# Spectral distortion constraints on photon injection from low-mass decaying
particles

Boris Bolliet1,2, Jens Chluba1 and Richard Battye1
1 Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The
University of Manchester, Manchester, M13 9PL, U.K.
2 Columbia Astrophysics Laboratory, Columbia University, 550 West 120th
Street, New York, NY, 10027, USA
boris.bolliet@gmail.comjens.chluba<EMAIL_ADDRESS>
(Accepted XXX –. Received December 2020)

###### Abstract

Spectral distortions (SDs) of the cosmic microwave background (CMB) provide a
powerful tool for studying particle physics. Here we compute the distortion
signals from decaying particles that convert directly into photons at
different epochs during cosmic history, focusing on injection energies
$E_{\mathrm{inj}}\lesssim 20\,\mathrm{keV}$. We deliver a comprehensive
library of SD solutions, using CosmoTherm to compute the SD signals, including
effects on the ionization history and opacities of the Universe, and
blackbody-induced stimulated decay. Then, we use data from COBE/FIRAS and
EDGES to constrain the properties of the decaying particles. We explore
scenarios where these provide a dark matter (DM) candidate or constitute only
a small fraction of DM. We complement the SD constraints with CMB anisotropy
constraints, highlighting new effects from injections at very-low photon
energies ($h\nu\lesssim 10^{-4}\,{\rm eV}$). Our model-independent constraints
exhibit rich structures in the lifetime-energy domain, covering injection
energies $E_{\mathrm{inj}}\simeq 10^{-10}\mathrm{eV}-10\mathrm{keV}$ and
lifetimes $\tau_{X}\simeq 10^{5}\,\mathrm{s}-10^{33}\,\mathrm{s}$. We discuss
the constraints on axions and axion-like particles, revising existing SD
constraints in the literature. Our limits are competitive with other
constraints for axion masses $m_{a}c^{2}\gtrsim 27\,{\rm eV}$ and we find that
simple estimates based on the overall energetics are generally inaccurate.
Future CMB spectrometers could significantly improve the obtained constraints,
thus providing an important complementary probe of early-universe particle
physics.

###### keywords:

Cosmology: cosmic microwave background – theory – observations.

##  1 Introduction

The average energy spectrum of the cosmic microwave background (CMB) is known
to be extremely close to that of a perfect blackbody at a temperature $T_{\rm
CMB}=2.7255\,{\rm K}$ (Mather et al., 1994; Fixsen et al., 1996; Fixsen,
2009). However, out-of-equilibrium processes lead to departures from the
Planckian spectrum (Zeldovich & Sunyaev, 1969; Sunyaev & Zeldovich, 1970;
Illarionov & Sunyaev, 1974; Danese & de Zotti, 1977), causing so-called CMB
spectral distortions (SDs). The presence of SDs has been tightly constrained
using COBE/FIRAS, ruling out significant episodes of energy release after
thermalization becomes inefficient at redshift $z\lesssim 2\times 10^{6}$
(Mather et al., 1994; Wright et al., 1994; Fixsen et al., 1996; Fixsen, 2009).
This provides strong limits on various early-energy release scenarios (e.g.,
Burigana et al., 1991; Hu & Silk, 1993a; Chluba & Jeong, 2014), and in the
future these limits are expected to be significantly improved with novel
spectrometer concepts such as PIXIE (Kogut et al., 2011; Kogut et al., 2016)
and its enhanced versions (PRISM Collaboration, 2014; Kogut et al., 2019),
including a possible mission concept for the ESA Voyage 2050 program (see
Chluba et al., 2019a). This promising perspective has spurred significant
interest in SD science in the last decade (e.g., Chluba & Sunyaev, 2012;
Sunyaev & Khatri, 2013; De Zotti et al., 2016; Chluba et al., 2019b, a; Lucca
et al., 2020). We refer to Chluba (2018) for a broad introduction to the
science of CMB spectral distortions.

Moreover, the results from ARCADE2 (Fixsen et al., 2011) and EDGES (Bowman et
al., 2018) have proven difficult to interpret with standard astrophysical
assumptions (e.g., Seiffert et al., 2011; Feng & Holder, 2018; Hardcastle et
al., 2020), and may be consistent with a brightness temperature in the
Rayleigh-Jeans tail of the CMB that is significantly larger than the one
corresponding to the COBE/FIRAS measurement (see Sect. 5.2.2 for discussion).
These low-frequency measurements of the background radiation with unexpected
feature are another motivation for the study of SDs.

There are two main ways of creating distortions. The most commonly considered
is due to the injection of energy, which leads to the heating of electrons,
generally causing the classical $\mu$\- and $y$-type distortions through
Comptonization, i.e., the repeated scattering of photons by free electrons
(Zeldovich & Sunyaev, 1969; Sunyaev & Zeldovich, 1970). The other is related
to directly adding or removing photons from the CMB (Hu, 1995; Chluba, 2015).
One concrete example is the SD caused by the cosmological recombination
process (see Sunyaev & Chluba, 2009, for an overview), primarily due to the
photons created in uncompensated atomic transitions of hydrogen and helium.
The corresponding signal is small but extremely rich (see Hart et al., 2020,
for recent calculations and forecasts). For the cosmological recombination
radiation, Comptonization only plays a minor role because around recombination
(and after) the Compton scattering time scale $t_{\mathrm{C}}$ is already much
longer than the expansion time scale $t_{H_{0}}$ (Rubiño-Martín et al., 2008;
Chluba & Ali-Haïmoud, 2016); however, decaying or annihilating particle
scenarios can in principle cause direct photon injection at earlier times,
requiring a careful thermalization treatment, which is expected to result in
signals that differ significantly from the classical Comptonization
distortions, in particular when occurring at redshift $z\lesssim 10^{5}$
(i.e., when $t_{\mathrm{C}}\gtrsim t_{H_{0}}$) and for photon energies
$h\nu\lesssim 10\times kT_{\rm CMB}(z)$ (see Chluba, 2015).

While the data from COBE/FIRAS has been extensively used to constrain energy
release processes, no comprehensive analysis of decaying particle scenarios
with photon injection has been carried out. The main goal of this paper is to
derive SD constraints on decaying particle scenarios that directly lead to the
production of photons at energies $h\nu\lesssim{20\,{\rm keV}}$. This upper
limit in energy is chosen for two reasons. First, to avoid complications from
high-energy processes associated with production of non-thermal electrons (see
discussion Sect. 3). Second, because at even higher energies most of the
effects on the CMB spectrum are indeed captured by treating the transfer of
energy to the baryons (e.g., Chluba, 2015), thus essentially mimicking pure
energy release (e.g., Ellis et al., 1992; Sarkar & Cooper, 1984; Hu & Silk,
1993b; McDonald et al., 2001, for classical references on the topic). For our
computations, the particles are assumed to be non-relativistic and cold, with
a constant lifetime parameter, injection energy and free abundance, which we
express relative to the dark matter (DM) density. Both decays in vacuum and
within the ambient CMB field, which gives rise to stimulated decay, are
considered. We furthermore take the effect of photon injection on the
ionization history into account, carefully accounting for the extra heating,
ionizations and collisional processes by modifying CosmoTherm (Chluba &
Sunyaev, 2012) and Cosmorec/Recfast++ (Chluba & Thomas, 2011). We use these
computations to create a comprehensive library of spectral distortion
solutions that are then translated into limits on the particle properties.
This extends the computations carried out by Chluba (2015), which focused on
single injections in time and energy, during the pre-recombination era,
bringing us one step closer to treating more general scenarios.

To derive constraints, we use the spectral distortion data from COBE/FIRAS,
marginalizing over a galactic foreground in the same way as Fixsen et al.
(1996). For injections at energies well below the maximum of the CMB
blackbody, we also consider the observations of EDGES (Bowman et al., 2018) as
an additional upper limit on the radio background, showing that this
significantly tightens the obtained limits for injection energies
$10^{-6}\,{\rm eV}\lesssim h\nu\lesssim\rm few\times 10^{-4}\,{\rm eV}$ (see
Fig. 27).

Further improvements on our limits can be achieved by considering the effects
of photon injection on the ionization history, since these can be constrained
by CMB anisotropy data, as measured by Planck (Planck Collaboration, 2018).
Here, in order to avoid computing the CMB anisotropy spectra and comparing
them with Planck for each ionisation history, we use the projection method
introduced by Hart & Chluba (2020) to estimate the constraints. We find that
for injection at very low ($h\nu\lesssim 10^{-4}\,{\rm eV}$) and high
($h\nu\gtrsim 13.6\,{\rm eV}$) energies the addition of Planck data tightens
the bounds on decays in the post-recombination era. See Sect. 5.2.3 for
details.

In that last part of this work, we translate our constraints to the parameter
space of axions and axion-like particles (ALPS), highlighting how SD
measurements can provide a sensitive probe of particle physics. These
particles are being considered as a possible DM candidate (see, for example
Marsh, 2016), given that the standard WIMP scenario is seeing increased
observational pressure from direct-detection and collider experiments. Our
constraints at masses $mc^{2}\gtrsim{\rm keV}$ are comparable to those
previously published (e.g. Cadamuro et al., 2011; Millea et al., 2015), but
unlike these previous works we use the full SD spectra, including late time
evolution during reionization and Lyman absorption to place our constraints,
rather than basing them on the approximate $\mu$ and $y$ estimates based on
heating. At lower masses, our constraints are not competitive with CAST
(Andriamonje et al., 2007); however, we emphasize that this is the first time
SD data is used to place an independent constraints in this part of the
parameter space, with data that predates many measurements by decades. We
close by briefly discussing other particle physics scenarios that can be
constrained with the SD library we provide here (see Sect. 4 and conclusions),
as a more detailed analysis is left to future work.

Our fiducial cosmology is assumed to be a spatially flat Friedmann-Lemaître-
Robertson-Walker universe with a cosmological constant and $T_{\rm
CMB}=2.726\,\mathrm{K}$ for the present CMB temperature, $Y_{\mathrm{P}}=0.24$
for the helium mass fraction, $N_{\mathrm{eff}}=3.046$ effective relativistic
species, $\Omega_{\mathrm{m}}=0.31$, $\Omega_{\mathrm{b}}=0.049$ for the
present density parameters of matter and baryons respectively, and Hubble
constant $H_{0}=67.5\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}$. Our results
are not strongly dependent on the specific value of the cosmological
parameters.

##  2 Computing photon injection distortions

In this section, we explain how the distortions created by photon injection
from decay processes are computed. The starting point is the standard
thermalization problem, which accounts for Compton (C), double Compton (DC)
and Bremsstrahlung (BR) interactions to evolve the CMB spectrum across time.
Schematically, the thermalization equation for the photon occupation number
$n_{\nu}$, with respect to time $\mathrm{d}\tau=\mathrm{d}t/t_{\mathrm{C}}$
where $t_{\mathrm{C}}$ is the Thomson scattering time
$t_{\mathrm{C}}=1/\sigma_{\rm T}N_{\rm e}c$, reads as

$\frac{\partial n_{\nu}}{\partial\tau}-Ht_{\mathrm{C}}\nu\frac{\partial
n_{\nu}}{\partial\nu}=\left.\frac{\mathrm{d}n_{\nu}}{\mathrm{d}\tau}\right|_{\mathrm{C}}+\left.\frac{\mathrm{d}n_{\nu}}{\mathrm{d}\tau}\right|_{\mathrm{DC}}+\left.\frac{\mathrm{d}n_{\nu}}{\mathrm{d}\tau}\right|_{\mathrm{BR}}.$
(1)

This problem is solved numerically using CosmoTherm (Chluba & Sunyaev, 2012),
which we modify by adding an explicit time-dependent source term on the right-
hand-side of Eq. (1), corresponding to photon injection. We also account for
extra ionizations of hydrogen and helium by hard photons as well as atomic
collision and recombination processes.

We consider the decay of a massive particle or an excited state of matter that
both lead to the injection of photons at frequency ${\nu_{\rm inj}}$. For the
decay of a particle with mass $m_{X}$ we assume that two photons are produced,
$X\rightarrow\gamma+\gamma$, such that ${E_{\rm inj}}=h{\nu_{\rm
inj}}=m_{X}c^{2}/2$. In contrast, for the decay of an excited state we have
$X^{*}\rightarrow X+\gamma$ and ${E_{\rm inj}}=h{\nu_{\rm inj}}=E^{\rm
ex}_{X}$, where $E^{\rm ex}_{X}$ is the particle’s excitation energy. The
efficiency of photon production per particle thus differs by a factor of two.
For simplicity we just refer to both cases as decaying particle scenarios.
We assume that the production of the decaying particle has happened before the
distortion era ($z\gtrsim 2\times 10^{6}$). The number density of the decaying
particle then evolves according to the exponential law

$\displaystyle\frac{{\,\rm d}\ln a^{3}N_{X}}{{\,\rm d}t}=-\Gamma_{X},$ (2)

governed by the decay rate $\Gamma_{X}$, or the lifetime ${t_{\rm
X}}=1/\Gamma_{X}$, of the particle. For the main discussion below we consider
vacuum decay, while Sect. 4.6 is dedicated to blackbody-stimulated decay for
which the time-dependence of the photon injection process in the low-energy
limit, i.e., ${E_{\rm inj}}\lesssim 0.1\,kT_{\rm CMB}(z)$, is modified. Note
that in Eq. (2), $a$ is the scale factor and $t$ the cosmic time of the FLRW
model.

Without decays, from Eq. (2) one has $N_{X}=N_{X,0}(1+z)^{3}$, where $N_{X,0}$
is the would-be number density today. The solution of the equation above then
reads $N_{X}(t)=N_{X,0}(1+z)^{3}\,{{\rm e}^{-\Gamma_{X}t}}$. This allows us to
define the photon source term for the occupation number $n_{x}=n(t,x)$ at
frequency $x=h\nu/kT_{\rm CMB}(z)$ in the form

$\displaystyle\left.\frac{{\,\rm d}n_{x}}{{\,\rm d}t}\right|_{\mathrm{inj}}$
$\displaystyle=\mathcal{G}_{2}\,{f_{\rm
inj}}\Gamma_{X}\exp\left(-\Gamma_{X}t\right)\times\frac{G\left(x,{x_{\rm
inj}},\sigma_{x}\right)}{x^{2}}.$ (3)

Here, we introduced the injection efficiency ${f_{\rm inj}}$ and the integrals
$\mathcal{G}_{k}=\int x^{k}/({{\rm e}^{x}}-1){\,\rm d}x$
($\mathcal{G}_{2}\approx 2.404$ and $\mathcal{G}_{3}\approx 6.494$). We assume
that the photons are injected in a narrow Gaussian with width $\sigma_{x}$
centered around ${x_{\rm inj}}(z)={E_{\rm inj}}/kT_{\rm CMB}(z)={x_{\rm
inj,0}}/(1+z)$, where ${x_{\rm inj,0}}={E_{\rm inj}}/kT_{\rm
CMB}=4.257\,[{E_{\rm inj}}/{{\rm meV}}]$ with $T_{\rm CMB}=2.726\,{\rm K}$. We
usually set $\sigma_{x}=0.05{x_{\rm inj}}$ in our computations, but the
results do not crucially depend on this choice. Our calculations are thus
applicable to cold non-relativistic relic particles such as axions or excited
internal states of cold dark matter.

Equation (3) is normalized such that the total number of photons injected at
any moment is given by

$\displaystyle\left.\frac{{\,\rm d}\ln a^{3}N_{\gamma}}{{\,\rm
d}t}\right|_{\mathrm{inj}}$ $\displaystyle={f_{\rm
inj}}\Gamma_{X}\exp\left(-\Gamma_{X}t\right)\equiv
f_{\gamma}\Gamma_{X}\frac{N_{X}}{N_{\gamma}},$ (4)

where $f_{\gamma}=2$ for decaying massive particles and $f_{\gamma}=1$ for
decaying excited states. This then implies

$\displaystyle{f_{\rm inj}}$
$\displaystyle=f_{\gamma}\,\frac{N_{X,0}}{N_{\gamma,0}}=\frac{\mathcal{G}_{3}}{\mathcal{G}_{2}}\,\frac{\epsilon}{{x_{\rm
inj,0}}}\,\frac{\rho_{X,0}}{\rho_{\gamma,0}}=\frac{\mathcal{G}_{3}}{\mathcal{G}_{2}}\,\frac{\epsilon
f_{\rm dm}}{{x_{\rm inj,0}}}\frac{\rho_{\rm cdm,0}}{\rho_{\gamma,0}}$
$\displaystyle\approx 1.31\times 10^{4}\,\,\frac{\epsilon f_{\rm dm}}{{x_{\rm
inj,0}}}\left[\frac{\Omega_{\rm cdm}h^{2}}{0.12}\right],$ (5)

where in the last steps we give the expression in terms of the dark matter
energy density, $\rho_{\rm cdm,0}$. Furthermore, we have $\epsilon=1$ for
decaying particles and $\epsilon=E^{\rm ex}_{X}/m_{X}c^{2}<1$ for excited
states. The excited state scenario is suppressed by a factor $\epsilon<1$ to
account for the reduction of the particle number density, which in both cases
is given by $N_{X,0}=\rho_{X,0}/m_{X}c^{2}$. Model-independent constraints are
then obtained for the effective dark matter fraction, $f^{*}_{\rm dm}=\epsilon
f_{\rm dm}$.

The injected photons can i) Compton scatter with electrons; ii) be absorbed in
a BR or DC event; iii) interact with the atoms in the Universe and iv) simply
remain in the CMB spectrum as a direct distortion. The processes i)-iii) all
lead to heating/cooling of the matter. This in turn causes $\mu$ and $y$-type
distortion contributions and indirectly changes to the ionization history,
depending on the epoch at which the injection happens. In addition, iii)
directly changes the ionization history, as we explain below.

###  2.1 Energy release histories

When studying decaying particle scenarios, it is instructive to first
understand the time-dependence of the injection process. The parameters
${f_{\rm inj}}$ and ${E_{\rm inj}}$ determine how much energy is added and at
which frequency. These do not directly affect the time-dependence of the
injection process, solely controlled by $\Gamma_{X}$. The energy release
history is directly obtained from Eq. (3) as

$\displaystyle\frac{{\,\rm d}\ln a^{4}\rho_{\gamma}}{{\,\rm d}t}$
$\displaystyle=\frac{\mathcal{G}_{2}}{\mathcal{G}_{3}}\,{f_{\rm inj}}{x_{\rm
inj}}\,\Gamma_{X}\exp\left(-\Gamma_{X}t\right)$ $\displaystyle=\epsilon f_{\rm
dm}\frac{\rho_{\rm
cdm,0}}{\rho_{\gamma,0}}\,\frac{\Gamma_{X}\exp\left(-\Gamma_{X}t\right)}{1+z},$
(6)

where ${x_{\rm inj}}={x_{\rm inj,0}}/(1+z)$ gives rise to a factor $1/(1+z)$.
Note that we also have $\rho_{\rm cdm,0}/\rho_{\gamma,0}\simeq 4.85\times
10^{3}\,[\Omega_{\rm cdm}h^{2}/0.12]$.

Figure 1: Energy release history for several lifetimes. For distortions
created by heating, the function already determines the form of the spectral
distortion, while for photon injection distortions a more complicated
interplay of injection time and energy is found. The dotted line illustrates
the effective energy release history when accounting for thermalization (see
Sect. 4.2)  Figure 2: The relationship between $\Gamma_{X}$ and the maximal
injection redshift ${z_{\rm X}}$. The solid black line shows the exact
numerical result, while the two other coloured curves are the approximations
in Eq. (7).

In Fig. 1, we show the energy release histories for several lifetimes. We
normalized all of them to a total energy release of $\Delta\rho/\rho=3\times
10^{-5}$, which corresponds to the 68% CL limit from COBE/FIRAS (see Section
5.2.1). Using the analytical approximations for the relationship between time
and redshift in the matter and radiation dominated era, we find that the
maximum of energy release occurs at redshift $z_{X}$ such that

$\displaystyle 1+z_{X}$ $\displaystyle\simeq\begin{cases}1.26\times
10^{4}\left[\frac{\Gamma_{X}}{10^{-11}\mathrm{s^{-1}}}\right]^{1/2}&\,\Gamma_{X}\gtrsim{\Gamma_{X}^{\rm
eq}}\\\\[2.84526pt] 2.17\times
10^{2}\left[\frac{\Omega_{\mathrm{m}}h^{2}}{0.15}\right]^{-1/3}\left[\frac{\Gamma_{X}}{10^{-14}\,{\rm
s}^{-1}}\right]^{2/3}&\,\Gamma_{X}\lesssim{\Gamma_{X}^{\rm eq}}.\end{cases}$
(7)

A comparison of these approximations with the exact result is shown in Fig. 2.
We note that the redshift of matter-radiation equality is given by $1+z_{\rm
eq}=\Omega_{\mathrm{m}}/\Omega_{\mathrm{rel}}\simeq 3.58\times
10^{3}\,[\Omega_{\mathrm{m}}h^{2}/0.15]$, and the decay rate at $z_{\rm eq}$
is ${\Gamma_{X}^{\rm eq}}\simeq{2\times 10^{-13}\,{\rm s}^{-1}}$. For
$\Gamma_{X}\lesssim 2.3\times 10^{-18}\,{{\rm s}^{-1}}$ (i.e., lifetime longer
than the age of the Universe), no maximum in the injection history is present
at $z>0$.

The total number of injected photon and energy release are obtained by
integrating Eq. 4 and 2.1 over time. One finds

$\displaystyle\left.\frac{\Delta N_{\gamma}}{N_{\gamma}}\right|_{\rm tot}$
$\displaystyle={f_{\rm
inj}}\,\left[1-\exp\left(-\Gamma_{X}t_{0}\right)\right]$ (8a)
$\displaystyle\left.\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\right|_{\rm
tot}$ $\displaystyle=\frac{\mathcal{G}_{2}}{\mathcal{G}_{3}}{f_{\rm
inj}}\,{x_{\rm inj,0}}\,\int_{0}^{t_{0}}\frac{\Gamma_{X}\,{{\rm
e}^{-\Gamma_{X}t}}}{(1+z)}\,{\,\rm d}t$ (8b)
$\displaystyle\approx\frac{\mathcal{G}_{2}}{\mathcal{G}_{3}}{f_{\rm
inj}}\,{x_{\rm
inj,0}}\times\begin{cases}\frac{\Gamma_{X}\,t_{0}}{2}&\text{for}\,\,\Gamma_{X}\,t_{0}\ll
1\\\\[2.84526pt] \sqrt{\frac{\pi}{2}}\,\frac{\Omega_{\rm
rel}^{1/4}}{\sqrt{\Gamma_{X}\,t_{0}}}&\text{for}\,\,\Gamma_{X}\,t_{0}\gg
1,\end{cases}$ (8c)

where $t_{0}=1/H_{0}=3.086\times 10^{17}\mathrm{s}/h\approx 4.41\times
10^{17}\,{\rm s}$ is the Hubble time. These two relations can be used to set
approximate initial conditions for ${f_{\rm inj}}$ once ${x_{\rm inj,0}}$ and
$\Gamma_{X}$ are chosen. (In our computations, we also take into account the
thermalization efficiency, as explained in Sect. 4.2.)

For distortions created by energy release, simple estimates for $\mu$ and $y$
distortions can be obtained by integrating the energy release history
multiplied by appropriate distortion visibility functions (e.g., Chluba, 2013,
2016). In particular, for distortions created by photon injection from a
decaying particle with short lifetime ($\tau_{X}\lesssim 10^{10}\,{\rm s}$) it
is crucial to consider the number of photons added by the injection process.
This can lead to both negative and positive $\mu$-distortions if injection
occurs for $z\gtrsim 3\times 10^{5}$ (Chluba, 2015), as we will see below. In
addition, for longer lifetime when the injection happens in the post-
recombination era, the transparency of the Universe to photons is a strong
function of their energy (e.g., Chluba, 2015, Fig. 10), and the simple $\mu$
and $y$ distortions are not appropriate to describe the final spectrum.

For $\Gamma_{X}\ll 1/t_{0}$, one furthermore finds that the particles are
essentially stable and the decay follows a power-law in redshift. This leads
to a quasi-universal distortion shape that mainly depends on the injection
energy and scales linearly with $\Gamma_{X}$ (see Sect. 4.4).

###  2.2 Interactions with atomic species

Around the recombination era, significant fractions of neutral hydrogen and
helium atoms in the ground-state form (Zeldovich et al., 1968; Peebles, 1968;
Seager et al., 2000). For photons injected at energies above the ionization
thresholds of hydrogen and helium, this strongly affects the distortion
evolution, as photo-ionization processes not only remove photons from the CMB
bands, but also affect the ionization history and lead to heating of the
medium. Accounting for both absorption and emission to the ground state, the
terms in the evolution equation of the photon occupation number Eq. (1),
related to hydrogen Lyman-continuum (Ly-c) transfer, can be written as

$\displaystyle\frac{1}{c}\,\frac{\partial n_{\nu}}{\partial t}\Bigg{|}_{{\rm
Ly{\textendash}c}}=N^{\rm eq}_{\rm 1s}\,\sigma_{\rm 1s}(\nu)\,{{\rm
e}^{-x_{\rm e}}}\,(1+n_{\nu})-N_{\rm 1s}\sigma_{\rm 1s}(\nu)\,n_{\nu},$ (9)

where $\sigma_{\rm 1s}(\nu)$ is the photoionization cross section111We neglect
the thermal broadening due to the motion of the atoms when computing the
photoionization cross section. of the ground-state, $N^{\rm eq}_{\rm 1s}$ is
the equilibrium population with respect to the continuum, $N^{\rm eq}_{\rm
1s}=N_{\rm e}\,N_{\rm p}f_{\rm 1s}(T_{\rm e})$, and $x_{\rm e}=h\nu/kT_{\rm
e}$. Here, $f_{\rm 1s}(T_{\rm e})$ is a temperature-dependent factor that
follows from Saha-equilibrium and detailed balance (e.g., see Seager et al.,
2000; Chluba & Sunyaev, 2007, for explicit expression). Note that these terms
are time dependent, in particular due to the evolution of the electron
temperature $T_{\rm e}$, which is solved simultaneously (see Chluba & Sunyaev,
2012). Similar terms arise for neutral and singly-ionized helium. Ionizations
from excited states remain negligible, since at any stage during recombination
the populations of the excited levels remain small (e.g., Rubiño-Martín et
al., 2006; Chluba et al., 2007). We also omit the effects of atomic
excitations from the ground-state (Lyman-series for hydrogen), as they do not
lead to a large energy transfer (e.g., Basko, 1981; Grachev & Dubrovich, 2008;
Hirata & Forbes, 2009; Chluba & Sunyaev, 2009b) and would only affect the CMB
spectrum in the distant Wien tail.

To account for the continuum transfer self-consistently is beyond the scope of
this paper; however, to capture the main effects, we approach the problem in
the following way: once significant absorption occurs, the re-emission process
is heavily suppressed because the Lyman-continuum escape probability is small
(Chluba & Sunyaev, 2007, 2009a). Therefore, photons escaping from the Lyman-
continuum will only lead to a very small distortion in the Wien tail of the
CMB blackbody, which we shall neglect here. We thus assume that photons
injected in excess of the CMB blackbody can be efficiently absorbed, but will
not be re-distributed or re-emitted. The kinetic energy of the liberated
electron is quickly thermalized by Coulomb interactions and thus leads to
heating of the medium. We will we show that for photons below a certain
critical energy this is a good approximation. We thus approximate Eq. (9) by

$\displaystyle\frac{\partial n_{x}}{\partial t}\Bigg{|}_{\rm Ly-c}\approx-
N_{\rm 1s}\sigma_{\rm 1s}\,c\,\Delta n_{x}$ (10)

only accounting for the absorption of the distortion $\Delta n(x)=n(x)-{n^{\rm
pl}}(x)$ with respect to the CMB blackbody, ${n^{\rm pl}}(x)=[e^{x}-1]^{-1}$.
This leads to a matter heating term

$\displaystyle\frac{\partial\rho_{\rm m}}{\partial t}\Bigg{|}_{\rm Ly-c}$
$\displaystyle\approx\frac{8\pi}{c^{2}}\int^{\infty}_{\nu_{\rm 1sc}}N_{\rm
1s}\sigma_{\rm 1s}\,h[\nu-\nu_{\rm 1sc}]\,\nu^{2}\Delta n_{\nu}{\,\rm d}\nu$
$\displaystyle=\frac{\rho_{\gamma}^{\rm pl}(T_{\gamma})}{G^{\rm
pl}_{3}}\int^{\infty}_{x_{\rm 1sc}}cN_{\rm 1s}\sigma_{\rm 1s}[x-x_{\rm
1sc}]\,x^{2}\Delta n_{x}{\,\rm d}x,$ (11)

where $x=h\nu/kT_{\gamma}$ and $\nu_{\rm 1sc}$ is the ionization frequency
(the subscript 1sc denotes 1s to continuum). In particular at low redshifts in
the post-recombination era, this term will be crucial, as it leads to a strong
thermal coupling of high energy photons to the matter, thereby creating
$y$-type distortions that would otherwise not appear. We also include the
extra ionizations into the rate equations for the ionization history
calculation using

$\displaystyle\frac{\partial N_{\rm 1s}}{\partial t}\Bigg{|}_{\rm Ly-c}$
$\displaystyle\approx-\frac{8\pi}{c^{2}}\int^{\infty}_{\nu_{\rm 1sc}}N_{\rm
1s}\sigma_{\rm 1s}\,\nu^{2}\Delta n_{\nu}{\,\rm d}\nu$
$\displaystyle=-\frac{N_{\gamma}^{\rm pl}(T_{\gamma})}{G^{\rm
pl}_{2}}\int^{\infty}_{x_{\rm 1sc}}cN_{\rm 1s}\sigma_{\rm 1s}\,x^{2}\Delta
n_{x}{\,\rm d}x,$ (12)

which again can lead to significant changes in the ionization history for
decay with energies above $\simeq 13.6\,{\rm eV}$.

####  2.2.1 Scattering of high energy photons by neutral atoms

It is well-known that for energies far above the ionization threshold of the
atoms, photons do not distinguish between free or bound electrons (Eisenberger
& Platzman, 1970; Sunyaev & Churazov, 1996; Houamer et al., 2020). Therefore,
the interaction cross section for scattering and photo-ionization essentially
become indistinguishable and approach the Compton scattering cross section for
free electrons. Treating this problem in full detail is far beyond the scope
of this paper. In particular for photons close to the ionization thresholds,
differences in the way energy and momentum are redistributed will affect some
of the scattering dynamics that in principle requires an independent Fokker-
Planck treatment.

However, one can estimate the photon energy above which the scattering by
bound electrons becomes similar to a Compton event by computing the energy
that is transferred to the electron. At high photon energies, in each
scattering event the energy transfer is dominated by electron recoil $\Delta
E\simeq(h\nu)^{2}/m_{\rm e}c^{2}$ (Sunyaev & Churazov, 1996). Equating this to
the ionization energy, yields the critical photon energies $h\nu_{\rm
crit}\simeq 2.6\,{\rm keV}$ for hydrogen, $h\nu_{\rm crit}\simeq 3.5\,{\rm
keV}$ for He I and $h\nu_{\rm crit}\simeq 5.3\,{\rm keV}$ for He II. Below
these energies, we can assume that for neutral atoms only photo-ionization
matters, while above, we may assume that neutral atoms directly contribute to
Compton scattering. In this case, a small correction to the Compton heating
should be added for bound electrons, as they lose the ionization energy
without actually heating the medium; however, we shall neglect this effect in
our computations. This will slightly affect the results for post-recombination
injection above these thresholds, however, not by order of magnitude.

###  2.3 Including collisions

Collisional processes can directly affect the recombination dynamics. This
usually has a minor effect in the standard calculation (Seager et al., 2000;
Chluba et al., 2010; Chluba & Ali-Haïmoud, 2016), but during the dark ages and
reionization in particular collisional ionization can become important (Chluba
et al., 2015). Since the photon injection scenarios considered here can lead
to significant heating, the contributions from collisions between free
electrons and ions are indeed found to become relevant for injections
associated with large heating (see Sect. 4.5.1).

To estimate the collisional ionization rates, $R_{{\rm 1sc}}^{\rm coll}$, due
to electron impact, we use the fits given for hydrogen and helium given by
Bell et al. (1983). At low temperatures, these rates are often simplified
(e.g., see expressions given in Chluba et al., 2015), but here we include the
more broadly applicable representations of Bell et al. (1983), which correctly
capture the decrease of the collisional coefficients at high temperatures.
This improvement is found to be noticeable during the dark ages. For
collisions connecting to the continuum, one has the net rate

$\displaystyle\Delta C_{{\rm 1sc}}^{\rm coll}\approx R_{{\rm 1sc}}^{\rm
coll}\,N_{\rm e}\,\left[N^{\rm eq}_{\rm 1s}-N_{\rm 1s}\right],$ (13)

where we only consider transitions from the ground-state. If requested, the
required rate equations are added to CosmoTherm to allow following the
ionization history. We note that here we use rates that account for thermal
population of electron and omit potential non-thermal electron contributions.

Collisions lead to extra ionizations, while collisional recombination is only
relevant in the pre-recombination era, since three-body interaction require
high densities. The liberated electron carries some excess kinetic energy,
which is quickly thermalized inside the baryonic plasma. However, the energy
required for the ionization event causes a net cooling of the baryons and is
added as a heat sink to the electron temperature equation.

##  3 Main domains for the solution

The main properties of the solutions can be understood by considering the fate
of photons injected at a single frequency and time. For the pre-recombination
Universe, this was already done previously (Chluba, 2015). Here, we extend
discussion to lower redshifts and also consider the energy exchange for
energetic photons more carefully, including ionizations and thresholds for the
production of non-thermal electrons. The resulting domains are summarized in
Fig. 4 and will be explained in the proceeding sections.

Figure 3: Absorption optical depth at low frequencies with and without
reionization. The medium becomes optically thin before reionization,
explaining the plateaus at $z\simeq 20-1000$.  Figure 4: Regions of large
opacity ($\tau\geq 1$) to free-free absorption (blue; see Sect. 3.1), hydrogen
and helium photo-ionization (purple; see Sect. 3.2), electron scattering
(bound and free; red; see Sect. 3.3) and non-thermal electron production
(magenta; see Sect. 3.3). Lines of constant energy in the dimensionless
variable ${x_{\rm inj}}=h{\nu_{\rm inj}}/kT_{\gamma}({z_{\rm inj}})={x_{\rm
inj,0}}/(1+{z_{\rm inj}})$ are shown as brown dashed lines. The pair-
production threshold is around the 1 MeV line. Non-thermal electron production
at $z\lesssim 10^{5}$ is negligible at photon energies $h{\nu_{\rm
inj}}\lesssim 20\,{\rm keV}$ or ${x_{\rm inj,0}}\lesssim 10^{8}$. The standard
ionization history obtained using CosmoRec/Recfast++ with and without
reionization module activated (see Sect. 3.1 for details) was used to compute
the domains. (HI and HeI denote neutral hydrogen and helium, HeII denotes
singly ionized helium.)

###  3.1 Photon absorption at low frequencies

At low frequencies, DC and BR become the dominant processes controlling the
solution. Together, these drive the photon occupation number towards a
blackbody at the electron temperature, ${n^{\rm pl}}(x_{\rm e})=1/[{{\rm
e}^{x_{\rm e}}}-1]$ with $x_{\rm e}=h\nu/kT_{\rm e}$. At low redshifts
($z\lesssim 10^{4}$), Compton scattering becomes extremely inefficient and it
is thus most important to ask at which frequency the medium becomes optically
thick to photon emission and absorption processes. We include both DC and BR
into the estimate, but BR dominates over DC emission at $z\lesssim 4\times
10^{5}$. Neglecting Compton scattering, at low frequencies we have the
evolution equation222For the estimates presented in this section, we neglect
any source terms and also the small changes to the blackbody induced by
differences in the electron and photon temperature. All these effects are
included in the main computation using CosmoTherm. for $\Delta n=n-{n^{\rm
pl}}(x_{\rm e})$

$\displaystyle\frac{\partial\Delta
n}{\partial\tau}\approx-\frac{\Lambda(x_{\rm e},\tau)}{x_{\rm e}^{3}}(1-{{\rm
e}^{-x_{\rm e}}})\Delta n,$ (14)

where ${\,\rm d}\tau=N_{\rm e}\,\sigma_{\rm T}\,c{\,\rm d}t$ determines the
Thomson scattering optical depth and $\Lambda(x_{\rm e})$ describes the photon
production (note the minus sign for absorption) rate by DC and BR, which
depends on the number density of free electrons and ions, as well as the
electron temperature (see Chluba & Sunyaev, 2012; Chluba, 2015, for more
details). For estimates, we use the standard solutions from CosmoRec/Recfast++
(Chluba & Thomas, 2011) for the ionization history, distinguishing scenarios
with and without reionization at $z\simeq 10$. To compute the BR emission
coefficient, we use BRpack (Chluba et al., 2020a) for the free-free Gaunt
factor. Reionization is modeled as in Short et al. (2020), with a refined
treatment of both singly- and doubly-ionized helium following the approach of
`CosmoSpec` (Chluba & Ali-Haïmoud, 2016).

The relevant characteristics of the solution to Eq. (14) are then determined
by the absorption optical depth (see also Chluba, 2015)

$\displaystyle\tau_{\rm abs}({\nu_{\rm inj}},{z_{\rm inj}})$
$\displaystyle\approx\int_{0}^{{z_{\rm inj}}}\frac{\Lambda\left(x_{\rm
e}(z^{\prime}),z^{\prime}\right)}{x_{\rm e}^{2}(z^{\prime})}\frac{N_{\rm
e}(z^{\prime})\sigma_{\rm T}c{\,\rm
d}z^{\prime}}{H(z^{\prime})(1+z^{\prime})}$ (15)

where ${z_{\rm inj}}$ is the injection redshift and ${x_{\rm inj}}=h{\nu_{\rm
inj}}/kT_{\gamma}({z_{\rm inj}})$ is determined by the injection frequency,
${\nu_{\rm inj}}$. Since we are considering late times, $T_{\rm e}$ can depart
significantly from $T_{\gamma}=T_{\rm CMB}(1+z)$. This effect has to be
included when carrying out the optical depth integral, and amounts to writing
$x_{\rm e}(z)={x_{\rm inj}}(z)\,\phi(z)$ with $\phi=T_{\gamma}/T_{\rm e}$
given by CosmoRec/Recfast++.

In Fig. 3, we illustrate the absorption optical depth for some examples. The
cases without reionization are similar to those presented in Chluba (2015),
but here we used BRpack for the BR Gaunt factors333We assumed that the
electron temperature never drops below $T_{\rm e}=1\,{\rm K}$ to avoid
unphysical contributions. and also keep all non-linear terms in $x_{\rm e}$.
Reionization significantly increases the free-free opacity at $z\lesssim 10$,
causing photons to be efficiently absorbed at $x\lesssim{\rm few}\times
10^{-8}$ for all redshifts. Quantitatively, this is shown in Fig. 4, where we
highlight the domain with $\tau_{\rm abs}\geq 1$ as a function of ${x_{\rm
inj}}$ and ${z_{\rm inj}}$. Photons injected inside this domain are quickly
converted into heat, sourcing $\mu$ and $y$-distortions but also hindering
electrons from recombining. Both aspects can be used to place limits on these
cases.

###  3.2 Absorption of photons at high frequencies

The photo-ionization optical depth for the hydrogen atoms can be computed by
(e.g., Chluba & Sunyaev, 2007)

$\displaystyle\tau_{\rm Ly-c}({\nu_{\rm inj}},{z_{\rm inj}})$
$\displaystyle=\int_{z_{\rm min}}^{{z_{\rm inj}}}\frac{N_{\rm
1s}(z^{\prime})\sigma_{\rm 1s}\Big{(}{\nu_{\rm
inj}}\frac{(1+z^{\prime})}{(1+{z_{\rm inj}})}\Big{)}\,c{\,\rm
d}z^{\prime}}{H(z^{\prime})(1+z^{\prime})},$ (16)

where $z_{\rm min}=\max[0,\nu_{\rm 1sc}(1+{z_{\rm inj}})/{\nu_{\rm inj}}]$.
Similar expressions apply for He I and He II (e.g., Chluba & Sunyaev, 2010).
To estimate when photo-ionization processes are important we again compute the
domains with $\tau\geq 1$. The corresponding regions for H I, He I and He II
are shown in Fig. 3. In contrast to the free-free process, for each species a
low- and high-energy boundary appears: the low-energy boundary is determined
by photons never being above the H I photon-ionization threshold of $\simeq
13.6\,{\rm eV}$ (curved region in the post-recombination Universe) or
redshifting below this threshold before significant absorption can occurs
(vertical part on the low energy side). The high-energy boundary is determined
by the fact that on their journey through the Universe the photons never reach
close enough to the corresponding continuum threshold to be absorbed.444When
computing the opacity, we only included redshifting along the photon
trajectory. In principle we should also add the effect of electron recoil
which is relevant at early times and high energies. This would change the
shape of the domain slightly, essentially tilting the vertical parts in Fig.
4. Broadly speaking, this means that photons injected at ${x_{\rm inj}}\gtrsim
10^{6}$ and $z\gtrsim 1$ do not lead to significant ionizations.

As also visible from Fig. 4, without reionization, the overall optically-thick
domain is a slightly bigger (dashed-purple line) since after recombination
neutral H I and He I atoms are abundant at all times. Including reionization,
allows the medium to become optically-thin to H I photo-ionization at
redshifts $z\lesssim 0.3$, although this transition depends on the specific
reionization model. The biggest region is determined by neutral helium, while
He II photo-ionization is only important in the pre-recombination era.

To be mainly absorbed in the H I Lyman-continuum, photons have to be injected
in a narrow region within a factor of $\nu^{\rm HeI}_{\rm 1sc}/\nu^{\rm
HI}_{\rm 1sc}\simeq 1.8$ of the Lyman-continuum threshold energy. However, the
full optically-thick H I photo-ionization region overlaps significantly with
the He I region (dotted-line visible within the He I domain) although He I
photo-ionization usually occurs more rapidly.

###  3.3 Energy exchange at high frequencies

The last aspect we are interested in is related to the cooling of particles at
high energies. In particular, we want to ask the question for which initial
photon energy a significant amount of non-thermal electrons is created,
requiring another treatment. To estimate this, two steps are necessary: we
first have to estimate for which energy of the electron cooling is dominated
by interactions with photons rather than Coulomb interactions. This defines
the kinetic energy threshold for electrons, $E_{\rm nt}$, to remain non-
thermal for a significant time. The next question then is what minimal energy
an injected photon produces a Compton electron above this threshold, $E_{\rm
nt}$.

To determine $E_{\rm nt}$, we simply have to compare the Coulomb scattering
rates with the Compton cooling rates at each redshift. In both cases, the
thermal particles provide the targets for the non-thermal electron to scatter
with, as the scattering between non-thermal particles has a very low
probability. The $e$-$p$ Coulomb scattering rates at a given temperature are
several orders of magnitudes lower than the $e$-$e$ scattering rates and are
thus neglected (Stepney, 1983; Dermer & Liang, 1989). For the $e$-$e$ Coulomb
scattering rates, we use expressions from Dermer & Liang (1989), which are
valid up to mildly relativistic temperatures of the thermal particles. We
first reproduced Fig. 1 of Dermer & Liang (1989) and then took the limit to
low temperatures. At kinetic energies of a few ${\rm keV}$, the energy
exchange rate obtained then becomes roughly independent of the plasma
temperature and is well approximated by

$\displaystyle\frac{{\,\rm d}\epsilon_{ee}}{{\,\rm d}\tau}$
$\displaystyle\approx-21.0\,\frac{\left(1+0.570\,\epsilon_{\rm
kin}-1.745\times 10^{-2}\epsilon_{\rm kin}^{2}\right)}{\sqrt{\epsilon_{\rm
kin}}}\,\left[\frac{\ln\Lambda}{20}\right],$ (17)

where we have used the Coulomb logarithm $\ln\Lambda=20$ as a fiducial value
and expressed the kinetic energy in units of the electron rest mass,
$\epsilon_{\rm kin}=E_{\rm kin}/m_{\rm e}c^{2}$. Equation (17) works well when
the kinetic energy of the projectile electrons is much larger than the typical
thermal energy of the background electrons, and should be valid up to $E_{\rm
kin}\simeq 10\,{\rm MeV}$. To lowest order, this agrees with the non-
relativistic result of Haug (1988, see Eq. 19 therein) but it roughly a factor
of 2 lower than what is given in Swartz et al. (1971).

As for the Compton scattering of photons by bound electrons, we can again
assume that above the threshold energies of $\simeq 2.6\,{\rm keV}$,
$3.5\,{\rm keV}$ and $5.3\,{\rm keV}$ (see Sect. 2.2.1) for the three atomic
species $e$-$e$ Coulomb scattering occurs in the same way whether the electron
is bound or free.555For $e$-$p$ scattering, the corresponding thresholds are
enhanced by a factor of $m_{\rm p}/m_{\rm e}\simeq 1836$, due to the reduction
of the recoil effect on protons. At energies below these thresholds, we also
expect collisional ionization and excitation to contribute, further increasing
the rate at which electrons loose their energy, however, these are neglected
for our estimates.

The $e$-$e$ Coulomb cooling rate has to be compared to the energy loss rate of
non-thermal electrons on the CMB blackbody. This can be approximated as
(Blumenthal & Gould, 1970)

$\displaystyle\frac{{\,\rm d}\epsilon_{e\gamma}}{{\,\rm d}\tau}$
$\displaystyle\approx-\frac{4}{3}\,\frac{(\gamma^{2}-1)}{N_{\rm
e}}\,\frac{\rho_{\rm CMB}}{m_{\rm e}c^{2}},$ (18)

where we neglected relativistic corrections (for additional approximations see
Sarkar et al., 2019) and set $\rho_{\rm CMB}\approx 2.7kT_{\rm CMB}N_{\rm
CMB}$. The Lorentz factor $\gamma$ furthermore yields
$\gamma^{2}-1=\epsilon_{\rm kin}(2+\epsilon_{\rm kin})$.

Figure 5: Domains of importance for production of non-thermal electrons. In
blue we show the domain in which an energetic electron cools by Compton
scattering of CMB blackbody photons. The magenta region marks the domain in
which an energetic photon can up-scatter an electron into the Compton cooling
domain, i.e., $E_{\rm kin}>E_{\rm nt,e}$. The energy required for the photon
is typically more than an order of magnitude larger than $E_{\rm nt,e}$.

In Fig. 5, we show the critical energy, $E_{\rm nt,e}$, at which the $e$-$e$
Coulomb cooling rate (i.e., Møller scattering) equals the Compton cooling
rate. For illustration, we show the difference when only considering Coulomb
scattering off of free electrons, which greatly underestimates the total loss
rate in the post-recombination era.

We next ask the question what initial photon energy, $E_{\rm nt,\gamma}$, is
required to produce a non-thermal electron above the critical energy $E_{\rm
nt,e}$ in a single Compton scattering event. To compute the corresponding
energy exchange we use CSpack (Sarkar et al., 2019, Eq. 16b). For $E_{\rm
nt,\gamma}\ll m_{\rm e}c^{2}$, to leading order this implies the condition
$E_{\rm nt,e}\approx E_{\rm nt,\gamma}^{2}/m_{\rm e}c^{2}$; however, Klein-
Nishina corrections become important at low redshifts, where $E_{\rm
nt,e}\gtrsim 1~{}{\rm MeV}$.

The domain above the critical photon energy, $E_{\rm nt,\gamma}$, is
illustrated in Fig. 5 and also Fig. 4 (magenta). Photons injected inside the
magenta regions would caused the production of non-thermal electrons, which
then through subsequent Compton scattering cause non-thermal distortion
corrections (Enßlin & Kaiser, 2000; Colafrancesco et al., 2003; Slatyer, 2016;
Acharya & Khatri, 2019). In the calculations presented here, we avoid the
production of non-thermal electrons by restricting ourselves to photon
energies $h{\nu_{\rm inj}}\lesssim{20\,{\rm keV}}$. This also avoids
complications related to the expected soft photon production by DC from the
high energy particle cascade (Ravenni & Chluba, 2020).

We furthermore point out that the Universe becomes transparent to high energy
photons for Compton interactions even in the pre-recombination era (outside of
red region, Fig. 4). This implies that for photon energies above $E_{\rm
nt,\gamma}$, no Compton scattering event may occur within a Hubble time.
However, many other processes (e.g., pair-production and photon-photon
scattering) become important in that regime (Svensson, 1984; Zdziarski &
Svensson, 1989; Chen & Kamionkowski, 2004; Padmanabhan & Finkbeiner, 2005),
but these refinements are avoided for the cases considered here.

###  3.4 Anticipating the final spectrum

Now that we determined all critical regions for photons injected at various
redshifts, we can already anticipate the main features of the solutions. For
this step, Fig. 4 provides considerable insight. The crucial aspect is that
for a given injection energy, the photon source moves along lines ${x_{\rm
inj}}(z)={x_{\rm inj,0}}/(1+z)$ (brown lines in Fig. 4). Assuming a certain
lifetime of the particle or excited state, one can determine the redshift at
which most of the photons are injected (see Fig. 2). Moving along the
corresponding trajectory ${x_{\rm inj}}(z)$ then explains what general
features the final spectrum will have.

For combinations of particle lifetimes and injection energies that mainly
target the white areas in Fig. 4, a spectral distortion that closely tracks
the time-dependence of the injection process is expected. In this case, the
direct constraint from spectrometers will apply and the distortion is not well
represented by a simple $y$\- or $\mu$-type distortion, but rather has a form

$\displaystyle\Delta I^{\rm inj}_{\nu}$
$\displaystyle\approx\frac{hc}{4\pi}\,\frac{f_{\gamma}\Gamma_{X}N_{X,0}\exp\left[-\Gamma_{X}t({z_{\rm
inj}})\right]}{H({z_{\rm inj}})},$ (19)

with $1+{z_{\rm inj}}={E_{\rm inj}}/h\nu\geq 1$. This expression can be
obtained as for recombination line emission, where extremely narrow line
injection is assumed (see Rubiño-Martín et al. (2006) or Masso & Toldra (1999)
for a different approach).

If the injection energy is ${x_{\rm inj,0}}\lesssim 10^{-8}$, we can assume
that all the energy is converted into heat. For lifetimes $\tau_{X}\lesssim
10^{13}\,{\rm s}$, this means that the standard $\mu$ and $y$-distortion
constraints from heating apply. For longer lifetimes, a $y$-distortion is
created; however, in this case the ionization history is also directly
affected and, hence, CMB anisotropy constraints apply, as we show below. For
injection at $z_{\rm inj}\gtrsim 3\times 10^{5}$, the resultant
$\mu$-distortion can be estimated analytically using the expressions from
Chluba (2015). If photons are injected primarily at ${x_{\rm inj}}<3.6$, this
can lead to $\mu<0$ (see Sect. 4.3).

If most of the photons are injected in the purple bands of Fig. 4, we again
expect that the energy is quickly converted into heat and hence causes
significant $\mu$ and $y$-type contributions. However, at this time also
direct ionizations of atoms will play a role which again can be constrained
using the CMB anisotropies. Finally, for photons mainly injected in the red
regions of Fig. 4, we expect a partially-Comptonized SD similar to $\Delta
I^{\rm inj}_{\nu}$ but broadened and with small $y$-distortion contributions
due to the associated heating. Focusing at the region ${x_{\rm inj}}\gtrsim
10^{6}$, we can also anticipate that only weak CMB anisotropy constraints can
be derived if injection primarily occurs at $z\lesssim 50$. In this case,
direct constraints from the X-ray background are expected to be more
stringent.

##  4 Distortions in decaying particle scenarios

In this section, we present the solutions for the photon injection problem
from continuous decay at various energies and particle lifetimes. The main
goal here is to illustrate the properties of the solutions with an eye on the
various physical processes. We numerically solve the photon injection problem
using CosmoTherm with the appropriate modifications to account for the effects
discussed above. The main character of the solution can be deduced from Fig. 4
as explained in Sect. 3.4.

Figure 6: Standard distortion signals expected from reionization and the
Hubble cooling. The standard $\mu$ and $y$ distortions for the COBE/FIRAS
limits are shown for comparison. Negative branches are shown as dashed lines.

###  4.1 Cooling and reionization distortion

To set the stage for the photon injection cases, we start with the standard
distortions created in $\Lambda$CDM by adiabatic cooling (Chluba & Sunyaev,
2012) and reionization (Hu et al., 1994). In the calculations presented below,
these signals have to be subtracted in order to reliably estimate the photon-
injection parameters. We do not include any extra heating from the dissipation
of acoustic modes or the cumulative contributions associated with Sunyaev-
Zeldovich effect from galaxy clusters (see Chluba, 2016, for overview) as
these do not affect the overall picture in terms of ionization history or
thermal history. The adiabatic cooling and reionization distortions obtained
with CosmoTherm are shown in Fig. 6 together with the standard $y$\- and
$\mu$-type distortion.

The adiabatic cooling process causes a small negative $\mu$\- and $y$-type
distortion with $\mu\simeq-3\times 10^{-9}$ and $y\simeq-5\times 10^{-10}$,
because the electrons, which are cooling faster than radiation, are
continuously extracting energy from the photons. For our computations of
photon injection distortions the adiabatic cooling process is included to
leave the ionization history at late stages comparable to the standard
CosmoRec computation. In Fig. 6, we can also observe that at very low
frequencies ($x\lesssim 0.1$ or $\nu\lesssim 6\,{\rm\,GHz}$), the standard
$\Lambda$CDM distortions departs notably from the analytic $\mu$ and $y$
formulae (see Sect. 5.2.1 for more details). This is related to free-free
absorption at late times, which has significant time-dependence (e.g.,
Illarionov & Sunyaev, 1974; Chluba & Sunyaev, 2012).
For the reionization distortion, a $y$-distortion with $y\simeq 10^{-7}$ is
created by the late-time heating at $z\lesssim 10$. The same heating leads to
a low-frequency free-free distortion which is visible at $x\lesssim 0.1$ (see
also Cooray & Furlanetto, 2004; Trombetti & Burigana, 2014), indicating that
$T_{\rm e}>T_{\gamma}$. In particular the low-frequency CMB spectrum can be
thought of as an electron thermostat.

Figure 7: Solutions for the spectral distortion created by photon injection
from decaying particles for several masses, $m_{X}$, and lifetimes,
$\Gamma_{X}$. The main features of the solutions can be understood in
combination with Fig. 4 (see Sect 3.4 and 4 for details). In all shown cases,
photo-ionization corrections were neglected and reionization is also not
included. All spectra are normalized such that
$\Delta\rho/\rho|_{\mathrm{inj}}=3\times 10^{-5}$. Negative parts of the
signal are shown as dashed lines.

###  4.2 Numerical spectral distortion results

In Fig. 7 we present several solutions for the spectral distortion created by
photon injection from decaying particles with varying masses and lifetimes.
For now, we omit the effect of atomic photo-ionization and reionization and
reintroduce them in the subsequent sections. All solutions are normalized such
that

$\displaystyle\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\Bigg{|}_{\rm inj}$
$\displaystyle=\int\frac{{\,\rm d}\ln\rho_{\gamma}}{{\,\rm
d}z}\,\mathcal{J}_{\rm bb}(z){\,\rm d}z.$ (20)

is fixed to $\Delta\rho/\rho|_{\mathrm{inj}}=3\times 10^{-5}$. Here, the
distortion visibility function $\mathcal{J}_{\rm bb}(z)\approx{{\rm
e}^{-(z/{z_{\mu}})^{5/2}}}$ with ${z_{\mu}}=1.98\times 10^{6}$ accounts for
the reduction of the distortion amplitude by thermalization processes,
efficient at $z\gtrsim{z_{\mu}}$ (see Sect. 4.3 for more details).

Starting with injections at low frequencies (${x_{\rm inj,0}}=2\times
10^{-5}$), we can see that the overall distortion is dominated by $\mu$ and
$y$-type contributions in the usual CMB bands. This is naturally expected from
the fact that for $z_{X}\gtrsim 50$ photons are mostly injected in the
optically thick BR absorption band (Fig. 4 without reionization), thus always
creating heating. At low frequencies, we can also notice the effect of free-
free emission, which rise the CMB spectrum to the temperature of the
electrons, which for long lifetimes can become noticeable in the EDGES band.
However, none of the direct decay photons are visible in the domains of
observational interest.

Moving to ${x_{\rm inj,0}}=10^{-3}$, we find solutions that are overall
similar to those for ${x_{\rm inj,0}}=2\times 10^{-5}$. However, for late
decay (i.e., rates $\Gamma_{X}<{\rm few}\times 10^{-14}\,{\rm s}^{-1}$), we
now notice the appearance of a direct injection distortion, with a shape
resembling Eq. (19) at $x\lesssim 10^{-3}$. This is even more visible for
${x_{\rm inj,0}}=10^{-1}$: redward of the emission peak, the slope is given by
$\Delta I_{\nu}\propto 1/H({z_{\rm inj}})\propto\nu^{3/2}$, while for
$\Gamma_{X}t_{0}<1$ the normalization scales as $\Delta\rho/\rho\propto
1/\nu_{\rm max}$ (see Fig. 7). Further increasing ${x_{\rm inj,0}}$, enhances
the visibility of this direct injection distortion. In particular, cases with
${x_{\rm inj,0}}\simeq 10^{-3}-10$ (or masses $m_{X}\simeq 0.5\,\mu{\rm eV}$
to $5\,{{\rm meV}}$) and long particle lifetimes, $\Gamma_{X}\lesssim{\rm
few}\times 10^{-14}\,{\rm s}^{-1}$ can be directly constrained with EDGES.

For ${x_{\rm inj,0}}\gtrsim 10-10^{3}$, the $y$\- and $\mu$-type contributions
and direct photon injection emission both play significant roles, and the
constraint on these cases are expected to be dominated by CMB spectrometer
measurements. For ${x_{\rm inj,0}}=10^{5}$, we observe that the distortion
response in the CMB bands becomes extremely small for the longest lifetimes,
since here we did not include any photo-ionization heating and the Universe
essentially is transparent for these energies in the post-recombination era
(Fig. 4 without purple region). Thus very weak distortion constraints are
expected in these cases. For the shortest lifetimes and ${x_{\rm
inj,0}}=10^{5}$, we can furthermore see that the distortion signal is given by
a negative $\mu$-distortion with significantly enhanced amplitude, reaching
$\mu\simeq 10^{-3}$ with our normalization condition. This interesting aspect
will be explained in Sect. 4.3 and is related to differences between
distortions sourced by pure energy release and photon injection (Chluba,
2015).

####  4.2.1 Effect of photo-ionization

As the next step, we include the effect of H I, He I and He II photo-
ionization on the distortion evolution. While it is clear that this will
change the distortions at high frequencies in all cases, this indirect effect
remains small until666In our computation, this transition is more gradual,
since we inject photons in a line with finite width. Furthermore, we slightly
smooth the continuum cross section close to the ionization threshold to ease
the numerical treatment. This does not have a major effect on the main
conclusions. ${x_{\rm inj,0}}\gtrsim 5.8\times 10^{4}$, corresponding to the
ionisation energy for hydrogen. Thus, as is clear from Fig. 4 in cases with
injection at ${x_{\rm inj}}\gtrsim 5.8\times 10^{4}/(1+{z_{\rm inj}})$ in the
post-recombination era, we expect significant extra heating from the absorbed
photons. Indeed, this is visible in Fig. 8, where the heating-related $y$-type
distortion is significantly enhanced over the case without photo-ionization
for ${x_{\rm inj,0}}=10^{5}$ (compare bottom and top rows of plots). Moreover,
an additional enhancement of the low-frequency distortion due to extra free-
free emission appears. Hence, it is clear that the distortion bounds for
decaying particles with rest mass $m_{X}c^{2}\gtrsim 2\times 13.6\,{\rm eV}$
will be noticeably tightened, once Lyman continuum absorption is taking into
account. However, partial transparency of the plasma to photons is restored at
${x_{\rm inj}}\gtrsim{\rm few}\times 10^{6}$ and $z\lesssim 10^{2}$, when the
injection happens so far above the ionization thresholds that no absorption
can occur until today (see Fig. 4).

We highlight that we also included the effect of the ionizations on the
recombination history, since the related modifications affect the thermal
contact between photons and baryons, as we explain below. This was achieved by
adding the corresponding rate equations to the CosmoTherm setup using
CosmoRec/Recfast++. When deriving the constraint in Sect. 5, we assumed that
the effects remain linear to leading order. We will discuss the validity of
this assumption below. We also mention again that here we do not include the
re-emission of photons by recombination processes. These introduce additional
spectral features in the Wien-tail of the CMB (e.g., Chluba & Sunyaev, 2009a),
outside of the regime that is currently directly constrained by COBE/FIRAS,
but otherwise should not affect the results significantly.

Figure 8: Similar to Fig. 7 but for ${x_{\rm
inj,0}}=\\{10^{4},10^{5},10^{6}\\}$ and with H I, He I and He II photo-
ionization taken into account. The differences are mainly visible at high
frequencies for cases with ${x_{\rm inj,0}}\gtrsim 5.8\times 10^{4}$ and
lifetimes $\Gamma_{X}\lesssim 10^{-13}\,{\rm s}^{-1}$. Including continuum
absorption leads to strong photon absorption at high frequencies and a
spectral distortion at low frequencies as discussed in the text. Negative
parts of the signal are shown as dashed lines.  Figure 9: Same as Fig. 7, but
including simple modelling for reionization in addition to H I, He I and He II
photo-ionization. In contrast to the case with only Lyman continuum absorption
included (Fig. 8), we can now observe differences for both very low (${x_{\rm
inj,0}}\lesssim 10^{-3}$) and high frequency injection (${x_{\rm
inj,0}}\gtrsim 10^{5}$) at long lifetimes, $\Gamma_{X}\lesssim 10^{-13}\,{\rm
s}^{-1}$. Negative parts of the signal are shown as dashed lines.

####  4.2.2 Effect of reionization

As a final illustration, we now also include the effect of reionization on the
distortion signal (see Fig 9). The most important difference with respect to
the previous cases is expected at low frequencies, since the free-free
emissivity of the plasma ($\propto N_{\rm e}N_{\rm p}$) is greatly enhanced
once reionization occurs. Indeed, by comparing Fig. 7 with Fig. 9, we notice
that for ${x_{\rm inj,0}}=2\times 10^{-5}$ and ${x_{\rm inj,0}}=10^{-3}$ the
$y$-distortion contribution for post-recombination injection is noticeably
enhanced. This just signifies the fact that the optically-thick domain, due to
free-free absorption, is increased (see Fig. 4, blue region, dashed versus
solid boundary) and conversion into heat is very efficient inside this region.
One important consequence of this is that the heating of electrons by soft
photon injection stays significant for a wider range of masses. Hence,
distortion and ionization history limits should become more stringent.

We note that in our computation the changes of the ionization history are
consistently propagated. Even if the general picture does not change, the
domains estimated in Fig. 4 for the standard ionization history are indeed
modified by these effects. We return to this point below (Sect. 4.5.1) when
discussing various effects related to collisional ionization.

###  4.3 Analytic description of the distortion in the $\mu$-era

Using the Green’s function method of Chluba (2015), we can in principle
describe the spectral distortions created by photon injection in the pre-
recombination era ($z\gtrsim 10^{3}$) analytically. For scenarios with short
lifetimes $\tau_{X}\lesssim 2\times 10^{8}\,{\rm s}$ ($z_{X}\gtrsim 3\times
10^{5}$), the signal is approximately given by a classical $\mu$-distortion
and an analytic treatment is straightforward. For energy release distortions,
it is well known that $\mu\approx
1.401\,\Delta\rho_{\gamma}/\rho_{\gamma}\big{|}_{\mu}$ (Sunyaev & Zeldovich,
1970), where $\Delta\rho_{\gamma}/\rho_{\gamma}\big{|}_{\mu}$ is the effective
energy release during the $\mu$-era

$\displaystyle\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\Bigg{|}_{\mu}$
$\displaystyle=\int\frac{{\,\rm d}\ln\rho_{\gamma}}{{\,\rm
d}z}\,\mathcal{J}_{\mu}(z){\,\rm d}z.$ (21)

with $\mathcal{J}_{\mu}(z)$ describing the $\mu$-distortion visibility (see
Chluba, 2016, for various approximations). Assuming that energy is only
released at $z\gtrsim 3\times 10^{5}$, one has
$\mathcal{J}_{\mu}(z)\approx\mathcal{J}^{*}_{\rm bb}(z)$, where the distortion
visibility function, $\mathcal{J}^{*}_{\rm bb}(z)$, can be approximated using
(Chluba, 2015)

$\displaystyle\mathcal{J}^{*}_{\rm bb}(z)$ $\displaystyle\approx
0.983\left[1-0.0381(z/{z_{\mu}})^{2.29}\right]{{\rm
e}^{-(z/{z_{\mu}})^{5/2}}}$ (22)

with ${z_{\mu}}=1.98\times 10^{6}$, or $\mathcal{J}^{*}_{\rm
bb}(z)\approx{{\rm e}^{-(z/{z_{\mu}})^{5/2}}}$ for even simpler estimates. We
note that for large initial distortions ($\mu\gtrsim 10^{-2}$), this
approximation is no longer valid and the visibility is significantly increased
(Chluba et al., 2020b), however, we do not consider these cases here as they
are not consistent with current observational bounds.

In contrast to energy release distortions, photon injection distortions need
to take the extra photons added by the injection process into account. In this
case, the $\mu$ parameter can be estimated using (see Eq. 15 of Chluba, 2015):

$\displaystyle\mu_{\rm inj}$ $\displaystyle\approx 1.401\int\left[{x_{\rm
inj}}-x_{\rm null}\,\mathcal{P}_{s}({x_{\rm
inj}},z)\right]\,\alpha_{\rho}\frac{{\,\rm d}\ln N_{\gamma}}{{\,\rm
d}z}\,\mathcal{J}^{*}_{\rm bb}(z){\,\rm d}z.$ (23)

Here, $\alpha_{\rho}=\mathcal{G}_{2}/\mathcal{G}_{3}\approx 0.370$ and $x_{\rm
null}=(4/3)\mathcal{G}_{3}/\mathcal{G}_{2}\approx 3.602$. We assume that the
injection of photons occurs at ${x_{\rm inj}}={x_{\rm inj,0}}/(1+z)$, with the
injection rate given by Eq. (4) in our case. In addition, the probability of
low-frequency photons being converted into heat is given by
$\mathcal{P}_{s}(x,z)\approx{{\rm e}^{-x_{\rm c}/x}}$, with the critical
frequency

$\displaystyle x_{\rm c}\approx 8.6\times 10^{-3}\,\sqrt{\frac{1+z}{2\times
10^{6}}}\,\sqrt{1+\left[\frac{1+z}{3.8\times 10^{5}}\right]^{-2.344}},$ (24)

which accounts for contributions from DC and BR.

Figure 10:  Analytic modeling in the $\mu$-era using Eq. (23) to compute the
chemical potential for various lifetimes and injection frequencies. Negative
branches are represented by dashed lines. For comparison we show the $\mu$
values obtained by fitting to the SDs generated using CosmoTherm. All
distortions are normalized such that $\Delta\rho/\rho|_{\mathrm{inj}}=3\times
10^{-5}$. The analytic approximations appear to work well for short lifetimes,
$\Gamma_{X}\gtrsim 10^{-8}\,{\rm s}^{-1}$.  Figure 11: Universal distortion
template for quasi-stable decay scenarios (i.e., $\Gamma_{X}\leq
10^{-20}\,{\rm s^{-1}}$). The results shown here are computed for
$\Delta\rho_{\gamma}/\rho_{\gamma}=3\times 10^{-5}$ and
$\Gamma_{X}=10^{-20}\,{\rm s^{-1}}$. Negative parts of the signal are shown as
dashed lines. For ${x_{\rm inj,0}}\leq 10^{-4}$, we can observe a noticeable
high-frequency $y$-distortion due to heating via free-free absorption at low
frequency. The case ${x_{\rm inj,0}}=10^{-10}$ essentially coincides with that
for ${x_{\rm inj,0}}=10^{-8}$ at all frequencies, but close to $x\simeq
10^{-8}$, where for ${x_{\rm inj,0}}=10^{-8}$ a direct injection distortion is
visible. For all cases with ${x_{\rm inj,0}}\leq 1$, the abrupt low-frequency
drop in the distortion redward of their respective peaks stems from free-free
absorption, while the upper boundary simply marks the end of photon injection
within the age of the Universe.

The most important difference to distortions created purely by heating is that
injections at $x\lesssim x_{\rm null}$ can cause a negative chemical potential
(Chluba, 2015). This is because the redistribution of photons over the full
CMB spectrum on average requires more energy than was added. In addition for
injection at $x\lesssim x_{\rm c}/8$ (Chluba, 2015), the photon absorption
process becomes so rapid that the added photons have no time to contribute to
the shaping of the spectrum at high frequencies, thus essentially generating
heat and again a positive chemical potential.

These aspects are recovered in Fig. 10, were we compare the analytic results
for $\mu$ directly with the solutions from CosmoTherm. For lifetimes
$\Gamma_{X}\gtrsim 10^{-8}$, the estimates reproduce the numerical result very
well. Both at very low and very high energies we find $\mu\simeq
1.4\,\Delta\rho/\rho\simeq 4.2\times 10^{-5}$ as expected from pure energy
release. At intermediate injection frequencies, $\mu$ becomes negative and is
significantly enhanced. This is because for fixed $\Delta\rho/\rho$, the
corresponding $\Delta N/N$ is enhanced by a factor of $f\simeq 2.7(1+{z_{\rm
inj}})/{x_{\rm inj,0}}$, which can become large. For example, for ${x_{\rm
inj,0}}=10^{4}$ and $\Gamma_{X}=10^{-8}\,{\rm s}$, we have ${z_{\rm
inj}}\simeq 4.1\times 10^{5}$ and hence $f\simeq 110$. Assuming pure photon
injection, we then have $\mu_{\rm inj}\approx-1.9\Delta
N/N\approx-1.9\,f\,\Delta\rho/\rho\approx-6.3\times 10^{-3}$, which is in very
good agreement with Fig. 10. This also explains the enhancements seen in Fig.
7 through 9 for cases with ${x_{\rm inj,0}}>10^{3}$. Needless to say, these
cases are already ruled out by COBE/FIRAS.

###  4.4 Universal distortion shapes for ultra-long lived particles

For extremely long lifetimes, significantly exceeding the age of the Universe,
i.e., $\Gamma_{X}\lesssim 2\times 10^{-18}\,{\rm s}^{-1}$, the decay process
never enters the exponential phase of the evolution (i.e., $\Gamma_{X}t_{0}\ll
1$), such that ${\,\rm d}\ln\rho/{\,\rm d}t\propto\epsilon f_{\rm
dm}\Gamma_{X}/(1+z)$ [cf., Eq. (2.1)]. In this case, a universal distortion
shape is obtained, which becomes only a function of ${x_{\rm inj,0}}$, such
that we may write

$\displaystyle\Delta I_{\nu}({x_{\rm inj,0}},\epsilon f_{\rm
dm},\Gamma_{X})\approx\epsilon f_{\rm
dm}\Gamma_{X}\,\Delta\hat{I}_{\nu}({x_{\rm inj,0}}).$ (25)

Here, the distortion template, $\Delta\hat{I}_{\nu}({x_{\rm inj,0}})$, is
obtained numerically for a sufficiently long lifetime (in practice we use
$\Gamma_{X}^{\rm ref}=10^{-20}\,\mathrm{s}^{-1}$). To obtain constraints on
quasi-stable particle decays we can use the universal distortion template to
determine $f_{\rm dm}$ from the data by simply rescaling the solution for
$\Gamma_{X}^{\rm ref}$.

The distortion created by extremely long-lived particle is essentially formed
by the combined action of photon injection, free-free absorption at low
frequencies and photo-ionization of neutral atoms at high frequencies.
Electron scattering can be neglected in terms of redistributing photon in
energy, but has to be included for the thermal balance with electrons. In Fig.
11, we show the obtained universal distortion template for various values of
${x_{\rm inj,0}}$. Injection at low frequencies leads to significant heating
while for ${x_{\rm inj,0}}>10^{-4}$ mostly the direct distortion becomes
visible. The free-free absorption edges are visible as abrupt rise of the
signal redward of the direct injection maxima for ${x_{\rm
inj,0}}=10^{-8},10^{-6},10^{-4},10^{-2}$ and $1$. For cases with ${x_{\rm
inj,0}}\simeq 10^{5}-10^{6}$ (not shown in the figure), photo-ionizations lead
to heating and, hence, increased $y$-distortions.

Figure 12: Ultra long-lived decaying particles for ${x_{\rm inj,0}}=10^{-6}$
and $10^{4}$. The solution in each case is scaled to match the overall
amplitude of the reference case with $\Gamma_{X}^{\rm ref}=10^{-20}\,{\rm
s^{-1}}$. For each ${x_{\rm inj,0}}$, the universal shape is obtained
independent of the lifetime once $\Gamma_{X}\lesssim 10^{-19}\,{\rm s}^{-1}$.

Figure 12 illustrates how the distortion approaches the universal distortion
shape for ${x_{\rm inj,0}}=10^{-6}$ and $10^{4}$ while increasing the
lifetime. The distortion shape freezes once $\Gamma_{X}\lesssim 10^{-19}\,{\rm
s^{-1}}$. One giveaway signature of the universal case is the abrupt drop of
the signal blueward of the direct injection peak, which is simply due to the
fact that the injection process is computed for a finite time. For cases with
lifetimes shorter than the age of the Universe, the onset of the exponential
decay phase is visible in the shape of the signal around the injection maximum
(blue and red lines).

In the upper panel of Fig. 12, we also show the spectra of the CMB at $T_{\rm
CMB}=2.725\,{\rm K}$ and for a $5000\,{\rm K}$ blackbody. For low-frequency
injection, the medium is indeed significantly heated and the direct injection
distortion can exceed the blackbody spectrum by many orders of magnitude.
While current direct distortion constraints do not exclude these cases, the
effect on the ionization history places more stringent limits in this regime.

We note here in passing that the computation for low-frequency injection
($x\lesssim 1$) becomes highly non-linear in the distortion itself. CosmoTherm
includes blackbody-induced stimulated scattering effects (e.g., Chluba &
Sunyaev, 2008, for analytic discussion) but these would need to be augmented
by terms $\propto\Delta n^{2}$ to obtain fully consistent results. This could
lead to Bose-Einstein condensation of photons (Zeldovich & Levich, 1969),
which we cannot include accurately in the current version of CosmoTherm. While
a detailed treatment is beyond the scope of this paper, the general results
are not expected to be changed by this omission. A recent related discussion
can be found in Brahma et al. (2020).

Figure 13: The pre-reionization history with heating from ultra soft photon
injection. Computation were done with CosmoRec/Recfast++. The dotted line
shows the free electron fraction in the case of no injection.

###  4.5 Effects on the ionization history

So far we focused on the final distortion as it would be observed today.
Another important effect of photon injection is the change associated to the
ionization history. For high energy decays, well above the ionization
thresholds of hydrogen and helium atoms ($E>1\,{\rm MeV}$), a high-energy
particle cascade is induced, leading to many secondary particles causing
atomic ionizations, excitations and heating (e.g., Shull & van Steenberg,
1985; Valdés et al., 2010; Slatyer, 2016). This problem has been studied
several times (e.g., Chen & Kamionkowski, 2004; Padmanabhan & Finkbeiner,
2005; Galli et al., 2009; Slatyer et al., 2009). Here we investigate
injections close to the ionization threshold and at low energies. Even in the
latter case, a significant effect on the ionization history can be observed
and, hence, can be constrained using the CMB temperature and polarization
anisotropy, as we explain now.

To have a significant effect on the ionization history and CMB anisotropy,
energy needs to be released at $z\lesssim 1400$, implying relevant lifetimes
$\Gamma_{X}\lesssim{\rm few}\times 10^{-13}\,{\rm s}^{-1}$. Earlier, the
plasma quickly adjusts to the extra energy input, but the ionization history
is hardly affected, such that the only witness of the injection process is the
spectral distortion (e.g. Chluba & Sunyaev, 2009a; Chluba, 2010). If we
consider injections at ${x_{\rm inj,0}}\lesssim 10^{-8}$ we can furthermore be
sure that the main effect is through heating of the medium. In Fig. 13, we
illustrate the results of this calculation for $f_{\rm dm}=10^{-6}$. The
curves were obtained with CosmoRec/Recfast++ (Chluba & Thomas, 2011) including
the extra heating source, Eq. (2.1), in the electron temperature equation.
Since we assume very soft photon injection, no direct ionizations or
excitations of atoms occurs. Thus, the main effect is a reduction of the
recombination rate due to hotter electrons. It is important to note that the
photo-ionization rates are not affected significantly, as was also discussed
in the context of heating from primordial magnetic fields (Chluba et al.,
2015). As expected (see Fig. 13), the heating by soft photon injection causes
a delay of recombination. This induces direct changes to the CMB anisotropies,
which we will constrain using an ionization history principal component
projection method (Hart & Chluba, 2020).

Looking at Fig. 4, even in cases with $\Gamma_{X}\lesssim{\rm few}\times
10^{-13}\,{\rm s}^{-1}$ we expect a gradual reduction of the effect on the
ionization history as we increase ${x_{\rm inj,0}}$ to about $\simeq 10^{-2}$.
This is because a diminishing fraction of the injected energy causes a direct
effect on the electrons, leaving the ionization history mostly unaffected.
This effect can be seen in Fig. 14, where ${x_{\rm inj,0}}=10^{-8}$ shows a
large effect that gradually reduces as ${x_{\rm inj,0}}=1$ is reached. This
behavior continues until ${x_{\rm inj,0}}\simeq 5.8\times 10^{4}$, which
corresponds to the H I Lyman continuum threshold, allowing for direct
ionizations from the ground state. In Fig. 14, we see a significant response
in ionization history caused by both heating and direction ionizations. Aside
from the scenarios for ${x_{\rm inj,0}}=1$, all other cases are already in
tension with current CMB data from Planck, showing how a combination of
anisotropy and distortion measurements provides complementary information
about particle physics.

Figure 14: Changes in ionization history with photon injection from a decaying
particle with $\Gamma_{X}=10^{-13}\,{\rm s}^{-1}$ and
$(\Delta\rho/\rho)_{\mathrm{inj}}=3\times 10^{-5}$ as a function of the
injection energy (upper panel). The corresponding distortion is shown in the
lower panel. The dotted line in the upper panel is case without photon
injection. The computations were performed with CosmoTherm. Figure 15: An
illustration of the modifications to the recombination history due to
collisions. We injected photons at ${x_{\rm inj,0}}=10^{-8}$ for
$\Gamma_{X}=10^{-14}\,{\rm s}^{-1}$ and
$\Delta\rho_{\gamma}/\rho_{\gamma}=3\times 10^{-5}$ switching the effect of
collisions on and off. For reference, we also show the standard ionization
history with reionization.

####  4.5.1 Importance of collisional processes

It turns out that collisional processes play an important role for the
evolution of the distortion and ionization history in particular when
significant injection occurs at very low frequencies. To illustrate the role
of collisions, in Fig. 15 we computed the ionization history for soft photon
injection switching the effect of collisions on and off. Due to the heating by
free-free absorption, collisional ionizations become important and lead to a
strong increase in the free electron fraction, which without collisional
ionizations shows a flat response in the freeze-out tail of recombination.

Figure 16: Non-linear responses in the ionization history around critical
injection frequencies for $\Gamma_{X}=10^{-14}\,{\rm s}^{-1}$ and
$\Delta\rho_{\gamma}/\rho_{\gamma}=3\times 10^{-5}$. For
$10^{-3}\lesssim{x_{\rm inj,0}}\lesssim 1.53\times 10^{-3}$ the free electron
fraction shows intermittent bursts due to photon injection, free-free
absorption and collisional processes.

In several of our computations, we included the effects of collisions to study
the changes in signals. In fact, when transitioning from very soft photon
injection to higher energies, we find a critical, highly non-linear behaviour
of the ionization history, mimicking a phase-transition in the recombination
mode. This is illustrated in Fig. 16, where we compute several histories for
$\Gamma_{X}=10^{-14}\,{\rm s}^{-1}$. The computations for these cases indeed
push the treatment within CosmoTherm to its limit. For injections at ${x_{\rm
inj,0}}\leq 10^{-4}$, as before, the heating leads to extra collisional
ionizations and a significant increase in the free electron fraction at
$z\lesssim 800$. Raising the injection energy to ${x_{\rm inj,0}}\simeq
10^{-3}$, we observe strong intermittent changes in $X_{\rm e}$, with episodes
of very high, followed by more moderate, levels of ionization. The general
intermittent behaviour with varying levels in the number of bursts continues
until ${x_{\rm inj,0}}\simeq 1.532\times 10^{-3}$ is reached, when the
recombination response becomes more moderate and smooth again.

What causes this erratic behavior? While the numerical treatment of this
transition is certainly challenging, we identified the interplay between
photon injection heating and increased collisional ionization as cause.
Photons are efficiently absorbed in the optically-thick regime of the free-
free process. Once crossing over to the optically-thin regime, the response in
the ionization history becomes very sensitive to the injection process.
Absorbed photons are converted into heat, which increases the ionization
fraction through collisions. This in turn increases the amount of free-free
absorption leading to a positive feedback loop. During the critical behavior,
we thus see alternating phases of strong free-free absorption and
recombination.

The onset of this new recombination mode depends on the selected lifetime and
also the amount of photon injection. Changing the lifetime modifies the
critical injection energies. Reducing the photon injection process causes the
non-linear response to stop, as no runaway collisional ionization phase
followed by increased free-free absorption is produced. We were furthermore
unable to produce the critical behaviour without explicitly following the
photon injection process and buildup of low-frequency distortions. Thus, the
full treatment implemented here in CosmoTherm is required.

Since the corresponding ionization histories are already in strong tension
with existing limits from Planck, we avoid the complications due to the
interesting non-linear physics by i) reducing the photon injection rate and
ii) omitting collisions in the main computation. Unless pushed to more extreme
cases with a significant rise in the baryon temperature ($T_{\rm e}\gtrsim
10^{4}\,{\rm K}$), the non-linear recombination mode is not usually excited,
such that this should not be a significant limitation for our main
conclusions.

###  4.6 Blackbody-stimulated decay

Until now, we have neglected the effect of stimulated decay; however, as we
shall see next, it can play an important role in modifying the phenomenology
of photon injection processes. By ‘stimulated decay’ we mean the enhancement
of the decay rate with respect to vacuum case due to the presence of
background photons. This process is only relevant if the coupling to the
photon field is direct, without an intermediate unstable mediator. The net
decay rate then depends on the Boltzmann terms

$\mathcal{F}=f_{X}(1+n_{\gamma})(1+n_{\gamma}^{\prime})-n_{\gamma}n_{\gamma}^{\prime}(1\pm
f_{X}),$ (26)

where $f_{X}$ is the distribution function of the decaying particle. The last
term reflects the inverse process, in which the ‘$+$’ applies for bosonic
particles and ‘$-$’ for Fermions. Again assuming that the particles are cold,
any broadening of the emission line due to thermal motion of the particle can
be neglected and $x\approx x^{\prime}\approx{x_{\rm inj}}(z)$. Collecting
terms, we then have

$\mathcal{F}\approx\begin{cases}f_{X}(1+2n_{\gamma})&\text{(Boson)}\\\
f_{X}(1+2n_{\gamma}+n^{2}_{\gamma})&\text{(Fermion)}\end{cases}$ (27)

with the implicit assumption $n_{\gamma}^{2}\ll f_{X}$. Below we will consider
the bosonic case as an example (see Alonso-Álvarez et al., 2020, for related
discussion in the context of ALPs). Therefore, the decay process in the
stimulated case has a rate
$\Gamma_{X}^{\mathrm{stim}}\approx[1+2n_{\gamma}(x_{\mathrm{inj}})]\Gamma_{X}$,
where $\Gamma_{X}$ is the vacuum decay rate. We note, however, that for the
fermionic case the effects could be significantly stronger, with the
enhancement essentially scaling like $\propto n_{\gamma}^{2}$.

For the ambient photon occupation number we use the Planck law,
$n_{\gamma}(x)=[{{\rm e}^{x}}-1]^{-1}$. For small distortions, this will be
extremely accurate down to redshift $z\simeq 10$, when one expects a
significant radio background to arise due to structure formation. In addition,
for large low-energy injection, one does expect the distortion itself to
induce further emission and potentially even violate the condition
$n_{\gamma}^{2}\ll f_{X}$. Hence, our calculations below are mostly for
illustration. For $x_{\mathrm{inj}}\gg 1$, there are no differences between
the stimulated and vacuum decay cases, since $n_{\gamma}(x_{\mathrm{inj}})\ll
1$. However, for $x_{\mathrm{inj}}\lesssim 1$, the photon occupation number,
$n_{\gamma}(x_{\mathrm{inj}})\approx(1+z)/x_{\mathrm{inj,0}}$, can be large
and lead to very different injection histories.

Taking stimulated decay into account, the time evolution of the particle
number density has same form as in vacuum, provided we change the time
coordinate using
$\mathrm{d}t_{\mathrm{stim}}=[1+2n_{\mathrm{Pl}}(x_{\mathrm{inj}})]\mathrm{d}t$,
which we compute explicitly in CosmoTherm. In this case, we have
$N_{X}^{\mathrm{stim}}(t)=N^{\rm vac}_{X}(t_{\mathrm{stim}})$, where $N^{\rm
vac}_{X}(t)=N_{X,0}\,(1+z)^{3}\exp(-\Gamma_{X}t)$ denotes the vacuum decay
solution. For instance, in the low-frequency regime, ${x_{\rm inj}}\ll 1$,
during radiation domination, we have

$\displaystyle t_{\mathrm{stim}}$
$\displaystyle\approx\frac{2\sqrt{2t}}{{x_{\rm
inj,0}}\Omega_{\mathrm{rad}}^{1/4}\sqrt{H_{0}}}\approx 2.0\times
10^{10}\,\frac{\sqrt{t}}{{x_{\rm inj,0}}}\,{\rm s}^{1/2}.$ (28)

Since
$\dot{N}_{\gamma}=-f_{\gamma}\dot{N}_{X}=f_{\gamma}\Gamma_{X}[1+2n_{\gamma}]N_{X}$,
it follows that the photon injection source term for the spectral distortion
can also be written in the same way as the vacuum case, provided one includes
the enhancement factor for the decay rate and uses
$N_{X}^{\mathrm{stim}}(t)=N^{\rm vac}_{X}(t_{\mathrm{stim}})$.

However, one must be cautious with the normalization condition. Indeed, for a
given $\Delta\rho/\rho|_{\mathrm{inj}}$ the value of $f_{\mathrm{inj}}$ in the
stimulated decay case differs from the vacuum case, affecting $f_{\rm
dm}^{*}=\epsilon f_{\rm dm}$. The modified values can be obtained using the
integral777The standard case is recovered for
$n_{\mathrm{Pl}}(x_{\mathrm{inj}})=0$, also implying $t_{\mathrm{stim}}=t$.

$\displaystyle\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\Bigg{|}^{\rm
stim}_{\rm inj}$
$\displaystyle=\int[1+2n_{\mathrm{Pl}}(x_{\mathrm{inj}})]\frac{{\,\rm
d}\ln\rho^{\rm vac}_{\gamma}}{{\,\rm
d}t}\Bigg{|}_{t=t_{\mathrm{stim}}}\mathcal{J}_{\rm bb}(z){\,\rm d}t$
$\displaystyle=\frac{\mathcal{G}_{3}}{\mathcal{G}_{2}}x_{\mathrm{inj,0}}\,\Gamma_{X}\int\frac{\mathcal{J}_{\rm
bb}(z)}{(1+z)}\,{{\rm e}^{-\Gamma_{X}t_{\mathrm{stim}}}}{\,\rm
d}t_{\mathrm{stim}}.$ (29)

The upper panel of Fig. 17 illustrates the initial conditions in terms of
$f_{\rm dm}^{*}=\epsilon f_{\rm dm}$ for
$\Delta\rho/\rho|_{\mathrm{inj}}=3\times 10^{-5}$. The dashed-dotted curve is
for vacuum decay, while the other curves are with stimulated decay for various
injection frequencies $x_{\mathrm{inj,0}}$ as labeled. As anticipated, there
is hardly any difference for large $x_{\mathrm{inj,0}}$. As
$x_{\mathrm{inj,0}}$ decreases, and the particles decay faster due to
stimulated effects, it is necessary to increase the particle abundance in
order to keep the same value of $\Delta\rho/\rho|_{\mathrm{inj}}$. This can be
understood when realizing that stimulated decays essentially reduce the
effective lifetime of the particle. Thus, for stronger stimulated effects,
injection occurs in the regime where thermalization is extremely effective
($z\gtrsim 2\times 10^{6}$) and the effective
$\Delta\rho/\rho|_{\mathrm{inj}}$ drops rapidly for a fixed value of $f_{\rm
dm}^{*}$.

Figure 17: Effect of stimulated decay on the injection process. Upper panel:
Allowed values of $f_{\rm dm}^{*}=\epsilon f_{\rm dm}$ satisfying the
condition $\Delta\rho/\rho|_{\rm inj}=3\times 10^{-5}$ as function of
$\Gamma_{X}$ for different values of ${x_{\rm inj,0}}$. Lower panel: Energy
release history for a lifetime $\Gamma_{X}=10^{-17}\,\mathrm{s}^{-1}$ and
several values of $x_{\mathrm{inj,0}}$. The maximum of the injection history
moves towards higher redshifts when decreasing ${x_{\rm inj,0}}$ because
blackbody-induced effects accelerate the decay.

The bottom panel of Fig. 17 shows the energy release history for a long-lived
particles with $\Gamma_{X}=10^{-17}\,\,\mathrm{s}^{-1}$ with stimulated decays
(solid lines) compared to the vacuum decay case (dashed dotted line). These
curves are normalized with respect to the total integral, and they include the
effect of the distortion visibility function, which exponentially suppresses
the energy injection in the temperature era, i.e., at $z\gtrsim 2\times
10^{6}$. In the vacuum decay case, the energy release history is independent
of the injection frequency and for the chosen long lifetime mostly occurs at
low redshifts, with a maximum at a redshift $z_{X}$ that can be estimated
using the approximation in Eq. (7). With stimulated decay, the energy release
history directly depends on ${x_{\rm inj,0}}$ because the effective decay
rate, $\Gamma_{\mathrm{stim}}$, becomes a function of the injection frequency.
The ratio $\Gamma_{\mathrm{stim}}/\Gamma_{X}$ is always larger than unity, so
the particles decay faster than in vacuum, which means that the energy
injection happens earlier, with lower ${x_{\rm inj,0}}$ corresponding to
earlier injection. Thus, while an injection was happening mostly in the post-
reionization phase in vacuum, the injection with stimulated decays may now
occur at an earlier time, depending on the injection energy.

In Fig. 18, we illustrate the distortions obtained when including stimulated
decay. These correspond to the cases shown in Fig. 17 for
$x_{\mathrm{inj,0}}=\\{10^{-2},1,10\\}$. For $x_{\mathrm{inj,0}}=10^{-2}$,
most of the injection occurs at the early stage of the $y$-era, while for
$x_{\mathrm{inj,0}}=1$, the recombination/post-recombination era is targeted.

Figure 18: SDs for stimulated and vacuum decay cases. The curves are for
$\Gamma_{X}=10^{-17}\,\mathrm{s}^{-1}$ and
$\Delta\rho/\rho|_{\mathrm{inj}}=3\times 10^{-5}$, with Lyman continuum
absorption and reionizaton included, but collisions switched off.

For relatively high injection energy (see bottom panel in Fig. 18), the
differences between stimulated and vacuum decay are less pronounced, as the
injection occurs roughly at the same time (i.e., post reionization, for
$\Gamma_{X}=10^{-17}\,\,\mathrm{s}^{-1}$) and with a similar time-dependence.
Nevertheless, we see a different slope for the distortions at frequencies
below the peak. As we explained in Sect. 4.2, for vacuum decay the scaling is
$\Delta I\propto x^{3/2}$. With stimulated decay at relatively high redshift
($z\gtrsim 10\,{x_{\rm inj,0}}$ such that $x_{\mathrm{inj}}(z)\ll 1$) the
amplitude of the injection is enhanced by a factor $1+2n_{\gamma}\simeq 2/x$
and hence the scaling of the distortion changes to $\Delta I\propto x^{1/2}$.

####  4.6.1 Critical lifetime for early injection

When the injection occurs at early time, in the $\mu$-era, for redshifts
$z>z_{\mu}=3\times 10^{5}$ we can again describe the distortion analytically,
following the procedure of Sect. 4.3. In the cases without stimulated decay,
this was for photon injection from decaying particles with $\Gamma_{X}\gtrsim
10^{-8}\mathrm{s}^{-1}$, irrespective of the injection frequency,
$x_{\mathrm{inj,0}}$. But in the stimulated decay case, even particles with
relatively low $\Gamma_{X}$ can decay at high redshift, provided
$x_{\mathrm{inj,0}}$ is small enough around the time where most of the decay
occurs.

To estimate the critical vacuum decay rate, we can simply solve the condition
$\Gamma^{\mu}_{X}t_{\mathrm{stim}}(x_{\mathrm{inj,0}},z_{\mu})\approx
10^{-8}\mathrm{s}^{-1}\,t(z_{\mu})$ at $z_{\mu}=3\times 10^{5}$. Using
$t(z_{\mu})\simeq 2.6\times 10^{8}\,{\rm s}$ together with Eq. (28) valid
during radiation domination, we then find $\Gamma^{\mu}_{X}\simeq 8.1\times
10^{-15}\,\mathrm{s}^{-1}\,{x_{\rm inj,0}}$. For
$\Gamma_{X}\gtrsim\Gamma^{\mu}_{X}({x_{\rm inj,0}})$, we can compute the SD
analytically using the expressions of Sect. 4.3, while for
$\Gamma_{X}<\Gamma^{\mu}_{X}({x_{\rm inj,0}})$, we use CosmoTherm.

####  4.6.2 Critical lifetime for late injection

To further limit the computational requirements, it is useful to estimate the
critical lifetime at which one can expect the distortion shape to become
universal. In the vacuum decay case, this was shown to be a good approximation
once $\Gamma_{X}\lesssim\Gamma_{X}^{\rm uni}=10^{-20}\,\mathrm{s}^{-1}$ (see
Sect. 4.4). The same reasoning applies when including stimulated decays simply
using $t_{\mathrm{stim}}(z=0)$ instead of $t_{0}$. As illustrated on Fig. 19,
the stimulated decay time $t_{\mathrm{stim}}(z=0)$ is the same as age of the
universe for $x_{\mathrm{inj,0}}\gtrsim 10$, i.e., roughly $1/H_{0}$. For
lower $x_{\mathrm{inj,0}}$, a simple derivation using the matter domination
relation between time and redshift allows one to find an approximation for the
asymptote, yielding $t_{\mathrm{stim}}(z=0)\approx 4/[H_{0}\Omega_{\rm
m}{x_{\rm inj,0}}]$ (see Fig. 19). Therefore, when including stimulated
effects, for decaying particles with $x_{\mathrm{inj,0}}\gtrsim 10$ we shall
still use $\Gamma_{X}^{\rm uni}=10^{-20}\,\mathrm{s}^{-1}$ as the lower
lifetime, while for particles with $x_{\mathrm{inj,0}}\lesssim 10$ we need to
solve for a wider range of lifetimes. Using fiducial values for $H_{0}$ and
$\Omega_{\mathrm{m}}$, we find $\Gamma^{\rm uni}_{X}({x_{\rm inj}})\approx
10^{-21}\,{\rm s}^{-1}\,x_{\mathrm{inj,0}}$.

Figure 19: Stimulated decay time at $z=0$ as a function of
$x_{\mathrm{inj,0}}$.

##  5 Constraints from COBE/FIRAS, EDGES and CMB anisotropies

In this section, we explain how we use existing data from COBE/FIRAS (Sect.
5.2.1) and EDGES (Sect. 5.2.2) to derive constraints on the decaying particle
scenarios from the previous section. In addition, we consider CMB anisotropy
limits from Planck using a principal component analysis method for changes to
the ionization history (Sect. 5.2.3). We present model-independent constraints
in Sect. 5.3 and then illustrate how to apply the limits to axion and ALP
scenarios in Sect. 5.4.

###  5.1 General setup for the distortion database

Since a single run of CosmoTherm can take several minutes, we generated a
database of distortion spectra for decaying particle scenarios. We can then
load and interpolate the spectra from our pre-computed database and compare
them with measurement of the CMB spectrum in order to extract constraints in a
relatively short time. The overall computational strategy of the distortion
spectra is summarized in Fig. 20 for cases with and without stimulated decay
effects. In the grey domains, a more detailed treatment of non-thermal
particle production is required. We avoid this regime by limiting the
injection energies when extracting constraints.

Figure 20: Strategies to solve the distortion without (upper panel) and with
(lower panel) stimulated decay effects.

As in Sect. 4.2, we have focused on three cases: i) ‘bare’, for which neither
the effects of photo-ionization, nor the effects of reionization were
included; ii) ‘lyc’, for which we included photo-ionizations; and iii)
‘lyc+reio’ where both reionization and photo-ionization were taken into
account. For each setup, we computed $O(10^{4})$ spectra using CosmoTherm for
$\approx 300$ values of the scaled injection frequency $x_{\mathrm{inj,0}}$,
logarithmically spaced between $10^{-8}$ and $10^{7}$, and $\approx 80$ values
of particle lifetime $\Gamma_{X}$, logarithmically-spaced between
$10^{-8}\,\rm{s^{-1}}$ and $10^{-20}\,\rm{s^{-1}}$. By interpolating the
database we can then obtain accurate spectra for any set of values
($x_{\mathrm{inj,0}},\Gamma_{X}$). The computational domain is modified when
considering stimulated decay effects, as illustrated for comparison in Fig.
20.

For $\Gamma_{X}>10^{-8}\,\rm{s^{-1}}$, i.e., short-lived particles, the
injection occurs at high redshift, in the $\mu$-era, and we can use the
analytical formulae of subsection 4.3 to predict the value of the chemical
potential corresponding to the requested para. For $\Gamma_{X}\lesssim
10^{-20}\,\rm{s^{-1}}$, i.e., long-lived particles, the lifetime is larger
than the age of the universe, and the universal SD shape described in 4.4 can
be used. The range of injection energy, i.e., $x_{\mathrm{inj,0}}$ is chosen
such that it covers the wide phenomenology associated with the regimes in
highlighted in Fig. 4. The modifications to the short and long-lifetime
regimes when including stimulated decay are also illustrated in Fig. 20.

We mention a few details relevant to the creation of the distortion database.
For baseline calculations with photons injected mostly at ${x_{\rm
inj}}=10^{-5}-150$, we require $\simeq 3000$ frequency points. Depending on
the injection frequency, the grid was extended at low or high frequencies. For
this we used a log-density with $300$ points per decade. Since we did not want
to perturb the background evolution too much, we included the standard heating
and cooling terms in all calculations. As mentioned above, the corresponding
distortion signal had to be subtracted from our result (see Fig. 6). Similar
statements apply to the contributions from reionization. Finally, the
calculated spectra for a given lifetime are then assumed to depend linearly on
the normalization parameter, ${f_{\rm inj}}$ or $f_{\rm dm}$. In detail, this
is not expected to be perfect, because changes to the ionization history can
lead to non-linear responses. However, the results are not affected
dramatically by this assumption, as we discuss below.

When computing cases with long lifetimes, we determine the redshift at which a
fraction (we chose one percent) of the total energy has been injected due to
the decay. We identify this redshift numerically by solving the energy release
history prior to the thermalization calculation and then use the result as our
starting redshift, if it is found to be higher than the decoupling redshift.
Otherwise, we chose the decoupling redshift in order to not miss some
important effects associated with photo-ionizations.

###  5.2 Data sets from various experiments

In this section, we briefly explain how we use data from COBE/FIRAS, EDGES and
Planck to constrain photon injection scenarios. The constraints will be
presented in Sect. 5.3 and 5.4.

####  5.2.1 COBE/FIRAS constraints on $\mu$ and $y$

The COBE/FIRAS monopole measurement spans frequencies between 68.05 GHz and
639.5 GHz ($1.20\leq x\leq 11.26$) in 43 bands888We used the 2005 release of
the COBE/FIRAS monopole spectrum measurement: firas_monopole_spec_v1.txt. This
dataset is absolutely-calibrated and, as of today, still provides the most
stringent constraints on the CMB spectrum (Mather et al., 1994; Fixsen et al.,
1996; Fixsen, 2003). Below, we use this data to derive constraints on photon
injection problems, but as a first step we repeat the calculation for the
constraints on $\mu$ and $y$.

Following Fixsen et al. (1996), the COBE/FIRAS constraints on $\mu$ and $y$
are obtained by fitting the monopole measurement with a blackbody law at a
pivot temperature $T_{0}$, the $\mu$ and $y$ distortions and a galactic
contamination term:

$I(\nu)=B(T_{0})+\Delta T\left.\frac{\partial B}{\partial
T}\right|_{T_{0}}+\mu\left.\frac{\partial
S_{\mu}}{\partial\mu}\right|_{T_{0}}+y\left.\frac{\partial S_{y}}{\partial
y}\right|_{T_{0}}+G_{0}g(\nu).$ (30)

Here, $G_{0}g(\nu)$ is the galactic contamination term with free parameter
$G_{0}$ and frequency dependence characterized by $g(\nu)$. The $\mu$ and $y$
distortions are proportional to the frequency dependent functions

$\frac{\partial
S_{\mu}}{\partial\mu}\Bigg{|}_{T_{0}}=-\frac{T_{0}}{x}\frac{\partial
B}{\partial T}\Bigg{|}_{T_{0}},\;\;\;\;\frac{\partial S_{y}}{\partial
y}\Bigg{|}_{T_{0}}=\left[x\coth\left(\frac{x}{2}\right)-4\right]T_{0}\frac{\partial
B}{\partial T}\Bigg{|}_{T_{0}}$ (31)

where $x=h\nu/kT_{0}$ and

$\frac{\partial B}{\partial
T}\Bigg{|}_{T_{0}}=\frac{B_{0}}{T_{0}}\frac{x\,{{\rm e}^{x}}}{{{\rm
e}^{x}}-1},\quad\mathrm{with}\quad
B(T_{0})=\frac{2h}{c^{2}}\frac{\nu^{3}}{{{\rm e}^{x}}-1}$

denoting the blackbody spectrum at temperature999When $h$, $c$ and $\nu$ are
expressed in standard units, one needs to multiply $B(\nu,T)$ by a factor
$10^{26}$ in order to obtain the intensity $I(\nu)$ in units of Jy/sr.
$T_{0}$. An alternative description of the $\mu$ distortion is

$\displaystyle\mu\,M(x)$ $\displaystyle=\mu\frac{\partial
S_{\mu}}{\partial\mu}\Bigg{|}_{T_{0}}\left(1-\frac{x}{\beta_{\mu}}\right),$
(32)

with $\beta_{\mu}=3{\cal G}_{2}/(2{\cal G}_{1})\simeq 2.1923$ and where photon
number conservation is enforced by $\int x^{2}M(x)=0$. However, the main
modification in this case is a shift in the monopole temperature and the final
results are not significantly affected by this choice. With this formula and
$\mu>0$, the $\mu$ distortion is negative at low frequency and changes sign at
$x=\beta_{\mu}$ or $\nu\simeq 124.5\mathrm{GHz}$ (magenta line on Fig. 21),
while with Eq. (31) it remains negative at all $\nu$ (blue line on Fig. 21).
We note that foregrounds such as synchrotron, anomalous microwave emission
(AME) and free-free are neglected as sub-dominant in the COBE/FIRAS analysis.
These can have important effects on the $\mu$-distortion constraints (Abitbol
et al., 2017), but a more careful reanalysis of the COBE/FIRAS data that also
includes valuable information of galactic foregrounds from Planck and uses
modern foreground separation methods (e.g., Rotti & Chluba, 2020) is beyond
the scope of this work.

Figure 21: Residuals, distortions and galactic contamination term for the
COBE/FIRAS monopole measurement. The green dotted line is the distortion
associated with a temperature shift of $\Delta T=100\mu\mathrm{K}$. For this
figure the pivot temperature is $T_{0}=2.725\,{\rm K}$. Figure 22:  Normalized
data covariance matrix
$Q(\left|\nu-\nu^{\prime}\right|)=C_{\nu\nu^{\prime}}/\sigma_{\nu}\sigma_{\nu^{\prime}}$
as used in the COBE/FIRAS analysis.

All the frequency-dependent functions in Eq. (30) are either fixed or depend
on the pivot temperature $T_{0}$, such that the monopole model is linear with
respect to the free parameters $\Delta T$, $G_{0}$, $\mu$ and $y$. Therefore,
one can use a simple $\chi^{2}$ fit to find the best-fitting parameters and
associated uncertainties, as was done in Fixsen et al. (1996). To ensure that
we use the COBE/FIRAS data in a consistent way to set constraints on photon
injection processes, we first attempted to reproduce the results for the
original analysis for $\mu$ and $y$. The covariance matrix between the 43
frequency bins is given in Fixsen et al. (1996) as
$C_{\nu\nu^{\prime}}=\sigma_{\nu}\sigma_{\nu^{\prime}}Q(\left|\nu-\nu^{\prime}\right|)$,
where the function $Q$ is tabulated in Section 3.3 of the original paper (also
see our Fig. 22). We write
$\chi^{2}=\mbox{\boldmath${D}$\unboldmath}^{T}C^{-1}\mbox{\boldmath${D}$\unboldmath}$
where ${D}$ is the vector of difference between measurement and model monopole
and $C$ is the covariance matrix. For the minimization we used either the
Levenberg-Marquardt algorithm via the curve_fit method of scipy or the Monte
Carlo Markov Chain method (MCMC). Both methods agree, nevertheless we note
that the MCMC method (which typically takes a few seconds to converge) is
always reliable while the curve_fit method fails for some cases when we study
photon injection constraints (see Table 1 for a compilation of all the
results).

Although overall our results on $\mu$ and $y$ are consistent with Fixsen et
al. (1996), we note minor differences which are probably associated with a
slightly different treatment of the galactic contamination term. For the
galactic contamination term $g(\nu)$ we used the data provided in the fourth
column of Table 4 of Fixsen et al. (1996) and fit for the amplitude $G_{0}$
(see dashed black line in Fig. 21). We find $T_{\rm CMB}=2.725\mathrm{K}\pm
10\mu\mathrm{K}$ (68%CL) when we fit for $T_{\rm CMB}$ and $G_{0}$, or for
$T_{\rm CMB},G_{0}$ and $y$. The quoted error here is purely determined by the
statistical noise. When we fit for $T_{\rm CMB},G_{0}$ and $\mu$ we find a
statistical uncertainty of $28\mu\mathrm{K}$ (68%CL) when we use Eq. (31) for
the $\mu$ distortion (blue line in Fig. 21) and $22\mu\mathrm{K}$ (68%CL) when
we use Eq. (32) (magenta line in Fig. 21). Fitting for $T_{\rm CMB},G_{0}$ and
$\mu$ or $y$ yields $\mu=(-1\pm 3.7)\times 10^{-5}$ (68%CL) and $y=(0.2\pm
3.9)\times 10^{-6}$ (68%CL), which is in good agreement with the values quoted
in Fixsen et al. (1996), namely $\mu=(-1\pm 4)\times 10^{-5}$ (68%CL) and
$y=(-1\pm 6)\times 10^{-6}$ (68%CL). Our statistical uncertainties on $\mu$
and $y$ are not sensitive to the choice for the pivot temperature
($T_{0}=2.728\mathrm{K}$ as suggested in Fixsen et al. (1996) or
$T_{0}=2.725\mathrm{K}$, the best-fitting temperature), or the expression used
for the $\mu$ distortion101010We understand that Fixsen et al. (1996) have
used Eq. (31) for the fit with a $\mu$ distortion, although it seems that Eq.
(32) was used in their Fig. 5..

To obtain the final COBE/FIRAS bounds on $\mu$ and $y$, the statistical
uncertainties are supplemented by systematic uncertainties, provided in Fixsen
et al. (1996). The systematic uncertainty for $\mu$ is $1\times 10^{-5}$
(68%CL) and for $y$ it is $4\times 10^{-6}$ (68%CL). Propagating the errors,
we find $\mu<7.7\times 10^{-5}$ (95%CL) and $y<1.1\times 10^{-5}$ (95%CL).
These translates into the energy bounds $(\Delta\rho/\rho)_{\mu}<5.5\times
10^{-5}$ and $(\Delta\rho/\rho)_{y}<4.5\times 10^{-5}$ each at 95%CL. Here, we
applied the relations $(\Delta\rho/\rho)_{\mu}=\mu/1.401$ and
$(\Delta\rho/\rho)_{y}=4y$. Had we considered only the statistical uncertainty
on $y$, the limit for the energy bound from $y$ would be
$(\Delta\rho/\rho)^{\mathrm{stat}}_{y}<3.1\times 10^{-5}$ (95%CL), which
motivates the normalization of the photon injection spectra that we computed.

Our results on $\mu$ and $y$ have to be compared with the well-known values
quoted in Fixsen et al. (1996), namely $\mu<9\times 10^{-5}$ (95%CL) and
$y<1.5\times 10^{-5}$ (95%CL). Our bound on $\mu$ is 15% tighter and our
2-$\sigma$ bound on $y$ is 30% tighter than the original ones. As mentioned
above, we believe the source of this differences is the treatment of the
galactic contamination term. Nevertheless, the differences are relatively
small and our treatment of the COBE/FIRAS seems satisfactory enough for it to
be used to set new constraints on photon injection processes. However, when
presenting results below, we only include the statistical uncertainties in the
analysis. It is not easy to estimate the systematic uncertainties for the wide
range of spectra obtained in the considered scenarios. This likely means that
our main constraints are uncertain by a factor of $\simeq 1.5-2$ due to
COBE/FIRAS systematic uncertainties.

Finally, we also mention the limits that are obtained when simultaneously
varying $\mu$ and $y$ (see last row of Table 1), which yields $\mu=(-3.6\pm
6.5)\times 10^{-5}$ and $y=(3.3\pm 6.9)\times 10^{-6}$ (68% CL). This implies
$(\Delta\rho/\rho)^{\mathrm{stat}}_{\mu}<9.3\times 10^{-5}$ and
$(\Delta\rho/\rho)^{\mathrm{stat}}_{y}<5.5\times 10^{-5}$ (95% CL) for the
energy injection in the $\mu$ and $y$ eras, respectively. Assuming continuous
energy release, e.g., from decaying particles, this can be interpreted at a
limit of $(\Delta\rho/\rho)^{\mathrm{stat}}_{\rm tot}<4.9\times 10^{-4}$
(95%CL) on the total energy release in the pre-recombination era, or about one
order of magnitude weaker than the individual $\mu$ or $y$ distortion limits.
Although here we only included statistical errors, adding the COBE/FIRAS
estimates for the systematic uncertainties does not modify the result
significantly.

####  5.2.2 EDGES measurement

The low-band antenna of EDGES covers frequencies in the range $50-100$ MHz
(Bowman et al., 2018). This frequency range probes the 21cm hyperfine
transition of neutral hydrogen from $z\simeq 27-13$. For a given frequency
$\nu$, or equivalently redshift $(1+z)=\nu_{21}/\nu$, within this range, EDGES
measures the brightness temperature associated with the global 21cm line,
$T_{21}(z)$. Theoretically, the signal can be modeled as (Zaldarriaga et al.,
2004; Pritchard & Loeb, 2012)

$\displaystyle T_{21}(z)$
$\displaystyle=\frac{T_{\mathrm{spin}}-T_{\gamma}}{1+z}(1-\mathrm{e}^{-\tau_{21}}),$
$\displaystyle\approx 0.023X_{\rm
HI}(z)\left[\frac{0.15}{\Omega_{\mathrm{m}}h^{2}}\frac{1+z}{10}\right]^{1/2}\left(\frac{\Omega_{\mathrm{b}}h^{2}}{0.02}\right)\left[1-\frac{T_{\gamma}}{T_{\mathrm{spin}}}\right],$

where $T_{\gamma}$ is the photon brightness temperature around the 21cm
transition, $T_{\mathrm{spin}}$ is the spin temperature of neutral hydrogen
(Field, 1959) and $\tau_{21}$ is the optical depth for the 21cm line, replaced
explicitly in the second line and assumed to remain small for the linear
expression to be a good approximation. Both $T_{\mathrm{spin}}$ and
$\tau_{21}$ depend on $T_{\gamma}$, since $\tau_{21}\propto
T_{\gamma}/T_{\mathrm{spin}}$ and

$T_{\mathrm{spin}}=\frac{x_{\mathrm{rad}}+x_{\mathrm{c}}+x_{\alpha}}{x_{\mathrm{rad}}T_{\gamma}^{-1}+x_{\mathrm{c}}T_{\mathrm{k}}^{-1}+x_{\mathrm{\alpha}}T_{\mathrm{\alpha}}^{-1}},\,\,\mathrm{with}\quad
x_{\mathrm{rad}}=\frac{1-\mathrm{e}^{-\tau_{21}}}{\tau_{21}}.$ (33)

Here, $T_{\mathrm{k}}$ is the kinetic temperature, $T_{\mathrm{\alpha}}$ the
Lyman-$\alpha$ brightness temperature, $x_{\mathrm{c}}$ and $x_{\alpha}$ are
the spin-flip rates due to atomic collisions and resonant scattering of
Lyman-$\alpha$ photons respectively (Wouthuysen-Field effect; Wouthuysen,
1952; Field, 1959). We refer to Venumadhav et al. (2018) and Fialkov & Barkana
(2019) for more thorough discussions on the EDGES results and 21cm physics,
and Panci (2020) for additional brief overview.

Bowman et al. (2018) reported a symmetric U-shaped absorption profile centred
at $78$ MHz, corresponding to 21cm absorption at $z\simeq 17$, with an
amplitude $\simeq-500$ mK and a full-width at half maximum of $20$ MHz. At
redshift $z\simeq 150-200$, Compton scattering keeps the spin and CMB
temperatures coupled. At lower redshifts, baryons and CMB are no longer
thermally coupled so that $T_{\mathrm{gas}}\propto(1+z)^{2}$ (adiabatic
cooling), while $T_{\gamma}\propto(1+z)$. Although Compton scattering may not
be efficient, collisional coupling ensures $T_{\mathrm{spin}}\simeq
T_{\mathrm{gas}}$ until $z\simeq 40$. Hence, at high redshift,
$T_{\mathrm{spin}}\simeq T_{\gamma}\simeq T_{\rm CMB}$ due to Compton
scattering, so no net effect from the 21cm line is expected until $z\simeq
150$. At lower redshift ($40<z<150$), the spin temperature is coupled to the
gas temperature and lower than $T_{\gamma}$, hence leading to an absorption in
the 21cm line. However, this corresponds to frequencies $35-10$ MHz, which are
below the EDGES band. When collisional coupling is inefficient at $z<40$, the
spin temperature is re-coupled to radiation and no absorption of the 21cm line
is expected. At cosmic dawn, $z<20$, the redshifted UV photons emitted during
star formations couple the neutral hydrogen to the gas via resonant scattering
of Lyman $\alpha$ photons and X-ray heating, $T_{\mathrm{spin}}\simeq
T_{\mathrm{gas}}<T_{\rm CMB}$ creating an absorption profile. At lower
redshift $z<13$, the gas becomes hotter than the background radiation due to
large heating and a 21cm emission signal is expected. Finally, no global
signal is expected when the Universe becomes fully reionised, as
$x_{\mathrm{HI}}\simeq 0$ at $z\lesssim 7$.

The absorption trough measured by EDGES is roughly a factor two deeper than
what is expected from standard astrophysics and cosmology, which gives
$T_{\mathrm{21}}\gtrsim-0.21\mathrm{K}$. It means that at $z\simeq 17$ either
the spin temperature is much lower than the standard expectation,
$T_{\mathrm{spin}}\simeq 7\mathrm{K}$, which is a possibility if, for
instance, the gas is cooled down non-adiabatically due to interacting dark
matter particles (e.g., Barkana, 2018; Muñoz & Loeb, 2018), or that the CMB
brightness temperature in the Rayleigh Jeans tail is much higher than
$T_{\gamma}=T_{\rm CMB}(1+17)\approx 50\,\mathrm{K}$ (e.g., Feng & Holder,
2018). Keeping the CMB temperature to the fiducial value, the spin temperature
would need to be $T_{\mathrm{spin}}\approx T_{\mathrm{gas}}\approx 3.5\,{\rm
K}$, to explain $T_{21}\lesssim-0.45\mathrm{K}$. Alternatively, keeping the
spin temperature to the fiducial value, the CMB temperature in the RJ tail
would need to be $T_{\gamma}\simeq 100\,\mathrm{K}$.

Here, we are interested in the second solution, i.e., an excess of photons in
the RJ tail of the CMB radiation with respect to the blackbody law. First
indications of a possible low-frequency excess stems from the measurements of
ARCADE at $\simeq{\rm few}\,{\rm\,GHz}$ (Fixsen et al., 2011; Seiffert et al.,
2011), and is also supported by a recent analysis of the LAW1 data at 40-80
MHz (Dowell & Taylor, 2018). Even before the EDGES measurement, it was already
suggested that the ARCADE signal could be the trace of some partially
Comptonized soft photon emission or injection from decaying or annihilating
particles (Chluba, 2015). A solution for the EDGES result along these lines
has been considered by several authors. For instance, Pospelov et al. (2018)
studied the decay of particles with milli-eV masses and lifetime longer than
the age of the Universe. Similarly, Brahma et al. (2020) considered models
with decaying particles to explain the EDGES observation.

Our main goal here is not to explain the EDGES result but rather to use the
measurement to derive constraints on various models. For our analysis of
photon injection constraints from CMB spectral distortions, we use the EDGES
data in a very simple way: we supplement the COBE/FIRAS measurements with one
data point at frequency $\nu_{\mathrm{EDGES}}=78\,\mathrm{MHz}$ (i.e.,
${x_{\rm inj,0}}\simeq 1.4\times 10^{-3}$), where the CMB brightness
temperature is simply $T\simeq(c^{2}/2k\nu^{2})\,I_{\nu}\lesssim 2T_{\rm CMB}$
at 68% CL, consistent with the above discussion. We do not attempt to model
low-frequency galactic and extra-galactic foregrounds carefully for this part
of the spectrum, but simply use this bound prima facie. This procedure is
adopted in Sect. 5.3 and 5.4 and provides a conservative constraint to the
considered scenarios, but for clarity, we also discuss COBE/FIRAS only limits.

####  5.2.3 Estimating limits from CMB anisotropy data

In addition to the constraints directly derived from COBE/FIRAS and EDGES, we
can consider changes to the CMB anisotropies caused by modifications in the
ionization history. In general this requires running a full MCMC analysis for
each of the models. In real-time this is not feasible as some of the
computations are quite time-consuming. However, possible departures away from
the standard recombination history are already tightly constrained using a
principal component (PC) analysis (Farhang et al., 2012; Farhang et al., 2013;
Calabrese et al., 2013; Planck Collaboration, 2015). Therefore, we can expect
tight limits on photon injection scenarios in the post-recombination and
recombination eras.

In Hart & Chluba (2020), improved PC constraints from Planck 2015 were
presented and a novel PC projection method that allows the derivations of
simple constraints was introduced, without the need to run a full MCMC
analysis for each ionization history. This method relies on the ionization
history mode functions, $E_{i}(z)$, which are computed in the variable
$\zeta(z)=\Delta N_{\rm e}/N_{\rm e}\ll 1$ around the standard ionization
history. Computing $\zeta_{\rm inj}(z)$ from our $N_{\rm e}(z)$ outputs and
then projecting this onto the eigenmodes $E_{i}(z)$ yields

$\displaystyle\rho^{\rm inj}_{i}=\int\zeta_{\rm inj}(z)\,E_{i}(z){\,\rm d}z,$
(34)

for the mode amplitudes. Comparing this to the Planck 2015 limits
$\rho_{1}=-0.08\pm 0.12$, $\rho_{2}=-0.14\pm 0.19$ and $\rho_{3}=-0.30\pm
0.35$, we can then estimate $f^{*}_{\rm dm}$ as follows. Since the covariance
of the PCs is by construction very close to diagonal (Hart & Chluba, 2020), we
find the limiting $f^{*}_{\rm dm}$ (95% CL) by solving111111In the first
version of this manuscript we had assumed a linear scaling of
$\rho_{i}^{\mathrm{inj}}$ with $f^{*}_{\rm dm}$ to simplify the determination
of the 95% CL limit. Here we propose an exact solution, using the root finding
method, which is slightly more computationally expensive but does not rely on
a linear scaling assumption between $\rho^{\rm inj}_{i}$ and $f^{*}_{\rm dm}$.

$\displaystyle 2\left[\sum_{i}\frac{\left[\rho^{\rm inj}_{i}(f^{*}_{\rm
dm})\right]^{2}}{\sigma(\mu_{i})^{2}}\right]^{-1/2}=1,$ (35)

where $\sigma(\mu_{i})=\\{0.12,0.19,0.35\\}$ denotes the measurement errors of
$\mu_{i}$.

Figure 23: Derived 95% CL dark matter fraction limit, $f^{*}_{\rm dm}=\epsilon
f_{\rm dm}$, for a range of lifetimes and ${x_{\rm inj,0}}\lesssim 10^{-8}$.
The CMB anisotropy limits were obtained with the PCA projection method, while
the other two are derived assuming $4y\lesssim 6\times 10^{-5}$ (95% CL) from
COBE/FIRAS. The dotted curve is obtained using Eq. (36), and the solid dashed
line using the total energy integral with the $y$-era visibility function. For
the CMB anisotropy constraints we considered a case where collisional
ionisations are not included (blue region) and a case where these are included
(red region). As can be seen, atomic collisions (see 2.3 and 4.5.1) are
driving post-recombination decay constraints ($\Gamma_{X}\lesssim
10^{-14}\,\mathrm{s}^{-1}$). We emphasize that these constraints are for very
low mass decaying particles ($m_{X}\ll 1\mathrm{eV}$) where all the injected
energy in the form of photons is converted into heat, in contrast with
scenario studied in (Poulin et al., 2017; Lucca et al., 2020) where the energy
is deposited and leads to direct ionisation.

Below we will estimate the CMB anisotropy constraints using Eq. (35). However,
for very low-frequency injection cases causing changes to $N_{\rm e}$ by pure
heating (e.g., see Fig. 13 for illustration of the ionisation history in these
cases), the corresponding constraint can be directly computed using
CosmoRec/Recfast++ and is illustrated in Fig. 23. For comparison we also show
the limit derived from the corresponding $y$-distortion, which was computed
consistently from the recombination output using

$\displaystyle y\approx\int\frac{k[T_{\rm e}-T_{\gamma}]}{m_{\rm
e}c^{2}}\,N_{\rm e}\sigma_{\rm T}c{\,\rm d}t$ (36)

and then assuming $4y\lesssim 6\times 10^{-5}$ (95% CL) [see previous
section]. For these low-frequency injection, where the energy is fully
converted into heat, the constraint obtained from CMB anisotropy data
supersedes the direct spectrometer constraint in all cases with
$\Gamma_{X}\lesssim 6\times 10^{-13}\,{\rm s}^{-1}$. The main reason is that,
due to the large excess of photons over baryons, it is a lot easier to perturb
the ionization history than to affect the energy spectrum of the CMB. This
illustrates the powerful complementarity of CMB anisotropy and spectrometer
constraints.

To obtain the PC projection limit in Fig. 23, we computed the extra heating
assuming all the injected photons are converted by free-free absorption. Then,
for a given lifetime, we solve Eq. (35).

We would like to highlight that in the low-redshift Universe not all the
injected energy creates a distortion signal. As indicated by Fig. 23, assuming
that all the energy reaches the CMB spectrum to create a $y$-distortion yields
significantly tighter limits for lifetimes $\Gamma_{X}\lesssim 10^{-15}\,{\rm
s}^{-1}$ (dashed blue line). This is because the thermal coupling between
electrons and photons reduces at $z\lesssim 200$ and a fraction of the energy
is indeed used to raise the electron temperature well above the CMB
temperature. We also mention that collisional-ionization turned out to be
highly relevant. Without collisions, the changes to the ionization history
were significantly weaker at $\Gamma_{X}\lesssim 10^{-14}\,{\rm s}^{-1}$. The
difference indeed reached two to three orders of magnitude at
$\Gamma_{X}\lesssim 10^{-16}\,{\rm s}^{-1}$ and highlights the importance of
modeling modifications to the ionization history accurately in these non-
standard scenarios (compare red and blue regions on Fig. 23).

We also note that with the same PC projection method we can estimate the
constraints on high-mass particle decays. In this case, we again simply ran
CosmoRec/Recfast++, but including the description of Chen & Kamionkowski
(2004) for the branching ratios into heating, ionization and excitation from
the decay products. The corresponding constraint on $f_{\rm dm}$ was found to
be in reasonable agreement with direct MCMC approaches (Poulin et al., 2017;
Lucca et al., 2020) given the differences in the computations of the energy
branching ratios and deposition efficiencies. Therefore, we believe that the
PC projection method provides an efficient way of estimating the CMB limit
from Planck. However, a more detailed comparison of the two approaches will be
the subject of future work.

Figure 24:  COBE/FIRAS constraints (95% CL) on the effective DM fraction for
decaying particle scenarios with short lifetimes. We use the analytic modeling
in the $\mu$-era given in Eq. (23) to derive these limits. For the longest
considered lifetime, this agrees very well with the full numerical result.

###  5.3 Model-independent constraints for various lifetimes

We now have all the ingredients to compute constraints on various decaying
particle scenarios. We start with model-independent limits and then use the
results for the discussion of specific scenarios. Given the CMB spectrum data
at $N$ frequencies, $\mathbf{I}^{\mathrm{data}}=(I_{\nu_{1}},..,I_{\nu_{N}})$,
we minimize $\chi^{2}=\mathbf{\Delta}^{t}C^{-1}\mathbf{\Delta}$, where $C$ is
the $N\times N$ covariance matrix and
$\mathbf{\Delta}=\mathbf{I}^{\mathrm{data}}-\mathbf{I}^{\mathrm{theory}}$. For
the theory model, we use

$I(\nu)=B(T_{0})+\Delta T\left.\frac{\partial B}{\partial
T}\right|_{T_{0}}+G_{0}g(\nu)+A_{\mathrm{PI}}\,\Delta
I^{\mathrm{PI}}(\nu;x_{\mathrm{inj,0}},\Gamma_{\mathrm{inj}}),$ (37)

and fit for the CMB temperature $\Delta T=T_{\rm CMB}-T_{0}$, the amplitude of
galactic contamination $G_{0}$, and the amplitude of the photon injection
spectral distortion $A_{\mathrm{PI}}$. The photon injection spectral
distortion $\Delta I^{\mathrm{PI}}(\nu;{x_{\rm inj,0}},\Gamma_{X})$ is
obtained at each frequency $\nu_{i=1..N}$ by interpolating the Cosmotherm
database (see section 5.1).

The constraint on the parameter $A_{\mathrm{PI}}$ can then be translated into
a constraint on the effective dark matter fraction $f^{*}_{\rm dm}$, as
follows. The spectra in the database are computed for
$\Delta\rho/\rho\big{|}_{\mathrm{inj}}=3\times 10^{-5}$ which corresponds to a
certain value $f_{\mathrm{inj}}^{\mathrm{ct}}$, via Eq. (20). The value of
$f_{\mathrm{inj}}$ corresponding to $A_{\mathrm{PI}}$ is then given by
$f_{\mathrm{inj}}=A_{\mathrm{PI}}f_{\mathrm{inj}}^{\mathrm{ct}}$ and the dark
matter fraction is then simply obtained using Eq. (2).

Figure 25:  COBE/FIRAS constraints (95% CL) on the effective DM fraction for
decaying particle scenarios with decay rates $\Gamma_{X}\leq 10^{-9}\,{\rm
s}^{-1}$. The upper panels show the limits without reionization and photo-
ionization effects included, while for the lower panels these are switched on.
See main text for discussion.

####  5.3.1 CMB distortion limits: short lifetimes

For particles that decay in the $\mu$-era, i.e., $\Gamma_{X}\gtrsim
10^{-8}\,\mathrm{s}^{-1}$, we can obtain constraints on $f^{*}_{\rm dm}$ based
on the COBE/FIRAS limit on the chemical potential $\mu_{\mathrm{COBE}}$. As
confirmed in Sect. 4.3, for these particles, the distortion is a simple $\mu$
distortion whose amplitude is given by Eq. (23) and is proportional to
$f_{\mathrm{inj}}$. Hence, the value of $f_{\mathrm{inj}}$ that corresponds to
$\mu_{\mathrm{COBE}}$ is given by $f_{\mathrm{inj}}=\mu_{\mathrm{COBE}}/\mu$,
where $\mu$ is computed with Eq. (23) at $f_{\mathrm{inj}}=1$, and the dark
matter fraction $f^{*}_{\rm dm}=\epsilon f_{\rm dm}$ is then obtained using
Eq. (2).

Figure 24 shows the constraints for several short lifetimes. The overall limit
weakens for shorter lifetimes, as expected from the fact that the distortion
visibility (and hence the distortion amplitude) drops. We can also observe the
frequencies at which the final distortion has $\mu\simeq 0$ due to the balance
between number and energy of the injected photons. In the intermediate regimes
between the nulls, the distortion is indeed negative, as explained in Sect.
4.3. Both at very low and very high injection energies, the constraint is
close to what would be expected from simple energy constraints on
$\Delta\rho/\rho$, but generally this estimate fails.

We mention that our computations do not include corrections for the evolution
of large distortions, which can become relevant to cases with
$\Gamma_{X}>10^{-6}\,{\rm s}^{-1}$ (Chluba et al., 2020b). In addition, for
the considered scenarios, the particles need not be DM particles and hence the
bound $\epsilon f_{\rm dm}<1$ can principally be avoided, as long as the
Universe remains radiation dominated. If considering DM as the source of the
photons, another interpretation of the results is related to the excited state
energy, with $\epsilon<1$, as we discuss below.

Figure 26:  Constraints on the effective DM fraction for decaying particle
scenarios (95% CL) for various combinations of data from COBE/FIRAS, EDGES and
Planck. The upper panels show the constraints for lifetimes $\Gamma_{X}\leq
10^{-11}\,{{\rm s}^{-1}}$ and various combinations of datasets when assuming
decay to occur in vacuum. The lower panels show several examples when
including stimulated decay. See main text for discussion.

####  5.3.2 CMB distortion limits: late injection

Next, we consider the COBE/FIRAS only limits on decaying particle scenarios
with $\Gamma_{X}\leq 10^{-9}\,{\rm s}^{-1}$. Our results are summarized in
Fig. 25. We already understand from the discussion in Sect. 4.2 that both
reionization and photo-ionization effects modify the resultant distortion
significantly. To illustrate the differences we compare the results for the
spectra obtained with and without these effects included. For $\Gamma_{X}\geq
10^{-11}\,{\rm s}^{-1}$, the corresponding constraints are not affected
significantly by these processes, while for $\Gamma_{X}\leq 10^{-11}\,{\rm
s}^{-1}$, the high-energy limits are tightened due to photo-ionizations. At
$\Gamma_{X}\leq 10^{-13}\,{\rm s}^{-1}$, we furthermore find the low-energy
constraint to tighten noticeably, simply because reionization affects the
free-free transparency (see Fig. 4).

Considering the case with $\Gamma_{X}=10^{-13}\,{\rm s}^{-1}$ (left panels of
Fig. 25), we see that the constraint shows significant structure around
${E_{\rm inj,0}}\simeq 0.05\,{\rm eV}$. This marks the domain in which the
final distortion at $z=0$ mainly appears redward of the COBE/FIRAS bands, but
above the regime where free-free absorption is highly effective in converting
the injected energy into heat. Remembering that the injection mostly occurs at
${E_{\rm inj}}={E_{\rm inj,0}}/(1+{z_{\rm inj}})$ then also explains why the
limits are strongest blueward of the COBE/FIRAS bands at $z=0$. Moving to
${E_{\rm inj,0}}>0.05\,{\rm eV}$, we observe a complicated pattern that
essentially reflects the variation in the shape of the distortion at $z=0$
across the COBE/FIRAS bands. Finally, once the H I Ly-c threshold is crossed
(visible in the lower panel of Fig. 25), the distortion limit tightens again,
as most of the photons are converted into heat, yielding noticeable $y$-type
contributions directly constrained by COBE/FIRAS.

For $\Gamma_{X}\leq 10^{-13}\,{\rm s}^{-1}$ (right panels of Fig. 25), we
observe a fairly self-similar constraint curve that gradually shifts across
${E_{\rm inj,0}}$ with varying lifetime. For the longest lifetime, we can see
that the significant drop in the limits at ${E_{\rm inj,0}}\simeq 3\times
10^{-4}\,{\rm eV}$ and the feature at ${E_{\rm inj,0}}\simeq 3\times
10^{-2}\,{\rm eV}$ directly coincide with the edges of the COBE/FIRAS bands.
The positions of these features move upward as the lifetime decreases,
reflecting the ${E_{\rm inj}}={E_{\rm inj,0}}/(1+{z_{\rm inj}})$ scaling for
the maximal injection energy. Overall, in both the low- and high- energy
limits only a very small value of $f^{*}_{\rm dm}$ is allowed when photo-
ionization and reionization are included. In particular on the high-energy
end, the constraint is weakened by several orders of magnitude without photo-
ionization effects. Significant holes in the limit from COBE/FIRAS alone exist
redward of the COBE/FIRAS band and below the H I Ly-c threshold energy. The
former will be tightened once EDGES data is included.

We also stress that naive estimate based only on energetic arguments are
generally inaccurate. When considering the low-energy limit of the constraint,
we observe that the limit becomes independent of the injection energy. These
asymptotes define the constraint obtained purely by considering
$\Delta\rho/\rho$ and hence $\mu$ and $y$ distortions. Extrapolating these
asymptotes to higher injection energies confirm our statement, which both
vastly over- and underestimated SD constraints with respect to these
asymptotes (e.g., see upper left panel of Fig. 25). Thus, the detailed
treatment of the thermalization problem presented here is required to obtain
reliable limits.

####  5.3.3 CMB distortion limits: quasi-stable particles

The constraints on $f_{\mathrm{dm}}$ for quasi-stable particles, i.e., for
decay rates