# Constraints on the spectral signatures of superconducting cosmic strings

Bryce Cyr1, Jens Chluba1 and Sandeep Kumar Acharya1
1Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The
University of Manchester, Manchester M13 9PL, U.K E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

If they exist, networks of superconducting cosmic strings are capable of
injecting copious amounts of electromagnetic energy into the background over a
broad range of frequencies. We study this injection both analytically, as well
as numerically using the thermalization code CosmoTherm. With our refined
analytic formalism, we update constraints from CMB spectral distortions by
following the injection of entropy, as well as energy, on the amplitude of the
$\mu$-distortion, leading to a significant improvement in those limits.
Furthermore, we utilize the full shape of the distorted spectrum from
CosmoTherm to include constraints from non-$\mu$, non-$y$ type distortions.
Additionally, we use the outputs for the ionization history and global 21cm
signal to derive and update constraints on string model parameters using
measurements from other datasets. Analysis of CMB anisotropies provides the
most stringent constraints, though with a slightly modified shape and strength
when compared to previous results. Modifications of the reionization history
provide new bounds in the high current domain, and we also find that the
observations of the low-frequency radio background probe a small region of
parameter space not explored by other datasets. We also analyze global $21$-cm
constraints, and find that the inclusion of soft photon heating plays a
crucial role, essentially removing any constraints in the considered parameter
domain. Spectral distortion measurements from COBE/FIRAS are covered by other
constraints, but our conservative forecast shows that a PIXIE-type satellite
would probe important unexplored regions of parameter space.

###### keywords:

Cosmology - Cosmic Microwave Background; Cosmology - Theory

††pubyear: 2023††pagerange: Constraints on the spectral signatures of
superconducting cosmic strings–B

##  1 Introduction

We are well into an age of precision cosmology, with detailed observations
being taken over much of the electromagnetic spectrum. At radio frequencies,
an anomalous background is being uncovered (Fixsen et al., 2011; Dowell &
Taylor, 2018) with no known astrophysical source. Meanwhile, data from the
epoch of reionization is slowly building up, with a first claimed detection of
the differential brightness temperature at cosmic dawn coming from the EDGES
experiment (Bowman et al., 2018). We also have exquisite measurements of the
cosmic microwave background (CMB) anisotropies from the Planck satellite
(Planck Collaboration et al., 2020), as well as the frequency spectrum from
COBE/FIRAS (Fixsen et al., 1996), which teaches us still further about our
thermal history. One of the aims of these cosmological observations is to help
us understand our origins. Here, we will consider how the spectral signatures
of a network of superconducting cosmic strings can be probed using these
observations.

Cosmic strings are a class of topological defect that may form at the
interface of cosmological phase transitions if the true vacuum manifold is
degenerate, and not simply connected (see Brandenberger, 1994; Hindmarsh &
Kibble, 1995; Vilenkin & Shellard, 2000, for comprehensive reviews). If they
form, cosmic strings (also known as line defects) are nearly one dimensional
objects with a small, but finite width. The interior of a string resembles the
state of the universe as it was in the false vacuum, consisting of a
condensate of scalar and gauge particles from this previously unbroken phase.
The detection (or non-detection) of cosmic strings in the various observations
give us pieces of information about our thermal history that would otherwise
be very challenging to infer.

String models exhibit a property known as scaling, where the macroscopic
properties of the string network can be described by one parameter. This
parameter is known as the string tension ($G\mu$), and is related to the
energy scale of the phase transition ($\eta$) through $G\mu\simeq G\eta^{2}$,
where $G$ is Newton’s gravitational constant. It has also been shown that some
symmetry breaking patterns can imbue the strings with superconductive
properties, leading to the generation of significant currents, $\mathcal{I}$,
as the string traverses through the plasma (see Witten, 1985; Ostriker et al.,
1986, for seminal examples). Superconducting cosmic string models are
typically described by these two parameters, $G\mu$ and $\mathcal{I}$.

As we will discuss below, a cosmic string network generally consists of a
small number of long strings which run through our Hubble patch at any given
time, as well as a distribution of smaller loops with curvature radii on all
scales up to some $\mathcal{O}(0.1)$ fraction of the Hubble scale. Although
these string loops act as seeds for density perturbations at early times,
detailed observations of the microwave background have relegated them to only
being a highly subdominant component of structure formation (Perivolaropoulos,
1995; Magueijo et al., 1996; Pen et al., 1997; Albrecht et al., 1997),
requiring $G\mu\lesssim 10^{-7}$. Even so, there are hints that the enhanced
gravitational effects of string loops could play a role in the formation of
supermassive black holes (Bramberger et al., 2015; Cyr et al., 2022).

Superconducting loops are capable of producing copious amounts of
gravitational (Vachaspati & Vilenkin, 1985) and electromagnetic radiation
(Vilenkin & Vachaspati, 1987; Garfinkle & Vachaspati, 1987; Vachaspati, 2008)
during every epoch after their formation. This has led to constraints on the
$G\mu$-$\mathcal{I}$ parameter space from a number of observations, including
primordial CMB spectral distortions (Ostriker & Thompson, 1987; Tashiro et
al., 2012a), anisotropies (Tashiro et al., 2012b), radio transient events (Cai
et al., 2012; Miyamoto & Nakayama, 2013), gamma ray bursts (Babul et al.,
1987), changes in the differential brightness temperature at cosmic dawn
(Brandenberger et al., 2019a; Brandenberger et al., 2019b), and more.

In this work, we revisit some of the constraints considered by these authors
using improved analytic estimates and the thermalization code CosmoTherm
(Chluba & Sunyaev, 2012). First, we recompute the spectral distortion
signature obtained by following an approximate Green’s function approach, as
described in Chluba (2016). Our analytic approach improves upon the previous
treatments in at least two ways. First, we compute the negative
$\mu$-distortion generated by direct photon (entropy) injection into the pre-
recombination plasma (Chluba, 2015), an effect that has been overlooked thus
far but significantly alters the overall constraints. Second, we include a
more precise treatment for the determination of the instantaneous spectrum of
emitted photons.

Following this, we implement the photon injection numerically into the
thermalization code CosmoTherm. This allows us to further strengthen our
constraints by analyzing the full shape of the string induced spectral
distortions, accounting for non-$\mu$, non-$y$ type deviations from the
COBE/FIRAS measurement (Fixsen et al., 1996). This implementation also allows
us to efficiently compare against other independent datasets. Using the
outputs from CosmoTherm, we generate constraints from the CMB anisotropies
(Komatsu et al., 2011; Planck Collaboration et al., 2020), the radio
synchrotron background (RSB) (Fixsen et al., 2011; Dowell & Taylor, 2018), the
EDGES experiment (Bowman et al., 2018), and the optical depth to reionization
as measured by the Planck Collaboration et al. (2020). We also generate a
conservative forecast to a PIXIE-type experiment (Kogut et al., 2011; Kogut et
al., 2016; Chluba et al., 2021), assuming a fiducial sensitivity to energy
release of $\Delta\rho/\rho=10^{-8}$ after foreground marginalization.

While the constraints we find from COBE/FIRAS are less stringent than other
datasets, it is important to stress that in principle, we have had access to
this spectral distortion data since the late 90s. Had a more sophisticated
analysis of the full shape of the COBE/FIRAS data been possible at that time,
the constraints that were derived would have been dominant for many years.
This highlights the legacy value that COBE/FIRAS still has today, and
showcases the constraining power that a next generation space-based
spectrometer such as PIXIE could obtain for exotic energy injection scenarios.

The rest of the paper is organized as follows: in Section 2, we describe the
loop distribution model that we implement, and discuss the microphysics of
gravitational and electromagnetic wave production. Section 3 discusses energy
release rates for the string network, and reviews the simple estimates for the
$\mu$ and $y$ parameters. Afterwards, in Section 4, we refine this estimate by
including entropy release contributions, which changes the analytic constraint
curve significantly. Section 5 describes how to implement this source term
into CosmoTherm, and derives a useful expression for the instantaneous
injection spectrum. In Sections 6 and 7, we discuss the output from CosmoTherm
and illustrate the constraints we derive from multiple different datasets. We
conclude in Section 8.

Throughout the analytic derivations, we use natural units where
$\hbar=c=k_{\rm b}=1$. When presenting the resultant spectra in Sections 6 and
onwards, we use more astrophysicist-friendly units. To minimize confusion, we
include a particle physics to astrophysics conversion dictionary in Appendix
B.

##  2 Number density of cosmic string loops

If the universe undergoes a phase transition in which the true vacuum manifold
permits cosmic string solutions, the Kibble mechanism states that a network of
these defects will form (Kibble, 1980, 1982). Importantly, Kibble’s mechanism
only guarantees the existence of long cosmic strings, with curvature radius
larger than the horizon scale. A second population of smaller, sub-horizon
loops is populated by the intersections and self-intersections of the long
strings.

Numerical simulations in which the widths of cosmic strings are neglected
(known as Nambu-Goto simulations) show that the distribution of sufficiently
large loops follow a scaling solution (Vanchurin et al., 2006; Martins &
Shellard, 2006; Ringeval et al., 2007; Lorenz et al., 2010; Blanco-Pillado et
al., 2014). We should note that another class of simulations (the so-called
Abelian-Higgs type) use field-theoretic input to resolve the cores of these
strings, and observe no significant loop production (Vincent et al., 1998;
Moore et al., 2002; Hindmarsh et al., 2009; Hindmarsh et al., 2017). This has
sparked numerous debates about the true nature of the string network on small
scales, with no consensus being reached as of yet.

One should not understate the numerical challenges that arise when attempting
to perform cosmological simulations over such a wide range of scales, from the
string core to the horizon at a given time. For this work, we assume that a
scaling loop distribution is formed as indicated by the Nambu-Goto
simulations.

###  2.1 Formation in the Radiation Era

We begin by examining the distribution of loops formed before matter-radiation
equality. In most regions of the $G\mu$-$\mathcal{I}$ parameter space, they
are the dominant source of primordial spectral distortions, which makes them a
useful case to study. Focusing now on the results from Nambu-Goto simulations,
one finds that the differential number density of loops (in physical
coordinates) with initial length $L_{\textrm{i}}$ is given by

$\displaystyle\frac{{\rm d}N}{{\rm
d}L_{\textrm{i}}}=\begin{dcases}\frac{\alpha}{t^{4-p}L_{\textrm{i}}^{p}}\hskip
62.59605pt(t\leq t_{\textrm{eq}})\\\
\frac{\alpha}{t_{\textrm{eq}}^{4-p}L_{\textrm{i}}^{p}}\left(\frac{t_{\textrm{eq}}}{t}\right)^{2}\hskip
35.56593pt(t>t_{\textrm{eq}}).\end{dcases}$ (1)

Simulations performed in Blanco-Pillado et al. (2014) yield $\alpha=0.18$ and
$p=5/2$. For $t>t_{\textrm{eq}}$ we simply redshift the loops which formed
during the radiation era, and discuss the formation and evolution of matter-
dominated loops in the next subsection.

Simulations indicate that at a given time, $t$, most string loops are formed
with roughly the same initial radius, given by some fraction
$\beta\approx\mathcal{O}(0.1)$ of the Hubble length, i.e.,
$L_{\textrm{i}}(t)\approx\beta t$. After formation, the loops oscillate with
period $T\approx L$, and develop transient substructures known as cusps and
kinks. Their decay (in particular, the cusps) releases energy in the form of
gravitational waves, photons, and exotic particles, causing the loop to shrink
in size. All loops radiate gravitational waves with an oscillation-averaged
power (Vachaspati & Vilenkin, 1985)

$\displaystyle P_{\textrm{g}}\simeq\Gamma_{\textrm{g}}G\mu^{2}\simeq 1.5\times
10^{18}\,\left[\frac{\Gamma_{\textrm{g}}}{100}\right]\left[\frac{G\mu}{10^{-11}}\right]^{2}\,\textrm{GeV}^{2},$
(2)

where $\Gamma_{\textrm{g}}$ is a normalization factor
$\simeq\mathcal{O}(100)$. In some symmetry breaking schemes, cosmic strings
can acquire an electromagnetic current, $\mathcal{I}$, and are said to be
superconducting (see Witten, 1985, for a seminal example).

A proper treatment of the current generation and dissipation on a cosmological
network of loops is beyond the scope of this work, and so for simplicity we
assume that all string loops carry the same time-independent current.
Superconducting string loops also generate sizeable electromagnetic bursts at
their cusps. This leads to an additional decay channel into photons (Vilenkin
& Vachaspati, 1987; Cai et al., 2012), with

$\displaystyle P_{\gamma}$
$\displaystyle\simeq\Gamma_{\gamma}\mathcal{I}\mu^{1/2}$ $\displaystyle\simeq
3.8\times
10^{18}\,\left[\frac{\Gamma_{\gamma}}{10}\right]\left[\frac{G\mu}{10^{-11}}\right]^{1/2}\left[\frac{\mathcal{I}}{10^{4}\,{\rm
GeV}}\right]\,\textrm{GeV}^{2},$ (3)

where $\Gamma_{\gamma}\simeq\mathcal{O}(10)$ depends on the precise geometry
of a loop. Superconducting strings gradually decay through these two main
channels, with a total rate that is determined by

$\displaystyle\Gamma\,G\mu$
$\displaystyle\simeq(P_{\textrm{g}}+P_{\gamma})/\mu\simeq\Gamma_{\textrm{g}}G\mu+\Gamma_{\gamma}\mathcal{I}\mu^{-1/2}.$
(4)

By comparing the power emitted in both gravitational and electromagnetic
waves, we can define a critical current,

$\displaystyle\mathcal{I}_{*}=\frac{\Gamma_{\textrm{g}}}{\Gamma_{\gamma}}G\mu^{3/2}\simeq
3.2\times 10^{3}\,{\rm
GeV}\,\left[\frac{\Gamma_{\textrm{g}}}{100}\right]\,\left[\frac{\Gamma_{\gamma}}{10}\right]^{-1}\left[\frac{G\mu}{10^{-11}}\right]^{3/2}.$
(5)

For a given string tension, $G\mu$, loops decay primarily into gravitational
waves if $\mathcal{I}<\mathcal{I}_{*}$, or into photons when
$\mathcal{I}\geq\mathcal{I}_{*}$. The dimensionless decay coefficient $\Gamma$
can then be expressed as

$\displaystyle\Gamma=\Gamma_{\textrm{g}}\left(1+\frac{\mathcal{I}}{\mathcal{I}_{*}}\right).$
(6)

We note that $\Gamma$ is a function of both the string tension and the
current.

Given the decay coefficient $\Gamma$, the loop size then shrinks as

$\displaystyle L=L_{\textrm{i}}-\Gamma\,G\mu(t-t_{\textrm{i}}).$ (7)

Noting that $t_{\textrm{i}}=L_{\textrm{i}}/\beta\leq t_{\rm eq}$, we then have
$L_{\textrm{i}}=(L+\Gamma\,G\mu\,t)/(1+\lambda)$, where $\lambda=\Gamma
G\mu/\beta$ is the decay-rapidity parameter, a measure of how long after
formation a given loop will exist before complete evaporation. Low values of
$\lambda$ describe long-lived loops, while for $\lambda>1/\beta$ the loops
decay within one oscillation, and the expressions for the oscillation-averaged
power should be re-examined. A loop forming at $t_{\rm i}$ decays at cosmic
time $t_{\rm decay}$ given by

$\displaystyle t_{\rm decay}=\left(\frac{1+\lambda}{\lambda}\right)t_{i}.$ (8)

From this, it is easy to derive that the total lifetime of any given loop is
$t_{\rm lifetime}=t_{\rm decay}-t_{\rm i}=t_{\rm i}/\lambda$. At any given
time, the initial loop lengths that contribute to background injections are
$L_{\rm i,min}\leq L\leq\beta t_{\rm i}$, where $L_{\rm i,min}=\beta\lambda
t/(1+\lambda)<\beta t$. As $\lambda$ increases we find that $L_{\rm
i,min}\rightarrow\beta t$, such that the largest initial loops vanish rapidly.

For typical choices of our parameters, $G\mu$ and $\mathcal{I}$, the loops are
long-lived, implying $t_{\rm decay}\gg t_{\textrm{i}}$, although we will also
discuss more extreme cases where fast loop decay is possible. Using
$L_{\textrm{i}}=(L+\Gamma\,G\mu\,t)/(1+\lambda)$, with Eq. (1) we can then
write

$\displaystyle\left.\frac{{\rm d}N_{\rm{loops}}}{{\rm d}L}\right|_{\rm
r}=\frac{\alpha\left(1+\lambda\right)^{3/2}}{t^{3/2}(L+\Gamma G\mu
t)^{5/2}}\times\begin{cases}1&(t\leq t_{\textrm{eq}})\\\
\left(\frac{t_{\textrm{eq}}}{t}\right)^{1/2}&(t>t_{\textrm{eq}}),\end{cases}$
(9)

where the distribution is defined for $0\leq L\leq L_{\rm max}(t)$. At $t\leq
t_{\rm eq}$, one has $L_{\rm max}(t)=\beta t$. However, for $t>t_{\rm eq}$,
the loops last sourced at $t=t_{\rm eq}$ have shrunken to $L_{\rm
max}(t)=\beta t_{\rm eq}[1+\lambda-\lambda t/t_{\rm eq}]<\beta t_{\rm eq}$ at
time $t$. This implies that the last loops sourced at $t_{\rm eq}$ only exists
at $t\leq t_{\rm end}\equiv t_{\rm eq}(1+\lambda)/\lambda$. Figure 1
highlights the region of parameter space which undergoes rapid decays (i.e.
$\lambda\geq 1/\beta$). For reference we also show the line for which $t_{\rm
end}\simeq t_{0}$ (i.e., $z_{\rm end}=0$) is equal to the age of the universe,
$t_{0}$, or $\lambda\simeq 4\times 10^{-6}$.

###  2.2 Formation in the Matter era

For loops formed in the matter-dominated era, a very similar picture can be
developed. Simulations show that these evolve according to

$\displaystyle\frac{{\rm d}N}{{\rm d}L_{\textrm{i}}}\simeq\frac{\alpha_{\rm
m}}{t^{2}L_{\textrm{i}}^{2}}.$ (10)

We will assume that the sourcing scale in the matter-dominated era is the same
as in the radiation dominated era, i.e., $\beta_{\rm m}t\simeq\beta\,t$. Then,
at $t$ the smallest loop scale is $L_{\rm min}(t)=\beta[t_{\rm
eq}-\lambda(t-t_{\rm eq})]$. This implies two regimes: at $t_{\rm eq}\leq
t\leq t_{\rm end}=t_{\rm eq}(1+\lambda)/\lambda$ the minimal loop length
fulfills $0\leq L_{\rm min}(t)\leq\beta t_{\rm eq}$, while at $t>t_{\rm end}$
one has $L_{\rm min}(t)=0$. The loop distribution is then defined at $L_{\rm
min}(t)\leq L\leq\beta t$.

To fix the normalization, $\alpha_{\rm m}$, we assume that at $t_{\rm eq}$ the
radiation-dominated loop distribution at the sourcing scale is the same as the
matter-dominated one. This gives the condition

$\displaystyle\frac{\alpha_{\rm m}}{t_{\rm
eq}^{2}L_{\textrm{i,eq}}^{2}}=\frac{\alpha}{t_{\rm
eq}^{3/2}L_{\textrm{i,eq}}^{5/2}}\quad\rightarrow\quad\alpha_{\rm
m}=\alpha/\sqrt{\beta}\approx 0.57.$ (11)

with $L_{\textrm{i,eq}}=\beta t_{\rm eq}$. With these expressions we then have

$\displaystyle\left.\frac{{\rm d}N_{\rm{loops}}}{{\rm d}L}\right|_{\rm
m}=\frac{\alpha_{\rm m}\left(1+\lambda\right)}{t^{2}(L+\Gamma G\mu t)^{2}}$
(12)

at $L_{\rm min}(t)\leq L\leq\beta t$ with $L_{\rm min}(t)=0$ at $t>t_{\rm
end}$.

##  3 Analytic Estimates - Energy Release

The injection of non-thermal photons into the microwave background before
recombination can lead to distortions in the frequency spectrum as the plasma
attempts to thermalize this new radiation. The time-dependence of photon
injection determines the spectral shape of this distortion, with roughly three
eras emerging (Chluba, 2015). At high redshifts ($z\gtrsim z_{\rm th}$, with
$z_{\rm th}\approx 2\times 10^{6}$), injections are fully thermalized through
a combination of Bremsstrahlung, Compton, and double Compton scattering,
producing an average (unobservable) temperature increase. At lower redshifts,
number changing processes freeze out. However, repeated Compton scattering
with the background electrons drives the CMB photons towards kinetic
equilibrium with a spectrum described by the Bose-Einstein distribution with a
small chemical potential ($\mu$) for $z\gtrsim 3\times 10^{5}$ (Illarionov &
Sunyaev, 1975; Burigana et al., 1991; Hu & Silk, 1993). At redshifts lower
than this, Compton scattering becomes less efficient and the distorted
spectrum may be analytically described by a $y$-distortion (Zeldovich &
Sunyaev, 1969).

The COBE/FIRAS instrument remains state-of-the-art when it comes to upper
bounds on these distortion parameters, providing $|\mu|<9\times 10^{-5}$ and
$|y|<1.5\times 10^{-5}$ at $2\sigma$ (Fixsen et al., 1996). Some improvements
have been discussed in Gervasi et al. (2008b); Bianchini & Fabbian (2022);
however, only in the future can we expect significant advances with PIXIE
(Kogut et al., 2011; Kogut et al., 2016), BISOU (Maffei et al., 2021), COSMO
(Masi et al., 2021), TMS (Rubiño Martín et al., 2020) or a spectrometer within
the ESA Voyage 2050 space program (Chluba et al., 2021).

As a network of superconducting cosmic strings decay, they are capable of
injecting both significant energy, and entropy, into the background radiation.
Previously, Tashiro et al. (2012a) computed analytic bounds on the string
tension and current by considering the $\mu$ and $y$ distortions caused by a
pure energy injection. As we will illustrate below, their analysis neglected a
factor of $\Gamma^{-1/2}$, an omission which yielded more stringent
constraints at high currents than we find here. Our $\mu$-distortion estimates
agree better with those presented in Miyamoto & Nakayama (2013); however, we
will show that the simple energy release arguments become inaccurate at late
times and for high currents, requiring a more detailed treatment of the
distortion evolution using CosmoTherm. Additional simple analytic estimates
for several astrophysical and cosmological observables can be found in
Miyamoto & Nakayama (2013).

In the following sections, we will further refine these analytic distortion
estimates by including the resultant entropy injection, yielding a novel shape
for the constraints on $G\mu$ and $\mathcal{I}$. Later on, we validate these
approximations by comparing against the full numerical solution, computed with
the thermalization code CosmoTherm. The numerical analysis allows us to go
beyond the simple $\mu$ and $y$ estimates, by comparing the full spectral
shape obtained with a cosmic string network to the residuals of the COBE/FIRAS
experiment.

Figure 1: Regions of parameter space affected by rapidly decaying loops. The
red area corresponds to $\lambda\geq 1/\beta\approx 10$ in which loops fully
evaporate within one loop oscillation time after their formation. The cusp
evaporation formalism may break down in this red region, and so constraints
given there should be taken with a grain of salt. The different contours
correspond to the surfaces of constant $z_{\rm end}$ for a given parameter
set. Regions to the right of, and above the contours represent parameter
values whose radiation loops will have completely evaporated by the indicated
$z_{\rm end}$.

###  3.1 Pure Energy Injection - Radiation Loops

To obtain analytic approximations for the amplitude of these distortion
parameters, we apply a Green’s function approach (method B of Chluba, 2016) in
which $\mu$ and $y$ are determined through

$\displaystyle\mu$ $\displaystyle\approx 1.401\int_{z_{\mu y}}^{\infty}{\rm
d}z\frac{1}{\rho_{\gamma}}\frac{{\rm d}Q}{{\rm d}z}\mathcal{J}_{\rm
bb}(z)\hskip 17.07164pty\approx\frac{1}{4}\int^{z_{\mu y}}_{z_{\rm rec}}{\rm
d}z\frac{1}{\rho_{\gamma}}\frac{{\rm d}Q}{{\rm d}z}.$ (13)

Here, we introduce the approximate distortion visibility function,
$\mathcal{J}_{\rm bb}(z)\approx{\rm e}^{-(z/z_{\rm th})^{5/2}}$ with $z_{\rm
th}\approx 2\times 10^{6}$, which models the smooth transition between the
$\mu$ and thermalization eras. We approximate the transition from $\mu$ and
$y$ eras by a simple step function at $z_{\mu y}=5\times 10^{4}$, the redshift
below which energy redistribution becomes inefficient. Finally, $z_{\rm
rec}\approx 10^{3}$ is the redshift of recombination, $\rho_{\gamma}$ is the
energy density in the CMB at redshift $z$, and ${\rm d}Q/{\rm d}z$ is the rate
of non-thermal energy (density) injection.

While more accurate estimates for the transition between $\mu$ and $y$ could
be obtained using various analytic methods (e.g., Khatri & Sunyaev, 2012;
Chluba, 2013; Chluba & Jeong, 2014), the precise inclusion of photon
production processes necessitates full numerical treatments with the efficient
scheme of CosmoTherm.

For superconducting strings, the energy release rate is given by averaging the
energy injection from one string over the entire loop distribution:

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|_{\rm r}$
$\displaystyle=\int_{0}^{L_{\rm max}}{\rm d}L\left.\frac{{\rm
d}N_{\rm{loops}}}{{\rm d}L}\right|_{\rm r}\,P_{\gamma}(L)$ (14a)
$\displaystyle L_{\rm max}$ $\displaystyle=\begin{cases}\beta t&(t\leq
t_{\textrm{eq}})\\\ \beta t_{\rm eq}[1+\lambda-\lambda t/t_{\rm eq}]&(t_{\rm
eq}<t\leq t_{\rm end}),\end{cases}$ (14b)

with $t_{\rm end}=t_{\rm eq}\,(1+\lambda)/\lambda$. Computation of this
integral gives an energy injection rate

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|_{\rm
r}=\frac{2}{3}\frac{\alpha\Gamma_{\gamma}}{G^{1/2}}$
$\displaystyle\frac{\mathcal{I}}{\Gamma^{3/2}G\mu}\frac{(1+\lambda)^{3/2}}{t^{3}}$
(15)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\begin{dcases}\left[1-\left(\frac{t_{\rm
eq}}{t_{\rm end}}\right)^{3/2}\right]&(t\leq t_{\rm eq})\\\
\left[1-\left(\frac{t}{t_{\rm end}}\right)^{3/2}\right]\left(\frac{t_{\rm
eq}}{t}\right)^{1/2}&(t_{\rm eq}<t\leq t_{\rm end}).\end{dcases}$

for loops created in the radiation-dominated era.

Figure 2: Fractional energy injection rate per logarithmic redshift bin from
the decay of superconducting cosmic strings forming in the radiation era. The
injection rate is roughly constant (in log-space), with an overall scaling
given by $\zeta_{\rho}=\mathcal{I}/\Gamma^{3/2}G\mu$ until the loops
completely evaporate. The evaporation redshift at which the last loop that
formed in the radiation era will decay can be determined by solving Eq. (8)
with $t_{i}=t_{\rm eq}$ for a given $G\mu-\mathcal{I}$ pair. Deviations from
an overall $\zeta_{\rho}$ scaling are expected near the evaporation redshift.
The colour scheme of the contours is set to match that of Figure 1.

The form of this heating rate is illustrated in Fig. 2. Due to the continuous
sourcing of loops at $L\simeq\beta t$, the heating occurs at all redshifts
until $t\simeq t_{\rm end}$, where the last loops injected at $t_{\rm eq}$
vanish, and heating ceases. In particular, in the $\mu$-distortion era, one
finds $\rho_{\gamma}^{-1}{\rm d}Q/{\rm d}\ln z$ to be redshift-independent. It
is clear from this figure that the smaller loops provide most of the heating.

As one can see from Eq. (15) and Fig. 2, a large value of $\lambda$ leads to a
decrease in energy injection efficiency. This is because a loop population
with a rapid decay rate cannot build up its number density. In other words,
there are simply fewer loops around at any given time to inject energy.
Naively, it looks as if the injection is amplified for large values of
$\mathcal{I}$. However, we remind the reader that $\Gamma\propto\mathcal{I}$
for large currents, leading to an overall decrement in the injection.
Physically, this is another consequence of a rapidly decaying population of
loops. However, in the very large current regime, one can also find regions
where $\lambda\gg 1$, such that the heating rate from radiation loops becomes

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|^{\rm high}_{\rm r}$
$\displaystyle\approx\frac{\alpha\Gamma_{\gamma}}{G^{1/2}}\frac{\mathcal{I}}{\Gamma^{3/2}G\mu}\frac{\lambda^{1/2}}{t^{3}}=\frac{\alpha\Gamma_{\gamma}}{G^{1/2}}\frac{\mathcal{I}}{\Gamma(G\mu)^{1/2}\beta^{1/2}}\frac{1}{t^{3}}$
$\displaystyle\approx\frac{\alpha\Gamma_{\gamma}}{G^{1/2}}\frac{\mathcal{I}_{*}}{\Gamma_{\rm
g}(G\mu)^{1/2}\beta^{1/2}}\frac{1}{t^{3}}=\frac{\alpha_{\rm
m}}{G}\frac{G\mu}{t^{3}}$ (16)

independent of $\mathcal{I}$ at $t\leq t_{\rm eq}$. Here we used
$\Gamma\approx\Gamma_{\rm g}\mathcal{I}/\mathcal{I}_{*}$ for the decay rate
and $\alpha_{\rm m}=\alpha/\sqrt{\beta}$. This expression is not valid for
$t\geq t_{\rm eq}$, as for $\lambda\gg 1$ the radiation loops fully evaporate
at $t_{\rm end}\approx t_{\rm eq}$.

One should note that the electromagnetic power generated by a cusp decay
($P_{\gamma}$) is an oscillation-averaged quantity, as the loop must undergo
at least one oscillation for cusp formation to take place. However, an
implication of $\lambda>1/\beta$ is that the loop should fully decay before a
single oscillation can even take place. We therefore urge the reader to be
cautious about statements made in regions of parameter space with
$\lambda>1/\beta$, which is indicated by the red region in Fig. 1.

Inserting the injection rate into Eq. (13), and noting that the background
energy density (in natural units) is $\rho_{\gamma}=(\pi^{2}/15)T^{4}$, we
find the following estimates for $\mu$ and $y$ from only the radiation loops

$\displaystyle\mu$ $\displaystyle\simeq 5.5\times
10^{-6}\left[\frac{\zeta_{\rho}}{10^{11}\,\rm GeV}\right]\hskip
56.9055pt(\lambda\ll 1)$ (17a) $\displaystyle y$ $\displaystyle\simeq
1.3\times 10^{-6}\left[\frac{\zeta_{\rho}}{10^{11}\,\rm GeV}\right]\hskip
56.9055pt(\lambda\ll 1),$ (17b)

where we introduced the variable $\zeta_{\rho}=\mathcal{I}/\Gamma^{3/2}G\mu$,
which describes the overall non-linear scaling of the total energy release
with $\mathcal{I}$ and $G\mu$, provided the loop distribution is not at their
evaporation redshift. Note that $\Gamma$ is a function of both parameters.

It can also be useful to phrase our constraints in terms of the fractional
energy injection into the background, $\Delta\rho_{\gamma}/\rho_{\gamma}$,
which is given by

$\displaystyle\left.\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\right|_{\rm
r}=\int_{z_{\rm rec}}^{\infty}{\rm d}z\frac{1}{\rho_{\gamma}}\left.\frac{{\rm
d}Q}{{\rm d}z}\right|_{\rm r}\mathcal{J}_{\rm bb}(z).$ (18)

The fractional energy release from superconducting cosmic strings over the
entire distortion window is therefore

$\displaystyle\left.\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\right|_{\rm
r}\simeq 8.9\times 10^{-6}\left[\frac{\zeta_{\rho}}{10^{11}\,\rm
GeV}\right]\hskip 42.67912pt(\lambda\ll 1).$ (19)

This is the expression one should use in deriving constraints based purely on
the total amount of energy release. However, it neglects the effects of
partial Comptonization and also photon injection, which can change the
character of the distortion significantly (Chluba, 2015).

###  3.2 Energy Injection from Matter Loops

At $t\lesssim t_{\rm eq}$, loops that are formed during the matter-dominated
era can contribute. To compute the energy release rate from these loops, we
can follow the same steps as above but instead average over the loop
distribution from the matter-era:

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|_{\rm m}$
$\displaystyle=\int^{\beta t}_{L_{\rm min}}{\rm d}L\left.\frac{{\rm
d}N_{\rm{loops}}}{{\rm d}L}\right|_{\rm m}\,P_{\gamma}(L)$ (20a)
$\displaystyle L_{\rm min}$ $\displaystyle=\left[\beta t_{\rm
eq}(1+\lambda-\lambda t/t_{\rm eq}),0\right]_{>}.$ (20b)

Here and below, we use the shorthand notation $[a,b]_{>}=\max[a,b]$ and
$[a,b]_{<}=\min[a,b]$. Computation of this integral gives an energy injection
rate

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|_{\rm m}$
$\displaystyle=\frac{\alpha_{\rm
m}\Gamma_{\gamma}}{G^{1/2}}\frac{\mathcal{I}\,(G\mu)^{1/2}}{\beta}\frac{1}{t^{3}}$
$\displaystyle\qquad\times\begin{dcases}\left[\frac{t}{t_{\rm
eq}}-1\right]&\quad(t_{\rm eq}\leq t\leq t_{\rm end})\\\ \left[\frac{t_{\rm
end}}{t_{\rm eq}}-1\right]&\quad(t_{\rm end}<t),\end{dcases}$ (21)

with $t_{\rm end}=t_{\rm eq}\,(1+\lambda)/\lambda$. Note that for
$\lambda\lesssim 4\times 10^{-6}$, all matter loops survive until today
($t_{\rm end}\geq t_{0}$).

Figure 3: Energy injection rate from loops which form in the matter dominated
era. The amplitude of the signal is primarily governed by the number density
of matter loops at any given time. A “knee" develops when $t_{\rm end}$ is
crossed for a given parameter set. The slowing of the growth of the injection
rate below that redshift is because the distribution now exhibits a source-
sink behaviour as the smallest loops fully evaporate, in contrast to just a
source before that point.

In Fig. 3 we illustrate the contribution to the heating rate from cosmic
string loops created during the matter dominated era. The overall amplitude of
the signal is governed primarily by two factors: the number density of the
distribution, and the smallest loop size, $L_{\rm min}$, as is evident from
Eq. (3.2). After matter radiation equality, $L_{\rm min}$ decreases as the
loops oscillate and decay, leading to a strong increase in the injection rate
as can be seen on the right hand side of the knees in Fig. 3. At $t_{\rm end}$
(the position of the knee), matter loops begin leaving the distribution as
they fully evaporate, leading to a slowing of the injection rate. Since the
distribution redshifts like matter, their contribution to the background
energy density grows linearly in $z$ relative to the CMB.

For large currents, we can again approximate the heating rate in the
$\lambda\gg 1$ regime:

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|^{\rm high}_{\rm m}$
$\displaystyle\approx\frac{\alpha_{\rm
m}\Gamma_{\gamma}}{G^{1/2}}\frac{\mathcal{I}\sqrt{G\mu}}{\beta\,t^{3}}\frac{1}{\lambda}\approx\frac{\alpha_{\rm
m}}{G}\frac{G\mu}{t^{3}}.$ (22)

This is identical to Eq. (3.1), which is expected since fundamentally nothing
special happens to the loop distribution at $t=t_{\rm eq}$. For $\lambda\gg
1$, all of the energy in the newly formed loops is immediately emitted. The
steady rise in fractional energy injection shown in the
$\lambda\rightarrow\infty$ contour of Fig. 3 is obtained because the energy
density in the loops redshifts as $a^{-3}$ while the CMB photons redshift as
$a^{-4}$.

We show the total contribution of both matter and radiation loops in Fig. 4.
Interestingly, the fractional injection rate converges at late times for a
wide range of parameters. This behaviour persists along contours of constant
$G\mu$ once $\mathcal{I}\gg\mathcal{I}_{*}$. In this current domination region
of parameter space, adjustments to $G\mu$ will linearly shift the amplitude of
the late time emission signal, while in the gravitational wave regime
($\mathcal{I}\leq\mathcal{I}_{*}$), the scaling of the signal is more
complicated. The overall degradation of the signal with high currents is due
to the fact that a smaller density of loops is available at any given time for
these models. For $\mathcal{I}\gg\mathcal{I}_{*}$, we have we have
$\zeta_{\rho}\propto\mathcal{I}^{-1/2}$, which encodes this rapid decay
effect.

Figure 4: The total energy injection rate from a string loop distribution. For
a constant $G\mu$, the late time injection rate converges for
$\mathcal{I}\gg\mathcal{I}_{*}$. In contrast, contours of current
$\mathcal{I}$ will experience amplitude shifts proportional to $G\mu$ in the
current-dominated regime.

With the injection function, we may now estimate the fractional energy release
from matter loops. For small $\lambda$ (equivalently, $t_{\rm end}\gg t_{\rm
eq}$), we find that from $z_{\rm rec}\leq z\leq z_{\rm eq}$, the energy
release is

$\displaystyle\left.\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}\right|_{\rm
m}\simeq 2.9\times 10^{-16}\left[\frac{\mathcal{I}}{10^{4}\,\rm
GeV}\right]\left[\frac{G\mu}{10^{-10}}\right]^{1/2}\hskip
14.22636pt(\lambda\ll 1).$ (23)

Recall that small values of $\lambda$ imply a long lifetime for a given loop.
Since most of the energy release comes from the smallest loops present in a
given distribution, matter loops will always source a subdominant contribution
to the primordial distortion signal relative to radiation loops for
$\lambda\ll 1$. As indicated in Fig. 3 and 4, matter loops begin to play more
of a role in pre-recombination injection for $\lambda\gtrsim 0.1$. While not a
factor for primordial spectral distortions, low redshift emission from the
matter loops can be constrained by the evolution of the ionization fraction.
This emission may also play a role in our understanding of the EDGES and
ARCADE-2 signals, as we will see below.

In Fig. 5 we show the constraints on energy injection in this picture using
the COBE/FIRAS data ($\Delta\rho/\rho\leq 6\times 10^{-5}$), as well as a
simple forecast from a PIXIE-like instrument ($\Delta\rho/\rho\leq 2\times
10^{-8}$). For COBE/FIRAS, the quoted energy release levels are consistent
with the $2\sigma$ errors on $\mu$ and $y$, while for the PIXIE-like setup we
assume that a penalty from marginalization over foregrounds (Abitbol et al.,
2017; Rotti & Chluba, 2021) is already included.

The constraints are nearly symmetric about the $\mathcal{I}=\mathcal{I}_{*}$
contour, with departures arising in the $\lambda\simeq 1$ regime due to
rapidly decaying loops. The work of Tashiro et al. (2012a) presented a flat
constraint in the $\mathcal{I}\geq\mathcal{I}_{*}$ region, a symptom of them
missing a factor of $\Gamma^{-1/2}$ in their equivalent expression of Eq.
(19). Our energy injection constraints are roughly consistent with the
analysis of Miyamoto & Nakayama (2013). In the next section, we include the
effects of entropy injection which cause further significant modifications to
these constraints, as can be seen in Fig. 11.

Figure 5: Solid lines indicate constraints on string parameters obtained by
requiring $\Delta\rho/\rho\leq 6\times 10^{-5}$ ($2\sigma$ for COBE/FIRAS),
and $\Delta\rho/\rho\leq 2\times 10^{-8}$ ($2\sigma$ forecast for a PIXIE-type
instrument). Statements in the $\lambda\geq 1$ region of the plot should be
viewed skeptically, as rapid decays may cause the injection framework to break
down. The dash-dotted lines are constraints obtained by $\mu$ distortions over
the redshift range $3\times 10^{5}\leq z\leq 2\times 10^{6}$, and may be
directly compared to the energy+entropy constraints illustrated in Fig. 11.

##  4 Analytic Estimate - Entropy Release

Energy release is not the only way one can disturb the CMB spectrum. An often
overlooked fact of distortion theory is that the production of additional
photons also perturbs the spectrum. In the $\mu$ era, the total distortion
should be expressed as (Hu, 1995; Chluba, 2015)

$\displaystyle\mu\approx
1.401\left[\frac{\Delta\rho_{\gamma}}{\rho_{\gamma}}-\frac{4}{3}\frac{\Delta
N_{\gamma}}{N_{\gamma}}\right].$ (24)

Therefore, models which introduce direct photon production (such as this one)
must necessarily take into account both energy and entropy injection when
determining constraints. In fact, this entropy injection is capable of
producing a net negative distortion signature. Of course, COBE/FIRAS is only
sensitive to $|\mu|$, but as we will see below, there are regions of parameter
space in which the constraints of Tashiro et al. (2012a) and Miyamoto &
Nakayama (2013) are significantly altered. There are also regions in parameter
space that cannot be easily estimated analytically, since strong
non-$\mu$/non-$y$ distortion signals are created (Chluba, 2015), requiring the
numerical treatment presented below.

To obtain a description for the photon source term, we adapt the work of Cai
et al. (2012) who derived the power emitted in photons per unit frequency, per
unit solid angle from a superconducting cosmic string cusp annihilation
(averaged over an oscillation time):

$\displaystyle\frac{{\rm d}^{2}P_{\gamma}^{\rm c}}{{\rm d}\omega{\rm
d}\Omega}\approx\left(\frac{\Gamma_{\gamma}}{3}\right)\mathcal{I}^{2}L.$ (25)

The cusp is a highly relativistic object, and so the radiation is heavily
beamed, with all emission in a solid angle $\Omega=(\omega L)^{-2/3}$. Here,
$\omega$ is the frequency of emitted radiation. Integrating Eq. (25) over
solid angle we find the instantaneous spectrum from a single loop

$\displaystyle\frac{{\rm d}^{2}E_{\gamma}^{\rm c}}{{\rm d}t{\rm
d}\omega}\approx\left(\frac{\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}L^{1/3}}{\omega^{2/3}},\hskip
42.67912pt\frac{{\rm d}^{2}N_{\gamma}^{\rm c}}{{\rm d}t{\rm
d}\omega}\approx\left(\frac{\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}L^{1/3}}{\omega^{5/3}}.$
(26)

Here, $E_{\gamma}^{c}$ and $N_{\gamma}^{c}$ are the total energy and number of
photons produced by a given loop. We introduced the factor of
$\Gamma_{\gamma}/3$ in these equations to properly match the total power
produced by a loop with Eq. (2.1).111This can be easily confirmed by
integrating over $\omega$ between $0$ and $\omega_{\rm max}$, which yields
${\rm d}E_{\gamma}^{\rm c}/{\rm
d}t\approx\Gamma_{\gamma}\mathcal{I}^{2}L^{1/3}\,\omega_{\rm
max}^{1/3}=\Gamma_{\gamma}\mathcal{I}\,\mu^{1/2}$, reproducing Eq. (2.1). The
numerical prefactor depends on the precise shape of a loop, and so the
$\Gamma_{\gamma}$ factor averages over the geometries of a network of loops.
Upon integration of the differential energy spectrum, one notices that the
total power is dominated by the highest frequency photons produced at a cusp.
Cai et al. (2012) have estimated this frequency to be

$\displaystyle\omega_{\rm max}\approx\frac{\mu^{3/2}}{\mathcal{I}^{3}L}.$ (27)

The cusp covers a finite spatial extent on the string, and so this result is
derived by requiring the total energy released in a cusp decay does not exceed
the rest-mass energy of this region.

###  4.1 Total emission from loops

To obtain the number density injection from all loops, one may first determine
the number of photons produced per unit time, and then average over the full
loop distribution. For our purposes, it will be useful to introduce the
dimensionless frequency as $x=\omega/T(z)$. The photon spectrum from cosmic
strings in principle extends to arbitrarily low energies222Plasma effects
(i.e., the plasma frequency or the Razin effect) become important at
frequencies well below the domain of interest.; however, low frequency photons
are at risk of being absorbed by the plasma and converted into heat (Chluba,
2015; Bolliet et al., 2021). The survival probability, $\mathcal{P}_{\rm
s}(x,z)\approx{\rm e}^{-x_{\rm c}/x}$ tells us how many photons survive as a
true entropy injection.

At high redshifts ($z\gtrsim 10^{5}$), we use the simple expression for the
critical frequency (Chluba, 2014)

$\displaystyle x_{\rm c}\approx 8.6\times 10^{-3}\sqrt{\frac{1+z}{2\times
10^{6}}}\sqrt{1+\left[\frac{1+z}{3.8\times 10^{5}}\right]^{-2.344}}.$ (28)

This frequency is determined through the balance of Compton, double Compton
(DC), and Bremsstrahlung (BR) effects. At later times ($z\lesssim 10^{5}$),
the absorption probability is mainly determined by the free-free process,
without much contribution from Compton scattering. In this regime, we estimate
the absorption probability by finding the frequency at which the free-free
optical depth is close to unity (Chluba, 2015; Bolliet et al., 2021).

The two regimes are merged in Fig. 6 over the relevant redshift range. At
recombination, the fraction of free electrons and protons drops substantially,
leading to a sharp decrease in the absorption probability which then remains
constant until reionization begins.

Figure 6: Evolution of the dimensionless critical frequency over redshift.
Photons produced with a frequency $x<x_{\rm c}$ are absorbed by the plasma
quickly after their creation, and therefore do not contribute significantly to
a photon number injection. The approximate expression in Eq. (28) is valid for
$z\gtrsim 10^{5}$, while below we use the pre-tabulated free-free absorption
frequency to estimate the absorption probability.

Integrating the right hand expression of Eq. (26) with the factor333Although
the free-free optical depth scales as $\tau_{\rm ff}\propto\ln(2.25/x)/x^{2}$
at low frequencies, we keep our estimates simple and just replace the critical
frequency with the free-free absorption frequency. This does not significantly
alter the illustrations that are presented below. For the final constraints,
we explicitly compute the distortion signal without these approximations.
$\mathcal{P}_{\rm s}(x,z)\approx{\rm e}^{-x_{\rm c}/x}$ yields the total
number of photons produced per cosmic string which will contribute to an
entropy injection

$\displaystyle\frac{{\rm d}N_{\gamma}^{c}}{{\rm d}t}$
$\displaystyle=\frac{\Gamma_{\gamma}}{3}\frac{\mathcal{I}^{2}L^{1/3}}{T^{2/3}}\int_{0}^{x_{\rm
max}}{\rm d}x\,\frac{{\rm e}^{-x_{\rm c}/x}}{x^{5/3}}$
$\displaystyle=\frac{\Gamma_{\gamma}}{3}\frac{\mathcal{I}^{2}L^{1/3}}{(x_{\rm
c}T)^{2/3}}\,\Gamma\left[\frac{2}{3},\frac{x_{\rm c}}{x_{\rm max}}\right].$
(29)

For most reasonable choices of $G\mu$ and $\mathcal{I}$, $x_{\rm c}\ll x_{\rm
max}$, where the incomplete gamma function is well approximated by
$\Gamma[2/3,x_{\rm c}/x_{\rm max}]\approx\Gamma[2/3]\approx 1.354$. However,
as is evident from Eq. (27), large values of $\mathcal{I}$ and small values of
$G\mu$ serve to decrease $x_{\rm max}$, implying that for those parameter
choices, most of the produced photons are below $x_{\rm c}$ and hence do not
contribute as entropy.

Figure 7: The general functional form of $\mathcal{J}_{\rm r}(l,\kappa)$ for
different $l$ values. For parameter values yielding $\lambda\ll 1$, the
$l\rightarrow\infty$ contour is representative of the emission from all loops
in the distribution before matter-radiation equality. Contours of smaller $l$
then show the contribution from correspondingly smaller loops, with $l$ = 1
representing emission from all loops with $L\leq\Gamma G\mu t$. The asymptotic
expressions for small and large $\kappa$ can be found in Eq. (A) and Eq. (A),
respectively. Alternatively, for a given $\lambda$, we note that $l=1/\lambda$
yields the total emission from all loops in the distribution.

With Eq. (4.1), we can compute the entropy injection rate from the loop
distribution in a similar way to the energy injection,

$\displaystyle\frac{{\rm d}N_{\gamma}}{{\rm d}t}=\int_{0}^{L_{\rm up}}{\rm
d}L\frac{{\rm d}N_{\rm loops}}{{\rm d}L}\frac{{\rm d}N_{\gamma}^{c}}{{\rm
d}t}.$ (30)

where $L_{\rm up}$ depends on the case of interest (i.e. matter/radiation
loops at a particular redshift). For clarity, let us first focus on loops
created and evolving in the radiation-dominated era (i.e., $t\leq t_{\rm
eq}$). The relevant integral then becomes

$\displaystyle\frac{{\rm d}N_{\gamma}}{{\rm d}t}$
$\displaystyle=\frac{\alpha\Gamma_{\gamma}}{3}\frac{\mathcal{I}^{2}(1+\lambda)^{3/2}}{(\Gamma
G\mu)^{7/6}\,t^{8/3}}\,\frac{\Gamma[2/3]}{(x_{\rm
c}T)^{2/3}}\times\mathcal{J}_{\rm r}(l_{\rm up},\kappa)$ (31a)
$\displaystyle\mathcal{J}_{\rm r}(l,\kappa)$ $\displaystyle=\int_{0}^{l}{\rm
d}l^{\prime}\,\frac{{l^{\prime}}^{1/3}}{(1+l^{\prime})^{5/2}}\,\frac{\Gamma\left[\frac{2}{3},\kappa\,l^{\prime}\right]}{\Gamma[2/3]},$
(31b)

where we have substituted $l=L/(\Gamma G\mu t)$ and $l_{\rm up}=L_{\rm
up}/(\Gamma G\mu t)$. We also used $x_{\rm c}/x_{\rm max}=\kappa\,l$ with
$\kappa=x_{\rm c}\,T\,\mathcal{I}^{3}\,\Gamma\,G\,t/\mu^{1/2}$.

Figure 7 shows the general form of $\mathcal{J}_{r}(l,\kappa)$. Heuristically,
$\Gamma[2/3,\kappa l^{\prime}]/\Gamma[2/3]$ acts as a normalized window
function which closes rapidly for $\kappa l^{\prime}\gtrsim 1$. Smaller loops
emit higher frequency photons, meaning that they are able to contribute to
entropy injections more efficiently than the large loops. This is reflected by
the fact that the argument of the window function depends on the length of any
particular loop, as we illustrate in Fig. 8.

Figure 8: The normalized window function at different redshifts for entropy
injections from cosmic strings for two different benchmark loop lengths. The
solid lines characterize the largest loops in the distribution, $L=\beta t$,
while the dash-dotted lines correspond to much smaller loops, with $L=\Gamma
G\mu t$, who will fully evaporate within one Hubble time of any given $t$. One
can clearly see that the window function for the larger loops closes off much
faster than for the smaller loops once the current becomes large enough. The
blue and green dash-dotted lines are overlapping in this figure, and for
$\mathcal{I}=10^{7.5}$ GeV the window function for the largest loops is
completely closed. For all parameter values displayed here $\lambda\ll 1$.

For $t>t_{\rm eq}$, we need to consider the evolution of loops created in the
radiation dominated era as well as newly created loops. For the former, two
changes occur. One is due to the change in the time-dependence of the
evolution, which leads to an extra factor of $(t_{\rm eq}/t)^{1/2}$ and the
other is due to the fact that the maximal loop scale now evolves from $\beta
t_{\rm eq}$ at $t=t_{\rm eq}$ to $\beta t_{\rm eq}(1+\lambda-\lambda t/t_{\rm
eq})$ at $t>t_{\rm eq}$. Put together, this yields

$\displaystyle\left.\frac{{\rm d}N_{\gamma}}{{\rm d}t}\right|_{\rm r}$
$\displaystyle=\frac{\alpha\Gamma_{\gamma}}{3}\frac{\mathcal{I}^{2}(1+\lambda)^{3/2}}{(\Gamma
G\mu)^{7/6}\,t^{8/3}}\,\frac{\Gamma[2/3]}{(x_{\rm c}T)^{2/3}}$
$\displaystyle\qquad\times\begin{dcases}\mathcal{J}_{\rm r}\left(\frac{t_{\rm
end}}{t_{\rm eq}}-1,\kappa\right)&(t\leq t_{\rm eq})\\\ \mathcal{J}_{\rm
r}\left(\frac{t_{\rm end}}{t}-1,\kappa\right)\left(\frac{t_{\rm
eq}}{t}\right)^{1/2}&(t_{\rm eq}<t\leq t_{\rm end}),\end{dcases}$ (32)

for the loops created in the radiation-dominated era. For loops created in the
matter-dominated era we similarly have

$\displaystyle\left.\frac{{\rm d}N_{\gamma}}{{\rm d}t}\right|_{\rm m}$
$\displaystyle=\frac{\alpha_{\rm
m}\Gamma_{\gamma}}{3}\frac{\mathcal{I}^{2}(1+\lambda)}{(\Gamma
G\mu)^{2/3}\,t^{8/3}}\,\frac{\Gamma[2/3]}{(x_{\rm c}T)^{2/3}}$ (33a)
$\displaystyle\qquad\times\begin{dcases}\mathcal{J}_{\rm m}\left(\frac{t_{\rm
end}}{t}-1,\frac{t_{\rm end}}{t_{\rm eq}}-1,\kappa\right)&(t_{\rm eq}<t\leq
t_{\rm end})\\\ \mathcal{J}_{\rm m}\left(0,\frac{t_{\rm end}}{t_{\rm
eq}}-1,\kappa\right)&(t_{\rm end}<t),\end{dcases}$
$\displaystyle\mathcal{J}_{\rm m}(l_{a},l_{b},\kappa)$
$\displaystyle=\int^{l_{b}}_{l_{a}}{\rm
d}l^{\prime}\,\frac{{l^{\prime}}^{1/3}}{(1+l^{\prime})^{2}}\,\frac{\Gamma\left[\frac{2}{3},\kappa\,l^{\prime}\right]}{\Gamma[2/3]}.$
(33b)

with $1/\lambda=\frac{t_{\rm end}}{t_{\rm eq}}-1$. Note that the integral over
the loop distribution is slightly different from the one in the radiation era.
Asymptotic forms for both $\mathcal{J}_{\rm r}(l,\kappa)$ and
$\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm b},\kappa)$ can be found in Appendix A.

The total entropy injection rate is given by the sum of Eq. (4.1) and Eq. (33)
in the relevant redshift regimes. We show the fractional injection rate in
Fig. 9. One major difference between energy and entropy injection is that here
we do not have a constant $\lambda\rightarrow\infty$ contour. This is because
at large currents, $\kappa\sim\mathcal{I}^{4}$, and for large $\kappa$ the
window function closes rapidly, as can be seen in Fig. 8. This shuts off all
emission above $x_{\rm c}$, causing a rapid decay of the signal. For
primordial spectral distortions, only the injection at $z\geq z_{\rm rec}$ is
relevant. However, it is evident from the figure that large fractions of
photons are injected after recombination, hinting at potential synergies with
experiments detecting excess emission at low frequencies such as ARCADE-2 or
21-cm surveys.

Figure 9: The fractional entropy injection at different redshifts. Along
contours of constant $G\mu$, increasing the current boosts the signal, until
the current becomes too high and the window function (see Fig. 8) rapidly
shuts off emission. For the brown contour, $z_{\rm end}$ marks the redshift at
which all radiation loops fully evaporate. All other sharp features are
inherited from $x_{\rm c}$.

Deep in the $\mu$ era ($3\times 10^{5}\leq z\leq 2\times 10^{6}$) energy
redistribution of the injected photons is still efficient through Compton
scattering. This allows us to quantify the $\mu$ distortion by computing the
relative energy and entropy injections during this era and using Eq. (24).
Figure 10 illustrates the net buildup of the $\mu$ distortion for some
examples of string parameters. The figure indicates that the amplitude of a
$|\mu|$ distortion can be boosted substantially compared to the case where
entropy injection is neglected, leading to more stringent constraints in some
regions of parameter space.

Figure 10: The buildup of a $\mu$ distortion from energy and entropy injection
for a range of cosmic string parameters. Solid lines represent positive $\mu$
(where energy injection is dominant), while dashed lines show negative $\mu$
(indicating entropy injection is the stronger process). The green contour
illustrates an interesting case where a transition takes place between strong
energy and strong entropy injection, which would lead to an anomalously low
total $\mu$.
Figure 11: Constraints ($2\sigma$) obtained requiring $|\mu|\leq 8.4\times
10^{-5}$ from COBE/FIRAS, and $|\mu|\leq 2.8\times 10^{-8}$ for a PIXIE-like
experiment. At high currents and string tensions, energy injection is
responsible for sourcing a positive $\mu$ distortion. However, as the string
parameters are lowered, we see a large strip of parameter space that is ruled
out due to excessive photon production, yielding a negative $\mu$. Finally,
PIXIE is capable of constraining further regions of parameter space from
energy injection at still smaller currents and tensions. Dashed-dotted lines
correspond to negative $\mu$ distortions.

In Fig. 11 we show the $\mu$ distortion constraints obtained by considering
both entropy and energy injection from a distribution of string loops. The
figure clearly shows a region of parameter space that is ruled out from
entropy injection but was missed in the analysis of Tashiro et al. (2012a) and
Miyamoto & Nakayama (2013). For PIXIE, the left and right solid contours
correspond to the usual energy injection constraints seen in Fig. 5, while the
central contour is novel, and comes from entropy injection. For the COBE/FIRAS
results, the sensitivity to $|\mu|$ is too weak to resolve the energy
injection constraint at lower currents. The energy injection constraints
appear weaker than in Fig. 5 simply because here we consider a smaller
redshift window in which the injection is active. This choice is because for
$z\leq 3\times 10^{5}$, injected photons are not efficiently redistributed and
hence Eq. (24) is not valid. To consider the spectral distortions produced
below this redshift, one has to go beyond the simple analytic picture and
solve the problem numerically. In the remainder of this work we compute and
analyze these numerical solutions using CosmoTherm.

##  5 Numerical Implementation in CosmoTherm

Figure 12: The instantaneous spectrum of photons produced by a network of
superconducting cosmic string loops as a function of the dimensionless
frequency $x=\hbar\omega/kT$. The left plot shows the response of the spectrum
at constant $G\mu$ and redshift, for variations of $\mathcal{I}$. On the right
we fix $G\mu$ and $\mathcal{I}$ to observe how the injection spectrum varies
at different redshifts. For increasing current and decreasing redshift, the
position of the knee moves to lower frequencies, indicating that fewer high
energy photons are produced by the network. The knee position $x_{\rm
k}=\omega_{\rm H}(t)/[\lambda l_{\rm c}T]$ by indicated as dashed vertical
lines.

In order to provide robust spectral distortion constraints, as well as to
validate our analytic treatment, we now aim to implement source terms from
cosmic strings into CosmoTherm (Chluba & Sunyaev, 2012), which is a state of
the art cosmological thermalization code that allows explicitly treating
multiple energy release and photon injection
scenarios.444www.Chluba.de/CosmoTherm Practically, CosmoTherm takes as input a
redshift-dependent injection function describing the change in photon
occupation number, and evolves it forward to determine the spectral distortion
signature. To determine the total source term, we average Eq. (26) over the
full loop distribution present at any given time. For clarity we first
consider injection in the radiation era (i.e., cases with $t\leq t_{\rm eq}$),
and then generalize to the matter-dominated era. Inserting the corresponding
loop distribution, we then have

$\displaystyle\left.\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega}\right|_{\rm r}$ $\displaystyle=\int_{0}^{L_{\rm max}}{\rm
d}L\,\frac{{\rm d}^{2}E_{\gamma}^{c}}{{\rm d}t{\rm d}\omega}\,\left.\frac{{\rm
d}N_{\rm loops}}{{\rm d}L}\right|_{\rm r}$
$\displaystyle=\left(\frac{\alpha\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}}{t^{3/2}}\frac{\left(1+\lambda\right)^{3/2}}{\omega^{2/3}}\int_{0}^{L_{\rm
up}(\omega)}\frac{{\rm d}L\,L^{1/3}}{(L+\Gamma G\mu t)^{5/2}},$ (34)

where the upper bound of the $L$ integral depends on $\omega$. Since the
maximal frequency of the emission for a loop of length $L$ is given by
$\omega_{\rm max}(L)=\mu^{3/2}/[\mathcal{I}^{3}L]$ (Cai et al., 2012), at a
given frequency $\omega$ this means that a limiting length,

$\displaystyle L_{\rm lim}(\omega)=\frac{\mu^{3/2}}{\mathcal{I}^{3}\omega},$
(35)

should not be exceeded, implying $L_{\rm up}=[\beta t,L_{\rm
lim}(\omega)]_{<}$. With the substitution $l=L/[\Gamma G\mu t]$ we then find

$\displaystyle\left.\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega}\right|_{\rm r}$
$\displaystyle=\left(\frac{\alpha\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}\,\left(1+\lambda\right)^{3/2}}{(\Gamma
G\mu)^{7/6}\,t^{8/3}}\times\frac{\mathcal{F}_{\rm r}(l_{\rm
up})}{\omega^{2/3}},$ (36)

where $\mathcal{F}_{\rm r}(l)=\mathcal{J}_{\rm r}(l,0)$ with various
approximations discussed in Appendix A. Since $l_{\rm up}$ scales strongly
with frequency, the emission spectrum steepens at high frequencies.

Also considering the evolution of loops from the radiation dominated era at
$t_{\rm eq}\leq t\leq t_{\rm end}$ we then find

$\displaystyle\left.\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega}\right|_{\rm r}$
$\displaystyle=\left(\frac{\alpha\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}\,\left(1+\lambda\right)^{3/2}}{(\Gamma
G\mu)^{7/6}\,t^{8/3}}\,\frac{1}{\omega^{2/3}}$ (37)
$\displaystyle\qquad\times\begin{dcases}\mathcal{F}_{\rm
r}\left(\left[\frac{t_{\rm end}}{t_{\rm eq}}-1,l_{\rm
lim}\right]_{<}\right)&(t\leq t_{\rm eq})\\\ \mathcal{F}_{\rm
r}\left(\left[\frac{t_{\rm end}}{t}-1,l_{\rm
lim}\right]_{<}\right)\left(\frac{t_{\rm eq}}{t}\right)^{1/2}&(t_{\rm
eq}<t\leq t_{\rm end}).\end{dcases}$

For the loops created in the matter-dominated era, we similarly find

$\displaystyle\left.\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega}\right|_{\rm m}$ $\displaystyle=\frac{\alpha_{\rm
m}\Gamma_{\gamma}}{3}\frac{\mathcal{I}^{2}(1+\lambda)}{(\Gamma
G\mu)^{2/3}\,t^{8/3}}\,\frac{1}{\omega^{2/3}}$ (38)
$\displaystyle\quad\times\begin{dcases}\mathcal{F}_{\rm m}\left(\frac{t_{\rm
end}}{t}-1,\left[\frac{t_{\rm end}}{t_{\rm eq}}-1,l_{\rm
lim}\right]_{<}\right)&(t_{\rm eq}<t\leq t_{\rm end})\\\ \mathcal{F}_{\rm
m}\left(0,\left[\frac{t_{\rm end}}{t_{\rm eq}}-1,l_{\rm
lim}\right]_{<}\right)&(t_{\rm end}<t),\end{dcases}$

with $\mathcal{F}_{\rm m}\left(l_{a},l_{b}\right)=\mathcal{J}_{\rm
m}\left(l_{a},l_{b},0\right)=\mathcal{F}_{\rm
m}\left(l_{b}\right)-\mathcal{F}_{\rm m}\left(l_{b}\right)$ while
$l_{b}>l_{a}$ and zero otherwise. The expression for $\mathcal{F}_{\rm
m}\left(l\right)$ is given in Eq. (A). This allows us to compute all the
required emission spectra.

As a simple example, let us approximate the spectrum of photons produced
before matter-radiation equality. Starting from Eq. (36), we can use the
approximation for $\mathcal{F}_{\rm r}(l_{\rm up})$ found in Appendix A to
obtain

$\displaystyle\left.\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega}\right|_{\rm r}$
$\displaystyle\approx\left(\frac{\alpha\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}(1+\lambda)^{3/2}}{\omega^{2/3}(\Gamma
G\mu)^{7/6}t^{8/3}}\frac{\mathcal{F_{\infty}}}{1+(l_{c}/l_{\rm up})^{7/6}},$
(39)

where
$\mathcal{F}_{\infty}=\Gamma\left(\frac{7}{6}\right)\Gamma\left(\frac{7}{3}\right)/\sqrt{\pi}\approx
0.6232$ and $l_{\rm c}\approx 1.314$ were determined by inspecting the
asymptotic regimes of $\mathcal{F}_{\rm r}(l_{\rm up})$ The approximation is
valid to within $10\%$ for $l_{\rm up}\gtrsim 0.1$, and we discuss better
approximations in the Appendix. For a given frequency and parameter set,
$l_{\rm up}$ can take on different values. Specifically,

$\displaystyle l_{\rm up}$ $\displaystyle=\begin{dcases}\frac{t_{\rm
end}}{t_{\rm eq}}-1=\frac{1}{\lambda}\hskip
28.45274pt&\left(\omega\leq\omega_{\rm H}\right)\\\ \frac{L_{\rm
lim}(\omega)}{\Gamma G\mu t}=\frac{1}{\lambda}\frac{\omega_{\rm
H}}{\omega}\hskip 28.45274pt&\left(\omega>\omega_{\rm H}\right),\end{dcases}$
(40)

where, $\omega_{\rm H}\equiv\omega_{\rm max}(\beta
t)=(G\mu)^{3/2}/[G^{3/2}\mathcal{I}^{3}\beta t]$. For most parameter values,
$\lambda\ll 1$, allowing us to simplify the spectrum as

$\displaystyle\left.\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega}\right|_{\rm r}$
$\displaystyle\approx\left(\frac{\alpha\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}(1+\lambda)^{3/2}}{\omega^{2/3}\,(\Gamma
G\mu)^{7/6}\,t^{8/3}}\,\begin{cases}\mathcal{F}_{\infty}&\;\omega\leq\omega_{\rm
H}\\\\[2.84526pt] \frac{\mathcal{F}_{\infty}}{1+(\omega/\omega_{\rm
k})^{7/6}}&\;\omega>\omega_{\rm H}\end{cases}$
$\displaystyle\approx\left(\frac{\alpha\Gamma_{\gamma}}{3}\right)\frac{\mathcal{I}^{2}(1+\lambda)^{3/2}}{\omega^{2/3}\,(\Gamma
G\mu)^{7/6}\,t^{8/3}}\frac{\mathcal{F}_{\infty}}{1+(\omega/\omega_{\rm
k})^{7/6}}.$ (41)

Here, we defined $\omega_{\rm k}=\omega_{\rm H}/[\lambda l_{\rm c}]$. Going
from the first to the second line, we note that for small $\lambda$, we have
$\omega\leq\omega_{\rm H}\ll\omega_{k}$. At low frequencies
($\omega\ll\omega_{\rm k}$), the spectrum decays as $\omega^{-2/3}$. As we
increase $\omega$, we encounter a knee in the spectrum at
$\omega\approx\omega_{\rm k}$ after which the spectrum decays much more
rapidly as $\omega^{-11/6}$. The position of this knee can be seen for
different values of the current and redshift in Fig. 12, where we illustrate
the exact spectrum which we compute numerically.

In Fig. 13 we separately show the contribution to the emission spectrum from
loops formed in the matter and radiation eras for $G\mu=10^{-11}$ and
$\mathcal{I}=4.2\times 10^{7}\,{\rm GeV}$. For $z\gg z_{\rm end}\simeq 10$,
the radiation loops dominate the contribution at all frequencies. As one
approaches $z\simeq z_{\rm end}$, the matter loops play a more important role,
except at the highest frequencies. This is because the highest frequency
photons are produced by the smallest loops, which still date back to the
radiation era until $z$ drops below $z_{\rm end}$.

Figure 13: Contributions to the emission spectrum from loops formed in the
matter era (dashed) and radiation era (solid) at different redshifts near
$z_{\rm end}$. At $z\simeq z_{\rm end}$, the radiation loops play a dominant
role only at the highest frequencies. At low frequencies, the contribution
from matter and radiation loops is roughly equal at $z\simeq 20$. Before this
time, matter loops are subdominant for the entire frequency range for the
chosen parameters.

To validate this approximation, we can readily integrate over frequencies to
find the total energy release rate

$\displaystyle\left.\frac{{\rm d}Q}{{\rm d}t}\right|_{\rm r}$
$\displaystyle=\frac{2}{3}\frac{\alpha\Gamma_{\gamma}}{G^{1/2}}\frac{\mathcal{I}}{\Gamma^{3/2}G\mu}\frac{(1+\lambda)^{3/2}}{t^{3}}\times\frac{3\pi\mathcal{F}_{\infty}}{7l_{\rm
c}^{1/3}\cos(3\pi/14)}.$ (42)

This matches the exact result found in Eq. (15) to better than $3\%$. This
small departure is caused by corrections to the total emission by means of our
approximation to $\mathcal{F}_{\rm r}(l)$. When using a refined approximation
for $\mathcal{F}_{\rm r}(l)$ in Eq. (53) and numerically integrating the
function find agreement with the exact result at the level of $\simeq 0.06\%$.

Figure 14: Snapshots of the distortion spectra $\Delta I=I_{\rm CT}-I_{\rm
pl}$ as output from CosmoTherm at different redshifts. Dashed lines represent
a deficit of photons (a negative distortion) when compared with a blackbody.
Top left: Output with no external heating sources. A small (negative)
distortion is generated through the adiabatic cooling of electrons, which
continuously extracts energy from the photon background. Top right: Inclusion
of a weak source term from string loops. A positive $\mu$ distortion is now
generated through injection of energy at early times. At late times, a low-
frequency spectrum of photons is generated. Bottom left: A string source with
$\mathcal{I}=\mathcal{I}_{*}$, which produces a sizeable $\mu$ distortion as
well as a low frequency excess. Bottom right: String source with a high
current. COBE/FIRAS is capable of constraining this model based upon entropy
injection.

For the photon number injection, the integral over frequency must be modulated
by the probability factor, ${\rm e}^{-\omega_{\rm c}/\omega}$, which makes it
more difficult to tract analytically. For applications we recommend simply
taking the integral numerically. As input, CosmoTherm requires a time-
dependent occupation number injection function. Schematically this can be done
for expressions (36)-(38) by noting

$\displaystyle\frac{{\rm d}n_{\gamma}}{{\rm
d}t}=\frac{\pi^{2}}{\omega^{3}}\frac{{\rm d}^{2}E_{\gamma}}{{\rm d}t{\rm
d}\omega},$ (43)

where $n_{\gamma}$ is the occupation number of the injected photons. We
validated our implementations for ${\rm d}^{2}E_{\gamma}/{\rm d}t{\rm
d}\omega$ in multiple regimes, finding excellent agreement with the exact
analytic expressions.

##  6 Numerical Results

After feeding in a source term of the form Eq. (43), CosmoTherm follows the
direct evolution of photons with frequency $x_{\rm min}\leq x\leq x_{\rm
max}$555We usually use $x_{\rm min}=10^{-8}$ and $x_{\rm max}=150$ with 4000
grid points.. Heuristically, this code solves the coupled evolution equations
for the photons and electrons on a finely spaced grid of redshift slices,
which allows us to analyze spectral data at any time. In principle, the source
term can take any form, making CosmoTherm a powerful and flexible tool to
quickly and accurately assess the validity of many beyond the standard model
(BSM) scenarios, such as decaying dark matter (Bolliet et al., 2021).

###  6.1 Spectral information

The main data product produced by CosmoTherm is the differential spectra
$\Delta I=I_{\rm CT}-I_{\rm pl}$, the difference between the numerically
computed spectrum with source functions and a pure blackbody. Figure 14
illustrates the buildup of spectral distortions for a range of different
redshifts and string parameters. In the top left panel, we note that even in
the absence of an external source term, a non-zero distortion is observed.
This comes from the fact that the adiabatic cooling of electrons is more rapid
than the photons ($T_{\rm e}\propto a^{-2}$ vs $T_{\gamma}\propto a^{-1}$). In
reality, $T_{\rm e}\simeq T_{\gamma}$ as the electrons are continuously heated
by the photons through the process of Compton cooling, which is effective
until $z\simeq 10^{2}$. This process of energy extraction from the CMB
produces the observed distortion in this panel.

The remaining three panels show benchmark models with values of $\mathcal{I}$
below, equal to, and above the critical current $\mathcal{I}_{\rm c}$. In each
of these three cases, we can observe the buildup of a low frequency background
of photons over time. Higher currents typically lead to stronger backgrounds,
though the background is only significant for $\omega\lesssim\omega_{\rm k}$
as can be seen by Fig. 12. This implies that low frequency data such as that
from the ARCADE-2 (Fixsen et al., 2011) experiment will not be sensitive to
arbitrarily large values of $\mathcal{I}$.

Positive $\mu$ and $y$ distortions are signified by an excess of photons at
high frequencies, and a decrement at low frequencies when compared to a
blackbody. For $\mathcal{I}=10$ GeV and $\mathcal{I}=\mathcal{I}_{*}$, the
distortion is primarily sourced by energy release, while for
$\mathcal{I}=10^{7}$ GeV, a strong entropy release generates a negative
distortion. Radio observations and CMB experiments probe complementary regions
of the induced cosmic string spectra, and with CosmoTherm we are able to
utilize constraints from both datasets simultaneously. See Fig. 17, for
additional illustration of the final distortion signals for these benchmark
scenarios.

Throughout this work we have chosen $G\mu=10^{-11}$ as a fiducial value for
illustrations. This choice has been made because these contours highlight all
of the relevant physical effects. Here, we would like to discuss the changes
one observes when varying $G\mu$, and assuming that $\lambda\ll 1$. First, we
note that the energy injection rate scales as ${\rm d}Q/{\rm
d}t\propto\mathcal{I}/G\mu$ in the gravitational decay regime (GDR,
$\mathcal{I}\ll\mathcal{I}_{*}$), and as ${\rm d}Q/{\rm
d}t\propto(G\mu)^{5/4}/\mathcal{I}^{1/2}$ in the electromagnetic decay regime
(EDR, $\mathcal{I}\gg\mathcal{I}_{*}$). This implies that the total energy
injection is maximized for $\mathcal{I}\simeq\mathcal{I}_{*}$, and that
reducing $G\mu$ causes a faster decay of the signal compared to increases in
$\mathcal{I}$ while in the EDR. This broadly explains the constraints
presented in Fig. 5.

In terms of direct photon production, the scaling is slightly more
complicated. In the GDR, the overall amplitude of the spectrum follows ${\rm
d}^{2}N_{\gamma}/{\rm d}t\,{\rm d}\omega\propto\mathcal{I}^{2}/(G\mu)^{7/6}$,
while in the EDR we have ${\rm d}^{2}N_{\gamma}/{\rm d}t\,{\rm
d}\omega\propto\mathcal{I}^{5/6}(G\mu)^{7/12}$. As expected, decreases of
$G\mu$ in the EDR cause a decrease in the amplitude. In contrast to the energy
injection case, the total number of photons produced is not maximized on the
$\mathcal{I}=\mathcal{I}_{*}$ contour, but instead increases with
$\mathcal{I}^{5/6}$ in the EDR. However, the constraints on direct photon
injection are sensitive to the precise frequencies of the produced photons.
The position of the spectral knee determines an effective upper cutoff to the
photon frequencies, and scales as $\omega_{\rm
k}\propto(G\mu)^{1/2}/\mathcal{I}^{3}$ in the GDR, and $\omega_{\rm
k}\propto(G\mu)^{2}/\mathcal{I}^{4}$ in the EDR. Thus, decreases in $G\mu$ and
increases in $\mathcal{I}$ rapidly degrade this effective cutoff frequency.
Once $\omega_{\rm k}\lesssim\rm{MHz}$, the vast majority of the photons are
produced outside of the sensitivity range of microwave and radio observations.

Figure 15: A comparison of the free electron fraction with and without the
direct production of ionizing photons from the string loops. When no ionizing
photons are included, all energy is injected in the form of heat. This energy
injection reduces the recombination rate and increases the collisional
ionization efficiency, leading to the mild deviation from the case with no
strings. When the direct production of ionizing photons is included, the
$X_{\rm e}$ history is strongly modified, which leads to much stronger
constraints. Figure 16: The free electron fraction for two different
treatments of ionizing photons. For the green curve, CosmoTherm directly
computes the number of ionizing photons produced by the strings, removing them
as they liberate electrons (see Bolliet et al., 2021, for details). This is
computationally expensive, and can systematically underestimate the number of
ionizations that take place. To produce our constraints we follow the
approximate treatment described by Chen & Kamionkowski (2004).

Figure 17: Left: A comparison of the final spectra that we would observe today
from our benchmark string parameters. Vertical lines indicate to frequencies
bands probed by CMB and radio experiments. Higher currents tend to build up
larger low frequency backgrounds, while parameters near the
$\mathcal{I}=\mathcal{I}_{*}$ contour deposit more raw energy/entropy into the
CMB, and are therefore more easily constrained by COBE/FIRAS. Right:
Ionization histories for the parameters. Departures from the standard history
are typically stronger for larger currents, until for a given set of
parameters, $\omega_{\rm k}\lesssim 13.6$ eV. In that case, ionizing photons
are produced with a greatly reduced efficiency, as indicated in Eq. (5).
Further details on this can be found in Fig. 20.

###  6.2 Treating recombination history changes

The emission produced by the cosmic string network also injects photons above
the ionization thresholds of hydrogen and helium. This causes modifications to
the recombination history of the Universe, which we treat approximately in
this work. Specifically, photons outside the computational domain have to be
added, leading to treatments discussed now.

For the first treatment, we only include the total heating from photons
outside of our computational domain, accounting for photons injected at both
$x\leq x_{\rm min}$ and $x\geq x_{\rm max}$, but do not consider ionizations
due to photons at frequency $x$ above the atomic ionization thresholds. The
energy density integrals are computed numerically at every time-step and then
converted into heating of the baryonic matter. Since hotter electrons
recombine less rapidly, this causes a delay of recombination.

In the second treatment, we more carefully consider the interactions of
photons above the atomic energy levels. For photons injected below the
Lyman-$\alpha$ line of hydrogen, we simply directly follow their evolution in
the distortion domain, computing the Compton heating internally.666In our
calculations we have $x_{\rm max}\simeq 150$, which means that at $z\lesssim
200$, the hydrogen Lyman-$\alpha$ line moves outside of the computational
domain. We neglect the related heating of photons between $x_{\rm max}$ and
$x_{\rm Ly-\alpha}$ at $z\lesssim 200$, which should have a minor effect. At
$x\geq x_{\rm Ly-\alpha}$, we integrate the total energy density of photons
and then use the method of Chen & Kamionkowski (2004) with refinements
according to Chluba (2010) to add heating and ionizations to the recombination
problem. At $z\geq 3000$, we assume all the injected high-frequency energy is
converted into heat. At $z\leq 3000$, we do not add any photons at $x\geq
x_{\rm Ly-\alpha}$, assuming that these get efficiently absorbed and
converted. A comparison of treatments one and two are shown in Fig. 15 with
the orange and green contours respectively.

In principle, we can also directly follow the evolution of photons in the
Lyman continuum using CosmoTherm, though this also has limitations. While this
process gives a more direct correspondence between the number of ionizing
photons produced, and the total amount of ionizations (see Bolliet et al.,
2021, for details), it misses out on an important reprocessing effect. Namely,
high energy photons ($x\gg x_{\rm Ly-\alpha}$) will both ionize and heat the
background. This heating of the background can introduce important secondary
ionizations, particularly in the high energy regime (e.g., Shull & van
Steenberg, 1985; Slatyer et al., 2009; Valdés et al., 2010) that can be missed
otherwise. Bolliet et al. (2021) utilized this Lyman continuum treatment for
generic decaying particle scenarios and at high energies found weaker bounds
compared to different approaches employed by Capozzi et al. (2023) and Liu et
al. (2021). The approximate treatment by Chen & Kamionkowski (2004) attempts
to account for this by partitioning a fraction of the energy above the
Lyman-$\alpha$ threshold to use for ionizations, excitation and heating. We
neglect the effect of excitations, which have been found to be minor (Galli et
al., 2013). A comparison of the free electron fraction for these two
treatments can be found in Fig. 16, where we see that the pure Lyman continuum
computation generically underestimates the effect.

Prior treatments of the ionization history in the presence of strings (Tashiro
et al., 2012b) utilized an incorrect spectral index for a large region of
parameter space, which we correct here. Additionally, their analysis
considered a simple photon counting procedure to derive their anisotropy
constraints, which misses out on the reprocessing effect mentioned above.
Ionization histories for our benchmark cases can be found in the right panel
of Fig. 17.

As mentioned above, our implementation is simplified and does not take into
account complications of high-energy cascades (e.g., Slatyer et al., 2009;
Hütsi et al., 2009; Slatyer, 2016), or most recently Liu et al. (2023a, b).
However, it allows us to approximately follow the evolution of both the
spectral distortions and the ionization fractions, which also can be used to
compute the related 21-cm signals, as outlined in Acharya et al. (2022). Our
treatment could be further improved by directly treating the ionizations
Bolliet et al. (2021) and also adding secondary energetic particles. This
would also allow us to go beyond the ’on-the-spot’ approximation (Chen &
Kamionkowski, 2004; Padmanabhan & Finkbeiner, 2005), but we leave a more
detailed exploration to future work.

###  6.3 Evolution at late stages

At $z\lesssim 500$, the effect of electron scattering on the evolution of the
spectral distortions starts to become very small. We can therefore omit the
broadening and shifting of distortions introduced at these late times. The
photon evolution equation then simplifies, and we only have to include the
$y$-distortion sources from differences in the electron and photon
temperature, the emission and absorption of photons by the free-free process
and the external photon source from cosmic strings. This greatly simplifies
the calculation, as the evolution of the photon distribution in each frequency
bin becomes independent. We confirmed that the results remain largely
unaffected by this simplification.

###  6.4 Soft photon heating and the global $21$-cm signal

As was recently discussed in Acharya et al. (2023), the presence of a
sufficiently steep radio background produces an important backreaction effect
on the $21$-cm differential brightness temperature ($\delta T_{\rm b}$) at
cosmic dawn. This brightness temperature is given by

$\delta T_{\rm b}=\frac{(1-{\rm e}^{-\tau_{21}})}{1+z}(T_{\rm s}-T_{\rm R}),$
(44)

where $T_{\rm R}$ is the temperature of the background at $21$-cm (usually
assumed to be solely the CMB), $\tau_{21}$ is the $21$-cm optical depth, and
$T_{\rm s}$ is the spin temperature. The spin temperature is a measure of the
ratio of hydrogen atoms in the triplet state relative to the singlet.

Multiple prescriptions for calculations the evolution of $T_{\rm s}$ can be
found in the literature (Furlanetto, 2006; Hirata, 2006; Venumadhav et al.,
2018). Overall, they can be expressed as

$T_{\rm s}^{-1}=\frac{x_{\rm R}T_{\rm R}^{-1}+x_{\rm c}T_{\rm
m}^{-1}+x_{\alpha}T_{\alpha}^{-1}}{x_{\rm R}+x_{\rm c}+x_{\alpha}}.$ (45)

Here, $T_{\rm R}$, $T_{\rm m}$, and $T_{\alpha}$ are the temperatures of the
radiation, matter, and the colour temperature of the Lyman-$\alpha$ radiation.
Additionally, $x_{\rm R}$, $x_{\rm c}$, and $x_{\alpha}$ are the radiative,
collisional, and Wouthuysen-Field couplings respectively.

During cosmic dawn, the dominant contribution to the spin temperature comes
from the kinetic motion of the hydrogen atoms. In Acharya et al. (2023), it
was shown that if an additional radio background is present with a
significantly steep spectral index, the hydrogen atoms are heated and the spin
temperatures rises. As the magnitude of $\delta T_{\rm b}$ is proportional to
$T_{\rm R}/T_{\rm s}$, this soft photon heating (SPH) dampens the expected
signal relative to the case where SPH is neglected.

Brandenberger et al. (2019b) considered the impact of superconducting strings
on $\delta T_{\rm b}$, but neglected this SPH. In addition, the estimates for
the enhancement of the 21cm signal did not account for the radiative transfer
effects and reduction of photons by free-free absorption. As a result, much of
the string parameter space was ruled out by taking the EDGES result as a
strict upper bound. Here, we find that by including soft photon heating, no
string models can be ruled out by taking the EDGES upper limit alone. Figure
18 shows how the inclusion of SPH prevents a particular cosmic string model
from being constrained by the EDGES measurement.

We also note that Hernández (2014, 2021) has studied the effects of cosmic
string wakes on the brightness temperature at $21$-cm and found that a network
can amplify the signal. In the case of superconducting cosmic strings, the
amplitude reduction from soft photon heating surpasses the potential
amplification from the effect of the wakes, meaning that our results do not
change. However, this effect is important for a network of non-superconducting
strings.

###  6.5 Reionization treatment

We give a brief description of our reionization modelling in this section. We
refer the readers to Acharya et al. (2022) for a more detailed discussion. The
evolution of the ionization fraction due to hydrogen and helium is given by,

$\displaystyle\dfrac{{\rm d}x_{\rm HII}}{{\rm d}t}$
$\displaystyle=\xi_{\text{ion}}(z)\dfrac{{\rm d}f_{\text{coll}}}{{\rm
d}t}-\alpha_{A}C\,x_{\rm HII}\,n_{\rm e}$ (46a) $\displaystyle\dfrac{{\rm
d}x_{\rm HeII}}{{\rm d}t}$ $\displaystyle=\xi_{\text{ion}}(z)\dfrac{{\rm
d}f_{\text{coll}}}{{\rm d}t}-\alpha_{A}C\,x_{\rm HeII}\,n_{\rm e},$ (46b)

where $x_{\rm HII}=\frac{n_{\rm HII}}{n_{\rm H}}$, $x_{\rm HeII}=\frac{n_{\rm
HeII}}{n_{\rm H}}$ are the hydrogen and helium ionized fractions respectively,
$\xi_{\text{ion}}$ is the ionizing efficiency parameter, $f_{\text{coll}}$ is
the matter collapse fraction, $\alpha_{A}$ is the case-A recombination
coefficient, $C\equiv\langle n_{\rm e}^{2}\rangle/\langle n_{\rm
e}\rangle^{2}$ is the clumping factor which is a function of gas density, and
$n_{\rm e}$ is the total electron number density. We use the fitting function
of Shull et al. (2012) for the clumping factor. The physics of this modelling
can be explained as follows. Once we have sufficiently massive dark matter
halos which can form galaxies, the photons emitted by these galaxies will
ionize their environment. Such massive halos are rare at higher redshifts (an
effect captured by the collapse fraction), leading to a few isolated
ionization bubbles. As more structure forms at lower redshifts, the number of
photon sources increases and reionization proceeds rapidly.

The ionizing efficiency parameter is given by

$\xi_{\text{ion}}=A_{\text{He}}f_{\star}f_{\text{esc}}N_{\text{ion}},$ (47)

where $A_{\text{He}}$ is a correction factor due to the presence of helium,
$N_{\text{ion}}$ is the number of ionizing photons per stellar baryon,
$f_{\text{esc}}$ is the fraction of ionizing photons escaping the host halo
and $f_{\star}$ is the star formation efficiency. Since $\xi_{\text{ion}}$ is
a degenerate combination of parameters, there are multiple ways to get the
same reionization history using different parameter choices.

For our fiducial model, we use the following combination of parameters
($N_{\text{ion}}$, $f_{\star}$, $f_{\text{esc}}$, $f_{\alpha}$, $f_{X}$) =
(4000, 0.1, 0.1, 1.0, 1.0). We include the heating of electrons due to
energetic X-ray photons through the expression given by Furlanetto (2006),

$\dfrac{2}{3}\dfrac{\epsilon_{X}}{k_{\text{B}}n_{\rm
H}H(z)}=10^{3}\text{K}\leavevmode\nobreak\
f_{X}\left[\dfrac{f_{\star}}{0.1}\right]\left[\dfrac{f_{X,h}}{0.2}\right]\left[\dfrac{{\rm
d}f_{\text{coll}}/{\rm d}z}{0.01}\right]\left[\dfrac{1+z}{10}\right],$ (48)

where $f_{X,h}\simeq(1+2x_{\rm e})/3$ is the fraction of X-ray energy
contributing to the heating (Chen & Kamionkowski, 2004), and $f_{X}$ is a
scaling factor.

Figure 18: The brightness temperature for a benchmark string model with and
without the inclusion of soft photon heating (SPH). As was recently discussed
in Acharya et al. (2023), the presence of a steep radio background increases
the spin temperature of the gas. This leads to a strongly subdued $\delta
T_{\rm b}$ relative to the estimation one would obtain by neglecting this
heating. The EDGES datapoint is plotted in green.

##  7 Constraints

To explore the $G\mu$-$\mathcal{I}$ parameter space, we run a finely spaced
grid of models to generate our numerical data, which directly produces the
likelihood values. We then obtain constraints from these CosmoTherm outputs
using observational data from COBE/FIRAS (Fixsen et al., 1996), CMB
anisotropies (Planck Collaboration et al., 2020), the radio synchrotron
background (RSB) (Fixsen et al., 2011; Dowell & Taylor, 2018), the EDGES
experiment (Bowman et al., 2018), and the optical depth to reionization as
measured by the Planck Collaboration et al. (2020). In addition, we also
forecast constraints from $\mu$, as well as non-$\mu$, non-$y$ type
distortions from a PIXIE-type experiment (Kogut et al., 2011). In order to
make contact with these observations, we analyze the output produced by
CosmoTherm using a rudimentary likelihood analysis. This typically involves
comparing the output produced with strings, to that without, as a test of the
null hypothesis. An exception to this is when comparing against measurements
of the radio synchrotron background, in which we compare to a best-fit power
law of the datapoints. In this section we describe in detail how we obtain the
constraint curves presented in Fig. 21

Figure 19: Comparison of the $2\sigma$ constraints found from COBE/FIRAS, as
well as the forecast for a PIXIE-type instrument, to the analytic predictions
seen in Fig 11. A numerical treatment shows that COBE/FIRAS is capable of
constraining much more than simple analytic treatments would suggest. With
current specifications, PIXIE would observe a strong $y$-type distortion from
reionization. To obtain a conservative estimate, we do not consider the
sensitivity of PIXIE to a primordial $y$-distortion, and instead focus on
limits from $\mu$ and non-$\mu/y$ type signatures.

###  7.1 CMB spectral distortions

The CMB spectrum can be well approximated by a Planck (blackbody) spectrum at
a temperature $T_{0}=2.7255\,{\rm K}$, with upper limits on the distortions of
the order $\Delta I/I\lesssim 10^{-5}-10^{-4}$ at frequencies $\nu\simeq
60-600\,{\rm GHz}$ from COBE/FIRAS (Fixsen et al., 1996). As we have seen
above, electromagnetic energy injection into the baryon-photon plasma heats
the electrons which in turn boosts the CMB photons creating distortions to the
CMB spectrum. Additionally, direct photon injection (entropy) can create
unique spectral patterns which strengthen constraints in some regions of
parameter space.

The current $2\sigma$ upper limit for the amplitude of the $\mu$ and $y$
parameters is $|\mu|\lesssim 9\times 10^{-5}$ and $|y|\lesssim 1.5\times
10^{-5}$ (Fixsen et al., 1996), which translates into a constraint on the
energy release as $\Delta\rho_{\gamma}/\rho_{\gamma}\lesssim 6\times 10^{-5}$,
where $\rho_{\gamma}$ is the CMB energy density today. Using CosmoTherm, we
can go beyond these simple $\mu$ and $y$ parameters by comparing the full
shape of the string-induced spectra to the residuals of the COBE/FIRAS
measurement. This allows a more precise determination of the validity of any
particular model by utilising all of the available data, therefore provides
stronger and more robust constraints when compared with previous analysis such
as the work by Tashiro et al. (2012a), or Miyamoto & Nakayama (2013). However,
it assumes that the marginalization over galactic foregrounds does not further
alter the constraints. In addition, we automatically deproject any
contributions that are degenerate with a simple shift in the CMB temperature.
This is achieved using a simple scalar product of the signal vector on the
COBE/FIRAS bands weighted by the inverse covariance matrix.

The COBE/FIRAS experiment measured the monopole of the background temperature
to exquisite precision. After subtracting the best fit blackboody from this
measurement, one is left with a series of residuals that are mostly consistent
with $0$ at the $1\sigma$ level. The lowest order monopole distortion expected
in standard cosmology is a $y$-type that comes from reionization at the level
of $\Delta I/I\simeq 10^{-7}-10^{-6}$ (Hill et al., 2015). This is far below
the sensitivity of COBE/FIRAS, and so we can be confident that any distortion
which exceeds the residuals comes from the cosmic strings and would therefore
be constrained. To compute the likelihood, we assume that the COBE/FIRAS
datapoints are uncorrelated, and perform two $\chi^{2}$ evaluations using the
residuals. The first using the output spectrum from CosmoTherm with a cosmic
string source term ($\chi^{2}_{\rm CS}$), and the second without the source
($\chi^{2}_{0}$). Finally, we compute $\Delta\chi^{2}=\chi^{2}_{\rm
CS}-\chi^{2}_{0}$ and require $\Delta\chi^{2}\leq 2$ for our $2\sigma$
constraint curves.

The sensitivity of a PIXIE-type instrument is high enough that it would see
the reionization $y$-distortion at more than $100\sigma$ (Abitbol et al.,
2017). In principle, this means that in order to claim a $y$-distortion
signature of cosmic strings from PIXIE, we would need to subtract off the
reionization signal at very high precision. The uncertainties in the model of
reionization implemented in CosmoTherm make this a tricky procedure to pull
off successfully. Therefore, we also choose to deproject the $y$-distortion
from the forecasted PIXIE residuals, and search for distortions of the $\mu$
type, and the non-$\mu$, non-$y$ type that are much cleaner. We perform our
likelihood analysis in the same way as for COBE/FIRAS, but with the
deprojected data. For the forecasting, we assume PIXIE has null residuals with
an effective (foreground-marginalized) sensitivity of $\Delta
I=5\,\rm{Jy/Sr}$, which roughly reproduces a $1\sigma$ sensitivity to
$\mu\simeq 1.4\times 10^{-8}$.

We show the results of this analysis in Fig. 19, and compare with the analytic
approximations computed in Section 4. It is important to note that the
analytic approximations consider only an integration from $3\times 10^{5}\leq
z\leq z_{\rm th}$, as it is impossible to analytically follow the entropy
injection constraints after this point. In contrast, the numerical result
considers the full evolution in the CMB distortion band down to redshift
$z=0$. We therefore expect some natural increase in the contours between the
analytic and numerical results. In general, the approximate constraints
computed from the negative $\mu$-distortion due to entropy release are rather
consistent with the numerical result. It is clear that the analysis of the
COBE/FIRAS residuals provides us with significantly increased constraining
power.

###  7.2 CMB anisotropies

The addition of energetic electromagnetic particles will ionize and heat the
background electrons during and after recombination, modifying the standard
recombination history. As a result, temperature anisotropies are damped while
polarization anisotropies are boosted (Adams et al., 1998; Chen &
Kamionkowski, 2004; Padmanabhan & Finkbeiner, 2005; Galli et al., 2009), and
these effects have been measured to great precision by CMB anisotropy
experiments (Komatsu et al., 2011; Planck Collaboration et al., 2020). There
have been several studies in past years which have improved our knowledge of
recombination physics (Zeldovich et al., 1968; Peebles, 1968; Seager et al.,
2000; Sunyaev & Chluba, 2009; Chluba & Thomas, 2011; Ali-Haïmoud & Hirata,
2011) and one can now compute the recombination history accurately using
publicly available codes such as CosmoRec (Chluba & Thomas, 2011).

Figure 20: $2\sigma$ limits on cosmic string parameters from the induced
changes to the free electron fraction $x_{\rm e}$. The coloured lines indicate
when $\omega_{k}=13.6$ eV. To the left of the lines, the spectrum of ionizing
photons produced by the loop network decays as $\omega^{-5/3}$, while to the
right it falls off much faster as $\omega^{-17/6}$. Consequently, the
constraints are greatly relaxed in the latter region.

To avoid the time-consuming computations that individual samples of the CMB
likelihood would entail, in this work, we use a direct projection method
developed in Hart & Chluba (2020). This is a principal component analysis
(PCA) method following the works developed in Farhang et al. (2012); Farhang
et al. (2013). For our case, with energy injections from string decay, we
compute the changes to the standard recombination history of the universe,
$\xi(z)=\Delta x_{\rm e}/x_{\rm e}$, using the recombination module in
CosmoTherm. We then compute the first three principal component coefficients
by projecting $\xi(z)$ onto the eigenmodes, $E_{i}(z)$, with the integral

$\mu_{i}=\int\xi(x)E_{i}(z)\,{\rm d}z.$ (49)

We use the covariance matrix of the $\mu_{i}$ obtained in Hart & Chluba (2020)
and compute the likelihood of the model assuming Gaussian statistics. We do
not include modifications to reionization in this setup, therefore, the
eigenmodes $E_{i}(z)$ are sensitive to changes in the ionization history only
at high redshifts $100\lesssim z\lesssim 4\times 10^{3}$ (Hart & Chluba,
2020). We treat the modification to reionization history separately which is
described below.

We show the $2\sigma$ constraints from CMB anisotropies in Fig. 20. Of the
datasets we consider, variations in the electron recombination history are the
most wide-sweeping and stringent. As mentioned above, the photons produced
from the strings heat the electrons (increasing collisional ionization rates
and quenching recombinations), as well as directly ionize hydrogen and helium.
Of these two effects, the production of photons with $\omega\geq 13.6$ eV are
the most potent (see Fig. 15). The spectrum of injected photons follows a
broken power law given by Eq. (5), which falls off very rapidly for
$\omega\gtrsim\omega_{\rm k}$. Therefore, a given parameter set with
$\omega_{\rm k}\lesssim 13.6$ eV will be much less efficient at ionizing the
background. However, as discussed in Sec. 6.4, soft photons can be efficiently
absorbed for $x\lesssim 10^{-5}$ (Fig. 6), a process that heats the electrons
and in turn ionize the neutral hydrogen. Since the total energy of photons
within $x\lesssim 10^{-5}$ is similar for certain choices of parameters, we
see the almost horizontal feature in $x_{\rm e}$ constraint at high currents.
We show contours of constant $\omega_{\rm k}$ in the figure, noting that
constraints are greatly reduced for sufficiently low values of the knee
frequency.

###  7.3 Radio synchrotron background (RSB) data

We use the RSB data from ARCADE experiment (Fixsen et al., 2011) and Dowell &
Taylor (2018). The ARCADE-2 experiment measured a RSB between 3-90 GHz. They
reanalyzed earlier data at lower frequencies and compiled it in their Table 4.
With this, they found a best fit power law with spectral index 2.6 and
temperature $T\simeq 24$ K at 310 MHz. In Dowell & Taylor (2018), the authors
redid this analysis using independent data points around $\simeq 40-80$ MHz
and found the best fit slope to be consistent with ARCADE but with slightly
higher normalization with $\simeq$ 30K at 310 MHz. In this paper, we use the
ARCADE data points within 3-90 GHz and the independent data points of Dowell &
Taylor (2018) within the 40-80 MHz band to compute the likelihood.

In the analysis of these datasets, the contribution from resolved extra-
galactic sources were not taken into account. To isolate the contribution to
the radio background from string decay, we add an irreducible extra-galactic
background to our solution obtained from CosmoTherm. The fitting function to
the minimal extra-galactic background (MEG) is given by (Gervasi et al.,
2008a),

$T_{\rm bg}(\nu)\simeq 0.23\,{\rm K}\left(\frac{\nu}{\rm GHz}\right)^{-2.7}.$
(50)

In contrast to the spectral distortion constraints, we compare the induced
radio background from a network of strings not to the CosmoTherm output with
no source term, but to a best-fit power law to the RSB data. With the
inclusion of the MEG, this power law is given by

$T_{\rm RSB}(\nu)\simeq 1.230\,{\rm K}\left(\frac{\nu}{\rm
GHz}\right)^{-2.555}.$ (51)

With this as our null hypothesis, we once again perform a $\chi^{2}$ analysis
against the string induced RSB as described in Sect. 7.1 to obtain our
constraints. This allows us to treat the RSB as a strict upper bound on the
amount of radio emission which can be produced by the string network. We note,
however, that for the RSB limits presented below, we do not add a penalty if
the total background is not reproduced by the sum of the MEG and our
distortion outputs. The excluded region is illustrated in Fig. 21.

###  7.4 Global $21$-cm measurements

We use the claimed detection by the EDGES collaboration (Bowman et al., 2018)
as a figure of merit to constrain the energy injection process from cosmic
strings. EDGES has claimed a detection of a 21-cm absorption feature with
$\delta T_{\rm b}\simeq-500$ mK originating from $z\approx 18$ and a 1$\sigma$
error of 200 mK. Recently, SARAS, an independent experiment, could not
reproduce this result (Singh et al., 2022). Therefore, our discussions on
constraints from global 21-cm measurements are broadly qualitative.

To constrain our energy injection cases, we demand that
$-500\leavevmode\nobreak\ {\rm mK}\lesssim\delta T_{\rm b}\lesssim 0$ at
$z=18$. For $\delta T_{\rm b}\leq-500$ mK, we use a Gaussian likelihood with
an error of 200 mK to quantify the tension with this data. We also penalize
cases with $\delta T_{\rm b}>0$ at $z=18$ by using a gaussian likelihood with
an error of 84 mK which is the value of $\delta T_{\rm b}$ at $z=18$ for our
fiducial 21-cm model without any energy injection.

We find that from the EDGES measurement alone, none of the models exhibit a
$2\sigma$ tension with the data. This is why the left panel of Fig. 21 does
not show a constraint curve. We comment that the soft photon heating effect
described in Acharya et al. (2023) indeed eliminates any regions in tension
with the EDGES measurement. Interestingly, we do find regions of parameter
space where $\delta T_{\rm b}\geq 0$, implying that the signal may be in
emission in the presence of a string network. This is also a direct
consequence of the soft photon heating effect. While we do make the choice to
penalize these models, the affected regions of parameter space have already
been ruled out at more than $2\sigma$ by other datasets.

###  7.5 Optical depth constraints

We use the optical depth measurement of Planck Collaboration et al. (2020)
with $\tau=0.0544\pm 0.0073$ to constrain our energy injection cases. The
significantly lower value of measured $\tau$ has sparked great interest and
has resulted in a shift of our understanding of the reionization epoch
(Kulkarni et al., 2019). However, these works use detailed hydrodynamical
simulations and include several physical effects such as a non-homogeneous
ionizing photon background which are difficult to capture in a simple analytic
setup.

In this work, we assume that energy injections modify the reionization
history, or $\tau$, only perturbatively. Since our fiducial reionization model
gives a $\tau=0.078$,777For this we assume the starting redshift of
reionization be $z=30$, though it does not change significantly with small
change to this starting redshift. we compute the difference of the optical
depth obtained with cosmic strings included, and then add a simple Gaussian
penalty to constrain the model. We checked that tuning the reionization model
parameters to more closely reproduce the measured $\tau$ value, does not alter
the constraints much. In Fig. 21, we show the optical depth constraints which
becomes dominant at high currents, i.e., $I\gtrsim 10^{7}$ GeV. As we
discussed above, at these high currents, the heating due to soft photons is an
important process that changes the ionization history of the universe
appreciably.

Figure 21: Left: A breakdown of the individual ($2\sigma$) constraints
analyzed by CosmoTherm. While changes to the ionization history are dominant
for the majority of the parameter space, late time effects such as an excess
radio background and changes to the optical depth to reionization can become
important for high currents. We also utilize the observation from the EDGES
(Bowman et al., 2018) experiment as a strict upper bound on the brightness
temperature at cosmic dawn, though we find no constraints on strings at the
$2\sigma$ level from that. Right: A combination of the constraints, as well as
a conservative forecast for a PIXIE-like instrument. Such an instrument would
probe a new wedge of parameter space by searching for a negative
$\mu$-distortion sourced by a strong entropy injection from the strings at
early times.

###  7.6 Summary of constraints

In the left panel of Fig. 21 we show the $2\sigma$ constraints obtained
through our likelihood analysis of the COBE/FIRAS, CMB anisotropy, radio
synchrotron background, EDGES, and optical depth measurements. It is clear
that of the datasets we analyzed, limits coming from CMB anisotropies are by
far the most stringent. Our updated limits using a full spectral analysis of
the COBE/FIRAS data are superseded by more recent observational results. It is
important to realize, however, that the COBE/FIRAS data has been available
since the mid 90s. If it had been used to its full extent at the time, these
constraints would have been relevant to the parameter space of superconducting
cosmic string models for many years.

Reionization and RSB constraints cover a region of high currents, which is
perhaps unsurprising as strong photon emitters in the late-time universe are
easier to detect. Interestingly, regions on the boundary of the RSB data may
offer a viable solution to this observed radio excess, as we discuss in a
forthcoming publication.

The right panel of Fig. 21 presents a joint likelihood analysis of the
datasets analyzed by CosmoTherm, alongside our simple forecast for a PIXIE-
type instrument. As described above, we make a conservative choice when
forecasting by considering the PIXIE sensitivity only to $\mu$, and non-$\mu$,
non-$y$ type distortions to avoid having to perform a careful subtraction of
the $y$-distortion induced by reionization. With proper foreground subtraction
and removal of this $y$-distortion, we expect a marginal improvement in the
PIXIE sensitivity to this cosmic string scenario. Importantly, PIXIE would be
capable of constraining an important region of parameter space that is
currently not covered.

Finally, in Fig. 22 we present a more complete illustration of the open
regions of parameter space for superconducting cosmic string models. In
addition to our work, we add constraint curves from Miyamoto & Nakayama (2013)
from pulsar timing array data, big-bang nucleosynthesis, and radio transients
that could be observed by the Parkes array. As a word of caution we mention
that they utilize a slightly different set of input parameters ($\alpha=0.1$
and $\Gamma_{\rm g}=50$) when computing their results, and so their curves
here should be taken as a rough estimate rather than a robust boundary. They
also consider a third form of energy release for the string network, through
so-called plasma dissipation. The analysis of the plasma dissipation
efficiency depends on many uncertain parameters, such as the velocity of any
given loop, the local plasma viscosity and more. We choose not to model these
effects in our work, but note that in Miyamoto & Nakayama (2013), that
deviations from our results seem to appear at very low values of the string
tension ($G\mu\simeq 10^{-18}$).

##  8 Discussion and Conclusions

Figure 22: A summary plot of the constraints analyzed in this work, as well as
limits obtained by Miyamoto & Nakayama (2013) on pulsar timing array
measurements, big-bang nucleosynthesis, and transient radio bursts which we
didn’t directly consider in our work. The work of Miyamoto & Nakayama (2013)
uses a slightly different string loop model, so their constraints should be
viewed as approximate boundaries when compared with ours.

In this work, we have investigated the spectral signatures produced by a
network of superconducting strings in place at early times ($z_{\rm form}\gg
10^{7}$). We have made improvements to the approximate analytic understanding
of the spectral distortions produced by this network, and strengthened these
results by performing full numerical solutions to the thermalization problem
using CosmoTherm.

Analytically, we have refined the previous estimates on primordial distortion
signatures made by Tashiro et al. (2012a) and Miyamoto & Nakayama (2013) by
including a non-trivial contribution to the $\mu$-distortion coming from
strong entropy injections. This negative $\mu$ contribution can be seen by
comparing Fig. 5, which neglects the entropy injection, to Fig. 11.
Additionally, in Sect. 5 we also develop a more sophisticated analytic
formalism for describing the instantaneous spectrum of photons produced by
such a string network, given in Eq. (37) and Eq. (38). These can readily be
applied to other physical scenarios to produce more robust analytic estimates.

We then developed a simple module in CosmoTherm to handle the injection and
processing of the cosmic string source function. The analytic approximations
were compared against the full numerical solution in Fig. 19, where we find
the analytics to be a conservative underestimate of the full constraining
power of spectral distortions. The extra constraining power achieved by
CosmoTherm is the result of being able to perform a full spectral analysis of
the COBE/FIRAS data (and the PIXIE forecast). Previous estimates relied on
constraining the specific distortion parameters, $\mu$ and $y$. With the full
spectral data from CosmoTherm, we are able to go beyond this by comparing the
string induced spectrum to the residuals of COBE/FIRAS, yielding constraints
on non-$\mu$ and non-$y$ type distortions. This full shape spectral analysis
will generically increase the constraining power of spectral distortions to
models of exotic energy/entropy injection, when compared against the simple
analytic estimates.

With the numerical implementation, we also gain access to precise spectral
information at virtually any redshift ($z\lesssim 10^{7}$), which allows us to
easily and efficiently derive constraints from other datasets. In the left
panel of Fig. 21, we show $2\sigma$ constraints from COBE/FIRAS (Fixsen et
al., 1996), CMB anisotropies (Planck Collaboration et al., 2020), the radio
synchrotron background (Fixsen et al., 2011; Dowell & Taylor, 2018), and the
optical depth to reionization as measured by the Planck Collaboration et al.
(2020). Additionally, we utilize the EDGES (Bowman et al., 2018) datapoint as
a strict lower limit on $\delta T_{\rm b}$ at $z\simeq 18$ when combining all
of our constraints. This analysis also presents an update and more robust
treatment compared to other analytic estimates presented by Miyamoto &
Nakayama (2013), and Tashiro et al. (2012b).

In Acharya et al. (2023) it was recently discovered that the presence of a
sufficiently steep soft photon background (spectral index $\gtrsim 2.5$ at
$\nu\lesssim 1$ GHz) can cause significant heating of gas in the late
universe, leading to an increase in the spin temperature $T_{\rm s}$. In turn,
this dampens the amplitude of the brightness temperature at cosmic dawn
($\delta T_{\rm b}$), relaxing the constraints derived from EDGES.
Brandenberger et al. (2019b) derived strong limits on the high current region
of the string parameter space using an analytic approximation which neglected
this new effect as well as overestimated the photon flux due to omission of
radiative transfer effects. We find that by including this effect, constraints
driven by EDGES disappear.

A summary of our constraints, as well as other datasets that we did not
include, can be seen in Fig. 22. As with most models of BSM physics, there
exists some degree of theoretical uncertainty that can be difficult to
quantify. The form of the string loop density distribution is well established
from Nambu-Goto simulations for non-superconducting strings (Blanco-Pillado et
al., 2014), but to our knowledge, large-scale simulations with superconducting
strings have not been performed (though recent progress has been made in Rybak
et al., 2023). Another simplifying assumption is that all loops carry the
same, time independent current from their formation until their eventual
decay. Current generation and dissipation on these loops will ultimately
depend on the local environment in which they propagate, and modelling of this
is beyond the scope of our work. We also note that Miyamoto & Nakayama (2013)
include an additional channel for the string decay through plasma dissipation.
This is a frictional effect which again depends on the dynamics of the local
environment, which we do not model. Based upon the work of these authors, it
appears that this effect may become important for very low string tensions
($G\mu\lesssim 10^{-18}$), and so we advise caution in the interpretation of
our constraints at that level.

In addition to these BSM related uncertainties, we have treated the injection
of energy and photons in an approximate manner (see Sect. 6.2). Similarly,
significant uncertainties exist in the modeling of reionization and the 21cm
signal, although the latter do not drive any constraint here. Finally, our
treatment of the CMB anisotropy likelihood (see Sect. 7.2) had the goal of
quickly exploring the range of models without a significant computational
burden. Similarly, marginalization over spectral distortion foregrounds will
have to be more carefully considered, in particular when assessing the
constraints from future CMB spectrometers. We leave these improvements to
future work, anticipating that some of the details may change, while leaving
the broad conclusions unaltered.

To conclude, superconducting cosmic strings offer an interesting and well-
motivated model that can probe particle physics from the top down. By hunting
for their signatures in different cosmological and astrophysical datasets, we
learn more about the phase transitions that may (or may not) have taken place
in the very early universe. CosmoTherm is a powerful and flexible tool capable
of uncovering the many spectral nuances of not just cosmic strings, but
virtually any scenario that injects energy or entropy into the background.
Here, we have focused on the derivation of constraints for cosmic strings, but
we plan to apply this toolbox to a wider array of physics beyond the standard
model in the future.

## Acknowledgements

We would like to acknowledge the initial work of Jiten Dhandha in setting up
the 21 cm and reionization modules in CosmoTherm. We would also like to thank
Richard Battye and Wenzer Qin for helpful comments and discussions on the
draft. This work was supported by the ERC Consolidator Grant CMBSPEC (No.
725456). JC was furthermore supported by the Royal Society as a Royal Society
University Research Fellow at the University of Manchester, UK (No.
URF/R/191023). BC would also like to acknowledge support from an NSERC-PDF.

## Data Availability

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

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##  Appendix A Entropy Injection Approximations

In order to compute the entropy injection from a network of cosmic string
loops, it was necessary for us to introduce the functions $\mathcal{J}_{\rm
r}(l,\kappa)$ and $\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm b},\kappa)$ in Eqs.
(31b) and (33b) respectively. Here, we investigate the asymptotic regimes of
these functions. These are also useful when describing the average emission
spectra caused by cosmic strings.

For $\kappa l\ll 1$, $\mathcal{J}_{\rm r}$ reduces to a simpler integral which
can be expressed in terms of a hypergeometric function as

$\displaystyle\mathcal{F}_{\rm r}(l)$ $\displaystyle=\mathcal{J}_{\rm
r}(l,0)=\int_{0}^{l}\frac{{l^{\prime}}^{1/3}{\rm
d}l^{\prime}}{(1+l^{\prime})^{5/2}}=\frac{3}{4}l^{4/3}\,_{2}F_{1}\left(\frac{4}{3},\frac{5}{2},\frac{7}{3},-l\right)$
$\displaystyle\approx\frac{\mathcal{F}_{\infty}}{1+(l_{\rm c}/l)^{7/6}}$ (52)

with
$\mathcal{F}_{\infty}=\Gamma\left[\frac{7}{6}\right]\Gamma\left[\frac{7}{3}\right]/\sqrt{\pi}\approx
0.6232$ and $l_{\rm c}=[(7/6)\,\mathcal{F}_{\infty}]^{-6/7}\approx 1.314$.

Figure 23: The form of $\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm b},\kappa)$ with
$l_{\rm a}=0$ and varying values of $l_{\rm b}$. Setting $l_{\rm a}=0$ is
equivalent to requiring that all radiation loops have fully evaporated. In
other words, we are likely deep in matter radiation when this is satisfied.
Approximate forms for small and large $\kappa$ can be found in Eq. (A) and Eq.
(A), respectively. For matter loops, $l_{\rm b}=1/\lambda$, meaning small
values of $l_{\rm b}$ may also be interpreted as the large $\lambda$ regime.

The approximation was obtained by inspecting the asymptotic form of the
solution at large $l$. The approximation does not capture the limit $l\ll 1$
very well, which is $\mathcal{F}_{\rm r}\approx(3/4)\,l^{4/3}$; however,
usually this regime does not contribute much and is only relevant to large
$\lambda$. A slightly improved approximation that is valid to better than
$1\%$ ($10\%$) precision at $0.1\lesssim l$ ($0.0002\lesssim l$) is

$\displaystyle\mathcal{F}_{\rm r}(l)$
$\displaystyle\approx\frac{\mathcal{F}_{\infty}}{1+(\alpha_{\rm f}\,l_{\rm
c}/l)^{(7/6)\,\beta_{\rm f}+\gamma_{\rm f}\ln(l_{\rm c}/l)}}$ (53)

with $\alpha_{\rm f}=0.9044$, $\beta_{\rm f}=0.9871$ and $\gamma_{\rm
f}=0.0164$, obtained from fitting the function. In numerical applications, we
simply pre-tabulate $\mathcal{F}_{\rm r}$ for $10^{-2}\leq l\leq 10^{2}$ and
use fourth order Taylor expansions outside of that domain.

In the opposite limit where $\kappa l\gg 1$, one finds

$\displaystyle\mathcal{G}_{\rm r}(l,\kappa)$ $\displaystyle=\mathcal{J}_{\rm
r}(l,\kappa\rightarrow\infty)\approx\frac{1}{\kappa^{1/3}}\int_{0}^{l}\frac{{\rm
d}l^{\prime}}{(1+l^{\prime})^{5/2}}\frac{{\rm e}^{-\kappa
l^{\prime}}}{\Gamma[2/3]}$
$\displaystyle\approx\frac{3}{4\Gamma[2/3]}\frac{1}{\kappa^{4/3}}$ (54)

For large values of $\kappa$ we find that $\mathcal{G}_{r}(l,\kappa)$ quickly
goes to zero as all of the produced photons are rapidly absorbed by the
plasma. Figure 7 shows the validity of these approximations.

We can do similar approximations for the matter loops. For $\kappa=0$, we have
$\mathcal{J}_{\rm m}(l_{a},l_{b},0)=\mathcal{F}_{\rm
m}(l_{b})-\mathcal{F}_{\rm m}(l_{a})$ with

$\displaystyle\mathcal{F}_{\rm m}(l)$
$\displaystyle=\int_{0}^{l}\frac{{l^{\prime}}^{1/3}{\rm
d}l^{\prime}}{(1+l^{\prime})^{2}}=\frac{3}{4}l^{4/3}\,_{2}F_{1}\left(\frac{4}{3},2,\frac{7}{3},-l\right)$
$\displaystyle=\frac{1}{3}\ln\left(\frac{1+l^{1/3}}{\sqrt{1-l^{1/3}+l^{2/3}}}\right)-\frac{l^{1/3}}{1+l}$
$\displaystyle\qquad\qquad+\frac{1}{\sqrt{3}}\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{l^{1/3}-2}{\sqrt{3}\,l^{1/3}}\right)\right].$
(55)

This function will be extremely important for the computation of the emission
spectrum in the matter dominated era.

Finally, for $\kappa\rightarrow\infty$ it is straightforward to show that
$\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm
b},\kappa\rightarrow\infty)\approx\mathcal{G}_{\rm m}(l_{\rm
b},\kappa)-\mathcal{G}_{\rm m}(l_{\rm a},\kappa)$, where

$\displaystyle\mathcal{G}_{\rm m}(l,\kappa)$
$\displaystyle\approx\frac{1}{\kappa^{1/3}}\int_{0}^{l}\frac{{\rm
d}l^{\prime}}{(1+l^{\prime})^{2}}\frac{{\rm e}^{-\kappa
l^{\prime}}}{\Gamma[2/3]}$
$\displaystyle\approx\frac{3}{4\Gamma[2/3]}\frac{1}{\kappa^{4/3}}\left(1+\mathcal{O}\left(\frac{1}{\kappa}\right)-\frac{{\rm
e}^{-l\kappa}}{(1+l)^{2}}\right)$ (56)

This allows a final simplification

$\displaystyle\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm
b},\kappa\rightarrow\infty)$
$\displaystyle\approx\frac{3}{4\Gamma[2/3]}\frac{1}{\kappa^{4/3}}\left(\frac{{\rm
e}^{-l_{\rm a}\kappa}}{(1+l_{\rm a})^{2}}-\frac{{\rm e}^{-l_{\rm
b}\kappa}}{(1+l_{\rm b})^{2}}\right).$ (57)

We show these expressions in Fig. 23. In Fig. 24 we illustrate the evolution
of $\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm b},\kappa)$ for small values of
$\lambda$ soon after $t_{\rm eq}$. In general, $l_{\rm a}$ and $l_{\rm b}$ are
related through

$\displaystyle l_{\rm a}=\begin{dcases}\frac{t_{\rm eq}}{t}(l_{\rm
b}+1)-1\hskip 56.9055pt(t_{\rm eq}<t\leq t_{\rm end})\\\ 0\hskip
106.69783pt(t_{\rm end}<t),\end{dcases}$ (58)

where $l_{\rm b}=1/\lambda$ at all times $t\geq t_{\rm eq}$.

Figure 24: The form of $\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm b},\kappa)$ with
$l_{\rm b}\rightarrow\infty$ (equivalently, $\lambda\rightarrow 0$). At
$t_{\rm eq}$ we have $l_{\rm a}=l_{\rm b}$, while afterwards $l_{\rm a}$
steadily decreases as the loops shrink in size. At constant $\kappa$, the
emission spectrum steadily rises until $t=t_{\rm end}$ when $l_{\rm a}=0$. In
reality, $\kappa$ increases as $l_{\rm a}$ decreases. Figure 8 shows the
redshift dependence of the window function which acts as a modulation on
$\mathcal{J}_{\rm m}(l_{\rm a},l_{\rm b},\kappa$).

##  Appendix B Particle Physics to Astrophysics dictionary

Astrophysical environments are often the ideal locations to probe particle
physics beyond the standard model. To simplify the notational burden, Particle
physicists prefer to work in natural units where $\hbar=c=k_{\rm b}=1$. This
allows for a simplification in which all units can be expressed in GeV to some
power. To recover real physical units at the end of a computation, one needs
only to insert appropriate values of $\hbar$, $c$, and $k_{\rm b}$ until the
desired units are acquired. Most of the results we present are expressed in a
dimensionless form, simplifying the interpretation. To get a more physical