# Soft Photon Heating: A Semi-Analytic Framework and Applications to $21$cm
Cosmology

Bryce Cyr1, Sandeep Kumar Acharya2 and Jens Chluba1
1Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The
University of Manchester, Manchester M13 9PL, U.K.
2Astrophysics Research Center of the Open University, The Open University of
Israel, Ra’anana, Israel<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The presence of an abundant population of low frequency photons at high
redshifts (such as a radio background) can source leading order effects on the
evolution of the matter and spin temperatures through rapid free-free
absorptions. This effect, known as soft photon heating, can have a dramatic
impact on the differential brightness temperature, $\Delta T_{\rm b}$, a
central observable in $21$cm cosmology. Here, we introduce a semi-analytic
framework to describe the dynamics of soft photon heating, providing a
simplified set of evolution equations and a useful numerical scheme which can
be used to study this generic effect. We also perform quasi-instantaneous and
continuous soft photon injections to elucidate the different regimes in which
soft photon heating is expected to impart a significant contribution to the
global $21$cm signal and its fluctuations. We find that soft photon
backgrounds produced after recombination with spectral index $\gamma>3.0$
undergo significant free-free absorption, and therefore this heating effect
cannot be neglected. The effect becomes stronger with steeper spectral index,
and in some cases the injection of a synchrotron-like spectrum ($\gamma=3.6$)
can suppress the amplitude of $\Delta T_{\rm b}$ relative to the standard
model prediction, making the global $21$cm signal even more difficult to
detect in these scenarios.

##  1 Introduction

Heating mechanisms in cosmology can take many different forms, and a rigorous
study by the community has rewarded us with a well-established understanding
of the thermal history of the universe. At very early times ($10^{3}\lesssim
z\lesssim 10^{6}$), sources of energy injection are well-known to produce
distortions to the frequency spectrum of the Cosmic Microwave Background (CMB)
(Zeldovich & Sunyaev, 1969; Sunyaev & Zeldovich, 1970; Illarionov & Sunyaev,
1974; Danese & de Zotti, 1982; Burigana et al., 1991; Hu & Silk, 1993; Chluba
& Sunyaev, 2012). These CMB spectral distortions are often characterized to be
of $\mu$\- or $y$-type, and measurements made by the COBE/FIRAS satellite in
the 90s has led to constraints on a variety of cosmological phenomena that
remain competitive even to this day.

At lower redshifts, around $z\simeq 10^{3}$, excess heating can manifest
itself as a delay to the recombination process. This can have adverse effects
on CMB anisotropy measurements [e.g. as made by the Planck Collaboration et
al. (2020)], as well as on the spectrum of cosmological recombination
radiation, allowing stringent constraints to be placed on models which disrupt
this process (Slatyer et al., 2009; Chluba, 2010; Finkbeiner et al., 2012;
Galli et al., 2013; Diamanti et al., 2014; Carr et al., 2021).

Another milestone in our thermal history, and the focus of this study, is the
epoch of cosmic dawn and reionization [$z\simeq\mathcal{O}(10)$], where
observations of redshifted $21$cm photon emission from spin-flip transitions
can be studied to infer the distribution of neutral hydrogen. Reionization has
seen a flurry of experimental and theoretical progress over the last decade,
allowing it to develop rapidly into a mature subfield of cosmology (Mellema et
al., 2013; Jacobs et al., 2015; DeBoer et al., 2017; Cohen et al., 2018;
Bowman et al., 2018; Singh et al., 2022). The central observable in this field
is known as the differential brightness temperature, $\Delta T_{\rm b}$, which
measures the contrast between the the temperature of the radiation bath at
$21$cm frequencies ($T_{\rm R}$), and an object known as the spin temperature,
$T_{\rm S}$ which measures the relative occupation of the spin triplet and
singlet states of neutral hydrogen (see Furlanetto et al. (2006); Pritchard &
Loeb (2012); Barkana (2016) for comprehensive reviews).

This differential brightness temperature can be probed in a sky-averaged sense
(a global signal, analogous to determining the CMB temperature), as well as by
studying its spatial fluctuations (similar to the measurement of CMB
temperature anisotropies). An initial detection of the global signal was
reported by the EDGES collaboration (Bowman et al., 2018) which indicated that
the $\Delta T_{\rm b}$ absorption trough was roughly twice as deep as was to
be expected when considering standard astrophysical and cosmological
evolution. Naturally, this spurred much development on the theory side to
elucidate how one could either lower $T_{\rm S}$ (Barkana, 2018; Muñoz & Loeb,
2018; Kovetz et al., 2018) or increase $T_{\rm R}$ (Feng & Holder, 2018;
Ewall-Wice et al., 2018; Brandenberger et al., 2019; Fialkov & Barkana, 2019).
Recently, the SARAS-3 experiment (Singh et al., 2022) reported results from
their radiometer experiment, ultimately refuting the claimed detection made by
the EDGES collaboration. While the observational study of the global signal
has undergone some tumultuous times, fluctuation experiments are well under
way, with the Hydrogen Epoch of Reionization Array (HERA, DeBoer et al.
(2017)), recently providing some truly impressive limits on the power spectrum
(Abdurashidova et al., 2023).

The surface of last scattering presents us with a snapshot of the primordial
perturbations, from which we have gleamed some of the most stringent
constraints on parameters in the standard cosmological model ($\Lambda$CDM).
In contrast, observations of the 21cm signal can be made at a variety of
redshifts, providing us with a sensitive tomographic probe of the evolution of
neutral hydrogen across cosmic time. Unlike the CMB, whose perturbations are
set mainly by linear physics, a reliable computation of $\Delta T_{\rm b}$
requires input from complex astrophysical modeling, often occurring in the
non-linear regimes of structure formation. Thus, in order to fully take
advantage of the impending mountain of data promised to us by 21cm
experiments, it is imperative that we understand the various ways in which
heating, both of the photon background and of the hydrogen gas, can occur.

In this work, we present a semi-analytic formalism to determine the heating
rate of the hydrogen gas at all times after recombination (including during
the epochs of cosmic dawn and reionization) in the presence of a radio (soft
photon) background. This effect was assumed small and neglected in previous
studies on the impacts of extra radio backgrounds in 21cm cosmology (e.g. in
Feng & Holder (2018); Ewall-Wice et al. (2018); Fialkov & Barkana (2019)).
Instead, we find that there are many scenarios in which this heating via soft
photon backgrounds produces a leading order effect on the determination of
$\Delta T_{\rm b}$, and can therefore not be neglected.

This mechanism, which we call soft photon heating, can be roughly summarized
as follows: the presence of a sufficiently steep soft photon background in
place before cosmic dawn ($z\gtrsim 20$) is capable of greatly lowering the
contrast between $T_{\rm R}$ and $T_{\rm S}$, in turn reducing the amplitude
of $\Delta T_{\rm b}$. The physical effect neglected in previous studies is
that of free-free absorption, which becomes extremely efficient at low
frequencies and can cause a significant boost to the gas temperature.

Recently (Acharya et al., 2023), we studied this effect using the (fully
numeric) thermalization code, CosmoTherm111www.chluba.de/CosmoTherm. This
tool, originally developed to perform detailed calculations of CMB spectral
distortions (Chluba & Sunyaev, 2012), has been augmented over the past years
to study a more versatile range of cosmological effects, including the global
21cm signal for both standard and exotic phenomena (Acharya et al., 2022).
Using this machinery, we produced constraints from a variety of observational
probes using CosmoTherm for decaying dark matter scenarios (Acharya et al.,
2023), superconducting cosmic strings (Cyr et al., 2023a, b), and more generic
broad-spectrum photon injections (Acharya et al., 2024).

Here, we provide a simplified set of coupled integro-differential equations
which must be solved to properly incorporate the effects of soft photon
heating. This formalism has been implemented in CosmoTherm for quite some time
as a way to simplify the computation at $z\lesssim 10^{3}$ (Chluba, 2015).
Here we further expand upon these approximations and illustrate some important
new effects related to soft photon heating. In addition, we provide an
efficient numerical scheme for solving the photon evolution equation, which
can readily be implemented into standard $21$cm codes (Mesinger et al., 2011;
Fialkov et al., 2015; Muñoz, 2023). After developing the formalism, we inject
soft photon backgrounds both quasi-instantaneously and continuously,
showcasing the non-trivial evolution of the spin, matter, and radiation
temperatures. Using these models, we study changes to $\Delta T_{\rm b}$ when
soft photon heating is switched on and off. We also provide a simple Jupyter
Notebook which one can use to qualitatively study this heating in the presence
of quasi-instantaneous injections222https://github.com/Bryce-
Cyr/Soft_Photon_Heating.

The rest of the paper is organized as follows. In Section 2, we briefly review
the various heating mechanisms relevant to the computation of $\Delta T_{\rm
b}$ finishing with a more qualitative discussion on the mechanism and
implications of soft photon heating. We proceed in Sec. 3 with a derivation of
the system of coupled integro-differential equations which describes the
effect, providing a useful numerical scheme. Section 4 describes the relative
(un)importance of soft photon heating when the CMB is the unique source of low
frequency photons. Next, in Section 5, we perform quasi-instantaneous soft
photon injections, with different injection redshifts and spectral indices,
studying the evolution of various quantities of interest to $21$cm cosmology.
This allows us to elucidate the conditions under which soft photon heating
cannot be ignored. Section 6 repeats this analysis for continuous injection
scenarios, motivated by decaying dark matter and cosmic string models. The
$21$cm power spectrum is discussed in Section 7, and we conclude in Section 8.

##  2 Heating mechanisms in 21cm cosmology

The differential brightness temperature is central to $21$cm observations, and
its amplitude is determined by the contrast between the radiation and spin
temperatures at a given redshift

$\displaystyle\Delta T_{\rm b}$ $\displaystyle=\frac{T_{\rm S}-T_{\rm
R}}{1+z}\left(1-{\rm e}^{-\tau_{21}}\right),$ (1)

where $\tau_{21}$ is the 21cm optical depth. This can be approximated as
(Barkana & Loeb, 2005; Mesinger et al., 2011),

$\displaystyle\Delta T_{b}\approx 27x_{\rm
H}(1+\delta)\left(\frac{1+z}{10}\frac{0.15}{\Omega_{m}h^{2}}\right)^{1/2}\left(\frac{\Omega_{b}h^{2}}{0.023}\right)$
$\displaystyle\left(\frac{H}{{\rm dv_{r}/dr}+H}\right)\left(1-\frac{T_{\rm
R}}{T_{\rm S}}\right)\hskip 5.69046pt{\rm mK},$ (2)

where $x_{\rm H}$ is the neutral hydrogen fraction, $\delta$ is the fractional
overdensity of the baryons, and ${\rm dv_{r}}/{\rm dr}$ is the velocity
gradient along the line of sight. Spatial fluctuations of $\Delta T_{\rm b}$
exist and represent a valuable area of study, but for the purpose of this
illustration we will focus on the case of a global signal, returning to the
fluctuations in Section 7. The amplitude of the differential brightness
temperature is thus proportional to $\Delta T_{\rm b}\propto x_{\rm
H}(1-T_{\rm R}/T_{\rm S})$. The radiation temperature is set by the intensity
of the photon background at $21$cm frequencies,

$\displaystyle T_{\rm R}(z)=\left.\frac{c^{2}}{2k_{\rm
b}}\frac{I_{\nu}(z)}{\nu^{2}}\right|_{\nu=\nu_{\rm 21}},$ (3)

where $c$ and $k_{\rm b}$ are the speed of light and Boltzmann constant
respectively, while $\nu_{\rm 21}$ is the rest frame 21cm frequency (1.42
GHz). In standard scenarios the intensity spectrum is simply set by the CMB,
while modifications are necessary when one introduces additional soft photon
(radio) backgrounds.

The spin temperature counts the relative number of hydrogen atoms in the spin
triplet state vs the singlet, and is often expressed as the equilibrium
balance of processes which can induce spin-flip transitions

$\displaystyle T_{\rm S}^{-1}=\frac{x_{\rm R}T_{\rm R}^{-1}+x_{\rm c}T_{\rm
M}^{-1}+x_{\alpha}T_{\alpha}^{-1}}{x_{\rm R}+x_{\rm c}+x_{\alpha}}.$ (4)

Here, $x_{\rm R}$, $x_{\rm c}$ and $x_{\alpha}$ are the radiative, collisional
and Wouthuysen-Field coupling coefficients respectively, $T_{\rm M}$ is the
matter temperature, and $T_{\alpha}$ is the colour temperature of the
Ly-$\alpha$ radiation field (see Pritchard & Loeb (2012) for a detailed
review, and Venumadhav et al. (2018) for a discussion on the precise form of
$x_{\rm R}$). The addition of extra radio backgrounds therefore directly
propagates in this way to $T_{\rm S}$.

At high redshifts ($z\gtrsim 150$), Compton scattering between the electrons
and background photons drive the temperatures of these two sectors to be
roughly equal, $T_{\rm M}\simeq T_{\rm CMB}$, strongly suppressing any
possible deviation of $\Delta T_{\rm b}$ from 0. At lower redshifts, Compton
scattering becomes inefficient, and the spin temperature (which is initially
tightly coupled to $T_{\rm M}$) deviates from $T_{\rm R}$, producing the well
known absorption trough feature in $\Delta T_{\rm b}$. At even later times,
the first stars begin to form, sourcing an abundant population of X-rays. This
hard photon injection provides an efficient heat source for the hydrogen gas,
driving the spin temperature back to $T_{\rm R}$ (Venkatesan et al., 2001;
Pritchard & Furlanetto, 2007; Zaroubi et al., 2007), marking an end to the
epoch known as cosmic dawn.

The mean free path of sufficiently strong X-rays ($E\gtrsim 1$ keV) can be
cosmological, implying that they are capable of depositing their energy at
distances far from their source. X-ray binaries produced from stellar remnants
appear to be a plausible source of heating around the time of cosmic dawn, and
the spectrum of X-rays produced by these binaries plays a major role in the
exact timing of of the $\Delta T_{\rm b}$ absorption trough, as discussed in
Fialkov et al. (2014); Pacucci et al. (2014).

Additional heating mechanisms have also been studied. For example, a number of
authors have considered cosmic rays as another source of early heat deposition
into the intergalactic medium (IGM). In a variety of works (Sazonov & Sunyaev,
2015; Leite et al., 2017; Yokoyama & Ohira, 2023), it was shown that these
cosmic rays can provide a significant increase to the gas temperature at high
enough redshifts to impact the observables relevant to cosmic dawn and
reionization. Overall, one expects a reduction in the amplitude of $\Delta
T_{\rm b}$ due to the increased gas temperature, as noted in Jana et al.
(2019).

More recently, Gessey-Jones et al. (2023) performed the first detailed
simulations of this setup and found that cosmic rays deposit most of their
energy nearer to the source than X-rays. In principle, this allows for
component separation between X-ray and cosmic ray heating by leveraging 21cm
power spectrum data and tomographic maps. In practice, however, the cosmic ray
scenarios still possess a large number of theoretical uncertainties which
makes performing detailed predictions difficult. In particular, both Leite et
al. (2017) and Gessey-Jones et al. (2023) found that modest variations in the
assumed spectral index of cosmic rays caused large deviations in the IGM
temperature at redshifts as low as $z\simeq 8$.

Around the time of the EDGES measurement (Bowman et al., 2018), it was thought
that the presence of an extra radio background with synchrotron-like spectral
index could provide large enhancements to $\Delta T_{\rm b}$ (Feng & Holder,
2018; Ewall-Wice et al., 2018; Brandenberger et al., 2019; Fialkov & Barkana,
2019) through an increase in $T_{\rm R}$. What was not realized at the time is
that the abundance of low frequency photons present in such a background can
cause significant heating of the IGM through efficient free-free absorption.
Heating through this effect is typically subdominant in standard scenarios
(e.g. when the CMB sets the amplitude of the low frequency photon background),
but this changes dramatically when a significant population of soft photons is
introduced. Thus, $T_{\rm R}$, $T_{\rm M}$, and $T_{\rm S}$ are not as
decoupled as originally thought, and must be evolved together to properly
account for this effect. It is this joint evolution which we call soft photon
heating.

In previous work, we studied this in a fully numerical setup, and found that
for a sufficiently energetic synchrotron background, the ratio of $T_{\rm
R}/T_{\rm S}$ was greatly suppressed relative to the case when soft photon
heating was neglected (Acharya et al., 2023). When examining a toy model of
the synchrotron spectrum which closely matched the ARCADE-2 (Fixsen et al.,
2011) and LWA-1 (Dowell & Taylor, 2018) radio background data, this effect
caused a dramatic suppression to the amplitude of $\Delta T_{\rm b}$.

In the following, we revisit this result and provide a semi-analytic scheme to
determine the non-trivial evolution of the matter temperature in the presence
of a soft photon background. After developing the formalism, we compute the
spin temperature evolution for a number of models, utilizing a simple
reionization module with a standard population of Ly-$\alpha$ fluxes. Details
of this module can be found in Acharya et al. (2022). Moreover, in order to
disentangle soft photon heating from other mechanisms, we neglect additional
modeling extensions of the early sources, such as the inclusion of cosmic ray
heating.

##  3 Soft photon heating: a semi-analytic approach

The transfer of energy and particle number between photons and electrons is a
setup which has been carefully studied in the context of CMB spectral
distortions. An intimate knowledge of this thermalization procedure is
necessary to understand which types of effects remain frozen in on the photon
spectrum, providing observational signatures one can search for. A deficit of
low-energy photons due to free-free absorption is one such signature, thus it
is perhaps unsurprising that our starting point comes from the spectral
distortion literature (Chluba, 2015).

A word on preliminaries: when referring to photon frequencies, we often make
use of the dimensionless form $x=h\nu/k_{\rm b}T_{\rm CMB}$, which has the
convenient property that it is redshift invariant, due to the identical
scalings of $\nu$ and $T_{\rm CMB}$ with $z$. The peak of a blackbody spectrum
has $x_{\rm peak}\approx 2.82$, while data on the radio synchrotron background
(Fixsen et al., 2011; Dowell & Taylor, 2018) sits at roughly $10^{-4}\lesssim
x\lesssim 10^{-1}$. We will often work in terms of occupation numbers instead
of intensities, which are related through

$\displaystyle I_{\rm bb}(x,T_{\rm CMB})$ $\displaystyle=\frac{2(k_{\rm
b}T_{\rm CMB})^{3}}{(hc)^{2}}x^{3}n_{\rm bb}(x),$ (5a) $\displaystyle n_{\rm
bb}(x)$ $\displaystyle=\frac{1}{{\rm e}^{x}-1}.$ (5b)

For a general (non-thermal) intensity spectrum, the same relation holds, such
that $I\propto x^{3}n(x)$ where $n(x)$ is a non-blackbody occupation number.
Keep in mind that $T_{\rm CMB}=T_{\rm CMB,0}(1+z)$, which causes the intensity
spectrum to retain a redshift dependence.

Additionally, we define an analogous form for photons frequencies at the
electron temperature $x_{\rm e}=x\,T_{\rm CMB}/T_{\rm M}$ which will be
computationally convenient later on. This quantity does inherit a redshift
dependence once the CMB and matter temperatures decouple at $z\lesssim 150$.
As a caution, this should not be confused with $X_{\rm e}$, which we will take
to define the free electron fraction. Finally, we will make use of the Thomson
scattering optical depth, ${\,\rm d}\tau=\sigma_{\rm T}N_{\rm e}c\,{\,\rm d}t$
as a proxy for time evolution. Here, $\sigma_{\rm T}$ is the Thomson
scattering cross section and $N_{\rm e}$ the free electron number density.

###  3.1 Free-free absorption at low frequencies

We begin our formulation of the photon evolution equation at times when
Compton scattering is quite inefficient at redistributing energy throughout
the photon spectrum (i.e. $z\lesssim 1500$). At these redshifts, the evolution
of the photon occupation number is primarily influenced by free-free emission
and absorption, and is given by (e.g., see Chluba, 2015)

$\frac{{\,\rm d}n}{{\,\rm d}\tau}\simeq\frac{\Lambda_{\rm BR}(\tau,x_{\rm
e})\,{\rm e}^{-x_{\rm e}}}{x_{\rm e}^{3}}\left[1-n({\rm e}^{x_{\rm
e}}-1)\right]+S(\tau,x).$ (6)

Here, $\Lambda_{\rm BR}(\tau,x_{\rm e})$ is the (dimensionless) Bremsstrahlung
emissivity coefficient and $S(\tau,x)$ is a source term that we will elaborate
upon briefly. A derivation of this equation from first principles, as well as
the details of $\Lambda_{\rm BR}(\tau,x_{\rm e})$ can be found in Appendix A.
The inefficacy of Compton scattering at $z\lesssim 1500$ allows us to neglect
mode couplings between different photon frequencies, greatly simplifying the
thermalization problem (Chluba, 2015).

We will be most interested in low-frequency departures from a blackbody, so it
is useful to consider only the part that deviates from a blackbody, $\Delta
n=n-n_{\rm bb}$, the so-called spectral distortion. The evolution of this
spectral distortion is

$\frac{{\,\rm d}\Delta n}{{\,\rm d}\tau}\simeq-\frac{\Lambda_{\rm BR}(1-{\rm
e}^{-x_{\rm e}})}{x_{\rm e}^{3}}\Delta n+\Delta S(\tau,x).$ (7)

The first term encodes the effects of free-free absorption and emission in the
presence of a distortion, governing the efficiency of soft photon heating. We
have absorbed the terms not dependent on the distortion into a newly defined
source term, $\Delta S$,

$\displaystyle\Delta S(\tau,x)$ $\displaystyle=S_{\rm
inj}(\tau,x)+\frac{k_{\rm b}}{m_{\rm e}c^{2}}(T_{\rm M}-T_{\rm
CMB})Y(x)+S_{\rm ff,bb}(\tau,x),$ (8) $\displaystyle Y(x)$
$\displaystyle=\frac{x{\rm e}^{x}}{({\rm e}^{x}-1)^{2}}\left[x\frac{{\rm
e}^{x}+1}{{\rm e}^{x}-1}-4\right].$

The first term is simply the rate of occupation number injection for whichever
low-frequency background we choose to study, $S_{\rm inj}={\,\rm d}n_{\rm
inj}/{\,\rm d}\tau$. The electron population cools faster than the photon
background, but at $z\gtrsim 150$ residual Compton scatterings are able to
keep $T_{\rm M}\simeq T_{\rm CMB}$. This energy extraction from the CMB
photons generically sources a distortion, which is enforced by the second term
in Eq. (8), with $Y(x)$ being the spectral parameterization of the
$y$-distortion. A full derivation of this effect is also presented in Appendix
A. Finally, the third term accounts for free-free emission and absorption off
of the blackbody itself. Its form is given by

$\displaystyle S_{\rm ff,bb}(\tau,x)=\frac{\Lambda_{\rm BR}(1-{\rm e}^{-x_{\rm
e}})}{x_{\rm e}^{3}}\left[\frac{1}{{\rm e}^{x_{\rm e}}-1}-\frac{1}{{\rm
e}^{x}-1}\right].$ (9)

From this expression it is evident that when the electron temperature is
higher than the CMB temperature, free-free effects lead to the emission of
photons, while photons are absorbed when $T_{\rm M}<T_{\rm CMB}$. Energy
transfer between these blackbody photons and the electrons is typically quite
small, as we will see in Sec. 4. Now, let us define the free-free optical
depth

$\displaystyle\tau_{\rm ff}=\int_{0}^{\tau}\frac{\Lambda_{\rm
BR}(\tau^{\prime},x_{\rm e})(1-{\rm e}^{-x_{\rm e}})}{x_{\rm e}^{3}}{\,\rm
d}\tau^{\prime}.$ (10)

By performing a change of variable to $\tau_{\rm ff}$ in Eq. (7), the
evolution equation is in a form that can easily be solved using integrating
factors. The result is (e.g., similar to Chluba, 2015)

$\displaystyle\Delta n(x,\tau_{\rm ff,obs})=\Delta n(x,0)\,{\rm e}^{-\tau_{\rm
ff,obs}}+{\rm e}^{-\tau_{\rm ff,obs}}\int_{0}^{\tau_{\rm ff,obs}}{\,\rm
d}\tau_{\rm ff}^{\prime}\,\Delta\tilde{S}\,{\rm e}^{\tau_{\rm ff}^{\prime}}.$
(11)

In this expression, $\tau_{\rm ff,obs}=\tau_{\rm ff}(z_{\rm obs})$ is the
free-free optical depth at some target redshift, and $\tau_{\rm ff,ini}=0$.
For convenience, the source term has also been slightly redefined

$\displaystyle\Delta\tilde{S}=\frac{x_{\rm e}^{3}}{\Lambda_{\rm BR}(1-{\rm
e}^{-x_{\rm e}})}\Delta S.$

Equation (11) illustrates how a spectral distortion injected at some initial
time [$\tau_{\rm ff}(z_{\rm ini})=0$] is attenuated as it propagates to later
times. Distortions which are sourced post-recombination with $x\gtrsim
10^{-4}$ do not exhibit a large amount of free-free absorption (Chluba, 2015;
Bolliet et al., 2020), as we will also show below. This situation changes
dramatically at lower frequencies due to the sharp increase in the absorption
cross section. For numerical schemes, Eq. (11) can be recast in a particularly
useful way

$\displaystyle\Delta n(x,\tau_{\rm ff,i+1})\approx\Delta n(x,\tau_{\rm
ff,i})\,{\rm e}^{-\Delta\tau_{\rm ff}}+\Delta\tilde{S}\left(1-{\rm
e}^{-\Delta\tau_{\rm ff}}\right),$ (12)

where $\Delta\tau_{\rm ff}=\tau_{\rm ff,i+1}-\tau_{\rm ff,i}$ is the size of a
given timestep. The condition for the validity of this iterative process is
that the source term $\Delta\tilde{S}$ must be roughly constant over a given
timestep, where $\Delta\tau_{\rm ff}$ need not be small. We have found that
redshift step sizes of $\Delta z\lesssim 0.5$ offer a good balance of
convergence to the full result (computed using CosmoTherm), and fast
runtimes333The reader is encouraged to check this themselves by running the
provided Jupyter Notebook..

###  3.2 Evolution of the matter temperature

The next step is to understand the response of the matter temperature to
(potentially sizeable) free-free absorptions. In the absence of interactions
with the photons, adiabatic cooling of the matter induces the well known
scaling $T_{\rm M}\propto(1+z)^{2}$. At early enough times ($z\gtrsim 150$),
Compton scattering off the background provides an efficient heat source for
the gas, enforcing $T_{\rm M}\simeq T_{\rm CMB}$. This only requires the
presence of the usual CMB, and produces a weak (negative) $y$-distortion to
the blackbody spectrum (Chluba, 2005; Chluba & Sunyaev, 2012). In the presence
of additional soft photon backgrounds, supplemental gas heating can occur and
the matter temperature evolves as

$\displaystyle\frac{{\rm d}T_{M}}{{\rm d}z}=\frac{2T_{M}}{1+z}+\frac{X_{\rm
e}}{1+X_{\rm e}+f_{\rm He}}\frac{8\sigma_{\rm T}\rho_{\rm CMB}}{3m_{\rm
e}c}\frac{T_{\rm M}-T_{\rm CMB}}{(1+z)H(z)}+\frac{{\rm d}T_{\rm ff}}{{\rm
d}z}.$ (13)

In this expression, $X_{\rm e}=N_{\rm e}/N_{\rm H}$ is the free electron
fraction, $f_{\rm He}\simeq 0.08$ is the helium fraction by number of nuclei,
$N_{\rm H}$ is the number density of hydrogen, $\rho_{\rm CMB}$ is the
(physical) energy density of the CMB at temperature $T_{\rm CMB}$, and $m_{\rm
e}$ is the electron mass. Note that a positive contribution, ${\,\rm d}T_{\rm
ff}/{\,\rm d}z>0$ corresponds to a cooling of the gas temperature, while a
negative contribution sources heating. The net effect caused by the
reprocessing of a soft photon background through free-free is

$\displaystyle\frac{{\,\rm d}T_{\rm ff}}{{\,\rm d}z}=-\frac{1}{(3/2)k_{\rm
b}N_{\rm H}(1+X_{\rm e}+f_{\rm He})}\frac{1}{a^{4}}$
$\displaystyle\frac{{\,\rm d}a^{4}\Delta\rho_{\rm ff}}{{\,\rm d}z},$ (14)

where the denominator represents the heat capacity of the medium, and ${\,\rm
d}(a^{4}\Delta\rho_{\rm ff})/{\,\rm d}z$ is the comoving change of the photon
energy density by the free-free part of the photon evolution equation. We
provide the derivation of the Compton heating term, as well as an explicit
form for this free-free heating term in Appendix B. For the latter, the final
result is

$\displaystyle\frac{1}{a^{4}}\frac{{\,\rm d}a^{4}\Delta\rho_{\rm ff}}{{\,\rm
d}z}=-\frac{\rho_{\rm CMB}}{\pi^{4}/15}\left(\frac{T_{\rm M}}{T_{\rm
CMB}}\right)^{3}\frac{\sigma_{\rm T}N_{\rm e}c}{H(1+z)}$ (15)
$\displaystyle\times\int_{0}^{\infty}{\,\rm d}x\,\Lambda_{\rm BR}(\tau,x_{\rm
e})\left(1-{\rm e}^{-x_{\rm e}}\right)\left[\frac{1}{{\rm e}^{x_{\rm
e}}-1}-\frac{1}{{\rm e}^{x}-1}-\Delta n(x)\right].$

The injection of a soft photon background is achieved through a large and
positive contribution to $\Delta n(x)$ at low frequencies. After injection,
these photons are reprocessed and absorbed by the medium, yielding important
contributions to the heating term, ${\,\rm d}T_{\rm ff}/{\,\rm d}z$.

###  3.3 Evolution of the electron fraction

The final piece of this puzzle requires us to understand the evolution of the
free-electron fraction in the presence of a soft photon background. Due to the
high optical depth for low-frequency photons prior to recombination, we are
primarily concerned with soft photon backgrounds generated at $z\lesssim
1500$. At these redshifts, helium recombination has almost entirely completed
(Switzer & Hirata, 2008; Kholupenko et al., 2007; Rubiño-Martín et al., 2008),
so we choose to neglect any effects associated with it. The evolution of the
free-electron fraction is determined by (Seager et al., 1999, 2000)

$\displaystyle\frac{{\rm d}X_{\rm e}}{{\rm d}z}=\frac{\big{[}\alpha_{\rm
H}X_{\rm e}^{2}N_{\rm H}-\beta_{\rm H}(1-X_{\rm e})B_{\rm H}\big{]}C_{\rm
H}}{(1+z)H(z)},$ (16)

where $\alpha_{\rm H}$ and $\beta_{\rm H}$ are the Case B recombination and
photoionization rates respectively, $B_{\rm H}={\rm exp}[-E_{\alpha}/k_{\rm
b}T_{\rm CMB}]$ (here $E_{\alpha}=10.2\,{\rm eV}$ is the energy of a
Ly-$\alpha$ photon), and the Peebles factor is given by

$\displaystyle C_{\rm H}=\frac{1+K_{\rm H}A_{\rm 2s1s,H}N_{\rm H}(1-X_{\rm
e})}{1+K_{\rm H}(A_{\rm 2s1s,H}+\beta_{\rm H})N_{\rm H}(1-X_{\rm e})}.$ (17)

In this expression, $A_{\rm 2s1s,H}=8.22458\,{\rm s}^{-1}$ is the decay rate
of the 2s level to the ground state, and $K_{\rm H}=\lambda_{\alpha}^{3}/8\pi
H(z)$ where $\lambda_{\alpha}$ is the wavelength of a Ly-$\alpha$ photon.
Additionally, the photoionization rate is related to the recombination rate
through

$\displaystyle\beta_{\rm H}=\alpha_{\rm H}(T_{\rm CMB})\left(\frac{m_{\rm
e}k_{\rm b}T_{\rm CMB}}{2\pi\hbar^{2}}\right)^{3/2}\,{\rm e}^{-E_{\rm
2s}/k_{\rm b}T_{\rm CMB}}.$ (18)

Here, $E_{\rm 2s}=3.4$ eV is the ionization energy from the 2s state. We make
a special note that $\beta_{\rm H}$ should be evaluated at the radiation
temperature, in contrast to $\alpha_{\rm H}$ which should be evaluated at
$T_{\rm M}$. While this does not represent the full physics of the
recombination and photonionization rates in a blackbody ambient radiation
field (e.g., Chluba et al., 2007; Grin & Hirata, 2010), it captures the
leading order dependence more correctly (Chluba et al., 2015). Thus, the
recombination rate coefficient is sensitive to the matter temperature, such
that higher $T_{\rm M}$ will lead to a higher frozen out fraction of $X_{\rm
e}$.

In practice, we evolve the electron fraction using Eq. (16) from a redshift of
$z=1500$ down to $z=50$. At this point, we utilize the reionization module of
CosmoTherm to further evolve $X_{\rm e}$, performing some simple modeling of
the first sources of hard photons, important around the epoch of cosmic dawn.
A full description of this module and its assumptions is given in Acharya et
al. (2022).

Equation (16) can be modified to also include the effects of collisional
ionizations within the gas. This form of ionization typically only becomes
important in extreme scenarios when $T_{\rm M}\gtrsim 10^{4}$ K (e.g., Chluba
et al., 2015). As we will see, this only happens in rather unphysical setups,
and we neglect these corrections from our analytic description. We make
further comments regarding this in Section 5.

Finally, source terms with significant emission above $E_{\rm inj}\gtrsim
10.2\,{\rm eV}$, will induce direct excitations and ionizations to the gas.
The usual partition in this case is that roughly $1/3$ of the energy injected
above Ly-$\alpha$ frequencies will contribute to direct ionizations of the
medium (Chen & Kamionkowski, 2004), with the rest of the energy contributing
to excitations and heating. For most of the injection scenarios considered in
this work, photons above this energy are never injected, allowing us to
neglect direct ionizations and focus solely on soft photon heating. However,
photon injections from decaying cosmic strings do possess a small fraction of
these hard photons, which we discuss further in Sec. 6.

Ultimately, the soft photon heating effect can be modeled by solving the
coupled system of evolution equations defined by Eqs. (7), (13), and (16).
Without the presence of a source term ($S_{\rm inj}=0$) our solutions
reproduce the standard cosmological evolution, which includes free-free
absorption and emission off of the CMB itself (this effect has always been
included in CosmoTherm). However, the inclusion of a soft photon background
can dramatically modify the matter temperature, which in turn induces strong
deviations to the expected $21$cm differential brightness temperature. In the
cases that follow, we have checked that this simple setup faithfully captures
the features of a more exact computation such as can be performed using
CosmoTherm.

##  4 Matter heating rates and the CMB

Before we consider exotic injections of a soft photon background, it is
instructive to understand the relative importance of the individual heating
rates in the standard scenario when only the CMB is present. The evolution of
the matter temperature proceeds as usual following the prescription set out in
Eq. (13), specifically, through Hubble expansion, Compton scattering, and
free-free absorption and emission. In Fig. 1 we show the redshift dependence
of each of these effects, where dashed lines indicate times when a particular
interaction heats the gas, while solid lines highlight periods of gas cooling.

Figure 1: Relative gas heating rates per logarithmic redshift interval in the
presence of the CMB. The Hubble, Compton, and Bremsstrahlung curves correspond
to the first, second, and third terms respectively in Eq. (13). Solid lines
indicate cooling of the ambient hydrogen gas, while the dashed correspond to
heating.

Focusing first on the Hubble and Compton contours, we see that at high
redshifts ($z\gtrsim 150$), their respective cooling and heating rates almost
completely compensate each other. It is during this period that residual
Compton scatterings between the electrons and background photons source a well
known (negative) $y$-type distortion, as the electrons continuously sap energy
from the CMB. At $z\lesssim 150$, the Compton interaction rate drops rapidly
and the matter temperature begins to evolve as $T_{\rm M}\propto(1+z)^{2}$
until reionization turns on at $z\simeq\mathcal{O}(10)$. At this point, the
matter is rapidly heated (primarily by X-ray sources), leading to the small
bump in the Compton heating rate around $z\simeq 20$ seen in Fig. 1. Shortly
afterwards, the gas reaches temperatures $T_{\rm M}>T_{\rm CMB}$, and Compton
cooling begins to occur. During this final phase, the electrons begin up-
scattering the background photons, producing a small (positive) $y$-type
distortion.

Figure 2: Left: Redshift evolution of the radiation, matter, and spin
temperatures in a standard $\Lambda$CDM scenario with no extra soft photon
background. Right: the (global) differential brightness temperature. The
amplitude of the absorption trough scales with $\Delta T_{\rm b}\propto x_{\rm
H}(1-T_{\rm R}/T_{\rm S})$.

Turning our attention now to the free-free effects, the first thing to notice
is that Bremsstrahlung typically plays a strongly subdominant role in the
evolution of $T_{\rm M}$ without the addition of an extra soft photon
background. At redshifts before recombination, the ionization fraction is
$X_{\rm e}\simeq 1$ and low frequency photons are rapidly absorbed and emitted
by the plasma. Note that free-free absorption will cause CMB photons (with
temperature $T_{\rm abs}=T_{\rm CMB}$) to be absorbed by the gas, while the
emission process will liberate a blackbody photon instead at the matter
temperature ($T_{\rm emit}=T_{\rm M}$). Thus, the small deviation of matter
and photon temperatures at early times leads to the weak (but steadily
growing) soft photon heating at $z\gtrsim 1000$444An examination of Eq. (15)
in the $\Delta n(x)=0$ regime also indicates that heating occurs whenever
$T_{\rm M}\lesssim T_{\rm CMB}$.. After recombination, $X_{\rm e}$ drops
rapidly which severely damps the free-free effects until reionization can
rejuvenate the fraction of free electrons. At this point, a familiar situation
occurs: soft photon heating boosts the gas temperature at an ever-increasing
rate, until $T_{\rm M}>T_{\rm CMB}$, at which point free-free emission begins
to provide a cooling contribution to $T_{\rm M}$.

The magenta contour represents the contribution to gas heating from the
standard reionization module implemented through CosmoTherm. Thus far we have
not accounted for any feedback from soft photon heating on the modeling
assumptions of the first astrophysical sources, though this is a rich avenue
for further study. As a result, in the following sections the heating curve
from reionization will not be altered between the different injection
scenarios.

The left panel of Fig. 2 showcases the familiar evolution of the radiation,
spin, and matter temperatures in this case, while the right panel displays the
predicted shape and amplitude of the differential brightness temperature,
$\Delta T_{\rm b}$. At the moment, there is no undisputed detection of the
global 21cm signal, though work by the EDGES and SARAS-3 collaborations have
provided us with a loose lower limit of $\Delta T_{\rm b}\geq-500$ mK (Bowman
et al., 2018; Singh et al., 2022). Current literature (Feng & Holder, 2018;
Ewall-Wice et al., 2018; Fialkov & Barkana, 2019) suggests that the presence
of a radio background at cosmic dawn will strongly enhance the depth of the
$\Delta T_{\rm b}$ absorption trough, making the signal much easier to detect.
Stringent constraints have been placed on models which enhance the background
temperature at $21$cm based on this lower limit (Brandenberger et al., 2019;
Mittal et al., 2021; Caputo et al., 2023). In the following two sections, we
will show how induced soft photon heating from sufficiently steep and bright
radio backgrounds can dramatically weaken these constraints, in some cases
rendering the global $21$cm signal even more difficult to observe.

##  5 Quasi-instantaneous injections

In this work, we consider the impact of extra soft photon backgrounds by
introducing a source term, $S_{\rm inj}={\,\rm d}n_{\rm inj}/{\,\rm d}\tau$
into the photon evolution equation. Generally speaking, the injection source
term can take any number of creative forms, of which several have been
extensively studied using CosmoTherm (e.g. in Chluba, 2015; Bolliet et al.,
2020). In Acharya et al. (2023) and Acharya et al. (2024), we presented a
dedicated numerical study of this effect by considering scenarios of decaying
dark matter particles, and those with a power-law time-dependence. We would
like to stress that some level of soft photon heating occurs even when no
extra photons are introduced, as highlighted in the previous section. In
addition, even in standard scenarios one can expect a noticeable amount of
soft photon production, e.g., from early structure formation with radio-loud
galaxies, an avenue that we plan to pursue in the future. As we show in this
section, the presence of a soft photon background can have a dramatic impact
on various cosmological observables.

To illustrate this effect, we begin by studying scenarios in which a soft
photon background is introduced in a quasi-instantaneous manner. This is meant
more to build intuition in understanding how the various heating rates of the
medium are affected, rather than to present a realistic, physically motivated
scenario. To achieve this, we present results based on four representative
injection redshifts, $z_{\rm inj}=3000,\,1000,\,500,\,100$, with each of these
highlighting a qualitatively different epoch in the thermal history. The
$z_{\rm inj}=3000$ and $z_{\rm inj}=1000$ scenarios capture the heating
dynamics when injection takes place before and during recombination. The dark
ages injection at $z_{\rm inj}=500$ provides insight into a time when $X_{\rm
e}$ is low, while at $z_{\rm inj}=100$ we can study what happens when soft
photons are sourced after $T_{\rm M}$ and $T_{\rm CMB}$ fully decouple.

We generically assume the soft photons follow a power law with spectral index
given by $\gamma$. To better understand the importance of this spectral slope
we compare and contrast between three choices of $\gamma$ at each injection
redshift. Our choices are $\gamma=3.0$, mimicking a free-free emission type
spectrum, $\gamma=3.6$ which is a typical synchrotron-type injection, and
$\gamma=3.3$ as an ad-hoc choice to understand intermediate slope power laws.
We define the amplitude of the soft photon background relative to the CMB as
$\Delta\rho/\rho$, and fix this quantity to be $10^{-6}$ for most of the
examples we showcase. At late times ($z\lesssim 10^{3}$), it is in principle
possible to take $\Delta\rho/\rho\simeq 6\times 10^{-5}$ (or even somewhat
larger) without violating constraints from CMB spectral distortions, though
these large amplitude injections over a quasi-instantaneous timescale lead to
other complications that we discuss in the following subsections. We have
found that $\Delta\rho/\rho=10^{-6}$ provides a good benchmark to highlight
the salient features of soft photon heating.

In the following subsections, we present a general prescription for the
injection of soft photon backgrounds, and present our main results for quasi-
instantaneous injections. After discussing these results, we highlight some
subtleties related to the rather unphysical nature of quasi-instantaneous
injections. We conclude by describing changes to the formalism when the soft
photon background is injected solely in the Rayleigh-Jeans tail ($x\ll 1$), a
scenario with specific applications to the observed radio synchrotron
background.

###  5.1 Generic soft photon backgrounds

The distinction between “soft” and “hard” photons is that soft photons should
not be capable of exciting or ionizing neutral hydrogen in the ground state.
Therefore, the injections that we perform are limited to frequencies where
$x<x_{\alpha}=E_{\alpha}/k_{\rm b}T_{\rm CMB}(z)$ to ensure no hard heating
takes place. Recalling that the intensity spectrum of an injected background
is related to its occupation number by Eq. (5a) and assuming that the
injection spectrum does not depend explicitly on time, we can parameterize the
soft photon occupation number by

$\displaystyle n_{\rm SPB}(\nu,\tau)=\zeta(\tau)\,f(\nu).$ (19)

For quasi-instantaneous injections, we choose
$\zeta(\tau)=A\,\Theta(\tau-\tau_{\rm inj})$, where $\Theta$ is the Heaviside
step function, $\tau_{\rm inj}$ the time of injection, and $A$ is the
dimensionless amplitude of the background at $\tau_{\rm inj}$ (and $\nu_{\rm
cut}$, defined below). This initial amplitude can be used to normalize the
total energy injection, as we will see shortly. The frequency dependence is
chosen to replicate a power law background,

$\displaystyle f(\nu)=\left(\frac{\nu}{\nu_{\rm cut}}\right)^{-\gamma}\,{\rm
e}^{-\nu/\nu_{\rm cut}},$ (20)

where $\gamma<4$ is the spectral index (as described earlier), and $\nu_{\rm
cut}$ is a high frequency cutoff which we impose to prevent the production of
hard photons. A cutoff such as this could reasonably be expected in certain
physical production processes (e.g. from the decay of a dark matter particle).
In general, the $\zeta(\tau)$ and $f(\nu)$ terms can take any form, but we
choose to focus on power law backgrounds for this work. The source term for
our quasi-instantaneous injections is then given by

$\displaystyle S_{\rm inj}(x,\tau)=A\,\left(\frac{x}{x_{\rm
cut}}\right)^{-\gamma}\,{\rm e}^{-x/x_{\rm cut}}\,\delta(\tau-\tau_{\rm
inj}).$ (21)

Here, we have transformed from $\nu\rightarrow x$. Additionally, $x_{\rm
cut}=E_{\rm cut}/k_{\rm b}T_{\rm CMB}$ is the dimensionless cutoff scale which
inherits a redshift dependence from $T_{\rm CMB}\propto(1+z)$. One can then
define the rate of comoving energy injection from this source term as

$\displaystyle\frac{1}{a^{4}}\frac{{\,\rm d}(a^{4}\rho_{\rm inj})}{{\,\rm
d}\tau}$ $\displaystyle=\frac{\rho_{\rm
CMB}}{\pi^{4}/15}\int_{0}^{\infty}{\,\rm d}x\,x^{3}\,S_{\rm inj}(x,\tau),$
$\displaystyle=\frac{\rho_{\rm CMB}}{\pi^{4}/15}\,A\,x_{\rm
cut}^{4}\,\Gamma(4-\gamma)\,\delta(\tau-\tau_{\rm inj})$ (22)

Note that this is the total energy injection rate of photons into the
background, which cannot be immediately identified with the heating rate of
the hydrogen gas (for this a Compton or Bremsstahlung interaction has to
occur). From here we can define the total fractional energy density increase
relative to the CMB, $\Delta\rho/\rho$. For injections taking place between an
initial and final redshift ($\tau(z_{\rm i})=0$ and $\tau(z_{\rm f})=\tau_{\rm
f}$), this is given by

$\displaystyle\frac{\Delta\rho}{\rho}=\int_{0}^{\tau_{\rm f}}{\,\rm
d}\tau\,\frac{1}{a^{4}\rho_{\rm CMB}}\frac{{\,\rm d}(a^{4}\rho_{\rm
inj})}{{\,\rm d}\tau}.$ (23)

For our quasi-instantaneous source term, assuming that $\tau_{\rm
inj}<\tau_{\rm f}$, we find a simple analytic form for the normalization
constant

$\displaystyle A$
$\displaystyle=\frac{\pi^{4}}{15\,\Gamma(4-\gamma)}\,\left(\frac{k_{\rm
b}T_{\rm CMB}(\tau_{\rm inj})}{E_{\rm
cut}}\right)^{4}\,\frac{\Delta\rho}{\rho}.$ (24)

For more general time-dependence, one can simply compute the normalization
condition using Eq. (23), where now the explicit time-dependence of ${\,\rm
d}\zeta(\tau)/{\,\rm d}\tau$ has to be specified. The form given in Eq. (24)
is particularly useful when one wishes to fix the fractional energy injection
$\Delta\rho/\rho$ of a background.

We must now specify some parameter combinations in order to illustrate various
interesting cases. We set the high frequency cutoff to be $E_{\rm cut}\simeq
0.235$ eV, which corresponds to $x_{\rm cut}(z=0)=10^{3}$, leaving us free
from hard photons. For the injection redshifts and spectral indices, we
consider $z_{\rm inj}=3000,\,1000,\,500,\,100$ and $\gamma=3.0,\,3.3,$ and
$3.6$ in what follows. As discussed above, we fix $\Delta\rho/\rho=10^{-6}$
and set the amplitude $A$ according to Eq. (24) for different spectral indices
and injection redshifts. This allows us to showcase the importance of soft
photon heating without violating constraints from CMB spectral distortions or
anisotropy measurements. This also has the side effect of greatly reducing the
amplitude of the soft photon background for injections occurring at lower
$z_{\rm inj}$, a point which we return to at the end of this subsection.

In Fig. 3, we present a detailed look at the heating rates from single
injections of a synchrotron ($\gamma=3.6$) spectrum at each of our target
redshifts. Starting from the top left, as expected the heating rates look very
similar to the vanilla $\Lambda$CDM scenario (Fig. 1), with the exception of
the spike around $z_{\rm inj}=3000$. At this high redshift, the pre-
recombination plasma is highly efficient at absorbing low frequency photons
(through free-free), while Compton scatterings quickly redistribute this newly
absorbed energy back into the photon bath, leading to no net deviation between
the electron and photon temperatures (though this causes a $y$-distortion with
$y\simeq\frac{1}{4}\Delta\rho/\rho$).

For injections taking place at $z_{\rm inj}\leq 1000$, the situation begins to
change quite dramatically. At $z_{\rm inj}=1000,\,500$ after an initial boost
in $T_{\rm M}$ near the injection point through soft photon heating, the
relatively efficient Compton interactions quickly bring the matter temperature
back to $\simeq T_{\rm CMB}$. However, the residual free-free heating rate
reaches a steady-state that is significantly enhanced compared to pre-
recombination injections and even exceeds the Compton heating for the case
with $z_{\rm inj}=500$. For $z_{\rm inj}=100$, Compton interactions have
almost completely frozen out and soft photon heating leads to a permanent
boost in the matter temperature when compared to the case of no injections.

Figure 3: Relative heating rates for quasi-instantaneous injections at a
variety of redshifts with spectral index $\gamma=3.6$. For pre-recombination
injections, the late-time heating rates roughly match those in Fig. 1. Post-
recombination injections lead to greatly enhanced Bremsstrahlung (soft photon)
heating. When free-free heating dominates over the reionization contour at
cosmic dawn ($z\simeq 20$), large corrections to the induced $\Delta T_{\rm
b}$ need to be taken into account.

Figure 4: Evolution of the matter temperature for free-free, synchrotron, and
$\gamma=3.3$ spectral indices. The inefficiency of Compton scattering at
$z\lesssim 150$ leads to a large offset in the matter temperature for
synchrotron-type injections. Deviations from the no injection scenario at
$z\simeq 20$ indicate the presence of soft photon heating relevant to global
$21$cm observations.

Figure 5: Left panels: A breakdown of the radiation, matter, and spin
temperatures for different injection redshifts spectral indices. Right panels:
The differential brightness temperature in the presence of each type of soft
photon background, both with soft photon heating, and by neglecting it by
setting ${\,\rm d}T_{\rm ff}/{\,\rm d}z=0$ in Eq. (13). When a soft photon
spectrum permeates the background, soft photon heating can only be neglected
if the background is sourced in the pre-recombination epoch, or if the
spectral index is $\gamma\leq 3.0$. In all other circumstances, heating
through free-free absorption will generically source large corrections to the
differential brightness temperature at cosmic dawn.

Figure 6: The same as Figure 5, but for $z_{\rm inj}=100$. The unphysical
spike comes from the non-adiabatic injection and immediate reprocessing of the
soft photon background.

Most important, however, is the fact that the secondary Bremsstrahlung bump
around redshift of $z\simeq 20$ can dominate over the heating sourced by
reionization. For pre-recombination injections, the free-free optical depth is
extremely high for low-frequency photons. This implies that the tail of an
injected soft photon background is quickly truncated at a relatively high
frequencies. For injections after recombination, this initial truncation takes
place at much lower frequencies. When reionization begins to turn on, $X_{\rm
e}$ rapidly increases and a second period of soft photon heating occurs as
free-free absorption once again becomes efficient. Therefore, a generic
consequence of post-recombination soft photon injections is this two stage
heating process, once near $z_{\rm inj}$, and once at the onset of
reionization, around cosmic dawn.

The plots in Fig. 4 show the deviation in the matter temperature over a wide
range of redshifts for $\gamma=3.0,\,3.3,\,$ and $3.6$ with different
injection times. The initial spike in the $z=1000,\,500$ scenarios happens as
a result of the quasi-instantaneous injection, after which $T_{\rm M}$ quickly
relaxes back to the CMB temperature due to Compton interactions. For $z_{\rm
inj}=100$, the initial temperature jump remains frozen in as the hydrogen
continues to cool adiabatically. The sharp increase in $T_{\rm M}$ around
$z\simeq 20$ comes from our reionization module, which can also trigger a
second phase of soft photon heating for the post-recombination injections as
mentioned above.

We present a detailed breakdown of the evolution of radiation, matter, and
spin temperatures for our different case studies in the left hand panels of
Figs. 5 and 6, focusing on the range $1\leq z\leq 200$. The amplitude and time
dependence of the differential brightness temperature is governed by the
contrast between the solid ($T_{\rm R}$) and dotted ($T_{\rm S}$) lines. We
present our computed $\Delta T_{\rm b}$ curves in the right hand panels,
comparing both with the vanilla $\Lambda$CDM setup with no soft photon
injections, as well as the predicted curves expected when one neglects soft
photon heating in the presence of such a background. These “No SPH” curves are
computed by simply setting ${\,\rm d}T_{\rm ff}/{\,\rm d}z=0$ in Eq. (13).

It is immediately evident from the first plot that for $z_{\rm inj}=3000$,
soft photon heating has a negligible impact on the late time evolution of the
matter temperature, as it does not deviate from the no-injection case
presented in Fig. 2. The radiation temperature at $21$cm frequencies begins to
receive a boost around $z=30$ due to the fact that initial truncation at
$z_{\rm inj}$ has redshifted down to the relevant wavelengths. Injections at
$z\gg 3000$ will not see any such bump, as the entirety of the injected
background at those redshifts would be quickly absorbed. The lack of free-free
heating is also evident in the top-right panel of Fig. 5, where the
differential brightness temperature with and without a ${\,\rm d}T_{\rm
ff}/{\,\rm d}z$ term completely overlap. This result matches nicely with the
conclusions of e.g. Feng & Holder (2018), who showed that large enhancements
to $\Delta T_{\rm b}$ are possible in the presence of a radio background.

The remainder of the plots in Figs. 5 and 6 tell a qualitatively different
story to what previous literature (Feng & Holder, 2018; Ewall-Wice et al.,
2018; Fialkov & Barkana, 2019) originally concluded. Deviations of the matter
temperature away from $\Lambda$CDM are observed due to the second phase of
soft photon heating for $z_{\rm inj}\leq 1000$. Extra thermal energy in the
gas due to soft photon heating acts to pump the occupation of the spin triplet
state, giving an additional boost to $T_{\rm S}$ at cosmic dawn. This increase
in spin temperature leads to a net reduction in the amplitude of $\Delta
T_{\rm b}\propto x_{\rm H}(1-T_{\rm R}/T_{\rm S})$, which in most
circumstances cannot reasonably be neglected. The centre of the $\Delta T_{\rm
b}$ absorption trough can also experience a slight offset depending on the
exact redshift that the second phase of soft photon heating starts to
dominate.

Plots in the right hand column of Figs. 5 and 6 show the magnitude of these
corrections for single injections with different spectral indices and
injection redshifts. It is clear that the effect is most prominent for
backgrounds with a steeper spectral index, produced after recombination. This
is intuitive, as it is due to the fact that a higher relative fraction of the
soft photon background is absorbed through free-free heating when the spectrum
is steeper.

For a free-free power-law ($\gamma=3.0$) injection, soft photon heating is
rather negligible, while for a synchrotron-like ($\gamma=3.6$) spectra, the
effect can reduce the amplitude of $\Delta T_{\rm b}$ even below the predicted
$\Lambda$CDM value, making it much harder to detect. The synchrotron case is
particularly interesting, due to indications from ARCADE-2 (Fixsen et al.,
2011) and LWA (Dowell & Taylor, 2018) that an unexplained radio synchrotron
background may be present. If such a background does exist and was produced at
$z\gtrsim 20$, our calculations indicate that the amplitude of the global
$21$cm differential brightness temperature could be far less than the standard
prediction ($|\Delta T_{\rm b}|\ll 100$ mK).

Finally, we should address why the amplitude of energy injections seems to be
decreasing with lower $z_{\rm inj}$. This can be seen most easily by looking
at the peak heating rates at $z_{\rm inj}$ in Fig. 3, or the steadily
decreasing amplitudes of $\Delta T_{\rm b}$ or $T_{\rm R}$ for $z_{\rm
inj}=1000,\,500,$ and $100$ in Figs. 5 and 6. This stems from the fact that we
have chosen to fix the ratio $\Delta\rho/\rho$, which implies that the initial
amplitude of soft photon backgrounds injected at later times is suppressed, as
can be seen in Eq. (24). If one instead chooses to fix $\Delta\rho$, this
subtlety can be avoided, but comes at the cost of introducing a more
complicated mapping to e.g. CMB spectral distortion constraints which are
sensitive to the fractional energy injection relative to the CMB.

###  5.2 Energetic photon backgrounds and collisional ionizations

Before moving on we would like to highlight an additional complication that
can arise when performing quasi-instantaneous injections with a brighter
background spectrum (i.e., a larger amplitude $A$). In the upper two panels of
Fig. 7, we show the matter temperature and ionization fraction evolution with
redshift for a single injection at $z_{\rm inj}=500$, and fractional energy
density $\Delta\rho/\rho=6\times 10^{-5}$. This value of $\Delta\rho/\rho$ was
chosen as it saturates the bound on exotic energy injections set by the
COBE/FIRAS measurements on CMB spectral distortions.

From the the upper left panel, we can see that for $\gamma=3.6$, the initial
spike in the matter temperature peaks at $T_{\rm M}(z_{\rm inj})\simeq 3\times
10^{4}$ K. As mentioned earlier, the threshold for when collisional
ionizations become important is roughly $T_{\rm M}\gtrsim 10^{4}$ (e.g.,
Chluba et al., 2015), which we clearly exceed for a brief time in this
scenario. The result of crossing this threshold is a simultaneous spike in the
$X_{\rm e}$ fraction, which soon relaxes to a new steady state value a
sizeable fraction higher than the standard $\Lambda$CDM value. For reference
we plot the $\Delta\rho/\rho=10^{-6}$ results in the bottom two panels,
showing that such a large perturbation in $X_{\rm e}$ is avoided for lower
energy injections.

While it is obviously important to not disturb the ionization fraction of the
universe in such a dramatic fashion, we would like to stress that this
situation is an artifact of the quasi-instantaneous nature of these soft
photon injections used in the illustrations here. For more physical scenarios,
such as the continuous injections we consider in Section 6, the soft photon
background builds up more slowly over time. This avoids large instantaneous
spikes in $T_{\rm M}$ by smoothing out the initial phase of soft photon
heating over a large range of redshifts. Therefore, it is reasonable to take
$\Delta\rho/\rho=6\times 10^{-5}$ for continuous injection scenarios without
violating constraints on $X_{\rm e}$. We note that in the post-recombination
era, this bound can still be exceeded as not all the injected energy is
reprocessed into $y$-type distortions. General scenarios therefore required a
detailed parameter study (Acharya et al., 2024).

Figure 7: Top panels: Evolution of the matter temperature and ionization
fraction over a wide range of redshifts for a single injection at $z_{\rm
inj}$ with $\Delta\rho/\rho=6\times 10^{-5}$. Bottom panels: Same as the top
but for a less energetic spectrum with $\Delta\rho/\rho=10^{-6}$. For the more
energetic case, the matter temperature briefly spikes to $T_{\rm M}\gtrsim
10^{4}$ K, inducing efficient collisional ionizations that strongly perturb
the $X_{\rm e}$ history.

###  5.3 A radio synchrotron background source term

In the previous subsections, we studied the impacts of soft photon heating
when one fixes the fractional energy release ($\Delta\rho/\rho)$ from an
injected background. This was useful to understand the relevance of different
spectral indices and injection redshifts. Here, we briefly illustrate how to
study these effects for a specified form of a radio synchrotron background,
i.e., when the amplitude $A$ of the background is fixed a priori. This is
motivated by measurements from the ARCADE-2 and LWA experiments which indicate
that an unexplained radio synchrotron background may exist.

Radio backgrounds such as this appear as deviations from the CMB spectrum in
the Rayleigh-Jeans tail ($x<1$). They are typically parameterized by a radio
brightness temperature, $T_{\rm amp}(\tau)$, pivot frequency, $\nu_{0}$, and a
(temperature) spectral index, $\beta$, in the form

$\displaystyle T_{\rm RSB}(\nu,\tau)=T_{\rm
amp}(\tau)\left(\frac{\nu}{\nu_{0}}\right)^{-\beta}.$ (25)

We note that $\gamma=\beta+1$ relates the temperature spectral index with the
$\gamma$ used in the previous subsections.The occupation number of the radio
background can be related to this parameterization through

$\displaystyle n_{\rm RSB}(x,\tau)=\frac{1}{x}\,\frac{T_{\rm
RSB}(x,\tau)}{T_{\rm CMB}(\tau)}.$ (26)

If we once again assume that the radio synchrotron background is injected
quasi-instantaneously at some $\tau_{\rm inj}$, we find a source term similar
to Eq. (20), namely

$\displaystyle S_{\rm RSB}=\tilde{A}\,\left(\frac{x}{x_{\rm
cut}}\right)^{-\gamma}\,{\rm e}^{-x/x_{\rm cut}}\,\delta(\tau-\tau_{\rm
inj}).$ (27)

Here, $\tilde{A}=(k_{\rm b}T_{\rm amp}/h\nu_{0})\,(x_{\rm
cut}/x_{0})^{-\gamma}$, which allows us to make use of the results from Sect.
5.1 with $A\rightarrow\tilde{A}$. Again, we modulate the spectrum by imposing
an exponential cutoff in frequency for $x_{\rm cut}<1$ to ensure we remain in
the Rayleigh-Jeans tail.

Figure 8: An overview of the heating mechanisms at play in the decaying dark
matter scenario. The specific parameters that we choose correspond to a model
which provides a good fit to the observed radio synchrotron background, as
discussed in Acharya et al. (2023). A decay rate of $\Gamma=2\times
10^{-14}\,{\rm s}^{-1}$ corresponds roughly to a redshift of $z\simeq 420$,
after which soft photon production no longer occurs. Qualitatively, the
results are similar to the case of quasi-instantaneous injection at $z_{\rm
inj}=500$, which we considered above.

Figure 9: The same breakdown as in Fig. 8, but for a superconducting cosmic
string scenario with $G\mu=10^{-12}$ and $\mathcal{I}=10^{6}$ GeV. In contrast
to the other cases considered in this work, a superconducting cosmic string
network is capable of producing an energetic spectrum of soft photons at all
redshifts, leading to consistent free-free heating. The parameters chosen in
this depiction represent a model that is also quite a convincing fit to the
ARCADE-2 and LWA data regarding the radio synchrotron background (Cyr et al.,
2023a).

This injection model consists of five parameters $\tilde{A}$, $\gamma$,
$x_{0}$, $x_{\rm cut}(<1)$ and $\tau_{\rm inj}$. As a simple example, let us
return to the radio synchrotron background that has been reported by the
ARCADE-2 and LWA experiments. In Eq. (6) of Acharya et al. (2023), we explored
a power-law fit to this background, which can be translated into our
parameterization yielding $\gamma=3.555$, $x_{0}(z=0)\simeq 0.0176$, and
$\tilde{A}\simeq 25.6$ in the case where $x_{0}=x_{\rm cut}$.

Strictly speaking, the injection source term given in Eq. (27) is valid in the
limit $x_{\rm cut}<1$, and useful when a radio brightness temperature, $T_{\rm
RSB}(\nu,\tau)$ is given. In contrast, the source given in Eq. (21) is valid
for $E_{\rm cut}\lesssim 10$ eV. The accompanying Jupyter Notebook
incorporates both of the formalisms for these source terms and allows one to
quickly and easily understand which types of low frequency backgrounds will
yield appreciable soft photon heating.

##  6 Continuous Injections

While quasi-instantaneous injections of a soft photon background provided us
with a useful tool to understand the underlying dynamics of free-free heating,
they are perhaps not the most physically relevant scenarios. To supplement our
understanding, we will briefly consider two scenarios in which a low-frequency
photon spectrum is built up over time.

###  6.1 Decaying dark matter

The first scenario we wish to consider is that of a decaying dark matter
particle (DP). In this setup, we imagine that there exists some fraction
$f_{\rm dm}$ of the dark matter which is unstable and decays into a broad
spectrum of photons555The role of dark matter decays into two photons has been
extensively studied in Bolliet et al. (2020).. The energy injection rate is
given by

$\displaystyle\frac{{\,\rm d}\rho_{\rm DP}}{{\,\rm d}t}=f_{\rm dm}\,\rho_{\rm
dm}\,\Gamma\,{\rm e}^{-\Gamma t},$ (28)

where $\rho_{\rm dm}$ is the dark matter energy density, and $\Gamma$ is the
decay rate (or inverse lifetime) of the unstable dark matter particle. Similar
to the previous section, the spectrum of decay products is assumed to follow a
power law with a high frequency cutoff,

$\displaystyle I_{\rm DP}=I_{\rm 0,DP}\,x^{3-\gamma}{\rm e}^{-x/x_{\rm inj}}$
(29)

with $x_{\rm inj}=x_{\rm inj,0}/(1+z)$. Studying this scenario allows us to
directly extend the results from the previous section to a setup with a non-
trivial time dependence. To fix the total energy release $\Delta\rho/\rho$
(and thereby determine $f_{\rm dm}$), we have to compute the normalization of
the injection spectrum (by integrating over $x^{3}{\,\rm d}x$) and also the
total energy release integral to determine the constant $f_{\rm dm}$. This
process can be carried out numerically. Further details of this model can be
found in Acharya et al. (2023), where we performed an in-depth numerical
analysis using CosmoTherm.

To illustrate the effects of soft photon heating, we select the parameters
$\gamma=3.6$, $\Delta\rho/\rho=10^{-6}$, $x_{\rm inj,0}=4035$, and
$\Gamma=2\times 10^{-14}\,{\rm s}^{-1}$ which corresponds to a decay redshift
of $z_{\rm dec}\simeq 420$, after which very few soft photons are produced.
This particular combination of parameters is chosen such that no hard photons
are emitted, and because the part of the spectrum that survives to late times
offers a good fit to the ARCADE-2 radio synchrotron background data.

A breakdown of the decaying particle scenario is presented in Fig. 8. The top
left panel shows that at early times, the soft photon injection follows a
power law in redshift as indicated by the steadily increasing Bremsstrahlung
heating. This heating rate is slightly reduced at recombination when the free
electron fraction drops, but continues heating at a roughly constant rate
until $z\simeq z_{\rm dec}$ when the majority of the dark matter particles
have decayed away. At this point, most of the soft photon injection stops, and
the situation becomes qualitatively very similar to the single injection
scenario with $z_{\rm inj}=500$ as presented in Fig. 3.

The remainder of the plots in Fig. 8 also tell a similar story to what we saw
with quasi-instantaneous injections near $z=500$, showing a strong departure
in both the spin and $\Delta T_{\rm b}$ temperatures from the $\Lambda$CDM
case. This leads us to the somewhat obvious conclusion that the most potent
effects of soft photon heating on $21$cm observations come from late time
free-free absorptions. Therefore, if a soft photon source turns off at some
point (as is the case in both the quasi-instantaneous and decaying particle
scenarios), the relevant observational effects come from the two heating bumps
at $z_{\rm inj}$ or $z_{\rm dec}$, as well as near cosmic dawn. For the
decaying particle scenario discussed here we also explicitly compared our
simplified treatment with the full numerical result computed using CosmoTherm,
finding very good agreement.

###  6.2 Superconducting cosmic strings

A setup in which soft photons are produced at all redshifts is given by the
superconducting cosmic string scenario, originally studied in Cyr et al.
(2023a, b). In this picture, cosmic strings are formed at the interface of a
cosmological phase transition at an arbitrarily high redshift, and endowed
through subsequent symmetry breakings with superconductive properties (see
Witten (1985); Ostriker et al. (1986) for seminal examples). On relatively
short timescales after formation of the strings, a scaling network is
achieved, with $\mathcal{O}(1)$ long strings running through the Hubble patch
at any given redshift (Kibble, 1980; Vilenkin & Shellard, 2000).

Alongside these long strings, a distribution of smaller, sub-Hubble loops is
constantly sourced as the network evolves (Vanchurin et al., 2006; Martins &
Shellard, 2006; Ringeval et al., 2007; Blanco-Pillado et al., 2014). These
smaller loops oscillate and are thought to slowly decay away into both
gravitational waves (Vachaspati & Vilenkin, 1985) and electromagnetic
radiation (Vilenkin & Vachaspati, 1987; Vachaspati, 2008; Cai et al., 2012).
The fraction of energy going into gravitational waves is controlled by the so-
called string tension parameter $G\mu$, where $G$ is Newton’s constant and
$\mu\sim\eta^{2}$ where $\eta$ is the energy scale of the phase transition
which produced the cosmic strings. Similarly, the strength of electromagnetic
emission is controlled by the amplitude of the current, $\mathcal{I}$, running
along the string.

The string-loop produced gravitational waves provide a possible explanation to
the recently detected stochastic background of detected by the pulsar timing
array consortium (Afzal et al., 2023; Antoniadis et al., 2023; Ellis et al.,
2024). Interestingly, the spectrum of electromagnetic radiation released
contains a large fraction of soft photons as can be seen in Fig. 12 of Cyr et
al. (2023b). Importantly, since a string network such as this would be
expected to survive until the present day, the persistent decay of cosmic
string loops ensures that new soft photons are constantly being pumped into
the background. Therefore, the soft photon heating process is extended
gradually over a wide range of redshifts. Importantly, the second bump in the
free-free heating rate at cosmic dawn is still present (this is simply due to
an increase in $X_{\rm e}$ around this time).

The details of the exact injection rate and fractional $\Delta\rho/\rho$ are
somewhat complicated, so we refer the interested reader to our dedicated
papers on the subject. In Cyr et al. (2023a) we found that a string tension of
$G\mu\simeq 10^{-12}$ and current of $\mathcal{I}\simeq 10^{6}$ GeV666Forgive
our diversion to natural units for this brief explanation. provided an
impressive fit to the radio synchrotron background data, similar to the
decaying dark matter scenario discussed above. We study the same model here in
Fig. 9.

As expected in the upper left plot of this figure, the free-free heating rate
is remarkably constant at early times, suffering only a small step down in
amplitude over the course of recombination. This rather democratic heating can
be appreciated by comparing the $T_{\rm M}$ plot in the upper right with that
of the decaying particle scenario. In the decaying particle setup, $T_{\rm M}$
is boosted around $z=400$, before soft photon heating turns off and the
temperature decays adiabatically with the usual $(1+z)^{2}$ scaling. In
contrast, the consistent heating from cosmic strings never allows this Hubble
cooling to become dominant, and the matter temperature at all times until
cosmic dawn follows a more photonic $(1+z)$ scaling. This is also clear by
comparing the slope of the dashed red curve in the bottom right plot with that
of the “No injection” radiation temperature.

As expected, soft photon heating plays a major role in dampening the amplitude
of $\Delta T_{\rm b}$ for this model. In fact, prior to our first papers
discussing soft photon heating, this model of superconducting strings was
ruled out as it produced a $\Delta T_{\rm b}$ far in excess of the EDGES lower
bound of $-500$ mK (Brandenberger et al., 2019). The fact that soft photon
heating has relaxed the parameter space of these models so significantly is
quite exciting in light of the fact that they are capable of providing a very
convincing fit to the observed radio synchrotron background (Cyr et al.,
2023a).

##  7 The 21cm power spectrum

In this section, we demonstrate the effects of soft photon heating on the 21cm
fluctuation power spectrum. Substituting the expression for $T_{\rm S}$ in
expression of $\Delta T_{\rm b}$ as in Sec. 2, we find

$\displaystyle\Delta T_{\rm b}$ $\displaystyle=27x_{\rm
H}(1+\delta)\left(\frac{1+z}{10}\frac{0.15}{\Omega_{m}h^{2}}\right)^{1/2}\left(\frac{\Omega_{b}h^{2}}{0.023}\right)$
(30) $\displaystyle\qquad\times\left(\frac{H}{{\rm
dv_{r}/dr}+H}\right)\left(\frac{x_{c}+x_{\alpha}}{x_{\rm
R}+x_{c}+x_{\alpha}}\right)\left(1-\frac{T_{\rm R}}{T_{\rm K}}\right)\hskip
5.69046pt{\rm mK},$

In the standard scenario, $T_{\rm R}=T_{\rm CMB}$ and $x_{c}\ll x_{\alpha}$ at
$z\lesssim 30$. Ignoring redshift space distortions, fluctuations in the
brightness temperature are sourced by fluctuations in the density contrast
($\delta$), as well as due to the spatially varying Lyman-alpha intensity, and
from fluctuations in $T_{\rm K}$ (from spatially varying adiabatic and X-ray
heating). Furthermore, during the reionization epoch, there are also
fluctuations from the inhomogeneous and evolving ionization fraction.

To compute the 21cm power spectrum, one needs to rely on simulations (Iliev et
al., 2006; Mellema et al., 2006) which can be computationally expensive. Semi-
numerical codes such as Mesinger et al. (2011); Murray et al. (2020); Fialkov
et al. (2015); Muñoz (2023) have been developed to speed up the computation so
that one can efficiently scan over a realistic range of astrophysical
parameters. In this work, we have used 21cmFAST to highlight the effect of
soft photon heating for the case of a continuous injection from dark matter
decay, with the same parameter values as considered in Sec. 6.

For this work, we use a 100 Mpc simulation box with 100 grid points which may
not be not realistic but suffices to showcase the importance of soft photon
heating. We turn on the USE-TS-FLUCTUATION flag which then performs spin
temperature fluctuations. The spin temperature, given in Eq. (4), is a
function of both the radiation and kinetic temperatures, and is computed at
each grid point in the simulation box.

We compute the source term $T_{\rm R}$ due to the non-thermal radio background
such that $T_{\rm R}=T_{\rm CMB}+\Delta T$. We then plug the extra source term
into the definition of $T_{\rm R}$ in 21cmFAST. Similarly, we add this source
term to the $T_{\rm K}$ (equivalently $T_{\rm M}$) equation given above. In
our computation, this source term is assumed to be position independent and
thus only depends on redshift. This allows us to simply shift the temperature
at each grid point in the simulation box by a redshift-dependent value. In
reality, there can be spatial fluctuations in the radio background, which are
expected to induce additional heating. This directly affects $T_{\rm S}$ and
the local value of $T_{\rm R}$, causing additional changes to the ratio
$\delta[T_{\rm R}/T_{\rm S}]\approx\frac{\bar{T}_{\rm R}}{\bar{T}_{\rm
S}}\left[\frac{\delta T_{\rm R}}{\bar{T}_{\rm R}}-\frac{\delta T_{\rm
S}}{\bar{T}_{\rm S}}\right]$. At leading order, these contributions are
expected to compensate each other; however, we do not go into these
complications here.

We also note that these source terms are computed using our fiducial
reionization model which may not precisely match the output of 21cmFAST or
other such codes. However, we expect this difference to not change the
qualitative features of soft photon heating, which is the effect we are aiming
to highlight here. We defer a more consistent calculation, involving
interfacing our code with 21cm codes, to the future. Furthermore, we have also
taken care to not include the non-thermal radio photons in the Comptonization
equation [Eq. (13)] between the CMB photons and background electrons.

In Fig. 10, we show the results (at $z=18$) for the 21cm power spectrum in a
decaying dark matter scenario with a decay rate of $\Gamma=2\times
10^{-14}\,{\rm s}^{-1}$, and a high frequency cutoff in the spectrum at
$x_{\rm inj,0}=4035$. We have used powerbox (Murray, 2018) to compute the
power spectrum. We see that the predicted power spectrum changes by orders of
magnitude when we include the effects of soft photon heating. The 21cm
fluctuations measure the temperature contrast between a hot object in a cold
background or vice versa. By increasing the radio background (and neglecting
soft photon heating), this contrast as well as the fluctuation signal
increases. When we include soft photon heating, the overall contrast is
reduced due to the increased background matter temperature. This can even push
the fluctuation signal below the standard case in which there are no
primordial radio backgrounds. This computation showcases the importance of
including soft photon heating within the analysis of the 21cm power spectrum
such as in Abdurashidova et al. (2022, 2023).

Figure 10: The power spectrum for 21cm brightness fluctuations at $z=18$ for
cases with and without the soft photon heating due to the presence of radio
background. We consider the case with $x_{\rm inj,0}=4035$ and $\Gamma=2\times
10^{-14}$ s-1 of our model which can produce the observed radio synchrotron
background.

While the discussion in this work assumes the presence of a radio background,
it is expected that there are localized sources of radio photons such as radio
galaxies. Recently, the authors in Sikder et al. (2024) studied the effect of
radio galaxies on the 21cm power spectrum. The radio photons emitted by
galaxies boost the power spectrum just as showcased in Fig. 10, making them
more detectable (or one can put strong upper limits on their abundance in case
of non-detection). Since radio galaxies are believed to be formed by accreting
black holes, it is reasonable to believe that they emit a broad spectrum of
energetic photons, in addition to radio photons, which can ionize their
surrounding. Following our work, one expects the radio photons to be absorbed
and heat their immediate surrounding. This can suppress the 21cm signal around
the location of such galaxies, and in some cases the absorption signal may
switch to an emission signal depending upon the physical details. Using a
simple estimate, the optical depth for radio photons turns out to be

$\tau_{\rm
ff}(x,z=20)=0.32\left(\frac{10^{-6}}{x}\right)^{2}\left(\frac{X_{\rm
e}}{0.001}\right)^{2}\left(\frac{g_{\rm ff}}{5}\right)\left(\frac{R}{\rm
1Mpc}\right),$ (31)

where $R$ is the distance that photons travel form source of origin and we
have assumed matter temperature to be 10 K at $z=20$ which follows from
standard model. It is interesting to note that the optical depth is of the
order of unity on Mpc scales for photons with frequency $x=10^{-6}-10^{-5}$
which contribute dominantly to soft photon heating. This would modify the
powerspectrum at $k=0.1-1$ Mpc-1 when taken into account. Implementing this
effect into codes like 21cmFAST should be similar to the treatment of X-ray
photons. We plan to follow this interesting avenue in future.

We would like to stress that in our simple demonstrations we have only
considered the build-up of an excess radio background and its associated soft
photon heating. Physical processes which produce higher energy quanta can
undergo more traditional “hard” photon heating, which in some cases can
dominate over the soft photon heating we have discussed here. In this case,
one can expect the effect on the global 21cm signal and its fluctuations to be
more more modest, since the ratio $T_{\rm R}/T_{\rm S}$ may stay more
moderate. While the soft photon heating process clearly demonstrates this
important interplay, from a more general perspective it is thus very important
to take the correlation of heating and enhancement of a soft photon background
into account. In other words, the same radio galaxies that could cause the
buildup of a potential radio background will also heat the medium, with the
balance of these two processes being important. A more detailed treatment of
radio photon sources and heating is thus necessary and an important future
direction for future work.

##  8 Discussion and Conclusions

In this article, we have presented a comprehensive look into the underlying
physical mechanism that mediates soft photon heating. The upshot of this
effect is that in the presence of a low-frequency photon background, free-free
absorption cannot be ignored, as it can lead to large corrections to the gas
temperature $T_{\rm M}$. In this work we have highlighted how these
corrections can cause leading order effects in $21$cm cosmology by disrupting
both the power spectrum and the global signal of the differential brightness
temperature, $\Delta T_{\rm b}$. We would like to stress that this effect is
universal, making it crucial to include as part of the standard toolbox for
the computation of theoretical predictions and forecasting within the $21$cm
cosmology community.

After performing a brief overview of other relevant heating mechanisms in
$21$cm cosmology, we presented the coupled set of integro-differential
equations that must be solved in order to properly incorporate the effects of
soft photon heating, namely, Eqs. (7), (13), and (16). The implementation of
this effect is currently fully treated in the numerical thermalization code,
CosmoTherm (Chluba & Sunyaev, 2012), and work is in progress that will bring
this functionality to other codes such as DarkHistory (Liu et al., 2020,
2023a, 2023b; Sun et al., 2023) in the near future. In the meantime, we have
released a simple standalone Jupyter Notebook that can be used to explore the
impact of soft photon heating for soft photon backgrounds using the injection
source terms given in Eqs. (21) and (27).

Sec. 5 was dedicated to the quasi-instantaneous injection of soft photon
backgrounds, where we considered injection redshifts at $z_{\rm
inj}=3000,\,1000,\,500,\,100$ and spectral indices $\gamma=3.0$ (free-free
like), $3.3$, and $3.6$ (synchrotron like). Two major conclusions were reached
in this section. First, if injection takes place in the pre-recombination
plasma ($z\gtrsim 1000$), the low frequency part of the spectrum where
$\tau_{\rm ff}(x,z)\gtrsim 1$ is quickly absorbed. The photons that survive
this initial absorption into the post-recombination universe exhibit little
free-free heating at the redshifts relevant to $21$cm observations, leading to
a large boost in the predicted $\Delta T_{\rm b}$, which comes purely from the
relative increase in $T_{\rm R}$ compared to the vanilla $\Lambda$CDM
scenario.

However, if injections take place after the universe is at least partially
recombined, we are met with two important phases of soft photon heating. The
first phase again takes place near the injection redshift, where a truncation
of the spectrum immediately occurs, but is less severe than in the case of
pre-recombination injection due to the reduced $X_{\rm e}$ fraction. The
second phase of soft photon heating takes place right around cosmic dawn. When
the first sources start to reionize the universe, $X_{\rm e}$ begins to
increase rapidly, and a large fraction of the remaining soft photon spectrum
is absorbed by the gas. The relative strength of this effect depends on the
spectral index of the injected photons. For $\gamma=3.0$, the effect is
largely negligible. Anything steeper than this, however, will introduce
important changes to observable quantities such as $\Delta T_{\rm b}$.

We also considered the more physical scenario in which a soft photon
background is built up gradually over time. In the decaying dark matter
scenario, we studied a fiducial decay rate of $\Gamma=2\times 10^{-14}\,{\rm
s}^{-1}$ for the particle. This causes soft photons to be injected into the
background at a continuous rate until a redshift of approximately $z_{\rm
dec}\simeq 420$, after which the injection stops. The results obtained in this
scenario were qualitatively very similar to what one would expect from a
quasi-instantaneous injection around $z_{\rm inj}=z_{\rm dec}$. We
additionally considered a model of soft photon emission from superconducting
cosmic strings. This model produces soft photons at all redshifts, providing a
consistently strong source of heating as evidenced by the upper left panel of
Fig. 9. As expected, models of continuous heating such as these do not possess
a characteristic initial period of soft photon heating (unlike the quasi-
instantaneous injection cases). Importantly, however, is that the consistent
production of radiation during the dark ages serves to fill up a soft photon
reservoir. The energy stored in this reservoir is then released near cosmic
dawn as the first sources begin turning on due to the rapid increase of
$X_{\rm e}$.

Soft photon heating also has important implications for 21cm power spectrum
calculations as can be seen in Fig. 10. For a homogeneous and isotropic soft
photon injection, this heating reduces the temperature contrast between the
hot sources and the background. Similar to the global case, this causes a net
reduction in the amplitude of the fluctuations, making the signal more
difficult to detect. However, the deposition of energy from soft photon
heating is expected to occur near to the source. Therefore, sources of
anisotropic soft photon injection may lead to new contributions to the power
spectrum that one could expect from, e.g., radio loud galaxies. We plan to
explore this avenue in a future study.

Recently, there have been papers which aim to provide constraints on the
presence of extra radio backgrounds during reionization and cosmic dawn
(Mondal et al., 2020; Abdurashidova et al., 2022; Bevins et al., 2022). To the
best of our knowledge, these works have not included the effects of soft
photon heating, or more generally a correlation between heating processes and
significant radio emission, which we have proven here can cause leading order
corrections to the differential brightness temperature, both on the global
signal and its fluctuations. Unfortunately, soft photon heating serves to
severely dampen the amplitude of $\Delta T_{\rm b}$ during this epoch. We
expect that once this effect is taken into account, the derived constraints on
radio backgrounds will be significantly relaxed.

Soft photon heating is a universal effect that takes place even in a scenario
where the CMB is the sole source of radio photons. While for standard
scenarios the effect is likely negligible, much uncertainty surrounds our
understanding of which low-frequency backgrounds may permeate the universe
throughout its thermal history. In this work, we elucidated the proper
interplay between these soft photon backgrounds, the matter temperature, and
the key observables relevant to $21$cm cosmology. Considering the potential
importance of soft photon heating, this effect should be implemented into
future studies on exotic energy injections, as well as in forecasting and
parameter inference for current and future experiments.

## Acknowledgements

BC would like to acknowledge support from an NSERC postdoctoral fellowship. SA
is supported by an ARCO fellowship. JC was supported by the ERC Consolidator
Grant CMBSPEC (No. 725456) and by the Royal Society as a Royal Society
University Research Fellow at the University of Manchester, UK (No.
URF/R/191023).

##  9 Data availability

The data underlying in this article are available in this article and can
further be made available on request.

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##  Appendix A Formalising the photon evolution equation

The evolution of the photon occupation number depends primarily on three
physical processes (and a potential source term) (Chluba & Sunyaev, 2012)

$\displaystyle\frac{{\,\rm d}n_{\rm x}}{{\,\rm d}\tau}=\left.\frac{{\,\rm
d}n_{\rm x}}{{\,\rm d}\tau}\right|_{\rm C}+\left.\frac{{\,\rm d}n_{\rm
x}}{{\,\rm d}\tau}\right|_{\rm DC}+\left.\frac{{\,\rm d}n_{\rm x}}{{\,\rm
d}\tau}\right|_{\rm BR}+\left.\frac{{\,\rm d}n_{\rm x}}{{\,\rm
d}\tau}\right|_{\rm Src}.$ (32)

A detailed discussion of the modelling of the source term for quasi-
instantaneous injections is given in Section 5, so we take the opportunity now
to briefly describe the effects of the other three standard processes.

###  A.1 Compton interactions

The first term encodes the Compton process, and describes the effects of
energy redistribution, in contrast to double Compton and Bremsstrahlung, which
are mainly responsible for the production and absorption of photons. The
Compton term is treated by the Kompaneets equation (Kompaneets, 1956)

$\displaystyle\left.\frac{{\,\rm d}n_{\rm x}}{{\,\rm d}\tau}\right|_{\rm
C}=\frac{\theta_{\rm e}}{x^{2}}\frac{\partial}{\partial
x}x^{4}\left[\frac{\partial n_{\rm x}}{\partial x}+\frac{T_{\rm CMB}}{T_{\rm
M}}n_{\rm x}(n_{\rm x}+1)\right].$ (33)

Here, we neglect relativistic effects as they present sub-percent level
corrections at the redshifts considered in this work. We also note that
$\theta_{\rm e}=k_{\rm b}T_{\rm M}/m_{\rm e}c^{2}$ where $m_{\rm e}$ is the
electron mass. By decomposing the photon spectrum into a blackbody part and a
distortion, $n_{\rm x}=n_{\rm bb}+\Delta n$, we find

$\displaystyle\left.\frac{{\,\rm d}\Delta n}{{\,\rm d}\tau}\right|_{\rm
C}=(\theta_{\rm e}-\theta_{\gamma})\frac{x{\rm e}^{x}}{({\rm
e}^{x}-1)^{2}}\left[x\frac{{\rm e}^{x}+1}{{\rm
e}^{x}-1}-4\right]+\mathcal{O}(\theta_{\rm e}\Delta n),$ (34)

where $\theta_{\gamma}=\theta_{\rm e}T_{\rm CMB}/T_{\rm M}$ (recall also that
${\,\rm d}n_{\rm bb}/{\,\rm d}\tau=0$). Here we neglect terms of order
$\theta_{\rm e}\Delta n$ and higher not because they are small ($\Delta n$ is
not necessarily smaller than $n_{\rm bb}$ for soft photon injections at small
$x$), but because at late times, the scattering $y$-parameter, leading to
redistribution of photons in energy, becomes unimportant (e.g., Fig. 1 of
Chluba, 2015). In addition, at the low $x$ values relevant to our analysis,
the timescale for free-free absorption is much more rapid than the Compton
process. Thus, even though $\Delta n$ becomes large at low frequencies, these
photons are absorbed by the medium before they have sufficient time to
redistribute their energy across the spectrum through Compton interactions.
The lowest order term in this expression corresponds to the second source term
given in Eq. (8).

###  A.2 Double Compton

At early times ($z\gtrsim 10^{5}$), double Compton scattering plays a crucial
role in fully thermalizing distortions that may be induced in the primordial
plasma. At late times, however, this process of $1\rightarrow 2$ photon
conversions becomes incredibly inefficient relative to that of free-free
absorption and emission. As a result, we neglect contributions from this
channel in our analysis, but the interested reader is referred to Sec. 2 of
Chluba & Sunyaev (2012). An in depth discussion of the double Compton process
can be furthermore found in (Ravenni & Chluba, 2020).

###  A.3 Free-free effects

Bremsstrahlung effects can be included into the photon evolution equation
following the procedure laid out by many authors (Rybicki & Lightman, 1979;
Lightman, 1981; Danese & de Zotti, 1982), taking the form

$\displaystyle\left.\frac{{\,\rm d}n_{\rm x}}{{\,\rm d}\tau}\right|_{\rm
BR}=\frac{1}{x^{3}}\left[1-n_{\rm x}({\rm e}^{x_{\rm e}}-1)\right]\times
K_{\rm BR}(x,\theta_{\rm e}).$ (35)

In this expression, $K_{\rm BR}$ is the Bremsstralung emission coefficient,
given by (Burigana et al., 1991; Hu & Silk, 1993)

$\displaystyle K_{\rm BR}(x,\theta_{\rm e})=\frac{\alpha\lambda_{\rm
e}^{3}}{2\pi\sqrt{6\pi}}\frac{\theta_{\rm e}^{-7/2}{\rm e}^{-x_{\rm
e}}}{(T_{\rm CMB}/T_{\rm M})^{3}}\sum_{\rm i}Z_{\rm i}^{2}N_{\rm i}g_{\rm
ff}(z_{\rm i},x,\theta_{\rm e}).$ (36)

Here, $\alpha\simeq 1/137$ the fine structure constant, $\lambda_{\rm
e}=h/m_{\rm e}c$ the Compton wavelength of the electron, as well as the charge
($Z_{\rm i}$) and number density ($N_{\rm i}$) of a particular atomic species.
The Gaunt factor ($g_{\rm ff}$) in the absence of Helium can be approximated
as777Note there is a typo in Eq. (31) of Chluba (2015), although the
implementation was not affected. (Draine, 2011)

$\displaystyle g_{\rm ff}(x_{\rm e},\theta_{\rm e})\approx 1+{\rm
ln}\left[1+{\rm exp}\left(\frac{\sqrt{3}}{\pi}\left[{\rm
ln}\left(\frac{2.25}{Z_{\rm H}x_{\rm e}}\right)+\frac{{\rm ln}\,\theta_{\rm
e}}{2}\right]+1.425\right)\right].$

Although a more accurate representation can be achieved with BRpack (Chluba et
al., 2020), this approximation suffices for our purposes. Inserting all of
this into Eq. (36), one finds

$\displaystyle\left.\frac{{\,\rm d}n_{\rm x}}{{\,\rm d}\tau}\right|_{\rm
BR}=\frac{{\rm e}^{-x_{\rm e}}}{x_{\rm e}^{3}}\left[1-n_{\rm x}({\rm
e}^{x_{\rm e}}-1)\right]\Lambda_{\rm BR},$ (37)

where we have now defined the Bremsstrahlung emissivity coefficient,
$\Lambda_{\rm BR}$ as

$\displaystyle\Lambda_{\rm BR}(\tau,x_{\rm e})=\frac{\alpha\lambda_{\rm
e}^{3}}{2\pi\sqrt{6\pi}}\theta_{\rm e}^{-7/2}N_{\rm p}\,g_{\rm ff}(x_{\rm
e},\theta_{\rm e}),$ (38)

with $N_{\rm p}$ being the proton number density. This matches with the
starting point in our computation, namely Eq. (6).

##  Appendix B Matter temperature evolution

Using the first law of thermodynamics, one can study the evolution of the gas
temperature in the presence of a photon background. The general expression is
given by

$\displaystyle\frac{{\,\rm d}T_{\rm M}}{{\,\rm d}z}=\frac{2T_{\rm
M}}{1+z}+\frac{1}{\chi\,a^{3}}\frac{{\,\rm d}Q_{\rm M}}{{\,\rm
d}z}-\frac{1}{\chi\,a^{4}}\sum_{i}\frac{{\,\rm
d}(a^{4}\Delta\rho_{\gamma,i})}{{\,\rm d}z},$ (39)

where $Q_{\rm M}$ is the direct (comoving) energy density injection into the
gas, $\Delta\rho_{\gamma}$ is the energy density in the various constituents
of a spectral distortion ($\Delta\rho_{\gamma}=\sum_{i}\Delta\rho_{\gamma,i}$,
with $\rho_{\gamma}=\rho_{\rm CMB}+\Delta\rho_{\gamma}$), and

$\displaystyle\chi=\frac{3}{2}k_{\rm b}N_{\rm H}(1+X_{\rm e}+f_{\rm He})$ (40)

is related to the heat capacity of the medium. We consider no direct energy
injections to the gas ($\\!{\,\rm d}Q_{\rm M}/{\,\rm d}z=0)$, and discuss
contributions from both Compton scattering and free-free.

###  B.1 Compton heating

As discussed above, Kompaneets equation governs to the evolution of the photon
occupation number in the presence of Compton scatterings between the
background and the matter sector. Neglecting again the relativistic
corrections, Eq. (34) takes a useful form

$\displaystyle\left.\frac{{\,\rm d}\Delta n}{{\,\rm d}\tau}\right|_{\rm
C}\approx\frac{(\theta_{\rm
e}-\theta_{\gamma})}{x^{2}}\frac{\partial}{\partial
x}x^{4}\left(\frac{\partial n_{\rm bb}}{\partial x}\right),$ (41)

where we have made use of the identity $\partial n_{\rm bb}/\partial x=-n_{\rm
bb}(n_{\rm bb}+1)$. For an isotropic photon distribution, the (physical)
energy density is

$\displaystyle\rho=\frac{(k_{\rm b}T_{\rm
CMB})^{4}}{\pi^{2}\hbar^{3}c^{3}}\int_{0}^{\infty}{\,\rm d}x\,x^{3}\,n(x).$
(42)

Thus, the photon energy injection rate is given by

$\displaystyle\frac{1}{a^{4}}\frac{{\,\rm d}(a^{4}\Delta\rho_{\rm C})}{{\,\rm
d}\tau}$ $\displaystyle=\frac{(k_{\rm b}T_{\rm
CMB})^{4}}{\pi^{2}\hbar^{3}c^{3}}\int_{0}^{\infty}{\,\rm
d}x\,x^{3}\,\frac{{\,\rm d}\Delta n_{\rm C}}{{\,\rm d}\tau},$
$\displaystyle=4\rho_{\rm CMB}(\theta_{\rm e}-\theta_{\gamma}),$ (43)

where $\rho_{\rm CMB}$ is the energy density of the CMB blackbody. The
conversion to redshift is given by ${\,\rm d}\tau=-[\sigma_{\rm T}N_{\rm
e}c/H(1+z)]{\,\rm d}z$, which upon substitution into Eq. (39) yields the well
known Compton heating term, the second term in Eq. (13).

###  B.2 Free-free heating

Free-free heating plays a negligible role without the presence of an
enhancement to the low frequency ($x\lesssim 1)$ part of the photon spectrum.
The free-free part of the injection can be seen by insertion of $n_{\rm
x}=n_{\rm bb}+\Delta n$ into Eq. (37), which yields the evolution equation and
source terms found in Sec. 3. This procedure provides

$\displaystyle\frac{{\,\rm d}\Delta n_{\rm ff,bb}}{{\,\rm d}\tau}$
$\displaystyle=\frac{\Lambda_{\rm BR}\left(1-{\rm e}^{-x_{\rm
e}}\right)}{x_{\rm e}^{3}}\left[\frac{1}{{\rm e}^{x_{\rm e}}-1}-\frac{1}{{\rm
e}^{x}-1}\right],$ (44a) $\displaystyle\frac{{\,\rm d}\Delta n_{\rm
ff,dist}}{{\,\rm d}\tau}$ $\displaystyle=-\frac{\Lambda_{\rm BR}(1-{\rm
e}^{-x_{\rm e}})}{x_{\rm e}^{3}}\Delta n.$ (44b)

The first expression carries information about free-free absorption and
emission off the blackbody, while the second does the same in the presence of
a distortion. Computing the energy density present in this expression yields

$\displaystyle\frac{1}{a^{4}}$ $\displaystyle\frac{{\,\rm
d}(a^{4}\Delta\rho_{\rm ff})}{{\,\rm d}z}=\frac{(k_{\rm b}T_{\rm
CMB})^{4}}{\pi^{2}\hbar^{3}c^{3}}\left(\frac{T_{\rm M}}{T_{\rm
CMB}}\right)^{3}\frac{\sigma_{\rm T}N_{\rm e}c}{H(1+z)}$ (45)
$\displaystyle\times\int_{0}^{\infty}{\,\rm d}x\,\Lambda_{\rm BR}(\tau,x_{\rm
e})\left(1-{\rm e}^{-x_{\rm e}}\right)\left[\Delta n(x)+\frac{1}{{\rm
e}^{x}-1}-\frac{1}{{\rm e}^{x_{\rm e}}-1}\right].$

Clearly, the presence of large enhancements to $\Delta n$(x) (such is the case
for extra radio backgrounds) can significantly boost the heating rate, leading
to a rise in the overall matter temperature.