events was calculated in Damour and Vilenkin (2000, 2001); Binetruy et al.
(2009) and is given by

$h_{b}(\ell,z,f)=A_{b}(\ell,z)f^{-q_{b}}.$ (79)

Here $f$ is the GW frequency at the observer, the index $b$ (for burst)
describes either cusps (c), kinks (k) or kink-kink collisions (kk),

$q_{b}=(4/3,5/3,2)\;{\text{for}}\;b={\text{(c,k,kk) }},$ (80)

and the amplitude $A_{b}\sim G\mu\ell^{2-q_{b}}(1+z)^{q_{b}-1}/r(z)$. For cusp
and kink bursts, the emission of GWs is concentrated in a beam of half-angle

$\theta_{m}\sim[f(1+z)\ell]^{-1/3}.$ (81)

Since $\theta_{m}$ scales as an inverse power of $f$, these individual GW
bursts should therefore be easier to detect with lower-frequency GW detectors
such as LISA Auclair et al. (2023a). Currently the non-observation of cusp and
kink bursts by the LVK collaboration Abbott et al. (2021b) puts an upper limit
on $G\mu$ given a loop distribution model, see below. The increased
sensitivity of 3G detectors will lead to tighter constraints on $G\mu$ from
bursts, however, as we will discuss below, these constraints are not
competitive with the SGWB constraints on $G\mu$.

Since loops are the main sources of GWs, it is crucial to understand their
distribution, namely $n(\ell,t)d\ell$, the number density of loops with length
between $\ell$ and $\ell+d\ell$ at time $t$. As a result of the self-
intercommutation of long strings, loops (with corresponding kinks) are formed
by the infinite string network. Furthermore, once formed, these loops
generally self-intersect many times and fragment into many smaller loops until
eventually all loops reach stable non-self-intersecting trajectories.
According to the Nambu-Goto equations, these stable loops evolve periodically
with period $T=2/\ell$, and thus oscillate and emit GWs (possibly other
radiation, see below) thus decaying. It is through this process that string
network looses energy, and reaches the attractor scaling solution.

Determining all the GW signatures of local cosmic strings therefore boils down
to understanding:

  1. 1.

how a loop radiates energy, and in particular $P_{j}^{{\rm{GW}}}$, the power
emitted into GWs of frequency $2j/\ell$ (the $j$th harmonic).

  2. 2.

the distribution $n(\ell,t)$ of non-self-intersecting loops of length $\ell$
at time $t$.

####  IV.1.1 Energy loss

At large $j$ the emission from cusp and kink bursts (and also kink-kink
collisions) dominates, and using the waveform given above, it follows that
Binetruy et al. (2009)

$P^{{\rm{GW}},b}_{j}\sim j^{-q_{b}}.$ (82)

Often this power-law behaviour is extrapolated down to $j=1$, though
gravitational backreaction effects may modify it see e.g. Blanco-Pillado and
Olum (2017); Blanco-Pillado et al. (2019a). The power emitted in each burst
type is then $\Gamma^{b}=\sum_{j=1}^{\infty}P^{\mathrm{GW},b}_{j}$, where the
$\Gamma^{b}$ are given in Siemens et al. (2006); Damour and Vilenkin (2001);
Ringeval and Suyama (2017). On a loop with $N_{c}$ cusps and $N_{k}$ kinks per
oscillation period, there are $N_{k}^{2}/4$ kink-kink collisions, and the
total GW power emitted is then

$\Gamma^{\rm{GW}}=N_{c}\Gamma^{c}+N_{k}\Gamma^{k}+\frac{N_{k}^{2}}{4}\Gamma^{kk}.$
(83)

Assuming that loops only decay into GWs, then it follows that the length of a
loop changes with time according to

$\frac{\differential\ell}{\differential
t}=-\Gamma^{\rm{GW}}G\mu\equiv\gamma_{d}.$ (84)

A non-self-intersecting loop formed with initial length $\ell_{i}$ at time
$t_{i}$ therefore has lifetime $\ell_{i}/\gamma_{d}$. Note that, in regions of
high curvature, particles and gauge boson radiation may also be emitted
Blanco-Pillado and Olum (1999); Olum and Blanco-Pillado (1999); Matsunami et
al. (2019); Auclair et al. (2020b); Hindmarsh et al. (2021b), leading to
modifications of Eq. (84), for example with terms that dominate at low
$\ell\sim w$ with $w$ the string width Blanco-Pillado and Olum (1999); Olum
and Blanco-Pillado (1999); Auclair et al. (2020b). A further consequence is
that cosmic strings become multi-messenger sources, emitting both GWs and
possibly a diffuse gamma ray background, see Santana Mota and Hindmarsh
(2015); Auclair et al. (2020b, 2023b); Hindmarsh and Kume (2023).

####  IV.1.2 Non-self-intersecting loop distribution

The general picture is that due to intercommutation of infinite strings (and
self-intercommutation of loops), loops are permanently produced over time
together with numerous kinks. The build up of kinks means that the strings
become more ‘wiggly’ (small-scale structure), and the resulting distribution
of kink angles as well as the inter-kink distance was determined in Copeland
and Kibble (2009) where it was shown that the backreaction of emitted GWs on
the shape of a kink is expected to be a crucial effect in order for this
interkink distance to scale. The loops produced eventually reach non-self-
intersecting trajectories, and it is the distribution $n(\ell,t)$ of these
loops at time $t$ which we are after.

This distribution is generally assumed to satisfy a Boltzmann equation which,
neglecting collisions between loops (and also self-intersections of loops), is
given by Copeland et al. (1998); Peter and Ringeval (2013)

$\evaluated{\partialderivative{t}}_{\ell}\quantity[a^{3}n(\ell,t)]+\evaluated{\partialderivative{\ell}}_{t}\quantity[\derivative{\ell}{t}a^{3}n(\ell,t)]=a^{3}{\cal{P}}(t,\ell)\,,$
(85)

where $a(t)$ is the scale factor and $\dot{\ell}$ is given in Eq. (84)
assuming only GR from loops. (See Auclair et al. (2020b, 2021) for solutions
of Eq. (85) including other forms of radiation, namely different
$\dot{\ell}$.) The source term on the right hand side is the loop production
function, with ${\cal{P}}\differential{\ell}\differential{t}$ the number of
non self-intersecting loops with length between $\ell$ and
$\ell+\differential{\ell}$ which are produced at times $t$ to
$t+\differential{t}$ by the infinite string network. Since the infinite string
network has been shown to scale in many simulations Bennett (1986a, b);
Bennett and Bouchet (1988, 1990); Allen and Shellard (1990); Allen and
Caldwell (1991); Ringeval et al. (2007); Blanco-Pillado et al. (2014) then by
dimensional analysis

$t^{5}{{\cal{P}}(t,\ell)}=f(\ell/t)\equiv f(\gamma).$ (86)

To determine the form of $f(\gamma)$ it is generally necessary to resort to
numerical simulations and analytical modelling. Indeed, since Nambu-Goto
simulations do not include effects from either gravitational radiation (GR) or
gravitational backreaction (GBR) on the string dynamics (important for
smoothing out kinks for example), they give a handle on $f(\gamma)$ only on
large scales $\gamma\gg\gamma_{d}$, where $\gamma_{d}$ was defined in Eq.
(84). Another approach is to use numerical simulations to determine the
distribution of non-self-intersecting loops at time $t$ and on scales
$\gamma\gg\gamma_{d}$, combined with analytical forms of $f(\gamma)$ including
GBR derived in Rocha (2008); Polchinski and Rocha (2006); Dubath et al.
(2008). The parameters in $f$ are then fixed so that the solution of 85 agrees
with numerical results on large scales.

In the following we consider two different loop distributions which we label
as Models A and B using the convention of Abbott et al. (2021b). In model A,
based on Blanco-Pillado et al. (2014), all loops are produced with initial
size $\gamma=0.1\gg\gamma_{d}$ at time $t$ so that $f=C\delta(\gamma-0.1)$,
with $C$ fit to Nambu-Goto simulations. In model B, based on Lorenz et al.
(2010), loops of all sizes are assumed to be produced with a power-law form
$f=C^{\prime}\gamma^{2\chi-3}\Theta(0.1-\gamma)\Theta(\gamma-\gamma_{c})$,
where the exponent is negative meaning that more smaller loops are produced.
The scale $\gamma_{c}\ll\gamma_{d}$ physically attempts to incorporate the
effects of gravitational back-reaction: below this scale, kinks are rounded
off and so the strings are straighter and less likely to intercommute and form
loops. Then, once $n(\ell,t)$ is determined from (85), $C^{\prime}$ and $\chi$
are fit to (different) numerical simulations Ringeval et al. (2007) on scales
$\gamma\gg\gamma_{d}$. While models A and B agree with each other (and by
construction with Nambu-Goto numerical simulations) for $\gamma\gg\gamma_{d}$,
they differ enormously on smaller scales, where model B has many smaller loops
Auclair et al. (2019); Auclair (2020).999Note that ${\cal{P}}(\ell,t)$ (or
rather its integral over $\ell$) in turn is related to the energy lost from
the infinite string network through the formation of loops. If the formation
of non-self-intersecting loops is assumed instantaneous (i.e. ignoring loop
fragmentation for instance) then model B has been criticised for non-
conservation of energy of the infinite string network in Blanco-Pillado et al.
(2019b). See also Blanco-Pillado and Olum (2020). Counter-arguments include
the fact that fragmentation is an important effect observed in many
simulations, which changes the energy conservation argument Auclair et al.
(2023b). Here we are agnostic regarding these two models.

Note that models A and B are both calibrated on Nambu-Goto simulations of
cosmic string networks. However, large scale field-theory simulations of
networks of Abelian-Higgs strings have also been developed Vincent et al.
(1998); Moore et al. (2002); Bevis et al. (2007, 2010); Daverio et al. (2016);
Correia and Martins (2019, 2020, 2021). Again these do not include
gravitational radiation, but allow for classical radiation into the scalar and
gauge fields. They observe that loops decay within a Hubble time to classical
radiation of scalar fields. In other words, $n(\ell,t)$ appears to be
essentially zero so that loops do not contribute to GWs. These results as yet
need to be understood further, and tied in with the results of Nambu-Goto
simulations. Indeed, note that some high resolution field theory simulations
of individual loops do show them evolving according to approximate Nambu-Goto
dynamics Matsunami et al. (2019); Hindmarsh et al. (2021b); Blanco-Pillado et
al. (2023). The discrepancy between these two sets of simulation results is
under debate (it may be down to a question of scales which must be resolved,
from the tiny string width $w$ to the Horizon size which differ by orders of
magnitude). We work with Nambu-Goto based results in the following.

####  IV.1.3 Stochastic GW background from local strings

The energy density in GWs observed at a given frequency $f$ today, sourced by
a local string network, can now be obtained by summing over the GW emission
produced from all the oscillating loops produced during the evolution of the
string network from its formation until today, and taking into account the
redshifting of frequencies from the moment of emission until today. In terms
of the number density of loops ${n}(\ell,t)$ Blanco-Pillado and Olum (2017)

$\mathrm{\Omega}_{\rm{GW}}(\ln
f)=\frac{16\pi(G\mu)^{2}}{3H_{0}^{2}f}\sum_{b=(c,k,kk)}\frac{N_{b}\Gamma^{b}}{\zeta(q_{b})}\;\sum_{j=1}^{\infty}j^{1-q_{b}}\int_{0}^{z_{f}}\frac{\mathrm{d}z}{H(z)(1+z)^{6}}~{}{n}\left(\frac{2j}{(1+z)f},t(z)\right)\,\,.$
(87)

Here $\zeta(q_{b})$ is the Riemann zeta function, $q_{b}$ is given in Eq.
(80), and $z_{f}$ is the redshift below which friction effects on the string
dynamics becomes negligible Vilenkin and Shellard (2000). An alternative but
entirely equivalent approach is to calculate the incoherent superposition of
burst signals from cusps, kink and kink-kink collisions emitted from all the
loops in the network, removing strong infrequent bursts that can be resolved
individually, see e.g. Abbott et al. (2021b).

Clearly the properties of the loop distribution ${n}(\ell,t)$ are crucial to
determine the SGWB spectrum. Since model B has 2 characteristic length scales
$(\gamma_{c},\gamma_{d})$, the resulting SGWB spectrum has two characteristic
frequency scales $f_{d}\sim H_{0}/\gamma_{d}$ and $f_{c}\sim
H_{0}/\gamma_{c}\gg f_{d}$. For model A, there is only $f_{d}$. As has been
discussed elsewhere, the spectrum is scale invariant at high frequency ($f\gg
f_{d}$ for model A, $f\gg f_{c}$ for model B) due to the properties of the
loop distribution in the radiation era. Its amplitude however is model
dependent, and is much larger for model B since this contains more small
loops.

In the following we assume standard flat $\Lambda$CDM cosmology with
Planck-2018 fiducial parameters, which determines $H(z)$ and $t(z)$. We also
fix $N_{c}=2$ and $N_{k}=0$ for which $\Gamma^{\rm{GW}}\simeq 57$, a value of
the order of that determined numerically for many different loop shapes
Vachaspati and Vilenkin (1985); Burden (1985); Garfinkle and Vachaspati
(1987); Blanco-Pillado and Olum (2017). Note that both the LVK and EPTA
collaborations considered $N_{k}$ as a free parameter together with $G\mu$
Abbott et al. (2021b); Antoniadis et al. (2023b). The explicit expressions for
$n(\ell,t)$ for the two models may be found for e.g. in the appendix of Abbott
et al. (2021b).

####  IV.1.4 Constraints from 3G detectors

Figure 2 shows $\Omega_{\rm{GW}}(f)$ for both models, calculated using (87).
For Model A (solid, thin line), $G\mu$ has been fixed to $G\mu=7.9\times
10^{-11}$, the value providing the best fit of the EPTA DR2 data set, assuming
strings are the unique source of the SGWB detected by PTA, see Antoniadis et
al. (2023b). In this case, strings formed with $G\mu=7.9\times 10^{-11}$ would
produce a signal in ET/CE with an SNR of more than $500$, providing a
wonderful opportunity for coincident detection and thereby a confirmation of
the origin of the SGWB in the PTA band. For model B, one the other hand, the
value providing the best fit of the EPTA DR2 data set would correspond to
$G\mu=2.5\times 10^{-11}$, giving a very loud signal in the Earth-based
interferometers band: indeed, such large tensions are already excluded by the
LVK data, for which the current constraint on Model B is
$G\mu\lesssim{\rm{few}}10^{-15}$ Abbott et al. (2021b). Therefore, for Model
B, we have chosen one example of signal detectable only within the Earth-based
interferometers band, with $G\mu=4\times 10^{-16}$ (dot-dashed
line).101010Note that there the spectrum is not flat since we are in the
regime $f_{d}\ll f\ll f_{c}$.

Figure 4 shows the SNR of local cosmic strings in CE and ET as a function of
$G\mu$, again for both models. As summarized in Table 2, for Model A, CE will
be able to detect strings with an SNR greater than the foreground from compact
binary mergers for $G\mu\gtrsim 3\times 10^{-14}$, whereas for ET, the
corresponding value is $G\mu\gtrsim 4\times 10^{-13}$. However, with a precise
estimation of the astrophysical foreground, CE and ET detectors might be able
to impose even more stringent constraints on $G\mu$ down to $10^{-16}$,
respectively few $10^{-15}$, for Model A, considering a SNR threshold of 5
(see Table 2). This depends in particular on how the frequency shape of the
SGWB spectrum differs from the one of the foregrounds. For comparison, the
current LVK constraints on Model A are $G\mu\lesssim 9.6\times 10^{-9}$,
whereas PTA constrains $G\mu\lesssim 10^{-11}$. Forecasts for LISA have shown
that it will be able to probe both models A and B, provided the string tension
$G\mu\gtrsim{\cal{O}}(10^{-17})$ Auclair et al. (2020a); Blanco-Pillado et al.
(2024). Thus, next generation ground-based interferometers will be competitive
with LISA.

| Model A | Model B
---|---|---
| SNR $>5$ | SNR $>\rm{SNR}_{\rm{AFG}}$ | SNR $>5$ | SNR $>\rm{SNR}_{\rm{AFG}}$
CE | $\log_{10}G\mu>-16.0$ | $\log_{10}G\mu>-13.5$ | $\log_{10}G\mu>-18.7$ | $\log_{10}G\mu>-16.9$
ET | $\log_{10}G\mu>-14.5$ | $\log_{10}G\mu>-12.4$ | $\log_{10}G\mu>-17.4$ | $\log_{10}G\mu>-16.2$
Table 2: For both loop density models A and B, we provide the value of the
string tension for which the SNR in the associated detector will be greater
than 5, or greater than the SNR of the astrophysical foreground (AFG) expected
in the detector (see Table 1 for details).

####  IV.1.5 Bursts searches

GW detectors will also search for individual loud linearly polarised
gravitational wave bursts from cusp/kink on nearby string loops. Their
observation, or non-observation, can put further constraints on $G\mu$ which
may — or may not — be competitive with the constraints coming from the SGWB
above. Furthermore, as loops are periodic in time, these bursts should repeat
periodically Auclair et al. (2023c), thus increasing their detectability. The
LVK O3 constraints on Models A and B from the non-observation of bursts is not
as competitive with that derived from the stochastic search Abbott et al.
(2021b), and the same is true for ET where the non-detection of bursts should
constrain $G\mu\lesssim 10^{-9}$ for model A and $G\mu\lesssim 10^{-11}$ for
model B PAp . The same is true for LISA: non-detection of these bursts would
constrain $G\mu\lesssim 10^{-11}$ for both models Auclair et al. (2023a).

####  IV.1.6 Beyond ‘standard’ strings

So far we have focused on ‘standard’ strings and $\Lambda$CDM cosmology. The
effects of non-standard cosmology, e.g. an early matter domination or kination
or a late secondary inflationary stage inside the radiation era, on such
strings are very significant and may break the scale-invariant properties of
the SGWB spectrum at high frequencies, see Gouttenoire et al. (2020, 2021) for
further details. Indeed the loop distribution is sensitive to the evolution of
$a(t)$, see Eq. (85).

Returning to standard cosmology, if the strings also radiate particles at
cusps and kinks, then this changes $\dot{\ell}$ and thus the loop distribution
on small scales Auclair et al. (2020b). As expected, the effect on the SGWB is
to cut off the spectrum at high frequencies, $f_{c}\sim(H_{0}\gamma_{d}/w)$
for cusps (the kink cutoff frequency is higher). Generally $f_{c}$ is much
larger than the Hz scale of ground-based 3G detectors for all $G\mu$ of
interest Auclair et al. (2020b). As mentioned above, particle radiation opens
up the possibility of other observational signatures beyond GWs from cosmic
strings, see Santana Mota and Hindmarsh (2015); Auclair et al. (2020b, 2023b);
Hindmarsh and Kume (2023).

If the strings become unstable at late times, for instance due to a second
phase transition creating monopole-antimonopole pairs, then this leads to a
low-frequency cutoff on the SGWB meaning that strings could evade the PTA
constraints Buchmuller et al. (2020b); Servant and Simakachorn (2024).Finally,
if strings carry currents, their evolution is altered (they no longer satisfy
the Nambu-Goto action), see Rybak et al. (2023) and references therein. In
Auclair et al. (2023d) it was shown that if currents are present only for a
certain amount of time between the instant of network formation and today,
then they can give a sizable SGWB in the 3G frequency band.

Figure 4: $\log_{10}$ SNR of the SGWB from a local cosmic string network, as a
function of the string tension $G\mu$. We assume $T_{\rm{obs}}=$ 1yr for both
CE and ET. The gray shaded region corresponds to SNR of the signal $<5$, while
the dotted lines correspond to the respective SNR value of the astrophysical
foreground from compact binary mergers in CE (blue) and ET (orange) detectors
(see Table 1).

###  IV.2 Global $U(1)$ strings

Due to their divergent energy per unit length
$\mu(t)\simeq\eta^{2}\ln(m_{\rm{scalar}}/H(t))$, understanding the evolution
of a network of global strings poses new challenges and uncertainties.
Furthermore, loops now radiate both GWs, radial modes, and Nambu-Goldstone
bosons (which in the following we refer to as axions), and the total rate of
energy loss can be written as
$\Gamma=\Gamma^{\rm{GW}}+\Gamma^{a}+\Gamma^{\rm{radial}}$. (See Drew and
Shellard (2022, 2023); Drew et al. (2023) for recent work on axion and radial
mode radiation from global strings, particularly in regions of high
curvature.) The main contribution to the energy loss is axion radiation, which
dominates over GW emission by a factor of
$\Gamma^{\rm{GW}}/\Gamma^{a}=G\mu^{2}/\eta^{2}$. Thus, the GW background from
a network of global strings is significantly suppressed compared to the axion
one, and for this reason we do not present it here (see, however, Gouttenoire
et al. (2020); Gorghetto et al. (2021a); Chang and Cui (2022); Gouttenoire et
al. (2021); Servant and Simakachorn (2023) for studies on the possibility to
detect GWs from global strings at NANOGrav, LISA and ET/CE). It is still
crucial, however, to understand the evolution of a network of global strings,
since its properties are required to understand the spectrum of axions
produced and thus the cosmological constraints which will be imposed on the
model (see section V for a concrete example).

For completeness, we very briefly highlight some of the significant
differences between local and global string evolution, and some still debated
points regarding global strings. Global strings satisfy the Kalb-Ramond as
opposed to the Nambu-Goto action Kalb and Ramond (1974); Davis and Shellard
(1988) from which, assuming the axion radiation force is small, it follows
that global string loops remain approximately periodic in time as in the
Nambu-Goto case, but with the logarithmic (renormalised) string tension
$\mu(t)\simeq\eta^{2}\ln(m_{\rm{scalar}}/H(t))$. Their lifetime is however
much shorter than that of local strings. There is a long standing debate over
the spectrum of radiation into axions into each harmonic mode $n$, namely
(using analogous notation to Eq. (82)) $P_{j}^{{a}}$ Harari and Sikivie
(1987); Davis (1985); Battye and Shellard (1997). The backreaction of axion
radiation on kinks and cusps may smooth off these features, and hence also
affect GR emission and thus the precise form of $q$ in the GW emission,
$P_{j}^{{\rm GW}}=j^{-q}$, see Chang and Cui (2022); Gorghetto et al. (2021a);
Baeza-Ballesteros et al. (2023) for recent results from numerical simulations
(in particular Baeza-Ballesteros et al. (2023) suggest $q$ is lower than the
standard $5/3$ value for cusps). The GW signal is additionally affected by the
behavior of the string network: several recent works Gorghetto et al. (2018,
2021b); Buschmann et al. (2022); Saikawa et al. (2024) find (logarithmic)
deviations from the scaling regime, implying that there can be more strings
per Hubble volume at late times in the network evolution (see Hindmarsh et al.
(2020) for a different perspective).

###  IV.3 Domain Walls

Domain walls (DWs) are receiving a revived interest currently, mainly because
they are efficient sources of GWs. As mentioned above, DWs arise whenever a
discrete symmetry is spontaneously broken leading to disconnected vacuum
manifold, $\Pi_{0}({\cal{M}})\neq 1$. The Standard Model (SM) does not lead to
DWs but they appear in a variety of SM extensions, remarkably in axion models.
In the early universe, phase transitions that break spontaneously a discrete
symmetry result in a cosmological DW network, a random array of walls formed
at all Hubble patches at the same “formation” time. DW networks were studied
in the past, and it is well known that they have a strong impact on cosmology,
see Vilenkin and Shellard (2000); Vachaspati (2010b) for reviews.

The evolution of the DW network is driven to an attractor ‘scaling’ regime
with statistically self-similar ‘scaling’, essentially described by having
$O(1)$ Hubble-sized walls in each Hubble volume at any time $t$. This leads to
an average network density

$\rho_{DW}=c\,\sigma\,H$ (88)

with $\sigma$ the DW tension and $c$ an $O(1)$ parameter that depends on the
model.111111This is a faster decay than that of a naive ’DW gas’ (an ensemble
of non-interacting DWs) which gain area (and so, mass) as $a(t)^{2}$ and would
dilute as $1/a(t)$. The scaling regime is sustained instead because the DWs
continuously loose energy by wall-wall reconnections and by scalar radiation.

The decay (88) is very slow, leading to the so-called “DW problem”. In
practice, this means that viable DW models must be equipped with some
annihilation mechanism121212Or, the DW tension $\sigma$ is tiny,
$\sigma\lesssim\text{MeV}^{3}$ Zeldovich et al. (1974); Lazanu et al. (2015).
However in this case the GW signal is also small and at ultra-low frequencies
Ramberg et al. (2023); Ferreira et al. (2023a).. Various annihilation
mechanisms are known, for instance i) a small explicit breaking of the
discrete symmetry, ii) an asymmetric population of the degenerate vacua at
formation Larsson et al. (1997) or iii) symmetry restoration Vilenkin (1981).
For simplicity, let us focus on the first mechanism.

Any explicit breaking leads generically to a vacuum non-degeneracy, $\Delta
V$, which acts as a pressure force that drives the DWs, favouring the amount
of volume occupied by the true vacuum. This is efficient roughly when the
Hubble rate decreases below a certain ‘annihilation’ temperature given by

$H(T_{\rm{ann}})\sim\frac{\Delta V}{\sigma}~{}.$ (89)

The details of the annihilation process, including the resulting GW amplitude
and spectrum, depend on the mechanism and require intensive numerical
simulations.

The evolution of DW networks has been computed in several models using field
theory numerical simulations, confirming that the scaling regime is attained.
Results on the produced GWs and their spectrum were confined until recently to
networks in the scaling regime, found to be well described by a broken power
law with exponents $3$ and $-1$ Hiramatsu et al. (2011, 2013, 2014); Krajewski
et al. (2021). While in this work we focus on GW production only during the
scaling regime, we notice that recent progress Kitajima et al. (2023); Chang
and Cui (2023); Ferreira et al. (2024) points to significant emission of GWs
throughout the annihilation phase, enhancing the signal by about 1 order of
magnitude. We comment below on how our results are affected by the
annihilation phase.

For the sake of performing searches and detection prospects, the most
important characteristics of the SGWB spectrum are the peak frequency and
amplitude. In a model independent fashion one can define, as for PTs, the
fraction of the total energy density in DWs at $T_{\text{ann}}$, namely

$\alpha_{\text{ann}}\equiv\frac{\rho_{\text{\tiny
DW}}}{3H^{2}M_{p}^{2}}\Big{|}_{\text{ann}}$ (90)

The GW spectrum is then given by (see e.g. Ferreira et al. (2023b))

$\displaystyle\Omega_{\text{\tiny GW,DW}}(f)h^{2}\simeq
10^{-10}\,\tilde{\epsilon}\left(\frac{10.75}{g_{*}(T_{\text{ann}})}\right)^{\frac{1}{3}}\left(\frac{\alpha_{\rm{ann}}}{0.01}\right)^{2}S\left(\frac{f}{f_{p}^{0}}\right),$
(91)

where $\tilde{\epsilon}\simeq 0.1-1$ is an efficiency parameter to be
extracted from numerical simulations Hiramatsu et al. (2014), and the peak
frequency is

$\displaystyle f_{p}^{0}\simeq
10^{-9}\,\text{Hz}\,\left(\frac{g_{*}(T_{\text{ann}})}{10.75}\right)^{\frac{1}{6}}\left(\frac{T_{\text{ann}}}{10\,\text{MeV}}\right).$
(92)

The function $S(x)$ describes the spectral shape of the signal. A common
parametrization is

$S(x)=\frac{4}{x^{-3}+3x},$ (93)

so that the spectral slopes at $f\ll f_{p}^{0}$ are $3$ as dictated by
causality (assuming radiation domination) and $-1$ and for $f\gg f_{p}^{0}$ as
found in most numerical analyses Saikawa (2017).131313Note that numerical
results for $\mathbb{Z}_{N}$ hybrid string-wall networks suggest that the
exponent can be flatter at high frequencies, decreasing with $N$ Hiramatsu et
al. (2013). The spectrum is cut off at frequencies larger than the
(redshifted) inverse of the wall width.

In the minimal $Z_{2}$ model above, one finds Ferreira et al. (2023b)

$T_{\text{ann}}\simeq\frac{5\,\text{MeV}}{\sqrt{c}}\left(\frac{10.75}{g_{*}(T_{\text{ann}})}\right)^{\frac{1}{4}}\left(\frac{\Delta
V^{1/4}}{10\,\text{MeV}}\right)^{2}\left(\frac{10^{5}\,\text{GeV}}{\sigma^{1/3}}\right)^{\frac{3}{2}},$
(94)

and

$\alpha_{\text{ann}}\simeq
c\,\sqrt{\frac{g_{*}(T_{\text{ann}})}{10.75}}\left(\frac{\sigma^{1/3}}{10^{5}\,\text{GeV}}\right)^{3}\left(\frac{10\,\text{MeV}}{T_{\text{ann}}}\right)^{2}.$
(95)

Figure 5: Expected SNR for DWs for all next generation GW detectors as a
function of $T_{\rm{ann}}$ and $\alpha_{\rm{ann}}$. The color grid corresponds
to the maximal SNR among all the detectors. The DW spectra are computed using
Eq. (91) with $\tilde{\epsilon}=0.7$. Dashed lines: contours with SNR = 5 for
each detector. Dotted lines: contour corresponding to the SNR value of the
respective astrophysical foregrounds in LISA, CE and ET (see section II.2). To
the right of the dotted lines, the DW SNR is above that of the foreground. For
the parameter space between the dotted and the dashed contours, detection can
still be possible depending on the particular frequency shape of the SGWB
signal. Note that the region $\alpha_{\text{ann}}\gtrsim 0.1$ is subject to
constraints from PBH overproduction Ferreira et al. (2024).

As shown in Fig. 5, future GW detectors will be a powerful probe of DW network
models. Their reach to the SGWB produced by a DW network is shown in terms of
the annihilation temperature $T_{\rm{ann}}$ and the fraction of the energy
density $\alpha_{\rm{ann}}$. As expected, ground based detectors are sensitive
to annihilation temperatures around $10^{8}$GeV. However, due to the quite
slow $1/f$ decay of the power spectra at high frequency, the range of
temperatures with high SNR extends to much lower values. In section V we
perform a more specific analysis, focused on the GW signals from axionic DW
networks in heavy axion-type QCD axion models.

Before moving to the next section, let us comment on the recent inclusion of
the annihilation phase Kitajima et al. (2023); Ferreira et al. (2024): this
can roughly be included by replacing $T_{\text{ann}}$ in the equations above
with the temperature at which the annihilating network stops radiating GWs,
$T_{\text{GW}}\sim 0.5~{}T_{\text{ann}}$, and thereby also replacing
$\alpha_{\text{ann}}$ with $\alpha(T_{\text{GW}})\sim 4\alpha_{\text{ann}}$.
Importantly, the region of large $\alpha_{\text{ann}}\gtrsim 0.1$ in Fig. 5 is
likely constrained by the overproduction of Primordial Black Holes (PBH) from
DW annihilation, see Ferreira et al. (2024), although a precise bound is
subject to theoretical uncertainties and model dependence.

##  V Case study: GW backgrounds from axions

Unlike LISA and PTAs, whose frequency ranges encompass the two fundamental
scales of the SM (the EW scale and the QCD scale respectively), the frequency
band of ground-based interferometers corresponds to temperatures that are much
larger than SM scales, roughly $T\sim(10^{7}-10^{9})~{}\text{GeV}$ (this range
of course depends on the typical timescale of the source, e.g. $\beta^{-1}$
for a first order PT given in Eq. 39, among other things, see above). Under
the (feasible) assumption that the Universe was reheated to such temperatures
after inflation, is there any well-motivated beyond the SM phenomenon which
could be probed by ground-based interferometers?

The QCD axion stands out with particular interest: it is motivated as a
solution to the strong CP problem of the SM, and a Dark Matter candidate. The
essential idea in axion models is the Peccei-Quinn (PQ) mechanism, consisting
in introducing a new global $U(1)_{PQ}$ symmetry with two properties: it is
spontaneously broken at a high scale $F_{a}$ and it is anomalous under QCD
Peccei and Quinn (1977a, b). The simplest realization of this mechanism is
effectively described by an additional complex scalar field

$\Phi=\phi\,e^{i\,a/F_{a}}$ (96)

(and possibly additional heavy color charged fermions Kim (1979); Shifman et
al. (1980) or extra Higgs doublets Dine et al. (1981); Zhitnitsky (1980),
which are not relevant for our discussion). This scalar acquires a VEV,
$\langle\phi\rangle\sim F_{a}$ leaving behind a Goldstone boson, the QCD axion
Weinberg (1978); Wilczek (1978), whose interactions with SM particles are
suppressed by the axion decay constant $F_{a}$. Astrophysical Raffelt (1990);
Miller Bertolami et al. (2014); Ayala et al. (2014); Dolan et al. (2022) (see
also Bar et al. (2020); Chang et al. (2018); Carenza et al. (2021)) and CMB
observations (see Notari et al. (2023) for a recent update) impose a lower
bound on $F_{a}\gtrsim(10^{7}-10^{8})\text{GeV}$ (stronger bounds apply in
specific models). In ‘post-inflationary’ scenarios (reheating temperature is
higher than $F_{a}$), the axion can be produced non-thermally and behave like
cold dark matter; requiring that the axion does not overclose the Universe
then leads to an upper bound on $F_{a}\lesssim(10^{10}-10^{11})~{}\text{GeV}$
Gorghetto et al. (2018, 2021b); Buschmann et al. (2022); Saikawa et al.
(2024). We thus see that the viable energy/temperature scale
$(10^{7}-10^{8})\text{GeV}\lesssim F_{a}\lesssim(10^{10}-10^{11})$ GeV has
significant overlap with the epoch that can be probed by ground-based
interferometers.

As sketched in Fig. 1, the PQ mechanism leads to two potential sources of GW
relic backgrounds from the early universe. The first is from the cosmological
$U(1)_{PQ}$ phase transition itself when the temperature $T$ was of order
$F_{a}$. We devote Sec. V.1 to this possibility.

The other potential source arises from the axionic topological defects that
follow after the PQ transition. The spontaneously broken $U(1)_{PQ}$ leads to
a network of global strings. At lower temperature,
$T\sim\Lambda_{\mathrm{QCD}}$, non-perturbative QCD effects induce a further
breaking of $U(1)_{PQ}$ down to a discrete subgroup $Z_{N_{\text{DW}}}$, and
domain walls appear. The integer $N_{\text{DW}}$ is the number of DWs attached
to each string (microphysically it is set by the assignment of PQ charges of
SM and/or heavy quarks). The subsequent evolution depends dramatically on
whether $N_{\text{DW}}=1$ or $N_{\text{DW}}>1$. For $N_{\text{DW}}=1$, the
string network is quickly annihilated at $T\sim\Lambda_{\mathrm{QCD}}$ (and a
tiny amount of GWs are produced). For $N_{\text{DW}}>1$, instead, the
annihilating effect from the DWs gets compensated and so one enters an epoch
with a hybrid string-wall network. An additional small breaking of $U(1)_{PQ}$
is needed to avoid DW domination. Since $U(1)_{PQ}$ is a global symmetry, very
minimal addition to the model (e.g., gravity) can provide the small vacuum
splitting $\Delta V$ for the DW epoch to reach an end Ferreira et al. (2023b).
The GW signal from axionic DWs is higher in heavy axion realizations of the PQ
mechanism (which have particular interest as they address the so-called axion
quality problem Ferreira et al. (2023b)). Remarkably, then, there is a class
of interesting QCD axion models with a DW epoch, and so, a sizeable GW signal.
We devote Sec. V.2 to them.

###  V.1 GWs from a first-order Peccei-Quinn phase transition

The spontaneous breaking of the PQ symmetry occurs via a PT, which can be
first order, depending on the microphysical implementation of the PQ mechanism
and on model parameters. In this case, bubble collisions and/or bulk fluid
motions can source a GW signal that is strong enough to be observable at
future ground-based interferometers. A strong first order PT can arise in a
variety of PQ models, the options ranging from supersymmetric embeddings of
the PQ mechanism to quasi conformal models, both based on weakly or strong
dynamics Von Harling et al. (2020), and based both on KSVZ-like Delle Rose et
al. (2020) or DFSZ-like Von Harling et al. (2020) models.

For simplicity we focus now on scenarios with an approximately conformal
dynamics. This fixes the potential for the PQ field to take the form

$V=\frac{1}{4}\lambda_{\phi}(\phi)\,\phi^{4},$ (97)

with $\lambda_{\phi}(\phi)$ a slowly varying function of $\phi$. There are two
well known implementations leading to (97): classically scale invariant models
where the potential (97) is induced radiatively à la Coleman-Weinberg (CW);
and models where the PQ symmetry is embedded in a strongly-interacting
conformal field theory with a nearly marginal deformation. Both ideas have
been mostly explored in relation to the EW phase transition (see Witten
(1981), and Iso et al. (2017) for a CW first order PT, Creminelli et al.
(2002); Randall and Servant (2007); von Harling and Servant (2018); Von
Harling et al. (2020); Agashe et al. (2020, 2021); Mishra and Randall (2024)
for the strongly interacting case). Here we focus instead on the PQ case, as
discussed in Von Harling et al. (2020) (see also Delle Rose et al. (2020)).

In both cases of interest, the first order PT can occur with a significant
amount of supercooling, meaning that it completes at temperatures much smaller
than the false vacuum energy $T\ll V_{0}(\varphi_{f})^{1/4}$. In this case,
the Universe undergoes a late phase of inflation, since the PQ field is stuck
in a local minimum for a long epoch. The transition thus occurs in an
inflationary background. As discussed in section III, in these cases the mere
occurrence of nucleation, given by the condition $\Gamma(T_{n})\simeq
H(T_{n})^{4}$ (or, more precisely, by Eq. 30), may not be sufficient to ensure
that the transition also completes. To ensure completion, the stronger
criterion given in Eq. 32 on the false vacuum volume decrease at percolation
must be imposed. We study when the criterion in Eq. 32 is fulfilled, and find
that this occurs in relevant regions of the parameter space. By accounting for
this criterion, we improve on the analysis of Von Harling et al. (2020). In
some regions of the parameter space, the first order PT completes without
dramatic supercooling; in this case, we have assessed completion of the PT by
imposing the percolation criterion, i.e. Eq. 31. On the other hand, in some
other regions of the parameter space of the PQ models considered in subsection
V.1.2, the false vacuum volume may start decreasing enough only after
percolation. In these cases, completion of the phase transition is
questionable. Nonetheless, we include this region of parameter space in our
analysis, by imposing a milder criterion, namely that the false vacuum
fraction decreases at some temperature, which can be lower than $T_{*}$. In
these cases, we identify this temperature with the completion of the PT. Note
that, in computing $\alpha$ (c.f. Eq. 54), we always use the temperature of
completion of the phase transition.

Furthermore, to overcome the difficulties associated to the computation of
$\beta/H_{*}$ in the strongly supercooled regime, we compute it at the
nucleation time rather than at percolation time $t_{*}$ (c.f. Eq. 39). We find
that, when $\beta_{n}/H_{n}\gtrsim 10$, nucleation and percolation/completion
are typically not far from each other, which justifies the approximation. For
lower values of $\beta_{n}/H_{n}$ (which we will only consider in subsection
V.1.2), percolation/completion can occur at temperatures that are
significantly lower than $T_{n}$ (in our examples, this separation will be at
most around one order of magnitude). Whenever $\beta_{n}/H_{n}\lesssim 10$,
our results may be considered as indicative estimates.

####  V.1.1 Weakly coupled

A strong first order PT can arise in a region of parameter space of the DFSZ
construction Dine et al. (1981); Zhitnitsky (1980), which features two Higgs
doublets $H_{1}$ and $H_{2}$ in addition to the complex PQ field $\Phi$ (the
same can occur in KSVZ-type constructions, although only in the presence of
extra scalars Delle Rose et al. (2020)). The potential for the real parts of
the $U(1)_{\text{EM}}$ components of these scalar fields ($h_{1},h_{2}$ and
$\phi$ respectively) is given by

$\displaystyle V=$
$\displaystyle\frac{\lambda_{\phi}}{4}(\phi^{2}-v^{2})^{2}+\frac{1}{2}h_{1}^{2}\left(\frac{\kappa_{1}}{2}\phi^{2}-\mu_{1}^{2}\right)+\frac{1}{2}h_{2}^{2}\left(\frac{\kappa_{2}}{2}\phi^{2}+\mu_{2}^{2}\right)-\frac{\kappa_{3}}{2^{\frac{n}{2}}}\phi^{n}h_{1}h_{2}+$
$\displaystyle\frac{\lambda_{1}}{4}h_{1}^{4}+\frac{\lambda_{2}}{4}h_{2}^{4}+\frac{\lambda_{12}}{4}h_{1}^{2}h_{2}^{2}.$
(98)

The value of $n$ depends on the PQ charges of $H_{1}$ and $H_{2}$, and does
not play an important role in our discussion ($n=2$ in the original proposal
Dine et al. (1981); Zhitnitsky (1980), $n=1$ can be easily arranged). All
couplings are real in the potential above ($\kappa_{3}$ can be made real by a
field redefinition). Note that with respect to the notation of section III,
here we have $v=\varphi_{t}$ and $\varphi_{f}=0$. In order for the potential
above to lead to viable EW symmetry breaking, the determinant of the resulting
mass matrix needs to be tuned, which fixes the parameter $\kappa_{3}$ in terms
of the other parameters:

$v^{2}\left(\kappa_{1}-2\frac{\mu_{1}^{2}}{v^{2}}\right)\left(\kappa_{2}+2\frac{\mu_{2}^{2}}{v^{2}}\right)\simeq
2\kappa_{3}^{2}\,.$ (99)

In order to simplify the analysis, one can neglect the dynamics of $h_{2}$ (it
tracks its minimum during the PT). In this case, the task simplifies to the
study of a phase transition in the two-dimensional field space of $h_{1}$ and
$\phi$, where however the loop corrections induced by $h_{2}$ are taken into
account.

At temperatures $T\gg v$, the fields are both at zero. As temperature is
lowered, a minimum in the $\phi$ direction appears, which is also the global
minimum of the two-field potential if the (tree-level) condition
$\lambda_{\phi}>1/\lambda_{1}(\mu_{1}/v)^{4}$ is imposed. We focus on the case
where the PT occurs along the $\phi$ direction (a two-step transition is also
possible, see Von Harling et al. (2020)). The corresponding potential is of
Coleman-Weinberg type when the mass parameters are small, i.e.
$\mu_{2}^{2},\lambda_{\phi}v^{2}\ll v^{2}$ (away from this limit, a weaker
transition is possible due to a loop-induced cubic term of the form $\sim
T^{3}\phi$). This leads to an effective potential of the form (97) with
$\lambda_{\phi}$ running logarithmically, due to loop corrections. For
simplicity here we focus on the case where $\lambda_{\phi}=0$ at tree level
and thus it arises only at loop-level. The minimum
$F_{a}\equiv\langle\phi\rangle$ is obtained by solving
$\lambda_{\phi}(\langle\phi\rangle)=-1/4\beta_{\lambda_{\phi}}(\langle\phi\rangle)$,
where $\beta_{\lambda_{\phi}}=\kappa_{2}^{2}/(8\pi^{2})$ is the beta function
of $\lambda_{\phi}$. The potential at the minimum is then

$V_{\text{min}}\simeq-\frac{\kappa_{2}^{2}}{128\pi^{2}}F^{4}_{a}\,.$ (100)

Note that, with respect to the notation of section III, we here set the zero
of the potential at the false vacuum, and
$V_{\mathrm{min}}=V_{0}(\varphi_{t})-V_{0}(\varphi_{f})$. Tunneling in such a
scenario has been first analyzed in Witten (1981), where it is shown to
proceed via $O(3)$-symmetric bubbles. In our case, the corresponding action is

$S_{B}=\frac{S_{3}}{T}\simeq
7.7\frac{\sqrt{\kappa_{2}(T)+2\lambda_{\phi}(T)}}{-\lambda_{\phi}(T)},$ (101)

where $\lambda_{\phi}(T)\sim-\beta_{\lambda_{\phi}}\ln(F_{a}/T)$. Given the
action above, the corresponding values of $\beta_{n}/H_{n}$ at nucleation can
be numerically computed, as a function of only two parameters $F_{a}$ and
$\kappa_{2}$ (under our simplifying assumptions on $\lambda_{\phi}$ and
$h_{2}$). The transition generically occurs with significant supercooling when
$\kappa_{2}\leq 1.5$. For larger values of $\kappa_{2}$ on the other hand, the
transition can occur during radiation domination, i.e. before the vacuum
energy dominates over the radiation background, when $F_{a}\simeq
10^{8}-10^{9}~{}\text{GeV}$.

When nucleation occurs in the supercooled regime, we find that the transition
may not complete if $\kappa_{2}$ is too small, even when the nucleation
condition $\Gamma(T_{n})\simeq H(T_{n})^{4}$ is fulfilled. Furthermore, even
when it completes, the completion temperature is very far from the nucleation
temperature when $\kappa_{2}\lesssim 1$. In order to more reliably assess the
reach of ground-based interferometers on this model, we thus focus on some
benchmark values of $1\lesssim\kappa_{2}\lesssim 2$ and
$10^{8}~{}\text{GeV}\leq F_{a}\leq 10^{10}~{}\text{GeV}$. The reheating
temperature after the transition (assuming instantaneous reheating) is
obtained by equating the radiation energy to the potential energy
$-V_{\text{min}}$, see eq. (33):

$T_{\text{reh}}\simeq 7\cdot
10^{7}~{}\text{GeV}\sqrt{\kappa_{2}}\left(\frac{F_{a}}{10^{9}~{}\text{GeV}}\right)\left(\frac{106.75}{g_{*}(T_{*})}\right)^{\frac{1}{4}},$
(102)

and the parameter $\alpha$ is then found by setting
$V_{0}(\varphi_{f})\rightarrow-V_{\mathrm{min}}$ and $T_{f}$ by the
percolation temperature $T_{*}$ in Eq. (54) (for the example values of
parameters that we consider below, the volume decrease condition is satisfied
at $T_{*}$). Notice that in the supercooling regime $T_{*}\ll T_{\text{reh}}$.
.

Figure 6: Spectra induced by first order PQ PT associated with a Coleman-Weinberg type potential, for the benchmark points of Table 3. Solid lines: strong ($\alpha>10$) PT; dashed lines: weak PT. Blue, gray, and black spectra have respectively $F_{a}$ values of $10^{8}$, $10^{9}$, and $10^{10}$ GeV. PLISCs corresponding to a SNR threshold of 5 are displayed for ET, CE, and LISA (including the expected galactic foreground). Additionally, we overlay the corresponding astrophysical foreground for each detector as dotted lines. $F_{a}$ in GeV | $\kappa_{2}$ | Nucleation occurs in: | $\beta_{n}/H_{n}$ | $\alpha$ |  $T_{\text{reh}}$ in GeV
---|---|---|---|---|---
$10^{10}$ | 1.6 | supercooled phase | $\simeq 19.4$ | $\gg 10$ | $\simeq 6.7\cdot 10^{8}$
$10^{9}$ | 1.5 | supercooled phase | $\simeq 22.6$ | $\gg 10$ | $\simeq 6.5\cdot 10^{7}$
$10^{9}$ | $1.9$ | radiation dominated phase | $\simeq 36$ | $\simeq 0.7$ | $\simeq 8.1\cdot 10^{7}$
$10^{8}$ | 1.3 | supercooled phase | $\simeq 21.4$ | $\gg 10$ | $\simeq 6.1\cdot 10^{6}$
$10^{8}$ | 1.85 | radiation dominated phase | $\simeq 42$ | $\simeq 0.3$ | $\simeq 8\cdot 10^{6}$
Table 3:  Benchmark points for the PQ phase transition in the weakly coupled
case.

Table 3 gives the values of $\beta_{n}/H_{n},\alpha$ and $T_{\text{reh}}$ for
some benchmark values of parameters. For these benchmark points, nucleation
occurs close to percolation. The SGWB spectra corresponding to these benchmark
points are shown in Fig. 6. When nucleation occurs in the supercooled phase,
the signal is sourced mostly by bubble and/or relativistic fluid shells
collisions, see section III. On the other hand, the signal is sourced mostly
by bulk fluid motion (sound waves and turbulence), when nucleation occurs
during radiation domination, as can be the case when the phase transition
completes rapidly below the critical temperature. Notice how at least part of
the SGWB spectrum is observable at ET/CE for $10^{8}~{}\text{GeV}\lesssim
F_{a}\lesssim 10^{10}~{}\text{GeV}$, when $\beta_{n}/H_{n}\lesssim O(10-100)$,
even for $\alpha\sim 0.1$.

####  V.1.2 Strong dynamics

The tunings required in the previous construction are avoided in a different
class of models, where a first order PT arises as a new strongly-interacting
sector (charged under QCD as well) undergoes confinement. This new sector
should have a global $U(1)_{PQ}\otimes SU(3)_{c}$ symmetry and an
$U(1)_{PQ}-SU(3)_{c}-SU(3)_{c}$ anomaly, therefore its constituents are
charged under QCD. These models can be investigated by means of holography,
when the strongly-interacting sector can be thought of as a gauge theory
$SU(N)$ in the large $N$ limit. In this picture, the cosmological evolution of
this new sector is similar to that of QCD: at high temperatures, the sector
can be thought of as a hot gas of “gluons” and “quarks”; below a certain
critical temperature a confined phase arises, with “hadrons” as degrees of
freedom. The associated breaking of conformal invariance can be described in
terms of a dilaton field $\mu$, which under appropriate circumstances is the
lightest field in the spectrum of the new sector. In this case, the
confinement phase transition can be studied as a single field problem (see
e.g. Baratella et al. (2019)).

The corresponding confinement scale $\Lambda_{c}$ and tunneling rates are
determined by a dilaton potential

$V_{\text{eff}}(\mu)=\frac{N^{2}}{16\pi^{2}}\lambda(\mu)\mu^{4},$ (103)

where $\lambda(\mu)=b_{0}(\ln(\Lambda_{c}/\mu)-1/4)$. The bounce action is
then given by (see Baratella et al. (2019))

$S_{B}\simeq\frac{24N^{2}}{b_{0}\log\left(\frac{T_{c}}{T}\right)}.$ (104)

In this case, it is possible to find analytical expressions for the PT
parameters $\beta_{n}/H_{n}$ and $\alpha$. Here we take $N=3$, as an example
value. We are thus left with two parameters only: $b_{0}$ and $F_{a}$,
similarly to the previous case.

The critical temperature $T_{c}$ is given by:

$T_{c}\simeq 1.4\cdot
10^{10}~{}\text{GeV}\left(\frac{F_{a}}{10^{10}~{}\text{GeV}}\right)b_{0}^{\frac{1}{4}}.$
(105)

The reheating temperature is easily found by matching the effective potential
at the minimum with the radiation energy density in the deconfined phase
Baratella et al. (2019), and is given by:

$T_{\text{reh}}\simeq
1.4\left(\frac{106.75}{g_{*}}\right)^{1/4}b_{0}^{1/4}~{}F_{a},$ (106)

where $g_{*}$ is the number of relativistic species at $T_{*}$.

The nucleation temperature $T_{n}$ is obtained by solving $\Gamma\approx
T^{4}e^{-S_{B}}=H^{4}$. Since the PT is typically strongly supercooled,
$H=(\pi^{2}g_{c}/90)^{1/2}T_{c}^{2}/M_{p}$ is constant and corresponds to the
Hubble rate during the phase of supercooling. Concretely, we have

$T_{n}=T_{c}\exp\left[-\frac{y}{2}\left(1-\sqrt{1-\frac{24N^{2}}{y^{2}b_{0}}}\right)\right]=T_{c}\exp[-\mathcal{N}_{e}]$
(107)

where $y\equiv\log(M_{p}/(aT_{c}))$, $M_{p}$ is the reduced Planck mass and
$a=\sqrt{\pi^{2}g_{*}(T_{c})/90}$ and we have defined the number of efolds
$\mathcal{N}_{e}$ above. The rate $\beta_{n}/H_{n}$ is then found to be:

$\left.\frac{\beta}{H}\right|_{n}=\left.T\frac{\mathrm{d}S_{B}}{\mathrm{d}T}\right|_{n}-4\approx\frac{24N^{2}}{\mathcal{N}^{2}_{e}b_{0}}-4.$
(108)

We vary $F_{a}$ between $10^{8}-10^{10}~{}\text{GeV}$ and $b_{0}$ between
$0.4$ and $1$. Similarly to the previous case, we find that the transition may
not complete even when the nucleation condition is satisfied, due to the
inflationary background. However, if the transition completes, according to
the volume decrease criterion Eq. 32, we find the temperature of completion to
be at most roughly one order of magnitude smaller than that of nucleation,
unlike the previous case. Furthermore, we checked that the transition
completes for the relevant values of $F_{a}$ whenever $\beta_{n}/H_{n}>5.2$ at
the nucleation temperature.

The value of $\alpha$ is always much larger than $10$ for any reasonable value
of $b_{0}$, i.e. the transition is always supercooled in the relevant range of
parameters. Forecasts on the regions of parameter space that will be probed by
GW observatories are shown in Fig. 7. In the hatched region the transition
does not complete. Close to that region, the GW signal is largest and very
loud at ET/CT for $F_{a}\sim 10^{8}-10^{9}~{}\text{GeV}$. Overall, the
transition remains observable at ET/CE for $b_{0}\lesssim 1$ and for $F_{a}$
up to $10^{10}~{}\text{GeV}$. Interestingly, in all this region of parameter
space the QCD axion can potentially be the totality of the Dark Matter,
depending on the specific model (see e.g.Gorghetto et al. (2021b)). In
particular, models with long-lived domain walls allow for axion dark matter
with $F_{a}$ down to $10^{8}~{}\text{GeV}$, otherwise $F_{a}\sim
10^{10}~{}\text{GeV}$ is needed. In the upper right corner of Fig. 7, the
astrophysical foregrounds at ET/CT are also large, and distinguishing a
primordial signal is more challenging, but still doable depending on the
signal frequency shape.

Figure 7: Left panel: $\frac{\beta}{H}$ at nucleation as a function of $F_{a}$
and $b_{0}$. The red line corresponds to $\beta_{n}/H_{n}=5.2$, which is the
threshold value below which the PT does not complete. Hence, the hatched
region of the parameter space is excluded. Right panel: $\log_{10}$ of the
maximum SNR among CE and ET. Each detector can detect the SGWB signal from the
PQ PT with SNR greater than 5, across all parameter space where the PT
completes. We overlay lines of constant reheating temperature $T_{\text{reh}}$
and constant $\beta_{n}/H_{n}$, showing the conversion from the parameters
$b_{0}$ and $F_{a}$. The contours corresponding to the SNR of the foreground
from compact binary mergers for each detector are displayed as dotted lines
(in the region enclosed by the contours, the foregrounds have higher SNR than
the signal). For $\beta_{n}/H_{n}\lesssim 10$, our results should be taken as
indicative estimates.

###  V.2 GWs from axionic Domain Wall networks

Beside the possibility of a first order PT, the PQ mechanism provides another
potential source of GWs. The spontaneous breaking of the global $U(1)_{PQ}$
symmetry leads to the formation of a network of topological defects, of hybrid
type (see section IV). First, global cosmic strings are formed at the PQ
transition, i.e. at $T\lesssim F_{a}$. As discussed in Section IV.2, these
strings rapidly evolve into a scaling regime (with possible logarithmic
corrections) and predominantly emit axions throughout their evolution. Then,
at a lower energy scale, a network of domain walls can form and emit GWs (see
section IV.3), as we now discuss.

####  V.2.1 The standard QCD axion

At $T\sim\text{GeV}$, non-pertubative QCD effects provide an explicit breaking
of the PQ symmetry, and therefore induce a potential for the axion field. The
original shift symmetry of the axion field is thus broken to a discrete
version, due to the periodicity of the QCD vacuum. The axion potential thus
exhibits a discrete $\mathbb{Z}_{N_{\text{DW}}}$ symmetry, $V(a)=V(a+2\pi
v/N_{\text{DW}})$, where $F_{a}\equiv v/N_{\text{DW}}$ and the domain wall
number, $N_{\rm{DW}}$, depends on the microphysical implementation of the
axion model (in particular on the PQ charges of SM and heavy fields). The
exact functional form of the axion potential does not play a particularly
important role in our discussion, and below QCD confinement it takes the form
(the actual form of the potential differs by $O(1)$, see Grilli di Cortona et
al. (2016))

$V_{\text{QCD}}(a)\simeq\kappa_{\text{QCD}}^{2}\Lambda_{\text{QCD}}^{4}\left[1-\cos\left(\frac{a}{F_{a}}\right)\right],$
(109)

where $\kappa_{\text{QCD}}\simeq\sqrt{m_{u}/\Lambda_{\text{QCD}}}$. The axion
mass at low temperatures is then $m_{a}\simeq
5.7~{}\text{meV}(10^{9}~{}\text{GeV}/F_{a})$. The residual axion discrete
symmetry is spontaneously broken once the axion starts oscillating, i.e. at
the temperature $T_{\text{DW}}$ when the condition $m_{a}(T)\simeq
H(T_{\text{DW}})$ is met.141414The axion mass rises quickly with temperature
above the confinement scale Borsanyi et al. (2016) and is constant at lower
temperatures. This condition is satisfied at $T_{\text{dw}}\simeq\text{GeV}$
for the values of $F_{a}$ of interest. Correspondingly, domain walls are
formed, attached to the previously formed strings. Their tension is
$\sigma\simeq 8m_{a}F_{a}^{2}$ (we neglect the small correction due to pion
mixing).

As described in section IV.3, such walls radiate GWs before disappearing
(either because $N_{\rm{DW}}=1$ or because $N_{\rm{DW}}>1$ but an additional
explicit breaking of the PQ symmetry is present). The signal would be peaked
in the $10^{-7}-10^{-9}$ Hz range, corresponding to the viable annihilation
temperatures of the network (see e.g. Ferrer et al. (2019)).

However, in the standard QCD axion cosmology described so far, the amplitude
of the signal would be too small for it to be detectable, even by the future
SKA. The obstruction is phenomenological: DW annihilation contributes to the
production of relic axions, which are stable and behave as Cold Dark Matter.
Requiring that these axions do not overproduce Dark Matter then implies that
DWs make up at most a fraction $\alpha_{\text{ann}}\lesssim 10^{-6}-10^{-9}$
of the energy density of the Universe at the time of their annihilation. This
leads to an unobservably small GW signal according to (91), see Hiramatsu et
al. (2013).

####  V.2.2 Heavy QCD axions

This conclusion can be dramatically altered in a certain class of QCD axion
models where the axion is much heavier than in the standard case. Indeed, a
heavy enough axion would decay to SM particles (gluons and/or photons) before
BBN, therefore leaving no dangerous relics behind.

Independently of the specific realisation, a heavy QCD axion scenario assumes
an additional contribution to the axion potential

$V_{a}=V_{\text{QCD}}(a)+\kappa^{2}_{\text{H}}\;\Lambda^{4}_{\text{H}}\left(1-\cos\frac{a}{F_{a}}\right),$
(110)

where $\Lambda_{\text{H}}\gg\Lambda_{\text{QCD}}$ and $\kappa_{H}\leq 1$
depending on details such as the fermion spectrum. Crucially, the additional
term should be aligned with the QCD potential (at least up to
$\Delta\theta\lesssim 10^{-10}$), otherwise the axion does not solve the
strong CP problem. This class of models is motivated by the so-called axion
quality problem Georgi et al. (1981); Holdom and Peskin (1982); Dine and
Seiberg (1986); Holman et al. (1992); Kamionkowski and March-Russell (1992);
Barr and Seckel (1992); Ghigna et al. (1992): the PQ mechanism is vulnerable
to additional sources of explicit symmetry breaking beyond QCD, that would
easily displace the minimum of the axion potential. The presence of these
explicit sources may be expected on theoretical grounds, since global
symmetries are expected not to be exact. The quality problem is quantitatively
important because of the smallness of the QCD scale compared to any other
higher scale which could provide additional PQ breakings. A heavy axion
scenario can significantly alleviate this issue, by raising the scale of the
axion potential, which thereby renders the PQ mechanism more robust against
additional sources of PQ symmetry breaking. Concrete models, often relying on
an additional confining dark sector that the axion couples to, face the
challenge of aligning the additional potential with QCD (see Holdom and Peskin
(1982) and Treiman and Wilczek (1978); Dimopoulos (1979); Tye (1981), Holdom
(1985); Flynn and Randall (1987); Choi and Kim (1999); Agrawal and Howe
(2018); Kitano and Yin (2021), Rubakov (1997); Gherghetta et al. (2016);
Gherghetta and Nguyen (2020) and Berezhiani et al. (2001); Dimopoulos et al.
(2016); Hook et al. (2020) for a partial collection of heavy axion models).

While the axion is not the Cold Dark Matter in this scenario, the hybrid
network of axionic topological defects can now offer an alternative probe, via
its GW signature Zambujal Ferreira et al. (2022); Ferreira et al. (2023b),
when $N_{\text{DW}}>1$ (which is a rather generic outcome in realistic
models). As the axion mass is now $m_{a}\simeq
10^{8}~{}\text{GeV}\left(\frac{10^{12}~{}\text{GeV}}{F_{a}}\right)\left(\frac{\Lambda_{\text{H}}}{10^{10}~{}\text{GeV}}\right)^{2}\kappa_{\text{H}}$,
the cosmological evolution of topological defects is shifted to higher
temperatures.

As usual, annihilation of the string-wall network is determined by the
aforementioned explicit breakings of the PQ symmetry, which can be captured by
an additional misaligned potential
$V_{b}\simeq-\mu_{b}^{4}\cos\left(\frac{N_{\text{b}}}{N_{\text{DW}}}\frac{a}{F_{a}}-\delta\right),$
where $N_{b}$ should be $1$ or co-prime with $N_{\text{DW}}$ in order to
induce annihilation. For illustration we take the representative values
$N_{b}=1,\kappa_{\text{H}}=1$ and $N_{\text{DW}}=6,\delta=0.3$ in the
following (see Ferreira et al. (2023b) for other choices). The annihilation
temperature is then

$\displaystyle T_{\text{ann}}\simeq
10^{7}\text{GeV}\sqrt{\frac{10^{11}\text{GeV}}{F_{a}}}\left(\frac{\Lambda_{\text{H}}}{10^{10}\text{GeV}}\right)\left(\frac{r}{0.003}\right)^{2},$
(111)

where the parameter $r\equiv\mu_{b}/\Lambda_{\text{H}}$.

Figure 8: Expected SNR of the DW SGWB from the heavy QCD axions for LISA and
3G Earth-based detectors, assuming $\kappa_{\rm H}=1$, $\alpha_{s}=0.1$,
$N_{\text{DW}}=6$ and $r=0.002$, corresponding to $\Delta\theta\simeq
2.5\times 10^{-12}$. The maximum individual SNR among detectors is depicted in
the color grid. Contour lines representing SNR = 5 for each detector are
displayed in dashed lines. Additionally, the contour corresponding to the SNR
value of the respective astrophysical foregrounds for LISA, CE and ET are
depicted as dotted lines: in the region on the left and below these lines, the
SNR of the foreground is larger than that of the cosmological signal (see
section II.2 for details on the expected foregrounds). Four exclusion zones
are overlaid: the three gray regions correspond to $\Lambda_{\rm H}>F_{a}$, DW
domination and $T_{a\rightarrow gg(\gamma\gamma)}<10\>\mathrm{MeV}$, while the
red region is the 95% exclusion region from the LVK O3 run Jiang and Huang
(2022).

Most importantly, it is now possible for domain walls to make up a large
fraction of the energy density of the Universe at the time of their
annihilation

$\alpha_{\text{ann}}\simeq
0.1~{}\left(\frac{F_{a}}{10^{11}~{}\text{GeV}}\right)^{2}\left(\frac{0.003}{r}\right)^{4}.$
(112)

Of course, $V_{b}$ also causes a displacement of the axion from the QCD
minimum by $\Delta\theta\simeq r^{4}/N_{\text{DW}}$, for a natural value
$\delta\sim O(1)$. Such a displacement induces electric dipole moments for the
neutron and the proton and upcoming measurements are able to probe down to
$\Delta\theta\lesssim 10^{-13}$ Abel et al. (2019); Filippone (2018).

Most of the energy density in domain walls is released into non-relativistic
quanta of the constituent field at annihilation (a small fraction goes instead
to GWs), i.e. into axions, which can then decay to light degrees of freedom;
for the model of interest these are SM gluons (if the decay occurs above QCD
confinement) and photons. The decay rate is $\Gamma_{a\rightarrow
gg(\gamma\gamma)}=1/(64\pi)(c_{g,\gamma}\alpha_{s,\text{em}})/(2\pi)^{2}m_{a}^{3}/F_{a}^{2}$,
where $c_{g}=1$ and $c_{\gamma}$ is a model-dependent coupling to photons (see
e.g. Zyla et al. (2020) for a review in standard DFSZ and KSVZ constructions)
and $\alpha_{s,\text{em}}$ is the (strong) electromagnetic coupling constant.
The decay temperature is found by setting $\Gamma\simeq H$:

$\displaystyle T_{a\rightarrow gg(\gamma\gamma)}$ $\displaystyle\simeq 3\cdot
10^{9}~{}\text{GeV}~{}\alpha_{s(\text{em})}~{}\kappa_{\text{H}}^{3/2}\left(\frac{\Lambda_{\text{H}}}{10^{10}~{}\text{GeV}}\right)^{3}\left(\frac{10^{11}~{}\text{GeV}}{F_{a}}\right)^{\frac{5}{2}},$
(113)

and is typically above QCD confinement for values of parameters of interest
for this work. Decay to photons is thus subleading with respect to decay to
gluons. When decay is not efficient at $T_{\text{ann}}$, non-relativistic
axions can come to dominate the universe before their decay becomes efficient.
By setting the initial energy carried by the non-relativistic quanta to equal
the energy density in domain walls at the time of annihilation, and
redshifting as $\sim a^{-3}$, one can determine the temperature at which non-
relativistic axions start dominating the universe

$\displaystyle T_{\text{MD}}\simeq 1.2\cdot
10^{7}~{}\text{GeV}~{}C_{1}\left(\frac{106.75}{g_{*,\text{ann}}}\right)^{\frac{1}{4}}\left(\frac{F_{a}}{10^{11}~{}\text{GeV}}\right)^{\frac{3}{2}}\left(\frac{\Lambda_{\text{H}}}{10^{10}~{}\text{GeV}}\right)\left(\frac{0.003}{r}\right)^{2},$
(114)

(see the appendix of Ferreira et al. (2023b) for more details). Of course an
intermediate phase of matter domination occurs only if $T_{a\rightarrow
gg(\gamma\gamma)}<T_{\text{MD}}$, that is when

$\Lambda_{\text{H}}\lesssim 7\cdot
10^{8}~{}\text{GeV}\left(\frac{F_{a}}{10^{11}~{}\text{GeV}}\right)^{2}\left(\frac{0.003}{r}\right),$
(115)

otherwise the axions decay as soon as they are radiated from the walls. If
this intermediate phase of matter domination occurs, then the amplitude of the
GW signal from DW annihilation (91) is suppressed by a redshift factor:

$\Omega_{\text{gw,MD}}h^{2}\simeq\Omega_{\text{\tiny
GW,DW}}(f)h^{2}\left(\frac{g_{*}(T_{a\rightarrow
gg(\gamma\gamma)})}{g_{*}(T_{\text{MD}})}\right)^{\frac{1}{3}}\left(\frac{T_{a\rightarrow
gg(\gamma\gamma)}}{T_{\text{MD}}}\right)^{\frac{4}{3}},$

and the peak frequency (92) is shifted to lower frequencies:

$\displaystyle f_{p,\text{MD}}^{0}\simeq f_{p}^{0}\left(\frac{T_{a\rightarrow
gg(\gamma\gamma)}}{T_{\text{MD}}}\right)^{\frac{1}{3}}.$ (116)

Finally, the amplitude and frequency of GW signal sourced by heavy axion DWs
is found by plugging (111) and (90) above into (91) and (92), and accounting
for matter domination when appropriate.

Fig. 8 shows the sensitivity reach of future detectors to this scenario
(dashed contours). We have fixed $r=0.002$ and $N_{\text{DW}}=6$ as
representative values, which correspond to $\Delta\theta=2.5\cdot 10^{-12}$,
within the reach of future electric dipole moment measurements. The gray-
shaded regions are excluded/constrained by (for further details, see Ferreira
et al. (2023b)): a) in the upper left corner, the axion effective theory is
not valid; b) in the lower right corner, the axion decay can alter BBN
predictions; c) for $F_{a}\gtrsim 10^{11}~{}\text{GeV}$ DWs dominate the
universe before their annihilation becomes efficient, most likely leading to a
universe incompatible with observations (see also the discussion in Vilenkin
and Shellard (2000)). (Note that the region $F_{a}\gtrsim 4\times
10^{10}~{}\text{GeV}$ is expected to be subject to constraints from PBH
production Ferreira et al. (2024).) The orange-shaded region is the current
exclusion limits by the LVK O3 Jiang and Huang (2022). In the left part of the
plot below the dotted contours astrophysical foreground are large and may thus
obstruct the detection of a cosmological signal. Nonetheless, in most of the
sensitivity region of ET, CE and LISA a cosmological signal from DWs is
distinguishable, and these detectors will thus extend the exploration of heavy
axion models very significantly, providing the exciting possibility of
detecting a signal correlated with measurement of electric dipole moments.

####  V.2.3 FOPT at ground-based detectors + DW signal at PTA

So far we have discussed GWs from a first order PQ phase transition in the
standard QCD axion model, and from domain walls in the heavy axion scenario.
One may wonder whether both signals can coexist in a heavy axion scenario. The
answer is positive, though it is hard to arrange for the peaks of both GW
signals to lie in the ground-based frequency band (see (111) and (106)). On
the other hand, a viable heavy axion model with $F_{a}\lesssim
10^{8}~{}\text{GeV}$ and $m_{a}\simeq\text{GeV}$ could have a first order PT
leading to a SGWB in the frequency band of ground-based interferometers, and
DW annihilation occurring around the QCD scale for appropriate values of
$\mu_{b}$, such that the corresponding GW signal would be peaked in the PTA
frequency band (and may provide an interpretation of the current detection).
In this case, the SGWB would be characterized by two peaks, one at nHz and the
other one at $1-10$ Hz, with the high frequency tail of the DW signal possibly
also observable at LISA. Several cosmological bounds apply to this scenario,
which may also be probed at laboratories and colliders, as discussed in
Ferreira et al. (2023b).

##  VI Conclusion

The aim of this review was to quantify the ability of 3G ground based
detectors, working together with LISA and SKA, to probe cosmological SGWBs
sourced by PTs in the early universe. PT and symmetry breaking are ubiquitous
in quantum field theories and particle physics beyond the SM, and as we have
discussed they can lead to copious GW production across different frequency
bands. Among the most interesting possible sources of GWs are (i) first order
PTs as reviewed in section III, and (ii) topological defects of different
kinds, as reviewed in section IV. Figure 2 shows examples of characteristic
spectra from these sources, together with the PLISCs of the detectors across
the frequency band, accounting for foregrounds.

Measuring a cosmological SGWB requires indeed a thorough understanding of both
the astrophysical foregrounds relevant to each detector (given their frequency
band and sensitivity), and the detection method applicable to each different
detector type. In section II.2 we have presented and summarised these points
for each of the detectors we consider here, namely CE, ET, LISA and SKA.
Furthermore we have explained how the SNR for a cosmological SGWB is
calculated for each. Our results are summarised in the PLISCs of figure 2, as
well as in table 1 where we highlight the relevant astrophysical foregrounds
for each detector and their corresponding SNRs. Our results provide realistic
sensitivities to cosmological signals for SKA and LISA. For ET/CE, rather than
deriving new sensitivity curves, we have used estimates of the astrophysical
foregrounds from Branchesi et al. (2023); Bellie et al. (2023) to understand
to what extent astrophysical foregrounds can limit the detection prospects of
cosmological backgrounds, see Fig. 3 and Fig. 5.

In section III, we have focused on GW production by a first order PT. We have
first reviewed the PT dynamics and defined the relevant stages of its
progression, and the associated parameters such as the PT duration and the
size of the bubbles at percolation. We have then reviewed the phenomena
associated with the PT capable of sourcing a SGWB, namely: bubbles and/or
relativistic fluid shells collisions for strongly first order PTs; bulk fluid
motion generated at percolation in the form of sound waves and (M)HD
turbulence for PTs of intermediate strength; and sound waves alone for weak
PTs. We have shown how the scaling of the SGWB signals with the source
parameters, provided by detailed analyses of the aforementioned sourcing
mechanisms, can be interpreted based on simple models for the source UETC,
namely, constant-in-time and stationary. We have then presented placeholders
for the SGWB spectra from the three generation mechanisms, highlighting the
assumptions behind them. Evaluating the SGWB signal from a first order PT is a
complicated endeavour, requiring particle physics and statistical physics as
well as non-equilibrium phenomena, fluid dynamics and cosmology, and further
progress is needed to clarify and overcome all the caveats that enter in this
process. Nevertheless, we have concluded by showing the reach of future 3G
detectors (as well as LISA and PTA with the SKA) to the compelling case of
SGWBs generated by first order PTs, based on the state of the art of our
knowledge. The result is presented in Fig. 3 as a scan on the SNR of the SGWB
signal in CE, ET, LISA and SKA, as a function of the parameter space of first
order PTs (characterised by the transition temperature $T_{*}$ and its
duration $\beta/H_{*}$, for different fixed strengths $\alpha=0.05,1,\gg 1$),
including the astrophysical foregrounds. The figure clearly shows the
potential of these detectors to probe first order PTs in full complementarity
with respect to the PT energy scale (represented by its temperature). It also
shows that, in a large region of parameter space, the signal will have a
higher SNR than that of the expected astrophysical foregrounds. For strong
first order PTs ($\alpha\gg 1$), some regions of parameter space are already
excluded by LVK O3 data, and others could account for the observed PTA signal.

In section IV we have focused on topological defects which may be formed in
PTs provided a symmetry is broken. Their existence depends on the properties
of the vacuum manifold ${\cal{M}}$, and the argument for defect formation is
based solely on causality: in other words, they may form whether the PT is
first order or not. For local $U(1)$ strings, whose tension is given by
$\eta^{2}$, where $\eta$ the energy scale of the PT, we reviewed in section
IV.1 the shape of the spectrum which covers orders of magnitude in frequency,
and whose amplitude is proportional to $(G\mu)^{2}$. As discussed,
understanding the detailed properties of the spectral shape still require
further theoretical understanding. In that section we have shown that, with a
precise estimation of the astrophysical foreground, CE should be able to probe
strings with tensions $G\mu\gtrsim 10^{-16}$ (Model A) and $G\mu\gtrsim
10^{-18.7}$ (Model B). For ET, one has respectively $G\mu\gtrsim 10^{-14.5}$
and $G\mu\gtrsim 10^{-17.4}$. Both ET and CE will be able to search for
individual bursts emitted by cosmic string cusps and kinks, but if none are
detected, then constraints on the tension of cosmic loops is less competitive
than those from the SGWB.

In section IV.3, we discussed domain walls, namely planar-like topological
defects which, unlike line-like strings, cannot exist indefinitely: indeed,
they must annihilate in order not to dominate the energy budget of the
universe. We briefly reviewed the properties of DWs and the SGWB spectrum they
produce. Figure 5 shows the expected SNR for domain walls for all next
generation GW detectors, as a function of the annihilation temperature
$T_{\rm{ann}}$ and $\alpha_{\rm{ann}}$, the fraction of the total energy
density in DWs at $T_{\rm{ann}}$. Ground based detectors in particular will be
sensitive to $T_{\rm{ann}}\sim 10^{8}$GeV, and $\alpha_{\rm{ann}}\gtrsim
10^{-2}$ (and up to $10^{-3}$ depending on the understanding of the
astrophysical foregrounds).

In section V we then focused on the PQ mechanism as a well-motivated beyond
the SM scenario that can potentially be probed by ET and CE (as well as LISA).
GWs can arise in this mechanism from the PQ phase transition, if it is of
first order, and/or from its topological byproduct, that is the axionic
string-domain wall network. First, we reviewed UV realizations where the PQ
sector is approximately scale-invariant, within weakly or strongly coupled
frameworks. The PT is then expected to be of first order, and typically
exhibits a significant amount of supercooling. The corresponding GW signal is
thus mostly sourced by bubble and/or relativistic shells collisions (although
bulk fluid motion can be efficient in certain regions of the parameter space)
and is particularly loud at ET/CE, since the scale of the PT $\sim F_{a}$ is
constrained to be around $10^{8}-10^{10}~{}\text{GeV}$ from cosmological and
astrophysical bounds on the QCD axion. We presented a forecast of the regions
of parameter space in strongly-coupled PQ models which will be probed by 3G
detectors in Fig. 7, whereas we presented spectra for benchmark points in the
case of weakly coupled models. Our analysis improves significantly over
previous work Von Harling et al. (2020), in that we have taken into account
more stringent conditions on the completion of PTs in vacuum dominated eras,
beside the calculation of the SNR of the signals. In both cases, we have shown
that significant regions of parameter space will be probed by 3G detectors.
This provides the exciting opportunity to explore the UV implementation of the
PQ mechanism, for values of $F_{a}$ where the QCD axion can be the Dark
Matter.

Secondly and independently of the order of the PQ phase transition, we have
investigated GWs from axionic domain walls. While the signal is too small to
be observable in the standard QCD axion scenario, we showed that DWs can
source a large GW signal in PQ models where the axion is much heavier than
usual, as can occur in UV models that include additional contributions to the
QCD potential. While UV realizations of these models require additional
ingredients, the scenario can potentially alleviate the axion quality problem,
while also leading to heavy DWs and large GW backgrounds. The relevant GW
abundance and frequencies depend on the axion mass and decay constant, and
additionally on the scale at which the axionic discrete symmetry is broken to
allow a viable cosmology without DWs at late times. Our results are presented
in Fig. 8 and show that the GW signal is louder than astrophysical foregrounds
in a large region of parameter space of these models. Very interestingly, the
region that can be probed via GWs in Fig. 8 also leads to observable electric
dipole moments for the neutron and the proton, thereby providing a
complementary channel to test heavy axion models.

Here we have focused on PTs as a well understood and physically well motivated
phenomenon that sources GWs. However, perhaps the most exciting perspective is
that, by probing early Universe temperatures as high as $\gtrsim
10^{8}~{}\text{GeV}$, 3G detectors may reveal imprints of totally new
phenomena which occurred in the primordial universe.

## Acknowledgements

We would like to thank José Juan Blanco-Pillado, Francesco Iacovelli, Michele
Maggiore, Gijs Nelemans, Alberto Roper-Pol, Géraldine Servant and Peera
Simakachorn for useful discussions and comments. CC is supported by the Swiss
National Science Foundation (SNSF Project Funding grant 212125). D.A.S is
grateful to CERN for hospitality whilst this work was in progress. The work of
F.R. is supported by the grant RYC2021-031105-I from the Ministerio de Ciencia
e Innovación (Spain). F.R. and O.P. acknowledge support from the Departament
de Recerca i Universitats from Generalitat de Catalunya to the Grup de Recerca
‘Grup de Fisica Teorica UAB/IFAE’ (Code: 2021 SGR 00649) and the Spanish
Ministry of Science and Innovation (PID2020-115845GB-
I00/AEI/10.13039/501100011033). IFAE is partially funded by the CERCA program
of the Generalitat de Catalunya. H.Q-L thanks Institut Polytechnique de Paris
for funding his PhD.

##  Appendix A Overlap Reduction Function

In this appendix we provide the the definition of the overlap reduction
function used in Sec. II.2 to compute the sensitivity of the GW detectors,
Allen (1996); Thrane and Romano (2013). The response of a detector located at
$\vec{x}$ to a SGWB is the correlation Thrane and Romano (2013)

$\displaystyle h(t,\vec{x})$
$\displaystyle=\int_{-\infty}^{+\infty}\mathrm{d}\tau\int\mathrm{d}^{3}yR^{\mu\nu}(\tau,\vec{y})h_{\mu\nu}(t-\tau,\vec{x}-\vec{y})$
(117)
$\displaystyle=\int_{-\infty}^{+\infty}\mathrm{d}f\int\mathrm{d}^{2}\Omega_{\hat{k}}\>R^{P}(f,\hat{k})h_{P}(f,\hat{k})e^{i2\pi
f(t-\hat{k}\cdot\vec{x}/c)},$ (118)

where in the second equality we have used Eq. (3), and $R^{P}(f,\hat{k})$ is
the detector response to a sinusoidal plane wave propagating along direction
$\hat{k}$: see Romano and Cornish (2017) for its expression for the different
GW detectors considered here. For two interferometers $I,J$ situated at
$\vec{x}_{I},\vec{x}_{J}$ respectively, and on using Eq. (4), it follows that

$\displaystyle\left\langle h_{I}(t)h_{J}(t^{\prime})\right\rangle$
$\displaystyle=\int\mathrm{d}^{2}\Omega_{\hat{k}}\int\mathrm{d}^{2}\Omega_{\hat{k}^{\prime}}\int_{-\infty}^{+\infty}\mathrm{d}f\int_{-\infty}^{+\infty}\mathrm{d}f^{\prime}$
$\displaystyle\quad\quad\left[\left\langle\tilde{h}_{P}(f,\hat{k})\tilde{h}_{P^{\prime}}^{*}(f^{\prime},\hat{k}^{\prime})\right\rangle
R_{I}^{P}(f,\hat{k})R_{J}^{P^{\prime}}(f^{\prime},\hat{k}^{\prime})^{*}\times
e^{i2\pi f\left(t-\frac{\hat{k}\cdot\vec{x}_{I}}{c}\right)}e^{-i2\pi
f^{\prime}\left(t^{\prime}-\frac{\hat{k}^{\prime}\cdot\vec{x}_{J}}{c}\right)}\right]$
(119)
$\displaystyle=\int\mathrm{d}^{2}\Omega_{\hat{k}}\int_{-\infty}^{+\infty}\mathrm{d}f\>\frac{1}{2}S_{h}(f)\times\frac{1}{8\pi}\sum_{P}R_{I}^{P}(f,\hat{k})R_{J}^{P}(f,\hat{k})^{*}e^{-i2\pi
f\frac{\hat{k}\cdot\vec{\Delta x}_{IJ}}{c}}\times e^{i2\pi
f\left(t-t^{\prime}\right)}$ (120)
$\displaystyle=\frac{1}{2}\int_{-\infty}^{+\infty}\mathrm{d}f\>S_{h}(f)\Gamma_{IJ}(f)e^{i2\pi
f\left(t-t^{\prime}\right)}$ (121)

where $\vec{\Delta x}_{IJ}=\vec{x}_{I}-\vec{x}_{J}$, and the factor of $1/2$
is due to the one-sided definition of $S_{h}$ in Eq. (4). The overlap
reduction function (ORF) $\Gamma_{IJ}(f)$ between the two detectors $I$, $J$
is thus given in terms of their respective response functions by

$\Gamma_{IJ}(f)=\frac{1}{8\pi}\sum_{P}\int\mathrm{d}^{2}\Omega_{\hat{k}}R_{I}^{P}(f,\hat{k})R_{J}^{P}(f,\hat{k})^{*}e^{-i2\pi
f\frac{\hat{k}\cdot\vec{\Delta x}_{IJ}}{c}}.$ (122)

Note that one often introduces the normalized ORF, defined as Allen and Romano
(1999); Thrane and Romano (2013); Romano and Cornish (2017)

$\gamma_{IJ}(f)=\frac{5}{\sin^{2}(\beta)}\>\Gamma_{IJ}(f),$ (123)

where $\beta$ is the opening angle of the interferometers, so that
$\gamma_{IJ}(0)=1$ for two co-located and co-aligned interferometers with the
same opening angle $\beta$. Figure 9 shows the normalized overlap reduction
function between the two Cosmic Explorer detectors with $\beta=\pi/2$, assumed
to be located at Hanford and Livingston sites. It vanishes starting from
$f\sim 80$ Hz and above, which limits the sensitivity of the CE network to
these frequencies.

Figure 9: Normalized overlap reduction function between the two Cosmic
Explorer detectors assumed to be located at Hanford and Livingston sites.

On setting $I=J$ is then straightforward to obtain the response function
$\mathcal{R}(f)=\Gamma_{II}(f)$ of one detector leading to

$\mathcal{R}(f)=\frac{1}{8\pi}\sum_{P}\int\mathrm{d}^{2}\Omega_{\hat{k}}\left|R_{I}^{P}(f,\hat{k})\right|^{2}.$
(124)

Note that for Earth-based interferometers, in the limit $fL/c\ll 1$, where $L$
is the interferometer arm length, the response function is frequency
independent as Romano and Cornish (2017)

$R_{I}^{P}(f,\hat{k})\simeq e_{ab}^{P}(\hat{k})D_{I}^{ab},$ (125)

where
$D_{I}^{ab}=\frac{1}{2}\left(\hat{X}_{I}^{a}\hat{X}_{I}^{b}-\hat{Y}_{I}^{a}\hat{Y}_{I}^{b}\right)$,
$\hat{X}$ and $\hat{Y}$ being unit vectors pointing in each interferometer arm
direction.

##  Appendix B Einstein Telescope SNR computation

In this appendix, we derive Eq. (19), used to compute the SNR of a
cosmological SGWB in ET.

Considering the triangular 10km ET configuration, the method for detecting the
SGWB that we adopt would consist in cross-correlating the data streams from
the three identical V-shaped interferometers labeled by $I,J\in[1,3]$. As a
consequence all overlap reduction function $\Gamma_{IJ}(f)$ ($I\neq J$) are
equal thanks to the integration over the sky in Eq. (122). Furthermore, since
these three interferometers are co-located, they will be affected by common
noise sources (such as seismic noise), causing their respective noise
realizations to be correlated. Consequently, one cannot use Eq. (8) to isolate
the signal component in Eq. (9). Instead, we utilize a “cross-correlation
excess” statistic, which presupposes knowledge of the cross-correlated noise,

$\hat{\mathcal{S}}_{IJ}=\int_{-\frac{T_{\mathrm{obs}}}{2}}^{\frac{T_{\mathrm{obs}}}{2}}\mathrm{d}t\int_{-\frac{T_{\mathrm{obs}}}{2}}^{\frac{T_{\mathrm{obs}}}{2}}\mathrm{d}t^{\prime}\left[d_{I}(t)d_{J}(t^{\prime})-\frac{1}{2}N_{IJ}(t^{\prime}-t)\right]Q(t-t^{\prime}),$
(126)

where

$\langle n_{I}(t)n_{J}(t^{\prime})\rangle=\frac{1}{2}N_{IJ}(t^{\prime}-t).$
(127)

It is then straightforward to show that

$\langle\hat{\mathcal{S}}_{IJ}\rangle=\frac{T_{\mathrm{obs}}}{2}\int_{-\infty}^{+\infty}\mathrm{d}f\>\Gamma_{IJ}(f)S_{h}(f)\tilde{Q}(f).$
(128)

To compute its associated SNR, we proceed by calculating the cross-correlation
of this statistic (in the weak-signal regime) between two pairs of
interferometers $(I,J)$ and $(K,L)$ (which may be identical),

$\displaystyle\left\langle\hat{\mathcal{S}}_{IJ}\hat{\mathcal{S}}_{KL}\right\rangle$
$\displaystyle=\int\mathrm{d}t_{1}...\mathrm{d}t_{4}\left\langle\left[d_{I}(t_{1})d_{J}(t_{2})-\frac{1}{2}N_{IJ}(t_{1}-t_{2})\right]\left[d_{K}(t_{3})d_{L}(t_{4})-\frac{1}{2}N_{KL}(t_{3}-t_{4})\right]\right\rangle
Q(t_{1}-t_{2})Q(t_{3}-t_{4})$
$\displaystyle=\int\mathrm{d}t_{1}...\mathrm{d}t_{4}\left[\left\langle
n_{I}(t_{1})n_{J}(t_{2})n_{K}(t_{3})n_{L}(t_{4})\right\rangle-\frac{1}{4}N_{IJ}(t_{1}-t_{2})N_{KL}(t_{3}-t_{4})\right]Q(t_{1}-t_{2})Q(t_{3}-t_{4})$
$\displaystyle=\frac{1}{4}\int\mathrm{d}t_{1}...\mathrm{d}t_{4}\left[N_{IK}(t_{1}-t_{3})N_{JL}(t_{2}-t_{4})+N_{IL}(t_{1}-t_{4})N_{JK}(t_{2}-t_{3})\right]Q(t_{1}-t_{2})Q(t_{3}-t_{4}),$
(129)

leading, in the frequency domain, to

$\left\langle\hat{\mathcal{S}}_{IJ}\hat{\mathcal{S}}_{KL}\right\rangle=\frac{T_{\mathrm{obs}}}{4}\int_{-\infty}^{+\infty}\mathrm{d}f\left[\tilde{N}_{IK}(f)\tilde{N}_{JL}^{*}(f)+\tilde{N}_{IL}(f)\tilde{N}_{JK}^{*}(f)\right]\left|\tilde{Q}(f)\right|^{2}.$
(130)

Here we have introduced the Fourier transform of the auto/cross-correlation
functions $N_{IJ}$, which we factorize as

$\tilde{N}_{IJ}(f)=\Gamma^{(n)}_{IJ}(f)P_{n}(f),$ (131)

where $P_{n}$ is the one-sided noise PSD of one V-interferometer defined in
Eq. (7), and we have introduced $\Gamma^{(n)}_{IJ}$ that accounts for the
correlation of the noise amongst detectors. It satisfies
$\Gamma^{(n)}_{II}(f)=1$, to recover the one-sided detector noise PSD $P_{n}$.
We now follow Branchesi et al. (2023) and consider that $\Gamma^{(n)}_{IJ}$ is
constant across frequency, taking $\Gamma^{(n)}_{IJ}(f)=0.2$, for $I\neq J$.

Contrary to the usual case of uncorrelated detector noises, one cannot simply
add up the SNR squared of the individual detector pairs as their associated
statistics is clearly correlated from Eq. (130). Instead, one can derive the
way to linearly combine these estimators in order to minimize the variance of
the resulting statistics while preserving their expectation value (unbiased
estimator). This is a well known problem already derived in Allen and Romano
(1999); Romano and Cornish (2017). To simplify the notation, we denote with
$A$ and $B$ interferometer pairs $(I,J)$ and $(K,L)$ (note again that $A$ and
$B$ do not need to be distinct) so that
$\hat{\mathcal{S}}_{A}=\hat{\mathcal{S}}_{IJ}$. We now define a statistic that
linearly combines the one from individual interferometer pairs $A$

$\hat{\mathcal{S}}_{\rm{cross}}\equiv\sum_{A=1}^{3}\lambda_{A}\hat{\mathcal{S}}_{A}.$
(132)

The $\lambda_{A}$ are chosen to minimize the variance of the estimator
$\langle\hat{\mathcal{S}}_{\rm{cross}}\rangle$ and are given by (see Eq.
(6.14) of Romano and Cornish (2017))

$\lambda_{A}=\left[\sum_{C=1}^{3}\sum_{D=1}^{3}(C^{-1})_{CD}\right]^{-1}\sum_{B=1}^{3}(C^{-1})_{AB},$
(133)

where we have introduced the inverse of the statistics covariance matrix,
$C_{AB}=\left\langle\hat{\mathcal{S}}_{A}\hat{\mathcal{S}}_{B}\right\rangle$
which, by symmetry, in the weak-signal regime, takes the form

$C=\begin{bmatrix}\mathcal{A}&\mathcal{T}&\mathcal{T}\\\
\mathcal{T}&\mathcal{A}&\mathcal{T}\\\
\mathcal{T}&\mathcal{T}&\mathcal{A}\end{bmatrix}.$ (134)

We can then use the fact that $C$ is symmetric (and so $C^{-1}$) to see from
Eq. (133) that $\lambda_{A}=\frac{1}{3}$ for all detector pairs $A$. The
estimator verifies
$\left\langle\hat{\mathcal{S}}_{\rm{cross}}\right\rangle=\left\langle\hat{\mathcal{S}}_{A}\right\rangle$,
with its explicit form given in Eq. (128). Its variance is then

$\left\langle\hat{\mathcal{S}}_{\rm{cross}}^{2}\right\rangle=\frac{\mathcal{A}+2\mathcal{T}}{3}.$
(135)

Thanks to our assumption that the cross-correlated noise is an order of
magnitude smaller than the auto-correlation power (namely
$\Gamma_{IJ}^{(n)}=0.2$ for $I\neq J$), we only keep the linear terms in
$\Gamma_{IJ}^{(n)}$ from Eq. (130) leading to,

$\mathcal{A}\simeq\frac{T_{\mathrm{obs}}}{4}\int_{-\infty}^{+\infty}\mathrm{d}f\>P_{n}^{2}(f)\tilde{Q}(f)^{2},$
(136)

and

$\mathcal{T}\simeq\frac{T_{\mathrm{obs}}}{4}\int_{-\infty}^{+\infty}\mathrm{d}f\>\Gamma^{(n)}_{12}P_{n}^{2}(f)\tilde{Q}(f)^{2}.$
(137)

The SNR can then be derived (in the weak signal regime) as

$\mathrm{SNR}\simeq\frac{\left\langle\hat{\mathcal{S}}_{\rm{cross}}\right\rangle}{\sqrt{\left\langle\hat{\mathcal{S}}_{\rm{cross}}^{2}\right\rangle}}=\sqrt{NT_{\mathrm{obs}}}\frac{\int_{-\infty}^{+\infty}\mathrm{d}f\>\Gamma_{12}S_{h}(f)\tilde{Q}(f)}{\sqrt{\int_{-\infty}^{+\infty}\mathrm{d}f\>\left[1+(N-1)\Gamma^{(n)}_{12}\right]P_{n}^{2}(f)\tilde{Q}^{2}(f)}},$
(138)

where we write the SNR for a general number of $N$ equivalent colocated
interferometers. For ET, $N=3$. Using the usual method from Allen and Romano
(1999); Romano and Cornish (2017), one can choose the filter function
$\tilde{Q}$ to maximize the SNR. One finds

$\tilde{Q}(f)=\Lambda\times\frac{\Gamma_{12}S_{h}(f)}{\left[1+(N-1)\Gamma_{12}^{(n)}\right]P_{n}^{2}(f)},$
(139)

where $\Lambda$ is an arbitrary normalization factor. Substituting this filter
into Eq. (138), one finally finds

$\mathrm{SNR}=\sqrt{2T_{\mathrm{obs}}\int_{0}^{+\infty}\mathrm{d}f\frac{N\Gamma_{12}^{2}S_{h}^{2}(f)}{\left[1+(N-1)\Gamma^{(n)}_{12}\right]P_{n}^{2}(f)}},$
(140)

leading to the effective strain noise PSD

$S_{\mathrm{eff}}=\sqrt{\frac{1+(N-1)\Gamma_{12}^{(n)}}{N\Gamma_{12}^{2}}}P_{n}$
(141)

written in Eq. (19) for the case of a triangular ET detector with $N=3$ and
$P_{n}=P_{\rm{ET-L}}/\sin^{2}(\pi/3)$. This should be compared with Eq. (A.7)
of Branchesi et al. (2023).

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