# Implications of the cosmological 21-cm absorption profile for high-redshift
star formation and deep JWST surveys

Shikhar Mittal  ID and Girish Kulkarni  ID
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted 2022 July 07. Received 2022 July 07; in original form 2022 March 17)

###### Abstract

Apart from its anomalously large depth, the cosmological 21-cm absorption
signal measured by the EDGES collaboration also has a shape that is distinctly
different from theoretical predictions. Models with non-traditional components
such as super-adiabatic baryonic cooling or an excess radio background explain
the depth of the observed profile, but still conspicuously fail to explain its
shape. In this paper, we quantify the requirements imposed by the EDGES
measurement on sources of Ly $\alpha$ and X-ray photons in the presence of
excess radio background at cosmic dawn. In extreme cases, the Ly $\alpha$ and
X-ray emissivities require to be enhanced by up to an order of magnitude
relative to traditional models. Furthermore, this enhancement needs to be
active only for a short duration. We find that under conventional assumptions
for the cosmic star formation rate density, standard stellar populations are
incapable of meeting these conditions. Only highly unusual models of massive
metal-free stars seem to provide a possible mechanism. Conversely, if the
sources of Ly $\alpha$ and X-ray photons are compelled to have standard
properties, the EDGES measurement puts strong demands on the cosmic star
formation rate density. This provides interesting falsifiable predictions for
high-redshift galaxy surveys enabled by James Webb Space Telescope (JWST). We
derive predictions for galaxy UV luminosity functions and number densities,
and show that a deep JWST survey with a limiting UV magnitude of
$m_{\mathrm{UV,lim}}=32$ would potentially be able to rule out the predictions
enforced by the EDGES measurement.

###### keywords:

galaxies: formation – cosmology: theory - dark ages, reionization, first
stars.

††pubyear: 2022††pagerange: Implications of the cosmological 21-cm absorption
profile for high-redshift star formation and deep JWST surveys–References

##  1 Introduction

The absorption signal in the low-frequency spectrum of the sky reported by the
EDGES collaboration (Bowman et al., 2018) has been widely considered to
potentially have a cosmological origin (Barkana, 2018; Mirocha & Furlanetto,
2019). But it is also famously known that the amplitude of this signal,
$\Delta T_{\mathrm{b}}=0.5^{+0.5}_{-0.2}\,$K at $z\approx 17.2$, is larger
than the expected cosmological value by a factor of two. This anomalously
large depth has received considerable attention in the literature. Many
studies explain the depth either by postulating excess cooling of cosmic
hydrogen due to novel physical mechanisms (eg. Meiksin & Madau, 2020; Li et
al., 2021; Mathur et al., 2022), or by bringing into play a part of the excess
radio background (ERB) observed at relevant frequencies (eg. Mittal &
Kulkarni, 2022). Some studies invoke even more exotic scenarios, such as a
modified Hubble parameter at cosmic dawn (Hill & Baxter, 2018; Costa et al.,
2018; Wang & Zhao, 2018; Xiao et al., 2019), new coupling terms in the spin
temperature (Widmark, 2019; Lambiase & Mohanty, 2020), or special classes of
dark energy and dark matter models (Mukhopadhyay et al., 2021).

Still, apart from the large depth, which in this work we explain by
incorporating excess radio background, the absorption profile measured by
EDGES also has peculiarly sharp bounds, at $z\sim 20$ and $z\sim 15$, and a
surprisingly flat bottom (Bowman et al., 2018). At the minimum, these unusual
features in the shape of the signal seem to require that a cosmic Ly $\alpha$
background must get established within 25 Myr from $z\sim 21$ to 19 and that
the cosmic gas content be heated above the background temperature within 50
Myr from $z\sim 16$ to 14. This aspect of the EDGES measurement has received
relatively less attention in the literature. For example, Kaurov et al. (2018)
argue that the rapid Ly $\alpha$ production required by the falling edge of
the signal suggests star formation was dominated by rapidly growing rare and
massive haloes. Vipp et al. (2021) claimed that such a falling feature could
be explained by warm dark matter models and put upper and lower bounds on the
mass of their particles. Population III motivated models by Mebane et al.
(2020) naturally achieved a reasonably flat trough, though the rise out of
absorption was not too rapid. In this work we aim to interpret the overall
shape of the EDGES signal, capturing not just its timing, width or depth but
also the sharp drop and rise.

It is possible that the signal measured by EDGES is not of a cosmological
origin. Indeed, Hills et al. (2018) argued against a cosmological origin of
this signal by pointing out that the EDGES inference results in unphysical
properties for the ionosphere and/or unexplained structure in the foreground
spectrum. The inferred absorption profile is also sensitive to the order of
the polynomial model used for the foregrounds, without a means to choose the
optimum order. More recently, Singh et al. (2022) used observations made with
the SARAS 3 (Shaped Antenna measurement of the background RAdio Spectrum)
experiment to rule out an astrophysical origin of the EDGES measurement at
93.5 per cent confidence. Still, in spite of these new developments, an
investigation into the shape of the absorption profile measured by EDGES
continues to be of interest. For instance, the new findings by Singh et al.
(2022) seem to remain in need of corroboration. First, the frequency band of
SARAS 3 covers only part of the the EDGES absorption profile, with the rising
part of the signal noticeably excluded. This makes it likely that any signal
consistent with the EDGES measurement gets partly or wholly fitted out by the
sixth-order polynomial foreground model used in the SARAS 3 analysis.
[Although it is important to note that Singh et al. (2022) have argued against
this possibility by showing that even with a partial coverage of 55–85 MHz,
the EDGES signal should be detected by SARAS 3 against a sixth-order
polynomial at more than two-sigma confidence.] Second, the choice of a sixth-
order polynomial foreground model by SARAS 3 is as problematic as the use of
the fifth-order linearised foreground model in the EDGES analysis by Bowman et
al. (2018). As Hills et al. (2018) pointed out, it is surprising that such
higher-order models are necessary to describe the foreground for which third-
order polynomials are usually sufficient (Bernardi et al., 2015). Third, it is
remarkable that with a fifth-order polynomial foreground, both SARAS 3 and
EDGES report residuals that are roughly identical in the rms amplitude. Until
a reliable explanation is found for this residual, the possibility of its
astrophysical origin remains open. It is also interesting to note that Bowman
et al. (2018) found the shape of the inferred absorption signal in the EDGES
data to be robust against different hardware modifications, data processing
pipelines, and calibration techniques (Mahesh et al., 2021). Newer high-band
measurements by the EDGES collaboration also seem to confirm this (Monsalve et
al., 2019). Further, it is important to note that more experiments are
currently being set up to verify the EDGES and SARAS 3 observation. These
include Large Aperture Experiment to Detect the Dark Ages (LEDA, Bernardi et
al., 2015, 2016; Price et al., 2018), Probing Radio Intensity at high-Z from
Marion (PRIzM, Philip et al., 2019), Radio Experiment for the Analysis of
Cosmic Hydrogen (REACH, de Lera Acedo, 2019), Sonda Cosmológica de las Islas
para la Detección de Hidrógeno Neutro (SCI-HI, Voytek et al., 2014) and the
Cosmic Twilight Polarimeter (CTP, Nhan et al., 2017; Nhan et al., 2019). If
only conventional seven-parameter models are used for the cosmological signal
in the analysis of these experiments, the analysis of data from these upcoming
experiments runs the risk of missing cosmological information. Finally, the
successful launch of James Webb Space Telescope (JWST) has opened a window of
opportunity for a rest-frame UV detection of sources of radiation at cosmic
dawn (Cowley et al., 2017; Williams et al., 2018; Kauffmann et al., 2020;
Mirocha & Furlanetto, 2019). A quantification of the implications of 21-cm
experiments for JWST surveys is therefore timely.

The major standard astrophysics components affecting the global 21-cm signal
for the period of cosmic dawn are the cosmic star formation rate (SFR), the Ly
$\alpha$ coupling via the Wouthuysen–Field effect (Wouthuysen, 1952; Field,
1958), and gas heating due to X-ray emission. This astrophysics is
conventionally described by 4 to 7 parameters (Mirocha et al., 2015; Cohen et
al., 2017; Monsalve et al., 2019; Mebane et al., 2020; Chatterjee et al.,
2021), usually chosen to describe the star formation efficiency (SFE), minimum
mass of dark matter haloes that host star formation, the normalization and
spectral index for the Ly $\alpha$ spectral energy distribution (SED), and the
normalization and spectral index for the X-ray SED. In this work we explore
extensions to these models if they are to also capture the shape of the
absorption profile.

We start by briefly mentioning key equations describing the 21-cm signal in
Section 2. In Section 3, we discuss the astrophysical requirements demanded by
the shape of the observed 21-cm absorption feature. The plausibility of these
requirements is discussed in Section 4, and Section 5 presents the consequent
implications for high-redshift galaxies. We discuss and summarise our findings
in Section 6. Our cosmological parameter values are
$\Omega_{\mathrm{m}}=0.315$, $\Omega_{\mathrm{b}}=0.049$,
$\Omega_{\Lambda}=0.685$, $h=0.674$, $Y_{\mathrm{p}}=0.25$,
$T_{0}=$2.725\text{\,}\mathrm{K}$$, $\sigma_{8}=0.811$ and
$n_{\mathrm{s}}=0.965$ (Fixsen, 2009; Planck Collaboration et al., 2016),
where $T_{0}$ and $Y_{\mathrm{p}}$ are the CMB temperature measured today and
primordial helium fraction by mass, respectively.

Figure 1: Our fiducial seven-parameter model for the 21-cm signal from M22. The left panel shows the evolution of the radio background temperature (dotted black curve), the hydrogen spin temperature (solid red) and the gas temperature (solid blue). The right panel shows the best-fitting model (solid green) in comparison with the EDGES measurement (dashed black), with a $50\text{\,}\mathrm{mK}$ uncertainty. The best-fitting values of the model parameters are given in Table 1. The goodness-of-fit of this model is $\chi^{2}/\mathrm{dof}=2.20$ (right tail $p$-value $8.7\times 10^{-13}$). The rms value of the residuals is $69\text{\,}\mathrm{mK}$. Table 1: Best-fitting parameter values for the seven-parameter model shown in Figure 1. Parameter | Description | Value
---|---|---
$f_{\star}$ | Star formation efficiency | 0.1
$T_{\mathrm{vir,min}}$ | Minimum virial temperature of star-forming haloes | $e4.25\text{\,}\mathrm{K}$
$f_{\alpha}$ | Normalization of the Ly $\alpha$ background | $0.985$
$E_{0}$ | Lower X-ray energy cut-off | $200\text{\,}\mathrm{eV}$
$w$ | X-ray spectral index | 1.5
$f_{\mathrm{X}}$ | Normalization of the X-ray background | $2.379$
$\zeta_{\mathrm{ERB}}$ | Strength of the ERB | 0.030

##  2 The 21-cm absorption profile

The 21-cm signal is a differential brightness temperature measured at low
frequencies against the radio background. This can be written as (Furlanetto
et al., 2006; Pritchard & Loeb, 2012; Barkana, 2016; Mesinger, 2019; Liu &
Shaw, 2020)

$\Delta
T_{\mathrm{b}}=\frac{T_{\mathrm{s}}-T_{\mathrm{r}}}{1+z}\left(1-\mathrm{e}^{-\tau_{21}}\right)\,,$
(1)

where $\tau_{21}$ is the optical depth for the 21-cm photon (Madau et al.,
1997), $T_{\mathrm{s}}$ is the hydrogen spin temperature, $T_{\mathrm{r}}$ is
the radio background temperature. For small 21-cm optical depth Equation (1)
becomes

$\Delta
T_{\mathrm{b}}=27\bar{x}_{\text{H\,{i}}}\left(\frac{1-Y_{\mathrm{p}}}{0.76}\right)\left(\frac{\Omega_{\mathrm{b}}h^{2}}{0.023}\right)\sqrt{\frac{0.15}{\Omega_{\mathrm{m}}h^{2}}\frac{1+z}{10}}\left(1-\frac{T_{\mathrm{r}}}{T_{\mathrm{s}}}\right)$\mathrm{mK}$\,,$
(2)

where $x_{\text{H\,{i}}}\equiv n_{\text{H\,{i}}}/n_{\text{H}}$ is the ratio of
number densities of neutral hydrogen (H i) and total hydrogen (H). The bar on
$x_{\text{H\,{i}}}$ represents a global average over the cosmic volume. At the
high redshifts that correspond to the cosmic dawn,
$H(z)=H_{0}\sqrt{\Omega_{\mathrm{m}}(1+z)^{3}}$, where $H_{0}$ is the Hubble’s
constant measured today. The signal is seen in absorption if $\Delta
T_{\mathrm{b}}<0$ and in emission if $\Delta T_{\mathrm{b}}>0$. The
temperature of the radio background is (Feng & Holder, 2018; Fialkov &
Barkana, 2019)

$T_{\mathrm{r}}(z)=2.725(1+z)\left[1+0.169\zeta_{\mathrm{ERB}}(1+z)^{2.6}\right]\,,$
(3)

where we have assumed a power-law ERB over the CMB with
$\zeta_{\mathrm{ERB}}=0.03$. Our ERB model is based on ARCADE2/LWA1
observations (Fixsen et al., 2011; Dowell & Taylor, 2018). The value of
$\zeta_{\mathrm{ERB}}$ quoted is required to explain the depth seen in EDGES
observation (Mittal et al., 2022, hereafter M22).

Figure 2: Our model M1. The left-hand side panels show the multiplicative
factors $\mathcal{M}_{\alpha}$ and $\mathcal{M}_{\mathrm{X}}$ for the Ly
$\alpha$ and X-ray emissivity, respectively, required to fit the flat-bottomed
shape of the EDGES measurement. The right-hand side panels show the
corresponding background intensities $J_{\alpha}$ and $J_{\mathrm{X}}$. Dotted
curves in all panels denote the fiducial model. This model presents the
extreme scenario in which the X-ray emissivity is held as close to the
fiducial model as possible. Note that this model explains the flat bottom of
the EDGES profile by mean of an evolution in the Ly $\alpha$ coupling.

The spin temperature is given by

$T_{\mathrm{s}}^{-1}=\frac{T_{\mathrm{r}}^{-1}+x_{\alpha}T_{\mathrm{k}}^{-1}}{1+x_{\alpha}}\,.$
(4)

where $x_{\alpha}$ is the Ly $\alpha$ coupling. Note that we have ignored the
collisional coupling as it is quite small in the redshift range of our
interest (Mittal & Kulkarni, 2020).

We have also approximated the colour temperature (Rybicki, 2006; Meiksin,
2006) by the gas temperature. The evolution of the gas temperature
$T_{\mathrm{k}}$ is determined by adiabatic cooling and X-ray heating
(Mesinger et al., 2011) so that

$(1+z)\frac{\mathrm{d}T_{\mathrm{k}}}{\mathrm{d}z}=2T_{\mathrm{k}}-\frac{2q_{\mathrm{X}}}{3n_{\mathrm{b}}k_{\mathrm{B}}H(z)}\,,$
(5)

where $k_{\mathrm{B}}$ is the Boltzmann constant, $n_{\mathrm{b}}$ is the
baryon number density and $q_{\mathrm{X}}$ is the volumetric heating rate by
X-rays, which is proportional to the specific intensity $J_{\mathrm{X}}$ of
X-ray photons. We ignore other heating/cooling processes such as Compton
(Weymann, 1965), Ly $\alpha$ (Mittal & Kulkarni, 2020) and change in internal
energy due to changing particle number as they are very small compared to
X-ray heating at the redshifts of our interest. Further details of our seven-
parameter model are given by M22.

Figure 3: Left panel: evolution of the radio background temperature (dotted
black), the spin temperature (solid red) and the gas temperature (solid blue)
in our model M1. Right panel: the corresponding 21-cm signal (solid blue), in
comparison with the EDGES measurement (dashed black), with a
$50\text{\,}\mathrm{mK}$ uncertainty (grey band). The rms value of the
residuals is $26\text{\,}\mathrm{mK}$, a dramatic improvement over the
fiducial model. Dotted curves in both panels show the fiducial model of Figure
1. The dashed red curve in the right panel shows the case where the Ly
$\alpha$ emissivity follows model M1 but the X-ray emissivity follows the
fiducial model.

Figure 1 shows the cosmic thermal evolution (left panel) and the 21-cm signal
(right panel) for the best-fitting values of the seven parameters of the model
described above, when the model is fit to the EDGES measurement, assuming a
constant $50\text{\,}\mathrm{mK}$ uncertainty across the frequency band (see
also Hills et al., 2018; Mirocha & Furlanetto, 2019; Chatterjee et al., 2021;
Hibbard et al., 2022). The best-fitting parameter values are listed in Table
1. The Ly $\alpha$ background is assumed to be set by a broken power law SED
of Population II stars (metallicity $=10^{-3}$) assuming a Scalo (1998)
initial mass function (IMF) (Leitherer et al., 1999; Barkana & Loeb, 2005).
The minimum virial temperature required is $e4.25\text{\,}\mathrm{K}$,
equivalent to a circular velocity of
$15.86\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and minimum halo mass of
$1.65\times 10^{9}\,\mathrm{M}_{\odot}h^{-1}$ (at $z=0$) assuming the mean
molecular weight to be $\mu=1.22$, for which molecular hydrogen cooling fuels
the star formation (Barkana, 2016). The X-ray background is assumed to be set
by the high mass X-ray binaries; the normalization is about 2.4 times that
suggested by extrapolating luminosity and SFR relation from Mineo et al.
(2012) for 0.2–$30\text{\,}\mathrm{keV}$. The ERB required is $0.5$ per cent
of CMB at $\nu_{\mathrm{obs}}=$1420\text{\,}\mathrm{MHz}$$, i.e., at $z=0$,
which is very well below the upper limit of $9.6$ per cent of CMB – $56.8$ per
cent of ARCADE2/LWA1 (Fixsen et al., 2011; Dowell & Taylor, 2018) – set by
LOFAR (Mondal et al., 2020).

While the model is able to explain the central redshift and frequency of the
observed absorption profile, the fit is quite poor at the falling and rising
bounds of the absorption profile. The high-redshift ‘fall’ in the profile
lasts for about 65 Myr in our best-fitting model, while the low-redshift
‘rise’ lasts for 97 Myr. (The corresponding numbers for the observed signal
are 25 Myr and 51 Myr, respectively.) Indeed, overall the best-fitting model
provides a very poor fit with a reduced-$\chi^{2}$ of $262/119$, which
corresponds to a $p$-value (right tail probability) of just $8.7\times
10^{-13}$. [For our $\chi^{2}$ calculation we use spectrally flat errors
$50\text{\,}\mathrm{mK}$ in magnitude motivated by noise level in the EDGES
measurement (Bowman et al., 2018).] The rms value of the residuals relative to
this model is $69\text{\,}\mathrm{mK}$.

In order to improve the fit at the falling edge of the profile, one seems to
require a sudden drop in the spin temperature, either by a sudden evolution in
the Ly $\alpha$ coupling or by a drop in the gas temperature (for which no
well-known mechanism exists). Similarly, the rapid rise in signal at $z\sim
15$ requires a sharp rise in the spin temperature, which is only possible by a
corresponding rapid increase in the gas temperature [possibly by X-ray heating
or exotic phenomenon such as cosmic ray heating (Bera et al., 2022)].

This problem persists even when we disregard the ‘non-standard’ ERB. In the
absence of an ERB, the model is no longer able to achieve the observed
amplitude, but the shape and the central redshift would remain relatively
similar to what is seen in Figure 1.

Figure 4: Our model M2. The left-hand side panels show the multiplicative
factors $\mathcal{M}_{\alpha}$ and $\mathcal{M}_{\mathrm{X}}$ for the Ly
$\alpha$ and X-ray emissivity, respectively, required to fit the flat-bottomed
shape of the EDGES measurement. The right-hand side panels show the
corresponding background intensities $J_{\alpha}$ and $J_{\mathrm{X}}$. Dotted
curves in all panels denote the fiducial model. This model presents the
extreme scenario in which the Ly $\alpha$ emissivity is held as close to the
fiducial model as possible. In contrast to Figure 2, this model explains the
flat bottom of the absorption profile by modifying the X-ray emissivity
evolution. Figure 5: Left panel: evolution of the CMB+ERB temperature (dotted
black), the spin temperature (solid red) and the gas temperature (solid blue)
in our model M2. Right panel: the corresponding 21-cm signal (solid blue), in
comparison with the EDGES measurement (dashed black), with a
$50\text{\,}\mathrm{mK}$ uncertainty (grey band). The rms value of the
residuals is $30\text{\,}\mathrm{mK}$, again a significant improvement over
the fiducial model. Dotted curves in both panels show the fiducial model of
Figure 1.

##  3 Direct consequences of a flat-bottomed 21-cm profile

Thanks to the simplicity of Equations (1) and (4), the most direct implication
of the flat-bottomed shape of the 21-cm absorption profile is a stringent
requirement on the evolution of the Ly $\alpha$ coupling and the gas
temperature. A non-trivial evolution in one or both of these quantities is
necessary if we are to improve the model fit shown in Figure 1. This non-
standard evolution in $x_{\alpha}$ and $T_{\mathrm{k}}$ can in turn arise due
to modified evolution of any of the several underlying processes, such as star
formation, or Ly $\alpha$ and X-ray emission. But before we investigate the
implications for these more fundamental quantities, it is instructive to
understand the primary implications for $x_{\alpha}$ and $T_{\mathrm{k}}$.

###  3.1 Ly $\alpha$ emission

To understand the requirements for the Ly $\alpha$ emission, starting from the
best-fitting fiducial model described above, we can give a time dependence to
Ly $\alpha$ specific intensity, keeping everything else fixed, to try and
reproduce the observed profile. Let the fiducial Ly $\alpha$ background be
$J_{\alpha}(z)$ and the required version be $J_{\alpha}^{\prime}(z)$. We
derive $J_{\alpha}^{\prime}(z)$ as follows: the observed signal $\Delta
T_{\mathrm{b}}^{\mathrm{obs}}$ corresponds to the spin temperature

${T^{\prime}_{\mathrm{s}}}^{-1}=T_{\mathrm{r}}^{-1}\left[1-$5.29\text{\times}{10}^{-3}\text{\,}\mathrm{m}\mathrm{K}^{-1}$\frac{\Delta
T_{\mathrm{b}}^{\mathrm{obs}}}{h\Omega_{\mathrm{b}}(1-Y_{\mathrm{p}})\bar{x}_{\text{H\,{i}}}}\sqrt{\frac{\Omega_{\mathrm{m}}}{1+z}}\right]\,,$
(6)

which is obtained by inverting Equation (2). Next, given Equation (4), we can
obtain the required Ly $\alpha$ coupling as

$x_{\alpha}^{\prime}=\frac{T_{\mathrm{r}}^{-1}-{T^{\prime}_{\mathrm{s}}}^{-1}}{{T^{\prime}_{\mathrm{s}}}^{-1}-T_{\mathrm{k}}^{-1}}\,.$
(7)

Since the Ly $\alpha$ coupling is directly proportional to the Ly $\alpha$
specific intensity we can write the required Ly $\alpha$ specific intensity as

$J^{\prime}_{\alpha}=\frac{x_{\alpha}^{\prime}}{x_{\alpha}}J_{\alpha}\,.$ (8)

Although theoretically it is easy to calculate $J^{\prime}_{\alpha}$ by the
above method, we must investigate if it is physically possible to have such a
background. Stated more concretely, how should the Ly $\alpha$ emissivity be
modified to get $J^{\prime}_{\alpha}$? We start by expressing the specific
intensity in terms of emissivity as

$\bar{J}_{\alpha}(z)=\frac{c}{4\pi}(1+z)^{2}\sum_{n=2}^{23}P_{n}\int_{z}^{z^{\mathrm{max}}_{n}}\mathcal{M}_{\alpha}(z^{\prime})\frac{\epsilon_{\alpha}(E_{n}^{\prime},z^{\prime})}{H(z^{\prime})}\,\mathrm{d}z^{\prime}\,,$
(9)

where $c$ is the speed of light and $\mathcal{M}_{\alpha}$ serves the purpose
of providing $z$ dependence to the normalization parameter $f_{\alpha}$, so
that $f_{\alpha}\to\mathcal{M}_{\alpha}(z)f_{\alpha}$. In the above equation
$\epsilon_{\alpha}$ is the comoving emissivity defined as the number of
photons emitted per unit comoving volume per unit proper time per unit energy
and $P_{n}$ is the probability with which a photon in the upper Lyman lines
will redshift to Ly $\alpha$ wavelength.

Our task boils down to evaluating the modifying factor
$\mathcal{M}_{\alpha}(z)$. Solving for $\mathcal{M}_{\alpha}$ is not
straightforward because the integration limits in Equation (9) are functions
of $z$. As a result, we can find this function by an iterative procedure in
which we update $\mathcal{M}_{\alpha}(z)$ by a factor of
$J_{\alpha}^{\prime}/\bar{J}_{\alpha}$ after each iteration starting with
$\mathcal{M}_{\alpha}(z)=1$.

We see that outside the absorption feature of EDGES, i.e. for $z>21$ and
$15>z$, the signal is almost 0, which should imply the same for
$\mathcal{M}_{\alpha}(z)$. At the centre of the profile, $z\sim 17$, our
current fitting already has the same depth as in the EDGES signal, which means
we must have $\mathcal{M}_{\alpha}(z=17)\sim 1$. At the high-redshift bound of
the profile, from $z\sim 18$ to 20, $\mathcal{M}_{\alpha}$ should be more than
1 and decrease to 0 towards higher redshifts. At the low-redshift bound of the
profile, there is only a small amount of improvement that can be derived from
the Ly$\alpha$ emissivity. This involves increasing $\mathcal{M}_{\alpha}$
until the Ly$\alpha$ coupling has saturated. Once the Ly$\alpha$ coupling has
saturated, $T_{\mathrm{s}}=T_{\mathrm{k}}$, subsequent improvement can only be
obtained by modifying the $T_{\mathrm{k}}$ evolution.

The modifying factor $\mathcal{M}_{\alpha}=\mathcal{M}_{\alpha}(z)$ obtained
using the above iterative procedure is shown in Figure 2 (top left panel). It
has the shape we anticipated above. The black dotted line,
$\mathcal{M}_{\alpha}=1$, represents the fiducial model. The top right panel
in Figure 2 shows the corresponding specific intensity (blue curve), in
comparison with the fiducial model (black dotted curve). Incorporating the
resultant specific intensity into our model leads to a significantly improved
fit to the EDGES measurement, as shown in the right panel of Figure 3 (dashed
pink curve). The rms value of the residuals is $49\text{\,}\mathrm{mK}$, a
dramatic improvement over the fiducial model.

In spite of this improvement, however, the model continues to perform poorly
at low redshifts ($z\lesssim 15.5$). The fit at $z\lesssim 15.5$ in the dashed
pink line can only be improved by increasing the gas temperature, which in our
model can be done by enhancing the X-ray heating rate. For this purpose we
introduce the function $\mathcal{M}_{\mathrm{X}}$, analogous to
$\mathcal{M}_{\alpha}$, so that
$f_{\mathrm{X}}\to\mathcal{M}_{\mathrm{X}}(z)f_{\mathrm{X}}$. Setting

$\mathcal{M}_{\mathrm{X}}(z)=1+\mathrm{e}^{-4(z-14.75)^{2}}\,,$ (10)

improves the model to the solid blue line as seen in the right panel of Figure
3. The evolution of $\mathcal{M}_{\mathrm{X}}$ is shown in the bottom left
panel of Figure 2, and the corresponding X-ray intensity is shown in the
bottom right panel of this figure. The model corresponding to Figure 2 has an
rms value of the residuals $26\text{\,}\mathrm{mK}$, which is a dramatic
improvement over the fiducial model. This model, which we label ‘M1’,
represents the direct consequence of the flat-bottomed shape of the 21-cm
absorption profile for the Ly $\alpha$ coupling, with minimal modification to
the X-ray heating relative to the standard model. In the next section, we will
examine if this requirement on the Ly $\alpha$ coupling is plausible.

It is worth stressing that in this model, the flat bottom of the absorption
profile does not arise from a fine tuning of heating and cooling processes, as
one might first guess. Instead, as we see in the left panel of Figure 3, the
flatness arises out of a finely tuned non-standard evolution of the Ly
$\alpha$ coupling.

###  3.2 X-ray emission

The above scenario with a maximum modification in Ly $\alpha$ emissivity
supplemented by a small change to X-ray emission for $z\lesssim 14$ improves
our model considerably. Here we consider an alternative scenario where we keep
the modification to Ly $\alpha$ background to a minimum, precisely for
$z\gtrsim 20$, but consider significant evolution of X-ray emission. The
evolutionary factor $\mathcal{M}_{\alpha}(z)$ is now a monotonically
increasing function for $z\gtrsim 20$ (see upper left corner of Figure 4)
which partly takes care of the sharp drop in the 21-cm signal. We design an
X-ray background for $z\lesssim 20$, which gives a flat bottom and a sharp
rise in the 21-cm signal. We write the X-ray specific intensity in terms of
X-ray emissivity modified by dimensionless function of $z$,
$\mathcal{M}_{\mathrm{X}}$ as

$\bar{J}_{\mathrm{X}}(E,z)=\frac{c(1+z)^{2}}{4\pi}\int_{z}^{z_{\star}}\mathcal{M}_{\mathrm{X}}(z^{\prime})\frac{\epsilon_{\mathrm{X}}(E^{\prime},z^{\prime})}{H(z^{\prime})}\mathrm{e}^{-\tau_{\mathrm{X}}(E,z,z^{\prime})}\mathrm{d}z^{\prime}\,,$
(11)

where $z_{\star}\sim 60$ is the redshift when the star formation starts (Naoz
& Barkana, 2008), and $\epsilon_{\mathrm{X}}$ and $\tau_{\mathrm{X}}$ are the
X-ray emissivity and optical depth for X-ray photons, respectively. To find
$\mathcal{M}_{\mathrm{X}}(z)$ we follow an iterative procedure as before: we
update $\mathcal{M}_{\mathrm{X}}(z)$ by a factor of
$J_{\mathrm{X}}^{\prime}/\bar{J}_{\mathrm{X}}$ after each iteration starting
with $\mathcal{M}_{\mathrm{X}}(z)=1$. Note that we should in principle choose
$\mathcal{M}_{\mathrm{X}}$ to be a function of both energy and redshift.
However, we find that it suffices to keep $E$ fixed at which $J_{\mathrm{X}}$
peaks, i.e., $E\sim$0.2\text{\,}\mathrm{keV}$$.

We first find the required form of X-ray specific intensity,
$J_{\mathrm{X}}^{\prime}$, which modifies the gas temperature evolution to
result in a rapid increasing 21-cm brightness that is consistent with the
EDGES measurement. We can estimate this from Equation (5). Since
$q_{\mathrm{X}}\sim J_{\mathrm{X}}$, we get

$J_{\mathrm{X}}^{\prime}=\frac{(1+z)\mathrm{d}T_{\mathrm{k}}^{\prime}/\mathrm{d}z-2T_{\mathrm{k}}^{\prime}}{(1+z)\mathrm{d}T_{\mathrm{k}}/\mathrm{d}z-2T_{\mathrm{k}}}J_{\mathrm{X}}\,,$
(12)

where $T_{\mathrm{k}}^{\prime}$ is the required gas temperature without
altering the Ly $\alpha$ coupling, which can be computed from

${T^{\prime}_{\mathrm{k}}}^{-1}={T^{\prime}_{\mathrm{s}}}^{-1}-\frac{T_{\mathrm{r}}^{-1}-{T^{\prime}_{\mathrm{s}}}^{-1}}{x_{\alpha}}\,,$
(13)

where $T^{\prime}_{\mathrm{s}}$ is the required spin temperature given in
Equation (6). The quantity $J_{\mathrm{X}}^{\prime}$ so obtained may or may
not be feasible. This assessment can be made by asking if there exists a
function $\mathcal{M}_{\mathrm{X}}$ such that it gives
$\bar{J}_{\mathrm{X}}=J_{\mathrm{X}}^{\prime}$, where $\bar{J}_{\mathrm{X}}$
is defined in Equation (11). Our iterative procedure answers this question.

Figure 4 shows the resultant modification factors for the Ly $\alpha$ and
X-ray emissivities, alongside the corresponding specific intensities. Relative
to Figure 2, $\mathcal{M}_{\alpha}$ now has small values, while
$\mathcal{M}_{\mathrm{X}}$ has a relatively stronger evolution. The consequent
thermal evolution and the 21-cm signal are shown in Figure 5. Similar to model
M1, this model too is a significant improvement over the fiducial model, with
a residual rms value of $30\text{\,}\mathrm{mK}$. This scenario represents the
direct empirical implication of the EDGES measurement for the X-ray heating,
with minimal modification to the fiducial Ly $\alpha$ coupling. We term this
model ‘M2’. Both models, M1 and M2, are much better at reproducing the
observed shape of the 21-cm absorption profile. Unlike M1, however, model M2
does invoke a balance between heating and cooling to explain the flat bottom.
This is achieved by means of a non-standard evolution of the X-ray heating
rate. We will discuss the plausibility of this scenario in the next section.

It can be noticed that models M1 and M2 represent two extreme scenarios. M1
allows the Ly $\alpha$ physics to deviate away from the fiducial picture but
admits only a minimal modification to the X-ray model. Conversely, M2 allows
the X-ray model to freely deviate away from the fiducial model but restricts
modifications to the Ly $\alpha$ physics to a minimum. One could possibly
investigate the degeneracies between these two models, thereby constructing a
family of models with flat-bottomed absorption profiles, but M1 and M2 already
allow us to study the plausibility of an EDGES-like flat-bottomed absorption
profile and outline its implications for high-redshift galaxies.

##  4 Astrophysical plausibility

The factors $\mathcal{M}_{\alpha}$ or $\mathcal{M}_{\mathrm{X}}$ derived in
previous sections are purely empirical. We now discuss if the values of these
factors required by the EDGES measurement are reasonable.

The non-trivial evolution of the Ly $\alpha$ emissivity seen in Figure 2 can
be obtained either by a similarly non-trivial evolution in the cosmic SFRD or
a change in the SED of the sources of UV photons. We will discuss
modifications to the SFRD below. For now, it is worth asking if a standard
SFRD evolution model could possibly result in the Ly $\alpha$ emissivity
required in our models M1 and M2. Previous literature worked with a fixed
number of Lyman series photons per baryon at all epochs (Barkana & Loeb,
2005). The cumulative Lyman series photons per baryon from newer population
synthesis models such as bpass (Binary Population and Stellar Synthesis;
Eldridge et al., 2017; Stanway & Eldridge, 2018) give somewhat higher values.
For example, consider a stellar population formed in an instantaneous
starburst, metallicity $Z=10^{-3}$ stars and a Kroupa et al. (1993) IMF (a
broken power law with slope $-1.3$ for masses $0.1$–$0.5\,\mathrm{M}_{\odot}$
and $-2.35$ for masses $0.5$–$300\,\mathrm{M}_{\odot}$). This gives $\sim
2\times 10^{4}$ Lyman series photons per baryon implying
$\mathcal{M}_{\alpha}\approx 2.06$.

Recently, Gessey-Jones et al. (2022) showed that Pop III stars could have much
brighter SEDs. A 2–$180\,\mathrm{M}_{\odot}$ power-law Pop III IMF with slope
$-0.5$ has $\mathcal{M}_{\alpha}\sim 6$. As a comparison, our extreme value
for $\mathcal{M}_{\alpha}$ is $\sim 8$ at about $z=16$, as we see in Figure 2.
It therefore appears that conventional Pop II IMFs are unlikely to be able to
produce the level of Ly $\alpha$ emissivity demanded by the 21-cm signal
unless, as we discuss below, the SFRD is boosted significantly. Pop III IMFs,
on the other hand, potentially can produce the required levels of Ly $\alpha$
emission (see also Schauer et al., 2019). For instance, the burst feature in
$\mathcal{M}_{\alpha}$ at $z=16$ could be the result of the peak in SED due to
Pop III stars of mass $\sim 5\,\mathrm{M}_{\odot}$ (Gessey-Jones et al.,
2022).

Possible sources of X-rays include X-ray binaries (XRBs) (Power et al., 2013;
Fragos et al., 2013b), inverse Compton scattering in supernova remnants (Oh,
2001), mini-quasars (Madau et al., 2004) and galactic winds (Meiksin et al.,
2017). Evidence suggests that quasars dominate at low redshifts while XRBs are
dominant at high redshifts $(z\gtrsim 5)$ (Fabbiano, 2006; Brorby et al.,
2016; Lehmer et al., 2016). Among the two types of XRBs, high-mass XRBs are
dominant over low-mass XRBs for $z\gtrsim 2.5$ (Fragos et al., 2013a).

The X-ray SED $\phi_{\mathrm{X}}$ (defined in units of number per unit energy
per stellar baryon, which for us would be such that $\phi_{\mathrm{X}}\propto
E^{-5/2}$) is normalized by specifying $C_{\mathrm{X}}=\int
E\phi_{\mathrm{X}}\,\mathrm{d}E$ (Mesinger et al., 2011, 2013). The quantity
$C_{\mathrm{X}}\equiv L_{\mathrm{X}}/\mathrm{SFR}$ is an observable that can
be measured from the relationship between the X-ray luminosity
$L_{\mathrm{X}}$ and the star formation rate (SFR). Measurements from nearby
star-forming galaxies suggest a linear relationship, i.e., a constant
$C_{\mathrm{X}}$ (Grimm et al., 2003; Gilfanov et al., 2004). In our previous
work (M22) we adopted the following $L_{\mathrm{X}}$–SFR relation for photon
energies 0.5–$8\text{\,}\mathrm{keV}$

$C_{\mathrm{X,fid}}=\frac{L_{\mathrm{X}}}{\mathrm{SFR}}\approx$2.61\text{\times}{10}^{32}\text{\,}\mathrm{W}$\left(\mathrm{M}_{\odot}\mathrm{yr}^{-1}\right)^{-1}\,,$
(14)

based on observation of HMXBs in star-forming galaxies out to distances of
$\lesssim$40\text{\,}\mathrm{Mpc}$$ (Mineo et al., 2012). Observations of
star-forming galaxies for $z\lesssim 1.3$ (Mineo et al., 2013) also suggest a
scaling relation consistent with Equation (14). In M22 and here, we extend the
relation in Equation (14) for photon energies 0.2–$30\text{\,}\mathrm{keV}$
(Mirocha & Furlanetto, 2019). The X-ray emissivity (in units of number per
unit time per unit energy per unit comoving volume) is then given by

$\epsilon_{\mathrm{X}}(E,z)=f_{\mathrm{X}}\phi_{\mathrm{X}}(E)\frac{\dot{\rho}_{\star}(z)}{m_{\mathrm{b}}}\,,$
(15)

where $m_{\mathrm{b}}$ is the average baryon mass and $\dot{\rho}_{\star}$ is
the comoving SFRD. The above emissivity then gives us the specific intensity
as in Equation (11).

Our fiducial value of $f_{\mathrm{X}}$, as mentioned in Table 1, is $2.379$.
In the language of Section 3.2, this value of $f_{\mathrm{X}}$ corresponds to
$\mathcal{M}_{\mathrm{X}}=1$, and Equation (14) corresponds to
$\mathcal{M}_{\mathrm{X}}=0.42$. At redshifts $z\gtrsim 1.3$, because of lack
of direct observations, $f_{\mathrm{X}}$ remains largely unconstrained.
However, evidence suggests that it is higher at higher redshifts (Mirabel et
al., 2011). An evolving $f_{\mathrm{X}}$ (power law in $1+z$) was constrained
by Dijkstra et al. (2012) based on the unresolved cosmic X-ray background
(Lehmer et al., 2012) assuming that this background is produced by HMXBs. But
this was applicable to much lower redshifts $(z\lesssim 2)$ compared to the
redshifts of our interests. At higher redshifts $(z\gtrsim 5)$ the unresolved
cosmic X-ray background puts only weak constraints on $f_{\mathrm{X}}$ (see
also Fialkov et al., 2016).

Not much is known about $f_{\mathrm{X}}$ for $z>14$. The only information for
these redshifts we have is by Fragos et al. (2013b) which is based on
population synthesis models. They gave fitting functions for HMXB luminosity
density for $0\leqslant z\leqslant 20$. Converting this to $C_{\mathrm{X}}$ we
find that $C_{\mathrm{X}}$ peaks at $z\approx 1.7$, decreases to about 0.025
at $z\approx 19$ before increasing slightly to $\sim 0.03$ for $z\sim 20$.
Translating this to our language, $\mathcal{M}_{\mathrm{X}}(z)$ varies from
about 0.7 at $z<2$ to $0.013$ at $z\sim 20$, far smaller than what is required
by the EDGES measurement (in either of our models M1 and M2, Figures 2 and 4).
It thus appears that obtaining rapidly increasing X-ray background of the sort
required by data is not plausible in conservative models. One way of
increasing $\mathcal{M}_{\mathrm{X}}$ is to again invoke Pop III HMXBs, which
can increase $\mathcal{M}_{\mathrm{X}}$ to values as large as $10^{3}$
(Mirocha et al., 2018).

It is worth pointing out that the low-redshift spikes in
$\mathcal{M}_{\alpha}$ and $\mathcal{M}_{\mathrm{X}}$ in Figures 2 and 4 are
connected to the rapid disappearance of 21-cm signal around $z\sim 15$. The
onset of reionization can also potentially explain this rapid disappearance.
Although reionization at such high redshifts will violate CMB constraints,
such a scenario could potentially lead to a picture in which
$\mathcal{M}_{\alpha}$ and $\mathcal{M}_{\mathrm{X}}$ level off or continue
rising at $z\lesssim 15$.

Figure 6: Our model M3. The left panel shows the $z$-dependent multiplication
factor $\mathcal{M}_{\star}$ required to fit the flat-bottomed shape of the
EDGES measurements with our fiducial value for stellar properties. The right
panel shows the corresponding comoving cosmic star formation rate density
(SFRD). The black dotted line in both panels shows our fiducial SFRD, which
has $f_{\star}=0.1$. Figure 7: Left panel: evolution of the CMB+ERB
temperature (dotted black), the spin temperature (solid red) and the gas
temperature (solid blue) in our model M3. Right panel: the corresponding 21-cm
signal (solid blue), in comparison with the EDGES measurement (dashed black),
with a $50\text{\,}\mathrm{mK}$ uncertainty (grey band). The rms value of the
residuals is $47\text{\,}\mathrm{mK}$. The green dotted curve is the fiducial
model of Figure 1.

##  5 Implications

###  5.1 Implications for high-redshift star formation

We see above that the EDGES measurement demands Ly $\alpha$ and X-ray
production mechanisms that are at the limits of what is reasonable. An
alternative therefore is that the cosmic star formation evolution is modified
relative to our fiducial model.

In M22 we worked with a constant value of SFE (defined as the fraction of gas
in the dark matter haloes which gets converted into stars) of $f_{\star}=0.1$.
However, in general $f_{\star}$ depends on feedback mechanisms, metallicity of
the collapsing gas, and the mass of the star forming haloes (Cohen et al.,
2017; Monsalve et al., 2019). In order to obtain a flat-bottomed absorption
profile, we follow a similar method as before and change $f_{\star}$ so that
it now has a time dependence: $f_{\star}\to\mathcal{M}_{\star}(z)f_{\star}$.
(We continue to hold the SFE independent of the halo mass. Also, the values of
other model parameters, including $f_{\alpha}$ and $f_{\mathrm{X}}$ are held
fixed to the fiducial values listed in Table 1.) Changing $f_{\star}$
inevitably affects both Ly $\alpha$ as well as X-ray backgrounds. As a result,
the calculation of $\mathcal{M}_{\star}$ is not as straightforward as that of
$\mathcal{M}_{\alpha}$ and $\mathcal{M}_{\mathrm{X}}$ in Models M1 and M2. We
use hit-and-trial method for this starting with
$\mathcal{M}_{\star}=\mathcal{M}_{\alpha}$. The result is shown in Figure 6.

We see that the 21-cm profile requires the SFE to drop rapidly to zero at
redshift higher than $\sim 20$. This is to prevent a Ly $\alpha$ background
from getting established too early. A rapid enhancement followed by a quick
drop in $f_{\star}$ relative to the fiducial model is then required at
redshift of $z\sim 20$–$19$ to reproduce the sharp falling edge of the
profile. Finally, the model requires a significant enhancement in $f_{\star}$
once more at $z\lesssim 17$ to obtain the required X-ray heating. The
resulting signal is a good fit, as shown in Figure 7. This model has a
residual rms value of $47\text{\,}\mathrm{mK}$.

This is a considerable improvement over the fiducial model. But this model is
still significantly worse than the models M1 and M2 in which the properties of
the Ly $\alpha$ and X-ray sources were varied. Thus it is hard to fit the
EDGES data for $17>z>15$ by only modifying the SFRD. This can be understood to
be because X-ray and Ly $\alpha$ have contrasting effects on the 21-cm signal.
Increasing SFRD for $17>z>15$ to get more Ly $\alpha$ and hence a better
fitting (by lower the signal) also increases X-ray heating at the same time
(in turn raising the signal). The effects are reversed if we consider lowering
the SFRD. Thus, if we leave the intrinsic properties of Ly $\alpha$ and X-ray
sources untouched and modify only the SFRD as shown in Figure 6 then the
signal shown in Figure 7 is the best that can be achieved.

It is also possible to get an equivalent model by changing the minimum mass of
star-forming haloes. This minimum mass in our fiducial model corresponds to a
virial temperature of $e4.25\text{\,}\mathrm{K}$, which is close to the limit
set by atomic hydrogen cooling. The required reduction in the SFRD at
$z\gtrsim 18$ necessitates a very high value of the minimum halo mass.
Conversely, the large SFRD required at the lower redshift end of the data can
be obtained by reducing the minimum mass by a factor of $\lesssim 2$.

###  5.2 Predictions for galaxy surveys

Given the required SFRD evolution, and with our assumption that the SFE is
independent of the halo mass, we can make predictions for the ultraviolet
luminosity functions (UV LFs) of high-redshift galaxies. As a measure of the
comoving number density of galaxies for a given luminosity, the UV LF has a
one-to-one correspondence with the halo mass function. The LF is traditionally
expressed as the comoving number density per unit absolute magnitude rather
than number density per unit luminosity. We construct the intrinsic LF as

$\mathrm{LF}=\frac{\mathrm{d}\varphi(M_{\lambda})}{\mathrm{d}M_{\lambda}}=\frac{\mathrm{d}n(M_{\mathrm{h}})}{\mathrm{d}M_{\mathrm{h}}}\frac{\mathrm{d}M_{\mathrm{h}}}{\mathrm{d}L_{\lambda}}\frac{\mathrm{d}L_{\lambda}}{\mathrm{d}M_{\lambda}}\,,$
(16)

where $\mathrm{d}n$ is comoving number density of haloes with mass between
$M_{\mathrm{h}}$ and $M_{\mathrm{h}}+\mathrm{d}M_{\mathrm{h}}$, $L_{\lambda}$
is the specific luminosity at a wavelength $\lambda$, and $M_{\lambda}$ is the
corresponding absolute magnitude. We consider the UV LF for
$\lambda=$1500\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$.

Figure 8: Left panel: galaxy UV luminosity function at $z\sim 17$ for
$M_{\mathrm{UV}}<-13$. Middle panel: number density of $z=17$ sources on the
sky. Right panel: predictions for an ultra-deep JWST survey with
$m_{\mathrm{UV,lim}}=32$. In all panels, dotted curves show our fiducial
models; solid blue curves show model M3.

The UV luminosity of a given star-forming halo can be assumed to track the
instantaneous SFR (e.g., Kennicutt, 1998; Madau & Dickinson, 2014) so that

$L_{\mathrm{UV}}=\dot{M}_{\star}\mathcal{L}_{\mathrm{UV}}\,,$ (17)

where $\dot{M}_{\star}$ is the SFR and
$\mathcal{L}_{\mathrm{UV}}=$8.695\text{\times}{10}^{20}\text{\,}\mathrm{W}\mathrm{Hz}^{-1}$\big{(}\mathrm{M}_{\odot}\mathrm{yr}^{-1}\big{)}^{-1}$
is the specific luminosity per unit SFR (Mirocha et al., 2016). We follow Sun
& Furlanetto (2016) to model the SFR as

$\dot{M}_{\star}=f_{\star}\dot{M}_{\mathrm{b,acc}}\,,$ (18)

where $\dot{M}_{\mathrm{b,acc}}$ is accretion of baryons onto dark matter
haloes. A halo’s baryonic mass accretion rate can be described by the
empirical relation (McBride et al., 2009; Sun & Furlanetto, 2016; Mirocha et
al., 2016)

$\dot{M}_{\mathrm{b,acc}}=\dot{M}_{0}\left(\frac{M_{\mathrm{h}}}{10^{10}\mathrm{M}_{\odot}}\right)^{a}\left(\frac{1+z}{7}\right)^{b}\,,$
(19)

where $\dot{M}_{0}=3\,\mathrm{M}_{\odot}$yr-1, $a=1.127$ and $b=2.5$.
Combining equations (17), (18) and (19) we get

$L_{\mathrm{UV}}=f_{\star}\dot{M}_{0}\left(\frac{M_{\mathrm{h}}}{10^{10}\mathrm{M}_{\odot}}\right)^{a}\left(\frac{1+z}{7}\right)^{b}\mathcal{L}_{\mathrm{UV}}\,.$
(20)

The corresponding apparent magnitude (in the AB system) is given by (Oke,
1974)

$m_{\mathrm{UV}}=-2.5\log_{10}\left[\frac{1}{4\pi}\left(\frac{L_{\mathrm{UV}}}{$1\text{\,}\mathrm{W}\mathrm{Hz}^{-1}$}\right)\left(\frac{d_{\mathrm{L}}}{$1\text{\,}\mathrm{m}$}\right)^{-2}\right]-56.1\,,$
(21)

where

$d_{\mathrm{L}}(z)=c(1+z)\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{H(z^{\prime})}\,,$
(22)

is the luminosity distance. The corresponding absolute magnitude is

$M_{\mathrm{UV}}=-2.5\log_{10}\left[\frac{1}{4\pi}\left(\frac{L_{\mathrm{UV}}}{$1\text{\,}\mathrm{W}\mathrm{Hz}^{-1}$}\right)\left(\frac{d_{10}}{$1\text{\,}\mathrm{m}$}\right)^{-2}\right]-56.1\,,$
(23)

where $d_{10}=$10\text{\,}\mathrm{pc}$$.

If we assume that $f_{\star}$ is independent of $M_{\mathrm{h}}$, then using
equations (20) and (23) we arrive at

$\frac{\mathrm{d}\ln M_{\mathrm{h}}}{\mathrm{d}M_{\mathrm{UV}}}=-\frac{2\ln
10}{5a}\,.$ (24)

Inserting the above into Equation (16) we get

$\mathrm{LF}=-\frac{2\ln 10}{5a}\frac{\mathrm{d}n}{\mathrm{d}\ln
M_{\mathrm{h}}}\,,$ (25)

where we will use the Press & Schechter (1974) form for the mass function
$\mathrm{d}n/\mathrm{d}\ln M_{\mathrm{h}}$. Because the above relation gives
us the LF as a function of halo mass we again use equations (20) and (23) to
eliminate $M_{\mathrm{h}}$ to get LF as a function of $M_{\mathrm{UV}}$. As a
consistency check, we used the above formalism to verify that our model
results in a UV LF consistent with observations from Bouwens et al. (2015).

For high-redshift galaxy surveys, such as those planned with the JWST or Nancy
Grace Roman Space Telescope (NGRST), a useful quantity that can be derived
from the LF is the galaxy count brighter than a certain limiting luminosity,
$L_{\mathrm{UV,lim}}$, per unit redshift per unit angular area on the sky.
This can be defined as (Mo et al., 2010)

$\frac{\mathrm{d}N}{\mathrm{d}z}=\frac{\mathrm{d}V}{\mathrm{d}z}\int_{L_{\mathrm{UV,lim}}}^{\infty}\frac{\mathrm{d}\varphi}{\mathrm{d}L_{\mathrm{UV}}}\mathrm{d}L_{\mathrm{UV}}\,,$
(26)

where the differential comoving volume element per square radian is given by
comoving area on the sky per square radian times the comoving differential
length, i.e., (Hogg, 1999)

$\mathrm{d}V=\left[(1+z)^{2}\,d^{2}_{\mathrm{A}}\right]\left[(1+z)\,c\,\mathrm{d}t\right]=\frac{c}{H(z)}\left(\frac{d_{\mathrm{L}}}{1+z}\right)^{2}\mathrm{d}z\,,$
(27)

where $d_{\mathrm{A}}$ is the angular diameter distance. Equation (26)
simplifies to

$\frac{\mathrm{d}N}{\mathrm{d}z}(M_{\mathrm{h,lim}},z)=\frac{c}{H(z)}\left(\frac{d_{\mathrm{L}}}{1+z}\right)^{2}\int_{M_{\mathrm{h,lim}}}^{\infty}\frac{\mathrm{d}n}{\mathrm{d}M_{\mathrm{h}}}\mathrm{d}M_{\mathrm{h}}\,,$
(28)

where $M_{\mathrm{h,lim}}$ is the limiting halo mass which gives luminosity
$L_{\mathrm{UV,lim}}$. Finally, to get galaxy count brighter than a limiting
luminosity per unit angular area on the sky we evaluate
$N(M_{\mathrm{h,lim}})=\int(\mathrm{d}N/\mathrm{d}z)\mathrm{d}z$.

Figure 8 shows our predictions. The left panel shows our prediction for the UV
LF at $z=17$ for two values of the SFE. Since in our modelling SFE depends
only on $z$ and not on halo mass, different curves are shifted horizontally by
just a constant. Using our best-fitting model parameters we show the UV LF by
a black dotted line for which SFE is 0.1, while solid blue curve corresponds
to a modified SFE, about 0.09 (as $\mathcal{M}_{\star}(z=17)\approx 0.9$). The
middle panel of Figure 8 shows the sky surface density of $z=17$ galaxies.
Note that the absolute (apparent) AB magnitude range in the figure,
$-25<M_{\mathrm{UV}}<-14\,(26<m_{\mathrm{UV}}<38)$, is mapped to halo mass
between $M_{\mathrm{h}}=10^{12}\,\mathrm{M}_{\odot}$ and
$10^{8}\,\mathrm{M}_{\odot}$. The absolute and apparent magnitude
corresponding to minimum star forming halo virial temperature are $-12.5$ and
$38.9$ at $z=17$, respectively.

The right panel of Figure 8 shows the evolution of the sky surface density
given a fixed limiting flux for which the apparent magnitude is
$m_{\mathrm{UV,lim}}=32$, corresponding to an ultra-deep survey using JWST
(Mason et al., 2015). The dotted curve in this figure corresponds to our
fiducial SFRD evolution, which requires extreme modification to the Ly
$\alpha$ and X-ray SEDs (via the $\mathcal{M}_{\alpha}$ and
$\mathcal{M}_{\mathrm{X}}$ factors of our models M1 and M2). The solid curve
shows the case corresponding to Figure 6, in which the stellar population is
held fixed to have rather conservative choices for the SED but an EDGES-like
flat-bottomed absorption profile is obtained via a non-trivial modulation of
the SFRD evolution. This SFRD evolution is imprinted in the observable
$\mathrm{d}N/\mathrm{d}z$.

It is interesting to note that the models shown in Figure 8 do not modify the
shape of the UV LF, but only its evolution. This can be compared with the
prediction by Mirocha & Furlanetto (2019), who studied models that also
modified the shape of the LF (see also Madau, 2018). Nonetheless, it is
interesting that Mirocha & Furlanetto (2019) also find that the EDGES
measurement requires an order-of-magnitude increase in galaxy number counts
relative to canonical extrapolations.

Figure 9 presents our results for surveys soon to be underway on the JWST and
two surveys proposed using the upcoming NGRST. The top panel in Figure 9 shows
our predicted $\mathrm{d}\mathcal{N}/\mathrm{d}z$ for model M3 as a function
of redshift, where $\mathrm{d}\mathcal{N}/\mathrm{d}z$ is the redshift
gradient of the cumulative count of galaxies brighter than the survey limit.
The bottom panel shows the excess relative to our fiducial model, i.e.,

$\left(\frac{\mathrm{d}\mathcal{N}}{\mathrm{d}z}\right)\cdot\left(\frac{\mathrm{d}\mathcal{N}}{\mathrm{d}z}\right)^{-1}_{\mathrm{fiducial}}\,.$

We have shown results for CEERS (Finkelstein et al., 2017), COSMOS-Webb
(Kartaltepe et al., 2021), JADES-Medium, JADES-Deep (Robertson, 2022), PRIMER
(Dunlop et al., 2016), WDEEP (Finkelstein et al., 2021), Roman-UDF (Koekemoer
& Group, 2021), and Roman-HLS (Wang et al., 2022). The survey areas and
limiting magnitudes are listed in Table 2. WDEEP, JADES-Deep, and JADES-Medium
on the JWST, and Roman-UDF on the NGRST, have the most optimistic
configurations for detecting the enhancements in galaxy counts predicted here.

Figure 9: Top panel: $\mathcal{N}^{\prime}$ for model M3. Bottom panel: $\mathcal{N}^{\prime}/\mathcal{N}^{\prime}_{\mathrm{fiducial}}$, where $\mathcal{N}^{\prime}=\mathrm{d}\mathcal{N}/\mathrm{d}z$ is redshift gradient of the cumulative count of galaxies brighter than the survey limit. The survey areas and limiting magnitudes are listed in Table 2. Table 2: Surveys with JWST (first six) and the Nancy Grace Roman Space Telescope (last two) considered in our analysis, with their area and limiting UV magnitude (AB system). Name | Area | Limit. mag. $(m_{\mathrm{UV}})$
---|---|---
CEERS | $100\,{\mathrm{arcmin}^{2}}$ | 26.5
COSMOS-Webb | $0.6\,{\mathrm{deg}^{2}}$ | 27.7
JADES-Medium | $290\,{\mathrm{arcmin}^{2}}$ | 29.3
JADES-Deep | $46\,{\mathrm{arcmin}^{2}}$ | 29.9
PRIMER | $144\,{\mathrm{arcmin}^{2}}$ | 29.0
WDEEP | $10.8\,{\mathrm{arcmin}^{2}}$ | 30.8
Roman-UDF | $1\,{\mathrm{deg}^{2}}$ | 30.0
Roman-HLS | $2000\,{\mathrm{deg}^{2}}$ | 26.0

##  6 Conclusion

The cosmological 21-cm absorption signal measured by the EDGES collaboration
has a distinctly different shape than predictions from standard models. In
this paper, we have developed models that provide far superior fits to the
EDGES measurement than standard models. These new models either invoke exotic
stellar populations or modulate the cosmic SFRD evolution in order to
establish a cosmic Ly $\alpha$ background within 25 Myr from $z\sim 21$ to 19,
and a cosmic X-ray background within 50 Myr from $z\sim 16$ to 14.

Scenarios involving exotic stellar populations have the advantage of the
cosmic SFRD evolution in these models being relatively standard. Our models M1
and M2 presented in Section 3 are two examples of such models. However, as
discussed in Section 4 above, these models seem to require extreme Population
III IMFs evolving over redshifts. It appears difficult to obtain the required
level of both Ly $\alpha$ as well as X-ray emission from models with standard
stellar populations.

On the other hand, in scenarios involving modifications to the cosmic SFRD,
more conservative assumptions about stellar populations are sufficient. The
model M3 presented in Section 5 is an example. This models the cosmic star
formation to rapidly drop to zero at redshifts higher than $z\sim 21$. This
leads to robust predictions for upcoming deep galaxy surveys using JWST or
NGRST. Combining data from such surveys with global 21-cm measurements should
be a rich source of information about galaxy formation as well as radiative
background at cosmic dawn.

Our three models represent extremes of their respective categories. Model M1
admits large modifications to the Ly $\alpha$ source properties but attempts
to keep the properties of X-ray sources to their conventional values.
Conversely, model M2 tries to get the maximum contribution from the X-ray
side. Model M3 retains standard assumptions for the Ly $\alpha$ and X-ray
sources and maximises the contribution from SFRD modifications.

There are obvious degeneracies between these models. These degeneracies could
perhaps be captured by developing parametric phenomenological models of the
radiative sources and the cosmic SFRD evolution. Indeed, without such
extensions to the standard models, it would not be possible to fit the EDGES
measurement to a satisfactory level at all. This would also prevent inferring
constraints on exotic physics models affecting the thermal properties of the
universe at cosmic dawn. One way to develop such extensions is to consider
families of fitting functions to the Ly $\alpha$ and X-ray emissivities
discussed in Section 3. Another possibility is to fold such possibilities in
end-to-end forward models of the signal such as those employed by the REACH
experiment (de Lera Acedo, 2019).

## Acknowledgements

We thank Basudeb Dasgupta, Nicole Drakos, Anastasia Fialkov, Thomas Gessey-
Jones, Avery Meiksin, Jordan Mirocha, Brant Robertson, Saurabh Singh, and
Sandro Tacchella for helpful discussions, and Joop Schaye for editorial
comments that improved this paper. It is also a pleasure to acknowledge
discussions with members of the REACH (Radio Experiment for the Analysis of
Cosmic Hydrogen) collaboration. GK is partly supported by the Department of
Atomic Energy (Government of India) research project with Project
Identification Number RTI 4002 and by the Max Planck Society through a Max
Planck Partner Group.

## Data availability

The data and code underlying this article will be shared on reasonable request
to the corresponding author.

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