac{\delta_{c}}{\sigma(z,M)}\right)$ (3.1)

where $\bar{\rho}_{m}$ is the background matter density, $\delta_{c}=1.686$ is
the critical density contrast, and $f(\nu)$ is a probability density given by

$\nu
f(\nu)=2A\left(1+\frac{1}{(a\nu)^{2p}}\right)\left(\frac{(a\nu)^{2}}{2}\right)^{1/2}\frac{\mathrm{e}^{-(a\nu)^{2}/2}}{\sqrt{\pi}},$
(3.2)

with the constants $a=0.707,\,p=0.3,$ and $A=0.322$. $\sigma(z,M)$ is a
smoothed mass variance over the scale $R(M)$ at redshift $z$, expressed as

$\sigma(z,M)^{2}=\frac{1}{2\pi^{2}}\int P(k,z)|W(kR)|^{2}\mathrm{d}k.$ (3.3)

Here the scale $R(M)$ is related to the minihalo mass $M$ by the relation
$M=4\pi R^{3}\bar{\rho}_{m}/3$, and $W(kR)$ is the Fourier transform of the
real space top-hat window function.

The mass functions for various cases are shown in Fig. 2. When we do not
consider the oscillons, the number of minihalos is smaller than in the CDM
case because the quantum pressure of ULAP suppresses the fluctuations on small
scales. On the other hand, when the oscillons are present, the matter power
spectrum is enhanced, which increases the number of minihalos compared to the
case without oscillons. We see that the mass function is even larger than that
in the CDM case for heavier minihalos.

Here, we consider the collapse time of each minihalo. Since the density
contrast is a random Gaussian variable with variance $\sigma(z,M)$ (3.3) , the
redshift at the halo collapse, $z_{\mathrm{coll}}$, also fluctuates. According
to the spherical collapse model, the halo collapses when the linear density
contrast $\delta(z,M)$ exceeds the critical value $\delta_{c}$, so the
collapse redshift is expressed as [42]

$1+z_{\mathrm{coll}}=\frac{\delta(z)}{\delta_{c}}(1+z).$ (3.4)

Using this relation, we can calculate the averaged collapse redshift as [25]

$\displaystyle\left\langle 1+z_{\text{coll}}\right\rangle(M,z)$
$\displaystyle=(1+z)\frac{\int_{\delta_{c}}^{\infty}\mathrm{d}\delta\,\frac{\delta}{\delta_{c}}\frac{1}{\sqrt{2\pi}\sigma(z,M)}\exp\left[-\frac{\delta^{2}}{2\sigma(z,M)^{2}}\right]}{\int_{\delta_{c}}^{\infty}\mathrm{d}\delta\,\frac{1}{\sqrt{2\pi}\sigma(z,M)}\exp\left[-\frac{\delta^{2}}{2\sigma(z,M)^{2}}\right]}$
$\displaystyle=\left.(1+z)\frac{e^{-x^{2}}}{\sqrt{\pi}x\operatorname{erfc}(x)}\right|_{x=\delta_{c}/\sqrt{2}\sigma(z,M)},$
(3.5)

where erfc($x$) represents the complementary error function. In the upper
panel of Fig. 3, we show $z_{\mathrm{coll}}$ as a function of $z$ for
$f_{\mathrm{ULAP}}=0.5$ and a fixed minihalo mass $M=10^{5}M_{\odot}$. Also,
in the lower panel, we plot $z_{\mathrm{coll}}$ with various minihalo masses
for a fixed ULAP mass $m_{a}=10^{-21}~{}\mathrm{eV}$. Without oscillons,
$z_{\mathrm{coll}}$ hardly depends on the minihalo mass. In the $\Lambda$CDM
model and ULAP model without oscillons, $\langle
z_{\mathrm{coll}}\rangle\simeq z$ because $\sigma(z,M)$ is small compared to
$\delta_{c}$. On the other hand, in the case of the ULAP model with oscillons,
$\sigma(z,M)$ is large compared to $\delta_{c}$, and thus, $\langle
z_{\mathrm{coll}}\rangle$ is much larger than $z$. In this case, the small
halos were formed at a much earlier epoch as we can see in the lower panel of
Fig. 3. In the following, physical quantities related to the halo will be
calculated using averaged collapse redshift, which is denoted by
$z_{\mathrm{coll}}$ for simplicity.

Figure 3: The collapse redshift $z_{\mathrm{coll}}$ (3.5) are shown as the
function of redshift $z$ in the region of our interest for
$f_{\mathrm{ULAP}}=0.5$. Line styles of the top panel is the same as in Figs.
1 and 2. On the bottom panel, line styles represent the different masses of
the minihalo. In the case of the model without oscillons, the values of the
collapse redshift are too close together to distinguish. For the CDM case,
only $M=10^{5}M_{\odot}$ was plotted since the change in mass did not change
$z_{\mathrm{coll}}$ so much.

###  3.2 Minihalo Profile

In order to calculate the 21 cm fluctuations, we need to know the distribution
of neutral hydrogen inside a minihalo, which absorbs and emits the 21 cm
background photons. First, we review the DM profile, which is the source of
the gas profile. N-body simulations show that the DM distribution is described
by the Navarro-Frenk-White (NFW) profile [9, 43],

$\rho_{\mathrm{DM}}(r)=\frac{\rho_{\mathrm{DM}0}}{\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}},$
(3.6)

where $r_{s}$ is the scale radius, and $\rho_{\mathrm{\mathrm{DM}0}}$ is the
central density. The scale radius is related to the virial radius
$r_{\mathrm{vir}}$ through the concentration parameter $y\equiv
r_{\mathrm{vir}}/r_{s}$, which is fitted by [44]

$y=\frac{14.8}{1+z_{\mathrm{coll}}}\left(\frac{M}{1.3\times
10^{13}h^{-1}M_{\odot}}\right)^{-0.14},$ (3.7)

where $h=0.68$ is the Hubble parameter in units of $100$ km/sec/Mpc. The
virial radius is calculated by the spherical collapse model [45] as

$r_{\mathrm{vir}}=0.784\,h^{-1}~{}\mathrm{kpc}\left(\frac{M}{10^{8}h^{-1}M_{\odot}}\right)^{1/3}\left(\frac{\Omega_{m0}}{\Omega_{m}(z_{\mathrm{coll}})}\frac{\Delta}{18\pi^{2}}\right)^{-1/3}\left(\frac{1+z_{\mathrm{coll}}}{10}\right)^{-1},$
(3.8)

where $\Delta\equiv 18\pi^{2}+82d-39d^{2}$. Here, $\rho_{m}(z)$ and
$\Omega_{m}(z)\equiv\rho_{m}(z)/\rho_{c}(z)$ are the matter density and
density parameter at redshift $z$, respectively. Using these parameters, the
central energy density is written as

$\rho_{\mathrm{DM}0}=\frac{M}{\int_{0}^{r_{\mathrm{vir}}}\left[y\frac{r}{r_{\mathrm{vir}}}\left(1+y\frac{r}{r_{\mathrm{vir}}}\right)^{2}\right]^{-1}\mathrm{d}^{3}r}.$
(3.9)

Next, we consider the halo gas profile, which is determined by the DM profile
in the halo. For simplicity, we make two assumptions: the gas within a halo is
isothermal and in hydrostatic equilibrium. Isothermal properties are
guaranteed by the fact that the gas in the halo is in virial equilibrium.
Then, we can define the virial temperature $T_{\mathrm{vir}}$ as $\langle
K\rangle=(3/2)k_{B}T_{\mathrm{vir}}$, where $\langle K\rangle$ is the time-
averaged kinetic energy per particle and $k_{B}$ is the Boltzmann constant.
From this relation, the virial theorem leads to

$\displaystyle T_{\mathrm{vir}}$ $\displaystyle=\frac{\mu
m_{p}}{2k_{B}}\frac{GM}{r_{\mathrm{vir}}}$ $\displaystyle=2.7\times
10^{4}\,h^{-1}~{}\mathrm{K}\left(\frac{M}{10^{8}h^{-1}M_{\odot}}\right)^{1/3}\left(\frac{\Omega_{m0}}{\Omega_{m}(z_{\mathrm{coll}})}\frac{\Delta}{18\pi^{2}}\right)^{-1/3}\left(\frac{1+z_{\mathrm{coll}}}{10}\right)^{-1},$
(3.10)

where $\mu=1.22$ is the mean molecular weight of the neutral hydrogen gas, and
$m_{p}$ is the proton mass [45]. On the other hand, hydrostatic equilibrium
means that the gas pressure $P(r)$ and gravitational force are balanced, that
is

$\frac{\mathrm{d}P(r)}{\mathrm{d}r}=-\frac{GM_{\mathrm{in}}(r)}{r^{2}}\rho_{g}(r),$
(3.11)

where $\rho_{g}(r)$ is the gas density at a distance $r$ from the center and
$M_{\mathrm{in}}(r)$ is the mass within a radius $r$ [46]. Combined with the
equation of state,

$P(r)=\frac{\rho_{g}(r)}{\mu m_{p}}k_{B}T_{\text{vir}},$ (3.12)

the gas density is obtained as

$\rho_{g}(r)=\rho_{g0}\exp\left[-\frac{\mu
m_{p}}{2k_{B}T_{\mathrm{vir}}}\left(v_{\mathrm{esc}}(0)^{2}-v_{\mathrm{esc}}(r)^{2}\right)\right],$
(3.13)

where $v_{\mathrm{esc}}(r)$ is the escape velocity of the halo given by

$v_{\mathrm{esc}}(r)^{2}=2\int_{r}^{\infty}\frac{GM(\tilde{r})}{\tilde{r}^{2}}\mathrm{d}\tilde{r}=\frac{2GM}{r}\frac{\log(1+y\frac{r}{r_{\mathrm{vir}}})}{\left[\log(1+y)-\frac{y}{1+y}\right]}.$
(3.14)

The central density $\rho_{g\mathrm{0}}$ is determined by requiring that the
ratio of the baryon mass to the total matter mass in the halo is equal to
$\Omega_{b}/\Omega_{m}$, which leads to

$\rho_{g0}=\frac{\Delta}{3}\frac{y^{3}e^{A}}{\int_{0}^{y}(1+t)^{A/t}t^{2}\mathrm{d}t}\frac{\Omega_{b}}{\Omega_{m}}\rho_{m}(z),$
(3.15)

where

$A\equiv\frac{3y}{\log(1+y)-\frac{y}{1+y}}.$ (3.16)

This gives us the number density of neutral hydrogen as

$n_{\mathrm{HI}}(r)=(1-Y)\frac{\rho_{g}(r)}{m_{p}},$ (3.17)

where $Y\simeq 0.25$ is the helium fraction.

The DM and gas profiles are shown in Fig. 4. It is seen that the gas profile
does not follow the DM profile in particular near the center. This contrasts
with the truncated isothermal sphere (TIS) model, which is commonly used in
other papers [47, 48]. We will discuss this point later.

Figure 4: Density profile of DM (left) and baryon gas (right) as a function of
the normalized halo radius at $z=10$ for $M=10^{5}M_{\odot}$. Line styles are
the same as in Figs. 1 and 2.

###  3.3 Spin Temperature

The 21 cm line photons are emitted and absorbed by the transition between the
ground state and the excited state of the hyperfine structure of hydrogen
atom. The spin temperature is defined using the number density ratio of the
excited state to the ground state [49]:

$\frac{n_{1}}{n_{0}}\equiv\frac{g_{1}}{g_{0}}\exp\left(-\frac{T_{\ast}}{T_{s}}\right),$
(3.18)

where $n_{1}$ and $n_{0}$ are the number density of hydrogen atoms in the
excited and ground states, $g_{1}=3$ and $g_{0}=1$ are the degrees of freedom
of the excited and ground states respectively, and $T_{\ast}\simeq
68~{}\mathrm{mK}$ is the temperature corresponding to the energy difference
between the two states.

The physical processes involved in the transition between the two states are
spontaneous emission, absorption, and stimulated emission. The transition rate
for spontaneous emission is given by $A_{10}=2.85\times
10^{-15}~{}\mathrm{s}^{-1}$. The transition rates of the other two processes
are determined by the details of three interactions; absorption and emission
of a CMB photon, particle collision, and the Wouthuysen-Field effect triggered
by a Ly-$\alpha$ photon. Assuming that all these three processes are in
equilibrium, the spin temperature can be written as a weighted average of the
three temperatures corresponding to the three processes,

$T_{s}=\frac{T_{\gamma}+y_{c}T_{K}+y_{\alpha}T_{\alpha}}{1+y_{c}+y_{\alpha}},$
(3.19)

where $T_{\gamma},\,T_{K},$ and $T_{\alpha}$ are the CMB photon temperature,
kinetic (or gas) temperature, and Ly-$\alpha$ color temperature respectively,
and $y_{c}$, $y_{\alpha}$ are the normalized deexcitation rate of particle
collision and Ly-$\alpha$ pumping, respectively [50]. Here, we use the virial
temperature $T_{\mathrm{vir}}$ as the gas temperature $T_{K}$, which is
justified when cooling is not efficient. Moreover, since there are few free
electrons at the time we are interested in, we consider only H-H collisions
for particle collisions. We can, then, write the normalized deexcitation rate
of particle collision $y_{c}$ as

$y_{c}=\frac{T_{\ast}n_{\mathrm{HI}}\kappa_{10}}{A_{10}T_{\mathrm{K}}},$
(3.20)

where $\kappa_{10}$ is the deexcitation rate of H-H collision, approximated by
[51]

$\kappa_{10}=3.1\times
10^{-11}\left(\frac{T_{K}}{1\mathrm{~{}K}}\right)^{0.357}\exp\left(-\frac{32\mathrm{~{}K}}{T_{K}}\right)~{}\mathrm{cm}^{3}\mathrm{~{}s}^{-1}.$
(3.21)

As for $y_{\alpha}$, we set $y_{\alpha}=0$ because, at the time we are
considering, the process of star formation depends on astrophysics and
contains large uncertainties. Thus, we do not consider the contribution from
Ly-$\alpha$, which primarily comes from stars.444Since it is often the case
that $T_{\alpha}\approx T_{K}$ [52], the weight of $T_{K}$ in the Eq. (3.19)
becomes larger when considering the Wouthuysen-Field effect. This will make
the spin temperature $T_{s}$ close to kinetic temperature $T_{K}$.  The spin
temperatures $T_{s}$ for various models are shown in Fig. 5. It is seen from
Fig. 5 that the spin temperature is nearly independent of $r$ when oscillons
are present. This is because large $z_{\mathrm{coll}}$ makes $y_{c}$ larger,
which leads to $T_{s}\simeq T_{\mathrm{K}}$. In the absence of oscillons, the
spin temperature is small compared to the CDM model due to the suppression of
fluctuation by ULAP, which leads to smaller $z_{\mathrm{coll}}$.

Figure 5: Spin temperature in a minihalo with $M=10^{5}M_{\odot}$ as a
function of the normalized halo radius at $z=10$. The difference in line
styles is the same as in Figs. 1 and 2.

###  3.4 Brightness Temperature

Now let us consider the effects of minihalos on the 21 cm radiation. The
observable related to the 21 cm radiation is the brightness temperature, which
is the temperature of the incoming flux when the radiation is assumed to be
blackbody radiation in the Rayleigh-Jeans limit. The brightness temperature of
the 21 cm line passing through a minihalo of mass $M$ with an impact parameter
$\alpha$ is estimated by solving the radiative transfer equation, which leads
to

$T_{b}\left(\nu,M,\alpha\right)=T_{\gamma}(z)e^{-\tau_{\mathrm{max}}(\nu,M,\alpha)}+\int_{-\infty}^{\infty}\mathrm{d}x\,T_{\mathrm{s}}(r)e^{-\tau(\nu,M,\alpha,x)}\frac{\partial\tau(\nu,M,\alpha,x)}{\partial
x},$ (3.22)

where $|x|=\sqrt{r^{2}-\alpha^{2}}$ is the length along the line of sight. The
partial optical depth $\tau(\nu,M,\alpha,x)$ and the total optical depth
$\tau_{\mathrm{max}}(\nu,M,\alpha)$ through a minihalo along the line of sight
are calculated by integrating the absorption coefficient as [53]

$\displaystyle\tau(\nu,M,\alpha,x)$
$\displaystyle=\frac{3c^{2}A_{10}T_{\ast}}{32\pi\nu_{\ast}^{2}}\int_{-x_{\mathrm{max}}}^{x}\frac{n_{\mathrm{HI}}(r)}{T_{s}(r)}\phi(\nu)\mathrm{d}x,$
(3.23) $\displaystyle\tau_{\mathrm{max}}(\nu,M,\alpha)$
$\displaystyle=\tau\left(\nu,M,\alpha,x_{\mathrm{max}}\right),$ (3.24)

where $\nu_{\ast}=1420~{}\mathrm{MHz}$ is the frequency of the 21 cm emission,
$x_{\mathrm{max}}=\sqrt{r_{\mathrm{vir}}^{2}-\alpha^{2}}$, and $\phi(\nu)$ is
the Doppler broadening line profile written as

$\phi(\nu)=\frac{1}{\sqrt{\pi}\Delta\nu}\exp\left[-\frac{\left(\nu-\nu_{\ast}\right)^{2}}{\Delta\nu^{2}}\right],\quad\left(\Delta\nu^{2}\equiv\nu_{\ast}\frac{2k_{B}T_{\mathrm{vir}}}{m_{p}c^{2}}\right).$
(3.25)

Since what we can observe is the difference between the brightness temperature
and CMB temperature, we define the differential brightness temperature with
respect to the CMB as

$\Delta T_{b}\equiv\frac{\langle T_{b}\rangle-T_{\gamma}(z)}{1+z},$ (3.26)

where $\langle T_{b}\rangle\equiv\int
T_{b}(\nu_{\ast},M,\alpha)\,\mathrm{d}A/A$ is the averaged brightness
temperature over the halo cross section $A\equiv\pi r_{\mathrm{vir}}^{2}$.
Furthermore, we take a weighted average of $\Delta T_{b}$ over the mass
distribution. The mean differential brightness temperature is then written as
[54]

$\overline{\Delta
T}_{b}=\frac{c(1+z)^{4}}{\nu_{\ast}H(z)}\int_{M_{\min}}^{M_{\max}}\Delta\nu_{\mathrm{eff}}\Delta
T_{b}A\frac{\mathrm{d}n}{\mathrm{d}M}\mathrm{d}M,$ (3.27)

where $\Delta\nu_{\mathrm{eff}}=[\phi(\nu_{\ast})(1+z)]^{-1}$ is the effective
redshifted linewidth, and $M_{\mathrm{max}}$ and $M_{\mathrm{min}}$ are
maximum and minimum minihalo masses, respectively. We set $M_{\mathrm{min}}$
as the Jeans mass [45]

$M_{\mathrm{min}}(z)=5.73\times
10^{3}\left(\frac{\Omega_{m}h^{2}}{0.15}\right)^{-1/2}\left(\frac{\Omega_{b}h^{2}}{0.02}\right)^{-3/5}\left(\frac{1+z}{10}\right)^{3/2}M_{\odot},$
(3.28)

and $M_{\mathrm{max}}$ is determined by the condition
$T_{\mathrm{vir}}(M)<10^{4}\,\mathrm{K}$, for which the star formation is not
effective due to the inefficient atomic gas cooling.

Since the spatial fluctuations of the minihalo distribution are proportional
to the density fluctuations [42], the fluctuations of the brightness
temperature should be written as a product of the density fluctuations and the
brightness temperature with a bias factor. From the above relations, we can
derive the root-mean-square (rms) of the 21 cm signal corresponding to
$3\sigma$ fluctuations as

$\left\langle\delta
T_{b}^{2}\right\rangle^{1/2}=3\sigma_{p}\left(\Delta\theta,\Delta\nu_{\mathrm{b}}\right)\bar{b}(z)\overline{\Delta
T_{b}},$ (3.29)

where $\sigma_{p}(\Delta\theta,\Delta\nu_{\mathrm{b}})$ is the mass variance
filtered over cylindrical pencil beam with the angular size $\Delta\theta$ and
frequency band width $\Delta\nu_{\mathrm{b}}$. Using comoving radius
$R=\Delta\theta(1+z)D_{A}(z)/2$ and length
$L\approx(1+z)cH(z)^{-1}(\Delta\nu_{\mathrm{b}}/\nu)$ of the beam,
$\sigma_{p}$ is expressed as [55, 54]

$\sigma_{p}^{2}=\frac{8D^{2}(z)}{\pi^{2}R^{2}L^{2}}\int_{0}^{\infty}\mathrm{d}k\int_{0}^{1}\mathrm{d}x\,\frac{\sin^{2}\left(\frac{kLx}{2}\right)J_{1}^{2}\left[kR\left(1-x^{2}\right)^{1/2}\right]}{x^{2}\left(1-x^{2}\right)}\left(1+fx^{2}\right)^{2}\frac{P(k)}{k^{2}},$
(3.30)

where $D(z)$ is a linear growth factor normalized by the condition $D(0)=1$,
$J_{1}$ is the Bessel function, and $f\approx\Omega_{\mathrm{m}}^{0.6}$ [56].
The factor $(1+fx^{2})^{2}$ is included because the peculiar velocities change
the effects of Hubble expansion slightly, which leads to the correction to the
cylinder length. $\bar{b}(z)$ is the flux-weighted bias factor defined by

$\bar{b}(z)\equiv\frac{\int_{M_{\min}}^{M_{\max}}\mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M}\mathcal{F}(z,M)b(z,M)}{\int_{M_{\min}}^{M_{\max}}\mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M}\mathcal{F}(z,M)},$
(3.31)

where $\mathcal{F}(z,M)=\langle T_{\mathrm{b}}\rangle A\sigma_{v}$ is the
effective flux from a minihalo, $\sigma_{v}$ is the velocity dispersion of a
minihalo, and

$b\left(z,M\right)=1+\frac{\nu(z,M)^{2}-1}{\delta(z)}$ (3.32)

is a halo bias, which represents how the minihalo distribution is biased
relative to the matter distribution [42].

##  4 21 cm signals and sensitivity

We evaluate the rms of the 21 cm fluctuation using Eq. (3.29) as shown in Fig.
6. The red hatched and blue hatched regions represent the sensitivity of the
SKA and Fast Fourier Transform Telescope (FFTT)-like observation,
respectively. The noise of these observations is given by [52, 57]

$\delta
T_{\text{noise}}=20~{}\mathrm{mK}\frac{10^{4}\mathrm{~{}m}^{2}}{A_{\mathrm{tot}}}\left(\frac{10~{}\operatorname{arcmin}}{\Delta\theta}\right)^{2}\left(\frac{1+z}{10}\right)^{4.6}\left(\frac{\mathrm{MHz}}{\Delta\nu_{\text{b}}}\frac{100h}{t_{\mathrm{int}}}\right)^{1/2},$
(4.1)

where $A_{\mathrm{tot}}$ is the effective collecting area, and
$t_{\mathrm{int}}$ is the integration time. Here, we set
$A_{\mathrm{tot}}=10^{5}~{}\mathrm{m}^{2}$ for the SKA-like observation, and
$A_{\mathrm{tot}}=10^{7}~{}\mathrm{m}^{2}$ for the FFTT-like observation. The
other parameters are set as $\Delta\theta=9~{}\mathrm{arcmin}$,
$\Delta\nu_{\text{b}}=1~{}\mathrm{MHz}$, and $t_{\mathrm{int}}=1000h$.

|
---|---
|
Figure 6: Root-mean-square of the 21 cm fluctuation as a function of redshift.
We have taken $f_{\mathrm{ULAP}}=0.01,\,0.1,\,0.5,\,1$ from left top to bottom
right. The regions shaded in light blue and magenta show the sensitivity curve
of SKA and FFTT-like observation. Line styles are the same as in Figs. 1 and
2.

From Fig. 6, one can see that, in the case of ULAP without oscillons, the 21
cm fluctuation signal becomes weaker as $f_{\mathrm{ULAP}}$ increases for a
given redshift. This is because ULAP suppresses the matter fluctuations on
small scales and reduces the number of minihalos that contribute to the 21 cm
signal. As the ULAP fraction decreases, the suppression becomes less effective
and the signal asymptotically approaches that in the CDM case. On the other
hand, in the presence of oscillons, the 21 cm signal changes non-trivially as
$f_{\mathrm{ULAP}}$ increases. There is no signal for $f_{\mathrm{ULAP}}=1$
because the fluctuations are amplified so much that there is no minihalo whose
virial temperature is smaller than $10^{4}~{}\mathrm{K}$, and the 21 cm
fluctuations are not generated. This is because we use $z_{\mathrm{coll}}$ for
$z$ in the formulas of $M_{\mathrm{min}}$ and $M_{\mathrm{max}}$, leading to
$M_{\mathrm{min}}>M_{\mathrm{max}}$, i.e. no mass range for halos inducing the
21 cm signal. We will discuss uncertainties of this procedure in Sec. 5. As
$f_{\mathrm{ULAP}}$ becomes smaller, the signals first become larger and
exceed the CDM case and then approach the CDM case, which is due to the
competing effects of the increasing fraction of minihalos contributing to the
21 cm fluctuations and the decreasing total number of minihalos as
$f_{\mathrm{ULAP}}$ decreases.

Figure 7: Parameter region where the 21 cm signal is observed in future
observation by SKA. The vertical and horizontal axes represent the ULAP decay
constant and ULAP mass, respectively. The black, magenta, and cyan lines
correspond to the sensitivity of SKA, half of the CDM signal, and twice as
large as the CDM signal, respectively. The region between the black and
magenta lines and the region surrounded by the cyan line can be distinguished
from the CDM signal. The blue shaded region is excluded by large-scale
structure observations. The red shaded region indicates the overabundance of
ULAP, $f_{\mathrm{ULAP}}>1$. Within the green shaded region, it is not
possible to obtain a signal distinguishable from the case without oscillons
because the power spectrum of oscillons is smaller than the CDM and ULAP power
spectrum at cut-off scale $k_{c}$,
$P_{\mathrm{osc}}(k_{c})<P_{\mathrm{CDM}}(k_{c})+P_{\mathrm{ULAP}}(k_{c})$. In
the yellow region of $m_{a}>4\times 10^{-20}~{}\mathrm{eV}$, the oscillons
decay before $z=10$, so their effects cannot be observed in the 21 cm lines.

Fig. 7 shows the parameter regions observable in future observations by SKA at
redshift $z=10$. We chose $z=10$ as the reference redshift because the signal
is large enough to be observed and reionization does not occur. The black
solid line shows the upper bound of the ULAP decay constant $F$ below which
the signal of 21 cm fluctuations is larger than the SKA sensitivity. The
magenta and cyan lines indicate the parameters with which the signal with
oscillons is twice and half of the CDM one, respectively. We roughly set the
criterion that we can distinguish the signal with oscillons from the CDM one
if it is larger than twice the CDM signal or smaller than half of the CDM
signal. Thus, the ULAP in the parameter region between the black line and the
magenta line, or surrounded by the cyan line, is considered to be
distinguished from CDM by the SKA observations in a rough estimation.

In addition to the observability of the signal of the 21 cm fluctuations, we
consider the following constraints. The blue shaded region is excluded because
the oscillon power spectrum is larger than $\Lambda$CDM one on large scale
$k<\mathcal{O}(1)~{}\mathrm{Mpc}^{-1}$,
$P_{\mathrm{osc}}(1/\mathrm{Mpc})>P_{\Lambda\mathrm{CDM}}(1/\mathrm{Mpc})$ [6,
58, 59, 5]. The red shaded region is constrained by the condition
$f_{\mathrm{ULAP}}\leq 1$. We should note that this limit depends on the
initial value of the ULAP. The initial value of the scalar field is set as
$\phi(t_{\mathrm{ini}})/F=12\pi$, but the limit becomes stricter if it becomes
larger. We cannot obtain a signal distinguishable from the signal in the case
without oscillons in the green shaded region because the power spectrum of
oscillons at $k_{c}$ is smaller than that of CDM and ULAP,
$P_{\mathrm{osc}}(k_{c})<P_{\mathrm{CDM}}(k_{c})+P_{\mathrm{ULAP}}(k_{c})$
(see Sec. 2.2). In the yellow shaded region, which corresponds to the region
of $m_{a}>4\times 10^{-20}$, the oscillons decay before $z=10$, so we cannot
apply our analysis.

##  5 Conclusion

In this paper, we evaluated the rms of the 21 cm fluctuations due to minihalos
in the presence of ULAP oscillons. Since oscillons amplify the fluctuations on
small scales, it has a non-trivial effect on the anisotropies of the 21 cm
background; if the ULAP with oscillons dominates DM $f_{\mathrm{ULAP}}\sim 1$,
the number of minihalos that contribute to the 21 cm fluctuations decreases
and the 21 cm fluctuations cannot be observed. On the other hand, for
$f_{\mathrm{ULAP}}\sim 0.1$, the signal is larger than that in the CDM case,
which could provide an evidence for the presence of oscillons if observed.

We should make some comments on the uncertainties of our analysis. We have
applied the NFW profile to the DM distribution in a halo, while others use the
TIS model, which assumes hydrostatic equilibrium between DM and baryons. This
assumption seems to be invalid, as can be read from Fig. 4. On the other hand,
the NFW profile may not describe the internal structure of a minihalo in the
presence of oscillons because the mass of the produced oscillons can be
comparable to the mass of the minihalo in our parameter region of interest.
This fact leads to more concentration of gas in the center of the minihalo and
results in larger 21 cm fluctuation signals.

Furthermore, we used the averaged collapse redshift $\langle
z_{\mathrm{coll}}\rangle$ in the calculation of the 21 cm fluctuation in
evaluating the physical properties of minihalos. However, it is more accurate
to consider the collapse redshift as a random variable which obeys the
Gaussian distribution. This leads to an underestimation of the 21 cm
fluctuation signal when $f_{\mathrm{ULAP}}\sim 1$ and oscillons are present.
With this taken into account, the 21 cm fluctuation signal may be non-zero
even when $f_{\mathrm{ULAP}}\sim 1$ in the presence of oscillons. Finally, we
only consider the case where the oscillons survive until $z\sim 10$, but if
they decay before then, the emitted ULAPs from oscillons, which behave like
warm DM, can suppress the small-scale fluctuations and provide another
constraint [60].

## Acknowledgements

We would like to thank Shin Kobayashi for helpful advice and Wakutaka Nakano
for kindly permitting us to use the numerical table of the decay rate of
oscillons.

This work is supported by the Grant-in-Aid for Scientific Research Fund of the
JSPS 20H05851(M. K.), 21K03567(M. K.), 20J20248 (K. M.) and 19J21974 (H. N.).
M. K. and K. M. are supported by World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan (M. K. and K. M.). K. M. is supported
by the Program of Excellence in Photon Science. H. N. is supported by Advanced
Leading Graduate Course for Photon Science.

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