# Imprints of fermionic and bosonic mixed dark matter on the 21-cm signal at
cosmic dawn.

Sambit K. Giri<EMAIL_ADDRESS>Aurel Schneider
<EMAIL_ADDRESS>Center for Cosmology and Theoretical Astrophysics,
Institute for Computational Science, University of Zurich, Switzerland.

###### Abstract

The 21-cm signal from the epoch of cosmic dawn prior to reionization consists
of a promising observable to gain new insights into the dark matter (DM)
sector. In this paper, we investigate its potential to constrain mixed (cold +
noncold) dark matter scenarios that are characterized by the noncold DM
fraction ($f_{\rm nCDM}$) and particle mass ($m_{\rm nCDM}$). As noncold DM
species, we investigate both a fermionic (sterile neutrino) and a bosonic
(ultralight axion) particle. We show how these scenarios affect the global
signal and the power spectrum using a halo-model implementation of the 21-cm
signal at cosmic dawn. Next to this study, we perform an inference-based
forecast study based on realistic mock power spectra from the Square Kilometre
Array (SKA) telescope. Assuming inefficient, yet non zero star formation in
minihaloes (i.e. haloes with mass below $10^{8}$ $M_{\odot}$), we obtain
stringent constraints on both $m_{\rm nCDM}$ and $f_{\rm nCDM}$ that go well
beyond current limits. Regarding the special case of $f_{\rm nCDM}\sim 1$, for
example, we find a constraint of $m_{\rm nCDM}>15$ keV (thermal mass) for
fermionic DM and $m_{\rm nCDM}>2\times 10^{-20}$ eV for bosonic DM. For the
opposite case of dominating cold DM, we find that at most 1% of the total DM
abundance can be made of a hot fermionic or bosonic relic. All constraints are
provided at the 95% confidence level.

††preprint: APS/123-QED

##  I Introduction

The cosmic dawn refers to the time before the epoch of reionization when the
first stars formed and the radiation from these stars percolated the Universe.
This stellar radiation induced spin flips in the neutral hydrogen atoms of the
primordial gas, leading to an absorption signal in the cosmic background
radiation at radio frequencies. Upcoming telescopes, such as the Square
Kilometre Array [SKA, 1, 2] will observe this signal, thereby revealing
information about the formation of first sources and the distribution of the
primordial gas at redshifts between about 10 and 25 [3].

The 21-cm signal at cosmic dawn is a promising probe for studying alternative
dark matter (DM) scenarios. It is particularly sensitive to DM decays or
annihilation [4, 5, 6], interactions between DM and standard model particles
[7, 8], or DM models yielding a suppression of the small-scale matter power
spectrum [9, 10, 11, 12, 13, 14]. While decay and annihilation scenarios
provide additional sources of radiation that may affect the 21-cm signal,
interacting DM models can lead to additional cooling channels, acting on the
temperature and the distribution of the gas. Models with suppressed power
spectra, on the other hand, are characterized by a smaller number of sources
and by a reduced clustering amplitude of the primordial gas.

In this paper, we investigate mixed dark matter scenarios featuring both a
perfectly cold and a warm or hot subcomponent. Depending on the composition of
the fluid (which is parametrized by the abundance ratio of hot to total dark
matter) this includes all cases from the standard warm DM scenario to a
$\Lambda$CDM model with an additional, hot relic particle. Note that mixed
dark matter models have been studied extensively in the literature primarily
in the context of the Lyman-$\alpha$ forest [15, 16], Milky Way satellites
[17, 18, 19], strong lensing systems [20], or the weak lensing shear signal
[21, 22]. Recently, it has also been applied to the global 21-cm signal at
cosmic dawn [10, 23] in context of the claimed first detection from the
Experiment to Detect the Global EoR Signal [EDGES, 24]. Since the
interpretation of the signal from the EDGES is highly disputed [see e.g. 25,
26], we do not discuss it further in this paper.

From a particle physics perspective, mixed dark matter scenarios can emerge
from a variety of different contexts. One popular option is an additional,
right-handed (sterile) neutrino species with a particle mass above the
electron volt (eV) range [27]. Such a fermionic particle can be produced in
the early Universe via freeze-in [28, 29] or decay production [30, 31], acting
as a hot/warm DM subcomponent similar to the left-handed neutrinos. Another
option is an ultralight axionlike particle [32, 33, 34]. Such bosonic
particles could be abundantly present in the early Universe, some of them
featuring suppressed structure formation due to a de Broglie wavelength of
astrophysical relevance. Finally, mixed dark matter scenarios with suppressed
power spectra can also arise if some of the DM interacts with another dark
relic or with particles from the standard model in the early Universe. Next to
the characteristic suppression in power, such models could also feature
remnants of acoustic oscillations [35, 36, 37].

In this paper we focus on mixed scenarios featuring either bosonic (axionlike)
or fermionic (neutrinolike) dark matter. In particular, we investigate the
imprint of these models on the spatial fluctuations of the 21-cm signal at
cosmic dawn. Although current interferometry-based radio telescopes, such as
the Low Frequency Array [LOFAR, 38], the Murchison Widefield Array [MWA, 39],
or the Hydrogen Epoch of Reionization Array [HERA, 40] have only provided
upper limits on the 21-cm power spectrum at cosmic dawn [see e.g. 41, 42], the
low frequency component of the upcoming SKA (SKA-Low) is expected to detect
the signal at high significance. In this paper, we therefore specifically
focus on a SKA-Low setup, forecasting the power of the SKA telescope to detect
or constrain fermionic or bosonic mixed dark matter.

Figure 1: The linear dimensionless matter power spectra for various models at
$z=0$. In the left- and right-hand panels, we focus on fermionic and bosonic
mixed DM models, respectively. The green lines show models with $f_{\rm
nCDM}=1$, i.e. pure warm DM with $m_{\mathrm{WDM}}=5$ keV and fuzzy DM with
$m_{\mathrm{FDM}}=10^{-20}$ eV. The orange lines correspond to the same mixed
DM models with $f_{\rm nCDM}=0.5$. In red and purple, we show models with
small fractions ($f_{\rm nCDM}=0.05$, $f_{\rm nCDM}=0.1$) and particle masses
($m_{\mathrm{WDM}}=0.1$ keV, $m_{\mathrm{FDM}}=10^{-24}$ eV). The CDM case is
shown as a blue solid line for comparison. Note that bosonic DM models are
characterized by steeper, more abrupt downturns compared to fermionic DM
models.

We begin by describing our parameterization for the DM and source scenarios in
Secs. II and III, respectively. Section IV is dedicated to the resulting 21-cm
global signal and power spectrum for mixed dark matter scenarios. In Sec. V,
we present a forecast study for the SKA telescope. Finally, we summarize our
findings in Sec. VI.

##  II  Mixed dark matter scenarios

In a two-component mixed dark matter (mixed DM) framework, the DM sector
consist of a perfectly cold plus a noncold component. By noncold DM we either
mean a hot or warm fermionic DM fluid with large free-streaming velocities or
a very light bosonic particle with wave properties affecting astrophysical
scales. The fraction of noncold DM is given by

$\displaystyle
f_{n\mathrm{CDM}}=\frac{\Omega_{\mathrm{nCDM}}}{\Omega_{\mathrm{nCDM}}+\Omega_{\mathrm{CDM}}}\
,$ (1)

where $\Omega_{\mathrm{nCDM}}$ and $\Omega_{\mathrm{CDM}}$ describe the
cosmological abundances of the two fluids. Note that $f_{\rm nCDM}=1$
corresponds to the fully warm or fuzzy DM scenarios, while $f_{\rm nCDM}\ll 1$
denotes the case of a $\Lambda$CDM model with a very subdominant, additional
noncold species.

Hereafter, the fermionic mixed DM scenario is denoted as WCDM, which stands
for warm and cold dark matter. Next to the warm-to-total fraction ($f_{\rm
WCDM}$), the model is characterized by the mass of the fermionic relic
($m_{\rm WDM}$). Following the historical convention, we use the mass
definition of a thermally produced particle. This mass is different from the
true particle mass in the case of nonthermal production in the early Universe.
Note, however, that there is a direct mapping between the different mass
definitions, as shown, e.g., by Ref. [43].

The fermionic mixed DM scenario is best motivated by the presence of an
additional sterile neutrino species. Sterile neutrinos could have been
produced in the early Universe via mixing with the active neutrino sector.
This process, known as Dodelson-Widrow production [28], is ruled out for
scenarios with $f_{\rm WCDM}=1$ [44, 43] but could be present if sterile
neutrinos only make up a fraction of the total DM sector. Other mechanisms
discussed in the literature are the resonant [or Shi-Fuller, 29] and the decay
production. Note that these scenarios may still be viable even for the case of
$f_{\rm nCDM}=1$, although large fractions of their parameter space have
recently been ruled out [e.g. 45, 46].

The bosonic mixed DM scenario is abbreviated as FCDM, standing for fuzzy and
cold dark matter. The model is characterized by the fuzzy-to-total DM fraction
$f_{\rm FCDM}$ and the bosonic particle mass $m_{\rm FDM}$. The bosonic
particle fluid forms a condensate in the early Universe, leading to a
suppression of the particle free-streaming process [32, 18]. As a consequence,
models with $f_{\rm FCDM}=1$ and tiny particle masses of order $m_{\rm
FDM}\sim 10^{-21}$ eV evade constraints from small-scale structure formation.
Note that bosonic particles of this mass range are characterized by their
wavelike nature that affects astrophysical scales and leads to the typical
small-scale power suppression of mixed DM models.

In Fig. 1, we show the dimensionless matter power spectra of fermionic (left
panel) and bosonic (right panel) DM benchmark models at $z=0$. The full WDM
and FDM cases with $f_{\rm nCDM}=1$, and with masses of $m_{\rm WDM}=5$ keV
and $m_{\rm FDM}=10^{-20}$ eV, respectively, are shown as green dashed-dotted
lines. These models exhibit a strong cutoff at large $k$ values compared to
the standard CDM model shown as a blue solid line. As expected, the cutoff is
significantly steeper for fuzzy DM compared to the warm DM model [18, 47].

The orange lines in Fig. 1 show the power spectra of WCDM (left) and FCDM
models (right) with $f_{\rm nCDM}=0.5$. The particle masses are $m_{\rm
WDM}=5$ keV and $m_{\rm FDM}=10^{-20}$ eV, i.e., the same as for the full WDM
and FDM cases shown in green. Due to the reduced noncold fraction, the visible
power cutoff is significantly shallower with some remaining power down to
small scales (high $k$ values).

Next to models featuring fairly strong downturns in the power spectrum, we
also show examples which are only slightly damped toward higher $k$ values.
These models are characterized by small particle masses ($m_{\rm WDM}=0.1$ keV
and $m_{\rm FDM}=10^{-24}$ eV) and small fractions ($f_{\rm nCDM}=0.05$ and
$f_{\rm nCDM}=0.10$), as shown by the red dashed-dotted-dotted and purple
dotted lines in Fig. 1. These models are qualitatively different from the ones
with higher fractions because they affect the source abundance at all scales
and yield modified power spectra at $k$ modes that are directly observable
with future radio interferometric surveys.

Note that throughout this paper, we use the Boltzmann solver class [48, 49] to
calculate the WCDM power spectra. For the FCDM power spectra, we rely on
axionCAMB [50], which is a modified version of CAMB [51] that includes the
option of an additional axionlike DM component.

##  III  Source model

One of the main reasons why the 21-cm signal can be used to distinguish
between cold and mixed DM models, is the change of source distributions
between the two models. The assumed properties of sources are therefore a
crucial model component, which we discuss in the present section.

###  III.1 Halo mass function, bias, and growth

We assume all photon sources to form within DM haloes. The halo number density
is described by the halo mass function obtained from the extended Press-
Schechter (EPS) approach,

$\displaystyle\frac{\mathrm{d}n}{\mathrm{d}\mathrm{ln}M}=-\frac{\bar{\rho}}{M}\nu
f(\nu)\frac{\mathrm{d}\mathrm{ln}\sigma}{\mathrm{d}\mathrm{ln}M}\ ,$ (2)

where $\nu=(\delta^{2}_{c}/\sigma)^{2}$ with $\delta_{c}=1.686$ and
$\bar{\rho}$ is the mean baryon density. The halo mass $M$ is given by
$M=(4\pi/3)\bar{\rho}(cR)^{3}$, where $c$ is a free parameter that has to be
fixed to simulations. We assume the first-crossing distribution $f(\nu)$ to
have the following form [52],

$\displaystyle f(\nu)=A\sqrt{\frac{2q\nu}{\pi}}(1+\nu^{-p})e^{-q\nu/2}\ ,$ (3)

with $A=0.3222$, $p=0.3$, and $q=1$. The variance is given by

$\displaystyle\sigma^{2}(R,z)=\int\frac{\mathrm{d}k^{3}}{(2\pi)^{3}}P_{\mathrm{lin}}(k)\mathcal{W}(k|R)\
,$ (4)

where we assume the window function to be described by the smooth-$k$ filter
[53]

$\displaystyle\mathcal{W}(k|R)=\frac{1}{1+(kR)^{\beta}}\ .$ (5)

Following Ref. [22], we assume $c=3.3$ and $\beta=4.8$, as these values
provide the best match to numerical simulations of a series of mixed DM
models.

WCDM
FCDM

Figure 2: The halo mass function of different fermionic (top panels) and
bosonic (bottom panels) mixed DM scenarios at redshift $z=$ 11 (left panels),
13 (middle panels), and 15 (right panels). The shown models are the same ones
shown in Fig. 1.

In Fig. 2, we plot the halo mass functions of the benchmark WCDM (top panels)
and FWDM (bottom panels) models shown in Fig. 1. The different columns
correspond to the three different redshifts $z=11$ (left), $13$ (middle), and
$15$ (right). As expected, the full WDM ($m_{\rm WDM}=5$ keV) and FDM ($m_{\rm
FDM}=10^{-20}$ eV) models exhibit a strongly suppressed halo abundance below
$M\sim 10^{9}$ $M_{\odot}$/h compared to the CDM model. The corresponding
mixed DM models (with $f_{\rm nCDM}=0.5$) also show suppressed halo numbers,
but the mass functions keep rising toward small halo masses. The two models
with low fraction ($f_{\rm nCDM}=0.05,\,0.1$) and small particle mass ($m_{\rm
WDM}=0.1$ keV, $m_{\rm FDM}=10^{-24}$ eV) are characterized by a low halo
number density over the full mass range. Compared to the CDM model, the halo
mass functions are shifted downwards while retaining a similar overall shape.
Even though these models do not reproduce the CDM halo number density at any
mass scale, they remain in agreement with low redshift observations [e.g. 19,
15, 54, 55].

Using the peak-background split model of Ref. [52], the halo bias can be
described within the EPS formalism. The halo bias is given as

$\displaystyle
b(M)=1+\frac{q\nu-1}{\delta_{c}(z)}+\frac{2p}{\delta_{c}(z)[1+(q\nu)^{p}]}\ ,$
(6)

where $q$ and $p$ have been defined above, and
$\delta_{c}(z)=\delta_{c}/D(z)$, with $D(z)$ being the growth function [see
e.g. Ref 56].  The halo bias is one of the key ingredients required to
calculate the power spectrum in the halo model [see eq. 35 in Ref. 57] and
changes for different nCDM models [e.g. 58].

To describe the growth of halos, we assume a halo mass accretion model based
on the EPS method. In this method, the accretion mass $M_{\mathrm{ac}}$ is
given by

$\displaystyle\frac{\mathrm{d}M_{\mathrm{ac}}}{\mathrm{d}z}=-\sqrt{\frac{2}{\pi}}\frac{M_{\mathrm{ac}}}{\sqrt{\sigma^{2}(QM_{\mathrm{ac}},0)-\sigma^{2}(M_{\mathrm{ac}},0)}}\frac{\mathrm{d}\delta_{c}(z)}{\mathrm{d}z}\
.$ (7)

The above differential equation can be solved assuming
$M_{\mathrm{ac}}(M,z_{f})=M$, where $z_{f}$ designates the final redshift of
interest. $Q$ is a free parameter. We choose $Q=0.6$ as it agrees with the
simulations in Ref. [59]. See Ref. [57] for a more detailed description of the
mass accretion model.

###  III.2 Stellar-to-halo mass relation

For now, we have established a model for the halo growth and distribution.
What remains to be modeled is the connection of these haloes to photon
emitting sources. Following Ref. [57], we assume the stellar-to-halo mass
relation,

$\displaystyle
f_{\star}(M)=\frac{2(\Omega_{b}/\Omega_{m})f_{\star,0}}{\left(\frac{M}{M_{p}}\right)^{\gamma_{1}}+\left(\frac{M}{M_{p}}\right)^{\gamma_{2}}}\times
S(M),$ (8)

which consists of a double power law multiplied by a small-scale term

$\displaystyle
S(M)=\left[1+\left(\frac{M_{t}}{M}\right)^{\gamma_{3}}\right]^{\gamma_{4}}$
(9)

that either leads to a boost or the a further suppression of the stellar-to-
halo mass relation below $M_{t}$.

For the double-power law of Eq. (8), we assume $M_{p}=2.5\times 10^{11}$
$M_{\odot}$/h, $\gamma_{1}=0.49$, $\gamma_{2}=-0.61$ [60]. These numbers
guarantee the recovery of the observed high-redshift luminosity functions of
Refs. [61, 62] assuming a $\Lambda$CDM universe.

The small-scale term of Eq. (9) provides a parametrization of the
uncertainties related to the stellar-to-halo mass relation at small mass
scales.  For large masses ($M_{t}\gtrsim 10^{9}$ M${}_{\odot}/h$), we have
constraints from high redshift ($6\lesssim z\lesssim 11$) observations of
luminosity functions abundance matched to the CDM haloes [60]. However, we
currently have no constraints on the population of galaxies residing in the
small mass haloes below $\sim 10^{9}$ M${}_{\odot}/h$.

Accounting for the constraints discussed above, we assume a stellar-to-halo
mass with free parameters $f_{\star,0}$ $M_{t}$, $\gamma_{3}$, and
$\gamma_{4}$. We furthermore restrict a hard limit of
$f_{\star}(M)\leq\Omega_{b}/\Omega_{m}$ imposed by the simple constraint that
the galaxy is a part of the total halo mass. Finally, we assume the stellar-
to-halo mass relation to drop to zero below $M_{\rm min}=5\times
10^{5}M_{\odot}$/h, which is motivated by the molecular cooling limit [63].
The molecular cooling limit corresponds to a virial temperature of 500 K [63],
which translates to a halo mass of $\sim 7\times 10^{5}M_{\odot}$/h and $\sim
3\times 10^{5}M_{\odot}$/h at redshift 10 and 20 respectively, following eq.
26 in Ref. [64]. Note, however, that this is an approximate relation in a CDM
universe and the conversion is tricky for nCDM universe [e.g. 65, 11, 10, 66].
Therefore for simplicity, we use a redshift independent $M_{\mathrm{min}}$ in
this work.

In Fig. 3, we provide three examples of possible stellar-to-halo mass
relations using the parametrization described above. The magenta solid line
shows the case of $\gamma_{4}=0$, resulting in a double power law shape all
the way down to the molecular cooling limit (DPL model). The sky blue dashed-
dotted line is characterized by a strong truncation around the scale where
atomic cooling becomes inefficient (TRUNCATED model). The corresponding
parameters are $M_{t}=7\times 10^{7}$ $M_{\odot}$/h, $\gamma_{3}=3$, and
$\gamma_{4}=-3$ [see Refs. 64, 67, for a justification of the truncation mass
used here]. The dark green dashed line finally provides an example of a
stellar-to-halo mass relation that is flattening toward small scales (FLOOR
model). Such a scenario could be realized via the efficient formation of first
generation (pop III) stars.  See Refs. [68, 69] for examples of such a model.
In this work, the FLOOR model is defined by the parameters $M_{t}=7\times
10^{7}$ $M_{\odot}$/h, $\gamma_{3}=3$, and $\gamma_{4}=0.2$.

Figure 3: The fraction of halo mass converted into photon emitting stars
plotted against the halo masses. We show the three models explored in this
study (see text).

The three source models DPL, TRUNCATED, and FLOOR presented in Fig. 3 are
examples that stand for our currently poor knowledge of the star-formation
process in minihaloes. Later in the paper, they are used as benchmark source
models for our SKA mock observations.

###  III.3 Spectral emission

After having established the connection between sources and haloes, we still
have to set up a parametrization for the spectral energy distribution (SED) of
the source emission. For both the SED of the Lyman-$\alpha$ ($\alpha$) and
x-ray ($X$) flux, we assume a power law

$\displaystyle I_{s}(\nu)=A_{s}\nu^{-\alpha_{s}}\ ,$ (10)

where $s=\\{\alpha,X\\}$. We fix the spectral indices to $\alpha_{\alpha}=0$
and $\alpha_{X}=1.5$, respectively. The normalization constants $A_{s}$ are
set such that

$\displaystyle\int^{\nu_{2,s}}_{\nu_{1,s}}I_{s}(\nu)=1\ .$ (11)

For the Lyman-$\alpha$ flux, the integration limits are given by
$\nu_{1,\alpha}=2.467\times 10^{15}$ Hz and $\nu_{2,\alpha}=3.290\times
10^{15}$ Hz, which correspond to the Lyman-$\alpha$ and Lyman-limit
frequencies. For the x-ray integration limits, we assume
$\nu_{1,X}=E_{0}/h_{P}$ and $\nu_{2,X}=2~{}\mathrm{keV}/h_{P}$, where $E_{0}$
is a free model parameter. See, e.g., Ref. [70] for a discussion on dependence
of the 21-cm power spectrum on $E_{0}$.

The number emissivity of UV photons between the Lyman-$\alpha$ and Lyman-limit
range is given by

$\displaystyle\epsilon_{\alpha}(\nu)=\frac{N_{\alpha}}{m_{p}}I_{\alpha}(\nu)\
,$ (12)

where $N_{\alpha}$ is the number of photons per baryon emitted in the range
between the Lyman-$\alpha$ ($\nu_{\alpha}$) and the Lyman limit
($\nu_{\mathrm{LL}}$) frequencies. The energy emissivity of x-ray photons is

$\displaystyle\epsilon_{X}(\nu)=f_{X}c_{X}\frac{I_{X}(\nu)}{\nu
h_{\mathrm{P}}}$ (13)

where $f_{X}$ is a free parameter of order unity, and $c_{X}$ is a
normalization factor constrained by observations. It is set to
$c_{X}=3.4\times 10^{40}$ ${\rm erg}~{}{\rm yr}~{}s^{-1}~{}M^{-1}_{\odot}$
based on the findings of Ref. [71].

FLOOR DPL TRUNCATED

Figure 4: Sky-averaged 21-cm differential brightness temperature
($\bar{dT}_{\mathrm{b}}$) for CDM and various WCDM (top panels) and FCDM
(bottom panels) models. We assume three different source models characterized
by a flat (FLOOR), a power law (DPL), and a truncated stellar-to-halo mass
relation at small scales (see Fig. 3).

##  IV 21-cm Signal

In this section, we provide a short summary of the 21-cm signal focusing on
the effects induced by a mixed DM sector. We discuss both the global signal
and the power spectrum with a primary focus on the latter. All calculations
are performed using the model presented in Ref. [57]. We only discuss some
general terminology here and refer to Ref. [57] for any details regarding the
formalism.

The observed 21-cm signal is given by the differential brightness temperature
$\delta T_{b}(\mathbf{x},z)$ which is a function of position $\mathbf{x}$ and
redshift $z$,

$\displaystyle\delta T_{b}(\mathbf{x},z)=T_{0}(z)x_{\rm
HI}(\mathbf{x},z)\left[1+\delta_{b}(\mathbf{x},z)\right]$
$\displaystyle\frac{x_{\alpha}(\mathbf{x},z)}{1+x_{\alpha}(\mathbf{x},z)}\left(1-\frac{T_{\rm
cmb}(z)}{T_{k}(\mathbf{x},z)}\right),$ (14)

where $T_{\rm cmb}$ is the cosmic microwave background (CMB) temperature,
$x_{\rm HI}$ the neutral hydrogen fraction, and $\delta_{b}$ describes the HI
gas perturbation field that is assumed to follow the dark matter. The
prefactor $T_{0}$ is given by,

$\displaystyle T_{0}(z)\approx
27\left(\frac{\Omega_{b}h^{2}}{0.023}\right)\left(\frac{0.15}{\Omega_{\rm
DM}h^{2}}\frac{1+z}{10}\right)^{1/2}{\rm mK}\ ,$ (15)

where $\Omega_{b}$, $\Omega_{\rm DM}$, and $h$ are the cosmological abundance
of baryons, the total DM ($\Omega_{\rm nCDM}+\Omega_{\rm CDM}$), and the
Hubble parameter. Throughout this study, we assume $\Omega_{b}=0.049$,
$\Omega_{\rm DM}=0.315$, and $h=0.673$ [72].

The radiation coupling coefficient is

$\displaystyle x_{\alpha}=\frac{1.81\times
10^{11}}{(1+z)}S_{\alpha}J_{\alpha}\ ,$ (16)

where $S_{\alpha}$ is a correction factor of order unity [calculated as
indicated in Ref. 73] and $J_{\alpha}$ is the redhsift and position-dependent
Lyman-$\alpha$ flux radiation field induced by the first stellar light. The
temperature of the gas $T_{k}$ is obtained via the differential equation,

$\displaystyle\frac{3}{2}\frac{\mathrm{d}T_{k}}{\mathrm{d}z}=\frac{T_{k}}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}z}-\frac{\Gamma_{h}}{k_{B}(1+z)H(z)},$
(17)

where $\rho$ is the total matter density field and $\Gamma_{h}$ is the
redhsift and position-dependent heating term induced by the x-ray radiation of
the first sources. Details on how to calculate $J_{\alpha}$ and $\Gamma_{h}$
can be found in Ref. [57] and references therein.

###  IV.1 Global signal

The global differential brightness temperature is obtained by averaging over
the spatial component of Eq. (IV). We refer to Ref. [3] for the details of
this calculation. Here we just want to note that the mean Lyman-$\alpha$ flux
radiation $J_{\alpha}$ and the mean x-ray heating term $\Gamma_{h}$ directly
depend on the star formation rate density (SFRD)

$\displaystyle\dot{\rho}_{\star}(z)=\int\frac{dM}{M}\frac{dn}{d\mathrm{ln}M}f_{\star}(M)\dot{M}_{\mathrm{ac}}(M,z)\
.$ (18)

The above integral depends on the halo mass function and the mass accretion
rate ($\dot{M}_{\mathrm{ac}}$). We assume the EPS method in Ref. [57] to model
the mass accretion. Both the halo mass function and the mass accretion rate
depend on the nature of DM. This means that any variation of the fraction
$f_{\rm nCDM}$ and mass $m_{\rm nCDM}$ parameters, which are introduced in
Sec. II will directly affect the global signal.

The high sensitivity of the global 21-cm signal to DM models featuring a
suppressed perturbations has been demonstrated multiple times in the past
[e.g. 74, 9, 75, 10, 12, 13, 76]. Here, we limit ourselves to qualitative
discussion including the DM and source models introduced in Sec. II and III
respectively.

FLOOR DPL TRUNCATED

Figure 5: Evolution of the power spectra over redshift at
$k=0.1~{}h/\mathrm{Mpc}$ for CDM and various benchmark WCDM (top panels) and
FCDM models (bottom panels). We again assume the three source models: FLOOR,
DPL, and TRUNCATED (left to right).

In Fig. 4 we plot the global differential brightness temperature for the WCDM
(top panels) and FCDM (bottom panels) models assuming the FLOOR (left panels),
DPL (central panels), and TRUNCATED (right panels) source models. The color
scheme is identical to the one used in Figs. 1 and 2. First, we notice that
the signal from the TRUNCATED source model is shifted toward lower redshifts
compared to the DPL and FLOOR models. This trend is a direct consequence of
the reduced stellar mass in small haloes. The same is true when comparing the
mixed DM models to the CDM case (shown in blue). The suppressed power spectra
lead to a delay in the buildup of sources that shifts the global 21-cm signal
to small redshifts. Furthermore, it is worth noticing that the FLOOR source
model provides the largest differences between the various DM models. This
indicates that different DM models can be distinguished best from each other
if the smallest haloes are populated with sources.

Finally, one should note that the DM and source models can conspire and give a
similar signal. For example, the DPL source model in CDM is nearly
indistinguishable to the FLOOR model in WCDM or FCDM scenarios, which shows
that parameters related to the DM sector and the source model can be
degenerate. We discuss more about these degeneracies in Sec. V.

###  IV.2 Power spectrum

The power spectrum of the spatially varying differential brightness
temperature field [given by Eq. (IV)] is calculated using the halo model of
cosmic dawn [57]. The model is build upon overlying flux profiles which
include spectral redshifting and look-back light-cone effects. It accounts for
the auto- and cross-spectra of all individual components (Lyman-$\alpha$
coupling, temperature, and gas density fields) and includes redshift-space
distortion effects at the linear level [77, 78]. The halo model of cosmic dawn
has been shown to agree with other predictions based on 21cmFAST [79, 80].

In terms of warm, fuzzy, and mixed DM, the 21-cm power spectrum is influenced
by two main effects: first, the number density and mass accretion rate of
sources at the smallest mass scales and, second, the direct modifications of
the matter power spectrum in the observable range. The latter is only relevant
for models with small mass ($m_{\rm nCDM}$) and fraction ($f_{\rm nCDM}$),
where the power spectrum is suppressed at scales of $k\lesssim 1$ h/Mpc.

Here we focus on the spherically averaged 21-cm power spectrum given by

$\Delta_{21}^{2}=\bar{\delta T_{\rm
b}}^{2}\frac{k^{3}}{4\pi^{2}}\left(\langle\delta_{21}^{2}\rangle+\frac{2}{3}\langle\delta_{21}\delta_{m}\rangle+\frac{1}{5}\langle\delta_{m}^{2}\rangle\right),$
(19)

where $\delta_{m}$ is the matter overdensity field. We have used Eq. (IV) with
the definition $\delta T_{b}=\bar{\delta T_{b}}(1+\delta_{21})$, with
$\bar{\delta T_{b}}$ being the globally averaged differential brightness
temperature. Note that the first term gives the 21-cm signal in real space and
the second and third terms in Eq. (19) are a consequence of averaging out the
angular dependence introduced by the redshift space distortion term [81]. See
Ref. [82] for a detailed study of redshift space distortion at cosmic dawn.

In Fig. 5, we plot the 21-cm power spectra as a function of redshift at
$k=0.1$ h/Mpc. The figure is structured in the same way as Fig. 4; i.e., the
top and bottom panels show the WCDM and FCDM scenarios, respectively, while
the three columns refer to the assumptions regarding the source models (FLOOR,
DPL, and TRUNCATED). All plots show the characteristic two-peaked shape of the
power spectrum at cosmic dawn, where the first peak refers to the
Lyman-$\alpha$ coupling and the second peak to the heating epoch. See, e.g.,
Refs. [83, 84, 85] for more discussion about this.

As a general effect visible in Fig. 5, we see that the positions of the
Lyman-$\alpha~{}$and heating peaks move toward smaller redshifts for mixed DM
models compared to CDM shown in blue. At the same time, the peaks are also
shifted to smaller redshifts when photon sources are less efficient, i.e. for
the case of the TRUNCATED models compared to the DPL and FLOOR model. This
trend again points toward potential degeneracies between DM and source models,
discussed before.

Finally, it is worth noticing that the different DM models seem to be best
distinguishable from each other when assuming the FLOOR model (as opposed to
the DPL and, more so, the TRUNCATED model). This is not surprising, since it
is the small halo mass scales that contain the most information about the
nature of dark matter. The more minihaloes are populated with sources, the
more it will be possible to constrain noncold DM models.

##  V  SKA forecast

After discussing the link between the 21-cm signal and the DM sector at the
level of individual benchmark models, we now perform a more detailed analysis
including the full model parameter space. The basic question we address is how
well the upcoming Square Kilometre Array (SKA) telescope will be able to
constrain the fermionic and bosonic mixed DM scenarios. For this reason, we
set up realistic mock data set from SKA-Low based on a $\Lambda$CDM cosmology
and assuming instrumental noise, sample variance, and foreground
contamination. With this at hand, we will perform a Markov Chain Monte Carlo
(MCMC) inference analysis of the 21-cm power spectrum, simultaneously varying
astrophysical and dark matter parameters. The final goal is to investigate to
what extent we can go beyond current constraints for warm, fuzzy, and mixed DM
models.

###  V.1 Mock observations

We start by presenting our mock observations for SKA-Low. We fully consider
the observational errors due to instrumental noise and sample variance, while
the foreground contamination is accounted for by applying cuts to the mock
data. This analysis is similar to previous forecast studies in the literature
[see e.g. Refs. 86, 70, 87, 88].

First, we simulate the interferometric data by considering the antennae
configuration of the first phase of SKA-Low 111The configuration used in this
work is taken from https://astronomers.skatelescope.org/wp-
content/uploads/2016/09/SKA-TEL-
SKO-0000422_02_SKA1_LowConfigurationCoordinates-1.pdf, which is currently
being constructed in western Australia. This phase of SKA-Low is planned to be
composed of $\sim 512$ radio antennae with a core area of 1 km diameter where
the antennae density will be high. The telescope configuration and observation
properties, such as e.g. collecting area, declination, system temperature
($T_{\mathrm{sys}}(\nu)$, where $\nu$ is the observed frequency), and
bandwidth, are summarized in Table 1.

Table 1: The properties of the first phase of SKA-Low and observation parameter values.  Parameters | Values
---|---
Total number of antennas | 512
Diameter of the core | 1 km
System temperature ($T_{\mathrm{sys}}$) |  $\left[100+60(\frac{\nu}{300\mathrm{MHz}})^{-2.55}\right]$ K
Effective collecting area | 962 $\mathrm{m}^{2}$
Declination | -30∘
Total observation time | 1000 h
Observation hour per day | 6 h
Signal integration time | 10 s
Bandwidth | 10 MHz
Figure 6: The 21-cm power spectrum for CDM universe at different redshifts
assuming the source models and error components used in the mock data set. The
purple, orange, and green lines correspond to the FLOOR, DPL, and TRUNCATED
source models, respectively. The color-shaded areas show the errors due to
instrumental noise and sample variance. The hatched areas indicate scale cuts
due to foreground contamination and noise. See text for more information.

For a symmetric radial distribution of antennae, the error on the power
spectrum due to instrumental noise can be calculated analytically [see e.g.,
90, 2]. However, the true antennae distribution is nonuniform. An
interferometry-based radio telescope records the complex visibilities, which
are proportional to the Fourier transform of the signal in the real space.
These visibilities are observed in the uv space. The telescope samples the uv
space due to the rotation of Earth. See, e.g., Ref. [91] for detailed
description of interferometric observations.

We use the Tools21cm code [92] to track the uv-space sampling (or uv coverage)
assuming a daily observation of 6 hours (see, e.g., Fig. 2 in Ref. [93] for
the simulated uv coverage of the first phase of SKA-Low). We assume an
integration time of 10 s, which is the time after which the observation is
recorded. For a detailed description of the method, see Refs. [94, 95].

Tools21cm outputs realizations of the instrumental noise for a given
interferometric telescope by taking into account the daily uv coverage. The
instrumental noise level is inversely related to the bandwidth of the
observations and the total observation time. We simulate observations assuming
1000 h exposure with SKA-Low. Previous studies have shown that with SKA-Low,
such an observation time will yield a sufficient signal-to-noise ratio (SNR)
to study the properties of cosmic dawn [e.g. 90, 2, 96, 70, 85, 82]. We
furthermore assume a bandwidth of 10 MHz. Within this bandwidth, the data will
not be significantly affected by the light-cone effect studied in Refs. [97,
98, 99]. The error from instrumental noise is calculated using the procedure
described in Sec. 3.2.1 of Ref. [100]. As part of this procedure, we
numerically simulate many realizations ($\sim 100$) of the instrumental noise
to estimate the standard deviation ($\sigma_{21}$) on the power spectrum at
various wave modes.

The error due to the sample variance is included using Eq. (9) in Ref. [90].
It depends on the field-of-view and therefore on the beam width of the
telescope. The size of the beam is given by the diameter of the antenna
stations, which is $\sim 35$ m for the first phase of SKA-Low.

The foreground contamination is not modeled in detail here. Instead, we assume
the foreground signal to reside in the foreground wedge [see e.g. 101, 102].
Following Refs. [103, 86], we apply a general, redshift independent cut at
$k=0.1$ h/Mpc. We also discard data from above $k=3$ h/Mpc. These scales are
expected to be unusable due to very large instrumental noise with our
instrumental setup [e.g. 82]. For our mock observations, we assume a
$\Lambda$CDM cosmology. This allows us to quantify the constraining power of
the 21-cm signal with respect to mixed DM model parameters.

Another important ingredient of our mock data is the source modeling. In order
to account for current uncertainties, especially at the low-mass range, we
include mocks for all three source models (FLOOR, DPL and TRUNCATED) shown in
Fig. 3. The FLOOR model corresponds to an optimistic case in which the small-
mass sources are highly efficient, whereas the TRUNCATED model shows a
pessimistic case with no sources below the atomic cooling limit ($\sim
10^{8}M_{\odot}$). The DPL model lies in between, allowing for sources below
the atomic cooling limit, but assuming very inefficient star-formation rates.

The mock data are set up assuming five different redshift bins between $z=11$
and 20, each of them with 15 data points equally separated in log space within
the range $k=0.1-3.0~{}$h/Mpc. These bins are selected to be broad enough to
minimize correlations between data points in redshift and $k$ space. This
considerably simplifies our analysis, as we do not have to deal with
nondiagonal terms in the covariance matrix during the inference analysis.

In Fig. 6, we show the 21-cm mock power spectra for the five redshift bins and
assuming the three different source models: FLOOR (dashed dark green lines),
DPL (solid magenta lines), and TRUNCATED (dotted-dashed sky blue lines). The
error due to instrumental noise and sample variance are indicated by the
color-shaded areas, which have been added in quadrature. While the former
induces a strong increase toward high wave numbers, the latter leads to a
gradual increase toward large scales, mainly visible at $k<0.1$ h/Mpc. The
hard scale cuts due to foreground contamination and noise are shown as hatched
gray areas.

WCDM |  FCDM
---|---
|

Figure 7: Forecast constraints on the particle mass and fraction of a
fermionic (left) and bosonic (right) mixed DM scenario. Different colors
correspond to different assumptions regarding the underlying source model of
the mock data set (TRUNCATED in sky blue, DPL in magenta, and FLOOR in dark
green). All constraints are provided at the 95% confidence level.

Accounting for all uncertainties, Fig. 6 shows that we can expect to observe
the 21-cm power spectrum up to $z\approx 20$ with 1000 h data from the first
phase of SKA-Low. The most interesting window, showing minimal noise
contamination, lies between $k\sim 0.1$ h/Mpc and $k\sim 0.3-1$ h/Mpc and
becomes narrower with increasing redshift. Note, furthermore, that at a given
redshift and $k$ value, the detectability of the signal will ultimately depend
on its strength, which is a function of cosmology and the underlying
astrophysics.

###  V.2 Inference analysis and constraints

Table 2: The prior range explored in the forecast study. We have four parameters ($f_{\star,0},\gamma_{4},f_{X},E_{0}$) to describe the photon sources. Our mixed DM models are given by two parameters ($m_{n\mathrm{DM}},f_{n\mathrm{CDM}}$).  Parameters | Prior range | FLOOR | DPL | TRUNCATED
---|---|---|---|---
$f_{\star,0}$ | $[0.1,0.5]$ | 0.3 | 0.3 | 0.3
$\gamma_{4}$ | $[-5,5]$ | 0.2 | 0 | -3
$f_{X}$ | $[0.05,0.5]$ | 0.2 | 0.2 | 0.2
$E_{0}$/eV | $[100,1000]$ | 500 | 500 | 500
log10(keV$/m_{\mathrm{WDM}}$) | $[-3,2]$ | -3 | -3 | -3
log10($f_{\mathrm{WDM}}$) | $[-3,0]$ | -3 | -3 | -3
log${}_{\mathrm{10}}(\mathrm{eV}/m_{\mathrm{FDM}}$) | $[18,26]$ | 18 | 18 | 18
log${}_{\mathrm{10}}f_{\mathrm{FDM}}$ | $[-3,0]$ | -3 | -3 | -3

As mentioned above, we use a MCMC framework to determine the constraints on
our model parameters $\pmb{\theta}$. The posterior distribution of
$\pmb{\theta}$, given the observational data $\pmb{x}$, is

$\displaystyle
p(\pmb{\theta}|\pmb{x})\propto\mathcal{L}(\pmb{\theta})\pi(\pmb{\theta})\ ,$
(20)

where $\pi(\pmb{\theta})$ is the prior probability of the model parameters,
and $\mathcal{L}(\pmb{\theta})=p(\pmb{x}|\pmb{\theta})$ is the likelihood
function. We use the log likelihood given by

$\displaystyle\log\mathcal{L}=-\frac{1}{2}\sum_{k,z}\left(\frac{\Delta^{2}_{\rm
21,ob}(k,z)-\Delta^{2}_{\rm 21,th}(k,z)}{\sigma_{21}(k,z)^{2}}\right)^{2},$
(21)

where $\Delta^{2}_{\rm 21,ob}$ and $\Delta^{2}_{\rm 21,th}$ are the observed
(mock) and modeled power spectra, respectively, and $\sigma_{21}$ is the
combined error on the data.

In terms of the modeling, we vary the six parameters $\pmb{\theta}=\\{f_{\rm
nCDM},\,m_{\rm nCDM},\,f_{\star,0},\,\gamma_{4},\,f_{X},\,E_{0}\\}$. Next to
the two dark matter parameters ($f_{\rm nCDM}$, $m_{\rm nCDM}$), we include
four source parameters in our MCMC analyses. They are related to the overall
amplitude ($f_{\star,0}$) and the small-scale behaviour ($\gamma_{4}$) of the
stellar-to-halo mass relation as well as the amplitude ($f_{X}$) and the
spectral range ($E_{0}$) of the x-ray flux. Other astrophysical parameters are
kept fixed for simplicity.  Previous works have shown that the signal is
sensitive to the atomic cooling threshold [e.g. 70]. As we study haloes below
the atomic cooling limit, $\gamma_{4}$ mimics the effect by suppressing star
formation in small mass haloes.

The prior ranges along with the mock values (for the three source models) are
provided in Table 2. Note that we assume uniform priors for the four
astrophysical parameters and uniform priors in log space for the two dark
matter parameters.

In Fig. 7, we plot the constraints on the WCDM (left panel) and the FCDM
(right panel) model parameters at 95 percent confidence level based on the
three different source models for the mock data (presented with different
colors). Note that all astrophysical parameters are marginalized over. Full
corner plots including posterior contours of all astrophysical parameters for
both WCDM and FCDM cases are provided in Appendix A.

Table 3:  Forecast constraints from special locations of the mixed DM parameter space: pure warm DM, pure fuzzy DM, CDM plus a subdominant fermionic relic, CDM plus a subdominant bosonic relic. All constraints are provided for the FLOOR, the DPL and the TRUNCATED source models. They are given at the 95% confidence level.  Source model: FLOOR |
---|---
Warm DM ($f_{\rm WCDM}\sim 1$) |  $m_{\rm WDM}\gtrsim 15$ keV
Fuzzy DM ($f_{\rm FCDM}\sim 1$) |  $m_{\rm FDM}\gtrsim 2\times 10^{-20}$ eV
CDM + hot fermionic relic | $f_{\rm WCDM}\lesssim 0.012$
CDM + hot bosonic relic | $f_{\rm FCDM}\lesssim 0.010$
Source model: DPL |
Warm DM ($f_{\rm WCDM}\sim 1$) |  $m_{\rm WDM}\gtrsim 15$ keV
Fuzzy DM ($f_{\rm FCDM}\sim 1$) |  $m_{\rm FDM}\gtrsim 2\times 10^{-20}$ eV
CDM + hot fermionic relic | $f_{\rm WCDM}\lesssim 0.012$
CDM + hot bosonic relic | $f_{\rm FCDM}\lesssim 0.011$
Source model: TRUNCATED |
Warm DM ($f_{\rm WCDM}\sim 1$) |  $m_{\rm WDM}\gtrsim 4$ keV
Fuzzy DM ($f_{\rm FCDM}\sim 1$) |  $m_{\rm WDM}\gtrsim 2\times 10^{-21}$ eV
CDM + hot fermionic relic | $f_{\rm WCDM}\lesssim 0.012$
CDM + hot bosonic relic | $f_{\rm FCDM}\lesssim 0.011$

The DM constraints behave as expected in the sense that the parameter space
with large $f_{\rm nCDM}$ and small $m_{\rm nCDM}$ in the top-right corner of
the plots is strongly ruled out. Very small values of $f_{\rm nCDM}$ (bottom
part of the plots), on the other hand, remain allowed independently of $m_{\rm
nCDM}$. The same is true for large values of $m_{\rm nCDM}$ (left-hand side of
the plots) that stay allowed independently of the fraction $f_{\rm nCDM}$.

Focusing on the different source models, constraints are weakest for the
TRUNCATED model (sky blue contours). This is expected as the model lacks
sources in small haloes, where the mixed DM models differ the most from
$\Lambda$CDM. The DPL and FLOOR models (magenta and dark green contours) yield
considerably stronger constraints, especially in the regime of large $f_{\rm
nCDM}$. Interestingly, they hardly differ between each other in their
constraining power. This points toward the possibility that minihaloes, as
long as they are populated with sources, are capable of providing stringent
constraints on DM parameters. It does not seem to matter much how large their
star-formation rate effectively is.

It is also worth noticing that in the regime of small $m_{\rm nCDM}$ (bottom
right part of the plots) for both WCDM and FCDM models, the constraints are
approximately independent of the source models (with all contours lying on top
of each other). This is due to the fact that models with small $f_{\rm nCDM}$
and $m_{\rm nCDM}$ exhibit halo mass functions that are modified over all mass
scales and are therefore not solely dependent on the abundance of sources in
minihaloes.

In the left panel of Fig. 7, we show the constraints given in Ref. [54] from
Lyman-$\alpha$ forest observations. The dashed and dash-dotted lines show the
constraints from SDSS data alone and from a combination of SDSS data with
XShooter, MIKE and HIRES data. This latter data set includes much smaller
scales and is, therefore, more constraining. See Refs. [54, 104] for a
discussion on the validity of these results.

Some of the constraints provided in Fig. 7 are considerably stronger than
current constraints from the literature. In the limit of pure WDM and FDM
($f_{\rm nCDM}\sim 1$), for example, we obtain constraints of $m_{\rm
WDM}\gtrsim 4,\,15,\,15$ keV and $m_{\rm FDM}\gtrsim 2\times
10^{-21},\,2\times 10^{-20},\,2\times 10^{-20}$ eV for the TRUNCATED, DPL, and
FLOOR model, respectively. In the limit of very small particle masses, on the
other hand, we obtain limits on the fraction of $f_{\rm WCDM}\lesssim 0.01$
and $f_{\rm FCDM}\lesssim 0.01$ which do not depend on the source model. All
these constraints are at 95% confidence level. A summary of these constraints
are provided in Table 3.

Some previous studies have included a theoretical error in their inference
pipeline to account for the limitations of current modeling methods [e.g. 86,
105, 106, 107]. In this paper, however, we neglect all modeling uncertainties
despite the fact that they are currently larger than the expected errors from
SKA-Low. Note that this is common practice for forecast studies. It means
that, until the SKA data are available, we expect the current prediction
techniques to improve to the level that the theoretical errors become
subdominant. In Appendix B, we nevertheless show the impact of a theoretical
error on the DM parameter constraints.

Here, we have quantified the constraining power of SKA-Low on mixed DM
scenarios provided the true Universe is filled with pure CDM. We have shown
that such as setup will lead to unprecedented constraints on the particle mass
and fraction of fermionic and bosonic noncold relics, especially for the case
that minihaloes are populated with stars. However, we cannot say anything
about constraints if the true model is not CDM but mixed DM instead. In that
case, we expect the observable signal from SKA-Low to be pushed to smaller
redshifts. Such a signal will be more difficult to interpret due to the strong
degeneracies between DM and source model parameters.

##  VI  Summary

In this paper, we have extended the framework from Ref. [57] to explore
nonstandard DM models using the 21-cm signal at cosmic dawn. We have focused
specifically on fermionic and bosonic mixed DM scenarios characterized by the
noncold particle mass $m_{\rm nCDM}$ and the mixing fraction $f_{\rm nCDM}$.
In the case of fermionic mixed DM, the noncold component could be made of a
sterile neutrino with mass in the eV or keV range. In the bosonic case, the
noncold component consists of a ultralight axion DM particle with mass in the
range of $\sim 10^{-18}-10^{-26}$ eV.

In terms of structure formation, mixed DM scenarios lead to modifications of
the matter power spectrum, the halo mass function, and the mass accretion
rates. This affects both the clustering of the gas as well as the properties
of the sources. As a result, the characteristic double-peak feature of the
21-cm power spectrum at cosmic dawn is pushed to smaller redshifts and
exhibits an increased amplitude compared to the standard $\Lambda$CDM model.

The main task of the paper consists of performing a forecast analysis based on
the low-frequency array of the upcoming Square Kilometre Array telescope (SKA-
Low). We carried out an inference analysis based on the Markov Chain Monte
Carlo (MCMC) method, varying two DM and four astrophysical parameters. For the
mock data set, we investigated three choices for the source modeling, a
pessimistic, neutral, and optimistic one. They differ with respect to the
assumed abundance of sources in minihaloes, going from zero to efficient star
formation below the atomic cooling limit.

For the SKA forecast analysis, we assume a $\Lambda$CDM universe,
investigating the potential of constraining mixed DM scenarios. In the case
that minihaloes are populated with stars, we obtained stringent constraints
that go far beyond current limits. If minihaloes are devoid of stars, the
constraints become weaker but remain more stringent than the current limits
from the Lyman-$\alpha$ data, Milky Way satellite counts, or strong lensing
studies. A summary of the constraints are provided in Fig. 7 and Table 3.

Interestingly, the DM constraints do not increase linearly with the number of
stars in minihaloes. We rather find that the constraints remain strong as long
as there is at least some star formation in minihaloes, even if the stellar-
to-halo ratio is strongly suppressed. This finding is encouraging because we
expect at least some stars to form in haloes between $10^{6}-10^{8}$
$M_{\odot}$ via the molecular cooling channel, even when assuming efficient
Lyman-Werner feedback process.

In summary, the results from this paper indicate that the SKA telescope has
the potential to significantly push current constraints on warm, fuzzy, and
mixed DM scenarios. However, the true strength of these constraints will
ultimately depend on whether the SKA will provide evidence for star-formation
in minihaloes below the atomic cooling limit. If this is not the case, then it
will be difficult to distinguish between an inefficient source models and a
Universe filled with warm, fuzzy, or mixed dark matter.

###### Acknowledgements.

This work is supported by the Swiss National Science Foundation via the Grant
No. PCEFP2_181157. All the MCMC runs were done using the emcee [108] package.
For analysis and plotting, we have used the following software packages: numpy
[109], scipy [110], matplotlib [111] and corner [112].

##  Appendix A  COMPLETE POSTERIOR DISTRIBUTION FROM THE MCMC RUNS

WCDM

FCDM

Figure 8: Corner plots showing the 95% confidence interval of the posterior
distributions on six parameter WCDM (top) and FCDM (bottom) models. As in the
main text, we assume the source models TRUNCATED (sky blue), DPL (magenta) and
FLOOR (dark green) for the mock data. The peaks of the 1D marginalized
probability distributions agree with the true values of the mock observations
for all cases.

The primary aim of this work is to study the capability of SKA in constraining
mixed dark matter (DM) models. Therefore, we only present the posterior
distributions of the DM parameters in the main text. However, while analyzing
the 21-cm signal observations, we can also constrain the astrophysical
processes driving the Lyman-$\alpha$ coupling and heating of the hydrogen gas
in the IGM during cosmic dawn.

In our MCMC analyses (Sec. V.2), we vary four astrophysical parameters
($f_{\star,0},\gamma_{4},f_{X},E_{0}$) along with the two DM parameters
($m_{\rm nCDM},f_{\rm nCDM}$). For completeness, we show the posterior
distributions on all parameters in Fig. 8. Note that all contours are shown at
the 95% confidence level.

With all three mock observations, which are the TRUNCATED (sky blue contour),
DPL (magenta contour), and FLOOR (dark green contour), we are able to
constrain the astrophysical parameters quite well. The peak of the 1D
marginalized probability distribution function of all the astrophysical
parameters lie very close to the mock values for both WCDM and FCDM scenarios.
This is an important cross-check validating the outcome of our inference
analysis.

The results of Fig. 8 show furthermore that with data from SKA-Low, it will be
possible to gain much better knowledge of the underlying astrophysical model.
In particular, it will be possible to derive strong constraints on the
stellar-to-halo mass relation and the x-ray efficiency during the epoch of
cosmic dawn.

##  Appendix B  IMPACT OF MODELING ERROR ON THE CONSTRAINTS

Figure 9: Forecast constraints on the WCDM model parameters assuming a
theoretical modeling error of 0%, 10% and 20% (magenta, brown and gray). Here
we assumed the DPL source model. All constraints are provided at the 95%
confidence level.

In many studies analyzing the 21-cm power spectrum, an additional modeling
error is introduced to account for the current limitations of signal
simulation methods [e.g. 86, 105, 106, 107]. Since our work is limited to a
forecast analysis, we do not include any modeling errors in the analysis
presented in the main text. This implicitly means that we expect current
prediction techniques to improve until the observations from the SKA telescope
will be available.

In this appendix, we study the impact of modeling errors for the case that the
improvement of prediction techniques does not proceed as expected. Similar to
previous methods from the literature, we empirically define the $m$ percent
modeling error as the standard deviation of $m$ percent at every wave-mode and
redshift. This error is appended in quadrature to the variance ($\sigma_{21}$)
shown in Eq. (21).

In Fig. 9, we show the constraints on the WCDM models, assuming a modeling
error of 0% (magenta), 10% (brown), and 20% (gray). As mock data set, we
consider the DPL source model. When the DM is purely noncold ($f_{\rm
WCDM}\sim 1$), the particle mass is constrained to be greater than 15, 9 and 6
keV for 0%, 10% and 20% modeling error respectively. In the other extreme case
where the particle mass is very small, we constrain the fraction of noncold DM
to be less than 0.012, 0.014 and 0.038 for 0%, 10% and 20% modeling error
respectively. These constraints are given at 95% confidence level.

We see that the constraints become weaker as we increase the modeling error.
However, even with 20% modeling error, we get competitive constraints on the
particle mass and fraction of mixed dark matter. Here we empirically assumed
the modeling error. In the future, we will compare our simulation framework
with other methods to better understand this error.

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