m_{1}^{2}v_{1}^{2}$.171717 Here, we implicitly assumed that there is no
additional mechanism to dial the potential energy of $\phi_{1}$ so that its
energy density at $\phi_{1}=0$ is at least from its potential energy
$\rho_{1}\sim m_{1}^{2}v_{1}^{2}$. The cancellation of $\rho_{1}$ will
generically require fine tuning.  This means that demanding a large
gravitational-wave signal from domain walls implies $\rho_{\text{rad}}\sim
T_{p}^{4}\ll m_{1}^{2}v_{1}^{2}\sim\rho_{1}$; $\phi_{1}$ needs to have a much
larger energy density than the radiation bath, implying inflation by vacuum-
energy domination. If $v_{1}\sim M_{\text{Pl}}$ instead, $\rho_{1}$ may be
smaller than $\rho_{\rm rad}$, and inflation does not seem to be required for
large GW signals. But the reheating by the dissipation of $\rho_{1}$ is
generically not efficient because of the Planck-suppressed dissipation rate of
$\phi_{1}$. As a result, there will be a prolonged period of matter domination
by $\phi_{1}$, violating our assumption that defects are formed and decay in a
radiation-dominated background, and their GW signal will be diluted. In the
next section, we will present a concrete model of inflation and revisit this
point on the inevitability of inflation.

##  6 Model Realization: Thermal Inflation

So far, we have discussed the evolution of an inflated string-bounded wall
network. This mechanism is generally applicable so long as a brief period of
inflation appears between two phase transitions. Note that $H_{\text{re}}$,
${\mu^{1/2}}$, and ${\sigma^{1/3}}$ are essentially free parameters coming
from different physics. In this section, we further restrict parameters in the
mechanism considered and propose a concrete model relating $H_{\text{re}}$ to
the mass of the field producing domain walls. Interestingly, measuring the
gravitational-wave signal provides a novel probe to model parameters, such as
the soft supersymmetry (SUSY) breaking scale.

We consider a scenario involving thermal inflation Yamamoto:1985rd ;
Lazarides:1985ja ; Lyth:1995hj ; Lyth:1995ka  driven by inflaton $\phi_{1}$,
which is also the field producing domain walls. The crucial mechanism here is
the interplay between a flat potential with a tachyonic instability near
$\phi_{1}=0$ and the thermal mass of $\phi_{1}$. Specifically, let us consider
the following potential

$V(\phi_{1})=-m_{1}^{2}\absolutevalue{\phi_{1}}^{2}+V_{\text{lift}}(\phi_{1}),$
(72)

in which $m_{1}$ is a small mass parameter and $V_{\text{lift}}(\phi_{1})$
denotes some lifting potential. Taking $m_{1}^{2}>0$, $\phi_{1}$ has a
tachyonic mass at the origin, but the potential is eventually stabilized by
$V_{\text{lift}}(\phi_{1})$ at $\phi_{1}=v_{1}$. The energy difference $\Delta
V$ between $\phi_{1}=0$ and $\phi_{1}=v_{1}$ is of order $m_{1}^{2}v_{1}^{2}$.
Thus, a large $v_{1}$ leads to large energy for $\phi_{1}$ if it is trapped at
the origin. To trap $\phi_{1}$, one can consider some coupling between
$\phi_{1}$ and the thermal bath. Then, thermal correction to the $\phi_{1}$
potential leads to a positive mass term of the form $V\supset
y^{2}T^{2}\absolutevalue{\phi_{1}}^{2}$, in which $y$ denotes some
dimensionless coupling, and $T$ is the temperature. The thermal mass
compensates for the tachyonic mass of $\phi_{1}$ and traps $\phi_{1}$ near the
origin so that $\phi_{1}$ can dominate the universe and drive a brief period
of inflation. This happens when the radiation density drops below the energy
density in $\phi_{1}$ at a temperature $T_{i}\approx\sqrt{m_{1}v_{1}}$. On the
other hand, cosmic expansion cools the universe, and $\phi_{1}$ can see its
vacuum potential when the temperature drops below $T_{f}\lesssim m_{1}/y$.
This terminates the thermal inflation. Hence, the number of $e$-foldings for
this thermal inflation can be estimated by

$N_{\text{inf}}\approx\ln(\frac{T_{i}}{T_{f}})\approx\frac{1}{2}\ln(\frac{y^{2}v_{1}}{m_{1}}).$
(73)

Using eq. 3, we determine the re-entry Hubble for cosmic strings as

$H_{\text{re}}\approx
e^{-2N_{\text{inf}}}H_{i}\approx\frac{1}{y^{2}}\frac{m_{1}^{2}}{M_{\text{Pl}}}$
(74)

in terms of the mass $m_{1}$. The wall tension $\sigma$ comes from the
trilinear coupling between $\phi_{1}$ and $\phi_{2}$ and is irrelevant for the
thermal inflation dynamics. We will thus treat ${\sigma^{1/3}}$ as an
independent model parameter. Then, it is possible to probe $m_{1}/y$ and
${\sigma^{1/3}}$ via its gravitational-wave signature.

The thermal inflation scenario provides a concrete example of the connection
between the a stage of inflation before the second phase transition and the
size of the gravitational wave signal. From the discussion in the previous
paragraph, we see that a long period of thermal inflation is naturally
realized when $T_{i}\gg T_{f}$ or $y^{2}v_{1}/m_{1}\gg 1$. Recall from eq. 70
that a lower domain-wall production temperature $T_{p}$ is preferred. In the
context of thermal inflation, this occurs at the end of thermal inflation when
$\phi_{1}$ rolls to its true vacuum, i.e., $T_{p}\sim T_{f}\sim m_{1}/y$.
Thus, $T_{p}\ll\sqrt{m_{1}v_{1}}$ means $yv_{1}^{1/2}/m_{1}^{1/2}\gg 1$, which
coincides with the condition for a period of inflation before phase
transition. On the other hand, the condition $yv_{1}^{1/2}/m_{1}^{1/2}\gg 1$
means $v_{1}\gg m_{1}$. A parametrically large $v_{1}$ typically means that
the lifting potential is sufficiently flat. Some examples of a flat lifting
potential include higher-dimensional operators and a SUSY Coleman-Weinberg
potential.

A flat potential is generally destroyed by the quartic coupling of $\phi_{1}$
generated by the renormalization group running $\sim y^{4}$. To protect the
flatness of the potential, we consider the following supersymmetric theory
with 4 chiral superfields charged under some $\textup{U}(1)_{\text{}}$
symmetry. They are $\phi_{1}$ with charge $1$, $\phi_{2}$ with charge $2$,
$\phi_{-2}$ with charge $-2$, and $\psi$ with charge $-1$. We also introduce
two more fields $X$ and $\bar{\psi}$ that are not charged under
$\textup{U}(1)_{\text{}}$. Let us consider the following superpotential

$W=X\quantity(\phi_{2}\phi_{-2}-v_{2}^{2})+\lambda\phi_{1}^{2}\phi_{-2}+y\phi_{1}\bar{\psi}\psi.$
(75)

The $F$ term of $X$ fixes $\phi_{2}$ and $\phi_{-2}$ on the $F$-flat direction
$\phi_{2}\phi_{-2}=v_{2}^{2}$. This breaks the $\textup{U}(1)_{\text{}}$
symmetry spontaneously and produces cosmic strings. Soft SUSY breaking,
parameterized by the scale $m_{\text{soft}}$, lifts the $F$-flat direction and
stabilize $\phi_{2}\approx\phi_{-2}\approx v_{2}$. When the
$\textup{U}(1)_{\text{}}$ symmetry is gauged, the $D$ term potential dominates
over the soft SUSY breaking, and it also stabilizes
$\phi_{2}\approx\phi_{-2}\approx v_{2}$. The same SUSY breaking can also
radiatively generate a flat potential for $\phi_{1}$ through the Yukawa
coupling $y\sim\order{1}$ to $\psi$ and $\bar{\psi}$ by

$V_{\cancel{\text{SUSY}},\absolutevalue{\phi_{1}}^{2}}\approx-
m_{\text{soft}}^{2}\absolutevalue{\phi_{1}}^{2}+\frac{y^{2}m_{\text{soft}}^{2}}{\quantity(4\pi)^{2}}\absolutevalue{\phi_{1}}^{2}\ln(\frac{\absolutevalue{\phi_{1}}^{2}}{\Lambda^{2}}),$
(76)

in which $\Lambda$ denotes a reference energy scale. This flat potential gives
a VEV of $\phi_{1}$ much larger than $m_{\text{soft}}$. Therefore, we may
treat the VEV of $\phi_{1}$ as an arbitrary parameter satisfying
$m_{\text{soft}}\ll v_{1}\ll v_{2}$. At the same time, when $\phi_{1}$ is near
the origin, $\psi$ becomes a light field in the thermal bath and can give
$\phi_{1}$ a thermal mass term $\approx
y^{2}T^{2}\absolutevalue{\phi_{1}}^{2}$ that stabilizes $\phi_{1}$ around the
origin at high temperatures. Then, using our conclusions from the previous
paragraph, we determine that the string re-entry Hubble scale is

$H_{\text{re}}\approx\frac{m_{\text{soft}}^{2}}{M_{\text{Pl}}}.$ (77)

As for the wall tension, soft SUSY breaking also induces a trilinear coupling
of the form

$V_{\cancel{\text{SUSY}},\text{tri.}}\approx-
m_{\text{soft}}\lambda\quantity(\phi_{1}^{2}\phi_{-2}+\text{h.c.})\approx-
m_{\text{soft}}\lambda v_{1}^{2}v_{2}\cos(\frac{2a_{1}}{v_{1}}).$ (78)

Then, the angular direction of $\phi_{1}$ obtains a cosine potential, and
domain walls are formed between the two degenerate vacua with a mass
$m_{a}\sim\sqrt{\lambda m_{\text{soft}}v_{2}}$. This give rises to a wall of
tensions $\sigma\sim m_{a}v_{1}^{2}\sim\sqrt{\lambda
m_{\text{soft}}v_{2}}v_{1}^{2}$. Note that while minimizing the potential for
$\phi_{1}$, we ignored the trilinear term; hence, to maintain consistency, we
must demand an upper bound on $\lambda\lesssim m_{\text{soft}}/v_{2}$.181818
While the trilinear coupling is numerically small according to this bound, its
small value is technically natural. Alternatively, one may consider a theory
with $\phi_{1}$ and $\phi_{\pm 3}$ and a higher-dimensional operators
$W\supset\phi_{1}^{3}\phi_{-3}$ to generate a naturally small domain wall
tension.  Instead of $\lambda$, $v_{1}$, and $v_{2}$, we use the wall tension
scale ${\sigma^{1/3}}$ and soft SUSY breaking scale $m_{\text{soft}}$ as model
parameters in presenting our results.

Figure 10:  Maximal gravitational-wave abundance and peak frequency observed
today as functions of the model parameters in the thermal inflation model
$m_{\text{soft}}$ and ${\sigma^{1/3}}$ in units of GeV: Blue solid lines show
$\evaluated{(f_{\text{peak}},\evaluated{\Omega_{\text{GW}}h^{2}}_{f_{\text{peak}}})}_{T_{0}}$
for fixed $m_{\text{soft}}$ with varying ${\sigma^{1/3}}$, and the orange
dashed curve is that for fixed ${\sigma^{1/3}}$ with varying
$m_{\text{soft}}$. The values of $m_{\text{soft}}$ and ${\sigma^{1/3}}$ are
shown near each line. Parameter space with too much gravitational wave is
shown as the shaded region. In plotting this figure, we assumed instantaneous
reheating. This turns out to be possible for most of the parameter space
considered except for a small region. More detailed discussions can be found
in appendix D.

Let us now focus on the gauged $\textup{U}(1)_{\text{}}$ case such that walls
entirely decay into gravitational waves, i.e.,
$\Gamma_{\text{tot}}=\Gamma_{\text{wall}}(\sigma)$, for which the maximal
fractional energy $\Omega_{\text{GW}}h^{2}$ and peak frequency
$f_{\text{peak}}$ today are determined by $m_{\text{soft}}$ and
${\sigma^{1/3}}$. In particular, the dependence of the peak frequency observed
today in terms on the thermal inflation parameters is

$\evaluated{f_{\text{peak}}}_{T_{0}}\propto m_{\text{soft}}^{2}\sigma^{-1/2},$
(79)

whereas the maximal abundance

$\evaluated{\Omega_{\text{GW}}h^{2}}_{T_{0},f_{\text{peak}}}\propto\sigma^{1/2}m_{\text{soft}}^{-1}.$
(80)

Due to the different power-law dependence, upon observing this particular GW
spectrum, the peak frequency and maximal abundance point to a particular
combination of $m_{\text{soft}}$ and ${\sigma^{1/3}}$. This is shown in fig.
10. Interestingly, the SUSY breaking scale $m_{\text{soft}}$ can be probed by
near-future gravitational-wave detectors for reasonably large wall tension. We
note that we assumed instantaneous reheating. If a significant matter-
dominated epoch follows the thermal inflation, the string re-entry Hubble size
$H_{\text{re}}$ becomes smaller than that estimated here, leading to a larger
$\Omega_{\text{GW}}h^{2}$. More remarks regarding non-instantaneous reheating
are provided in appendix D.

##  7 Conclusion and Discussions

In this paper, we discussed how domain walls bounded by inflated cosmic
strings can produce large gravitational-wave (GW) signals. The mechanism
relies on two phase transitions, the first one producing cosmic strings and
the second one producing domain walls, and an epoch of inflation between them.
This inflation can be either the primordial cosmic inflation or a second stage
of inflation. The crucial mechanism here is that while the domain walls may be
topologically unstable, they remain dynamically stable so long as no cosmic
string terminates their boundaries. In other words, although $G\to H\to K$ is
the full symmetry-breaking pattern, because boundary defects are inflated
away, each local Hubble patch does not see cosmic strings and cannot
appreciate the full breaking pattern. The breaking $H\to K$ then appears to
produce stable domain walls until boundary defects associated with $G\to H$
re-enter the horizon. After string re-entry, the full pattern emerges, and the
network starts to annihilate. We demonstrated this with a simple two-field
model as shown in eq. 1 that follows a breaking pattern
$\textup{U}(1)_{\text{}}\to\mathbb{Z}_{2}\to\emptyset$.

At first, we gauged $\textup{U}(1)_{\text{}}$ to suppress the energy loss due
to the radiation of Nambu-Goldstone bosons (NGBs) from boundary strings. In
particular, we considered the case where the walls’ total energy is larger
than the strings’ energy when the strings re-enter the horizon around
$t\approx H_{\text{re}}^{-1}$. In this case, the string-wall network undergoes
three stages of evolution: 1) walls enter a scaling regime while strings are
frozen outside the horizon, 2) the string-wall network begins oscillations
when cosmic strings re-enter, and 3) the network rapidly decays. We estimated
the gravitational-wave spectrum from domain wall dynamics in each of these
stages and provided the power-law dependence (eq. 37) and the approximate peak
frequency (eq. 38) of the spectrum, as well as the maximal gravitational-wave
abundance (eq. 39) as observed today. A particular feature of gauging the
$\textup{U}(1)_{\text{}}$ symmetry here is the existence of gauge strings.
These gauge strings produce more gravitational-wave signals than their global
string counterparts due to the absence of the decay channel into NGBs. This
leads to the flat shoulder below the sharp domain wall peak as shown in fig.
7.

We then discussed the case in which the $\textup{U}(1)_{\text{}}$ symmetry is
not gauged and show that it can still lead to novel gravitational-wave signals
in section 4. The radiation into NGBs changes the spectrum from both the
cosmic disks (walls bounded by inflated string loops) and cosmic rings
(string-bounded walls formed due to the reconnection of cosmic belts). These
can produce extra power-law suppressions to the spectrum, and the frequencies
at which the power-law dependence changes (eqs. 62 and 65) can be used to
infer the string tension scale in spite of the low-lying GW spectrum from
strings.

We provided a few more benchmark spectra in section 5. Since there needs not
be a relation between the wall tension $\sigma$, the string tension $\mu$, and
the string re-entry Hubble scale $H_{\text{re}}$, the gravitational-wave
signature from these inflated string-bounded walls can peak at a wide range of
frequencies as shown in fig. 1. We also compared the GW spectrum in our
scenario with other GW spectra by various cosmological or astrophysical
sources and showed that our GW spectrum has a distinct spectral shape. A
comparison between this study and a previous study is made in section 5.3. We
then argued in section 5.4 that to have large gravitational-wave signals from
domain walls, a period of inflation before their production, as considered in
this work, seems to be inevitable.

In section 6, we discussed a concrete model relating $H_{\text{re}}$ to the
mass of the wall-producing field $\phi_{1}$ by thermal inflation. In this
model, only two free parameters $(m_{1},{\sigma^{1/3}})$ appear, and they can
be constrained by the peak frequency and amplitude of the GW signal. This
model is more naturally realized with SUSY, and observing this GW signal can
probe the soft SUSY breaking scale $m_{\text{soft}}$ along with the wall
tension scale ${\sigma^{1/3}}$ as shown in fig. 10.

One of the most exciting directions for future studies is to cross-check the
estimated spectra presented in this paper against numerical simulations. In
particular, the following points should be checked:

  * •

Cosmic belts are assumed to reconnect efficiently and follow a scaling law. Is
this a reasonable assumption? What are the dominant configurations of the
resulting breakoff topological defects such as cosmic rings?

  * •

We assumed that string-wall defects mainly oscillate to dissipate energy into
gravitational waves or NGBs. However, self-intersection during such
oscillations may also provide additional damping on the system. It would be
interesting to check if such an effect is important and how it may modify the
GW spectrum.

  * •

Given the scaling regime of the cosmic-belt network, we assume that cosmic
rings produced from these connections have a size controlled by $H_{p}^{-1}$,
and the distribution of the size is logarithmically even in the range of
$H_{\text{re}}^{-1}<H_{p}^{-1}<\Gamma_{\text{tot}}^{-1}$. Are the size indeed
$\sim H_{p}^{-1}$, and are the upper and lower bounds for the size reasonable?

  * •

During reconnection of the inflated string-bounded wall network, small
structures on the defects, such as kinks and cusps, may also produce bursts of
gravitational waves or NGBs. How significant are these contributions in the
scenario proposed in this paper? Also, will small structures on the wall
modify the UV part of the GW spectrum as discussed in section 3.5? Our naïve
model only takes into account GW radiation in the fundamental mode. It would
be interesting to see whether higher-order harmonics due to the oscillation of
the defect network alter the spectrum significantly.

  * •

Upon reconnection, cosmic strings may bring domain walls very near each other.
This may prompt the formation of Y-junction-like structures on the cosmic
belt. It is possible that these Y-junction-like structures can zip the cosmic
belts. In this study, we implicitly assumed that the violent oscillation of
the domain wall prevents this zipping process from destroying the cosmic belt
network. Does this emerge in numerical simulations? If so, under what
conditions is this a good assumption?

  * •

We also assumed that inflated cosmic strings re-enter around the same cosmic
scale $H_{\text{re}}$. Is this true if the phase transition happens during
inflation? As for strings reaching the scaling regime before they are inflated
away, it is likely that this characteristic scale $H_{\text{re}}$ follows some
distribution. This could smear the spectrum as $H_{\text{re}}$ takes various
possible values in different Hubble patches. In this case, is it still
reasonable to talk about some characteristic scale $H_{\text{re}}$, or is the
smearing effect too strong?

There are possible future studies besides numerical simulation. It would be
interesting to see if more complicated symmetry-breaking patterns can result
in model-dependent features in the GW spectrum. For instance, one of the best-
motivated global $\textup{U}(1)_{\text{}}$ symmetry would be the PQ symmetry,
for which the NGB radiation may be affected by the non-zero mass of the QCD
axion due to the explicit PQ breaking by QCD anomaly. The axions radiated from
the string-wall network may be dark matter. As these walls once occupied a
significant portion of the Hubble patch, it would also be interesting to ask
how they might impact the matter power spectrum, and whether smaller-scale
structures seeded by these large walls can be formed and detected. If these
defects couple to the Standard Model, it may also be worth investigating
whether current or near-future collider and beam dump experiments can probe
these couplings and what novel cosmological signature they may induce such as
that proposed in Long:2014mxa ; Long:2014lxa . If inflation happens between
phase transitions that produce other kinds of topological defects, such as
monopoles and strings, their GW signal may also beg for further investigation.
Lastly, in our mechanism, the causality plays an important role in the
evolution of topological-defect networks and producing a feature in the GW
spectrum. It, hence, would be enlightening to learn if such mechanism can be
applied in other contexts.

###### Acknowledgements.

We thank David Dunsky and Andrew Long for the enlightening comments on our
draft and Daniel Figueroa, Sungwoo Hong, Sung Mook Lee, Hitoshi Murayama, and
Jun’ichi Yokoyama for discussions. The work of K.H. was supported by Grant-in-
Aid for Scientific Research from the Ministry of Education, Culture, Sports,
Science, and Technology (MEXT), Japan (20H01895), and by World Premier
International Research Center Initiative (WPI), MEXT, Japan (Kavli IPMU). Y.B.
and L.T.W. are supported by the Department of Energy grant DE-SC0013642.

##  Appendix A Evolution of Domain Wall from Nambu-Goto Action

In this appendix, we discuss how we estimate the domain wall evolution using
the Nambu-Goto action and justify our claim that when
$E_{\text{wall}}>E_{\text{str}}$, the characteristic frequency of the wall
dynamics is set by $k\sim r_{0}^{-1}$ in which $r_{0}$ is its initial size.
For a semi-classical string and wall, its Nambu-Goto action reads

$S=-\sigma\int\differential^{3}\zeta\;\sqrt{\gamma}-\mu\int\differential^{2}\zeta\;\sqrt{-\Upsilon},$
(81)

in which $\zeta$ denotes the world-volume coordinates on the wall, $\gamma$
and $\Upsilon$ are the determinants of the induced metric on the wall and
string respectively, defined as

$\gamma\equiv\absolutevalue{g_{\mu\nu}\partialderivative{X^{\mu}}{\zeta^{a}}\partialderivative{X^{\nu}}{\zeta^{b}}},$
(82)

in which the $g_{\mu\nu}$ is given by the FLRW metric $\differential
s^{2}=\differential t^{2}-a(t)^{2}\differential\mathbf{r}^{2}$.

Considering a hemispherical wall bounded by a string with a radius $r$, its
worldvolume is most conveniently parameterized as
$X^{\mu}=(t,r(t),\theta,\phi)$ in spherical coordinates with

$\partialderivative{X^{\mu}}{t}=\quantity(1,\dot{r},0,0)^{\mu},\quad\partialderivative{X^{\mu}}{\theta}=\quantity(0,0,1,0)^{\mu},\quad\partialderivative{X^{\mu}}{\phi}=\quantity(0,0,0,1)^{\mu},$
(83)

while fixing the gauge of the worldvolume coordinate on the wall to be
$\zeta^{a}=\quantity(t,\theta,\phi)^{a}$. The effective action becomes

$S=-2\pi\int\differential t\;\sigma a^{2}r^{2}\sqrt{1-a^{2}\dot{r}^{2}}+\mu
ar\sqrt{1-a^{2}\dot{r}^{2}}.$ (84)

Assuming that $E_{\text{str}}\ll E_{\text{wall}}$, we may drop the term
proportionate to $\mu$ governing the string dynamics and find that the
variation of the action by $r\to r+\trigbraces{\delta}r$ is

$\quantity(-2\pi\sigma)^{-1}\trigbraces{\delta}S=\int\differential
t\;\trigbraces{\delta}r\quantity[2a^{2}r\sqrt{1-a^{2}\dot{r}^{2}}]-\trigbraces{\delta}\dot{r}\quantity[\frac{a^{4}r^{2}\dot{r}}{\sqrt{1-a^{2}\dot{r}^{2}}}].$
(85)

In the flat spacetime limit with $a(t)=1$, this reduces to an equation of
motion for $r(t)$

$\derivative{t}(\frac{r^{2}\dot{r}}{\sqrt{1-\dot{r}^{2}}})=-2r\sqrt{1-\dot{r}^{2}},$
(86)

which can be solved by

$r(t)=r(0)\cdot\text{cd}\quantity(\frac{t}{r(0)},-1),$ (87)

in which $\text{cd}(u,m)$ denotes the Jacobi elliptic function. This function
has an angular frequency $\omega=2\pi\quantity(4K(-1)r(0))^{-1}\approx
1.2r(0)^{-1}$, in which $K(m)$ denotes the complete elliptical integral of the
first kind. Hence, one can see that the wall radius oscillates with a
characteristic frequency comparable to $k\sim r(0)^{-1}$.

Alternatively, one may consider a cylindrical wall of height $L$ bounded by a
circular string of a radius $r$ at the endpoints such that the wall’s world-
volume is parameterized as $X^{\mu}=(t,r(t),\theta,z)$ in cylindrical
coordinates. Then, by choosing $\zeta^{a}=\quantity(t,\theta,z)^{a}$ on the
worldvolume, the effective Lagrangian becomes

$S=-2\pi\int\differential t\;\sigma a^{2}rL\sqrt{1-a^{2}\dot{r}^{2}}+\mu
ar\sqrt{1-a^{2}\dot{r}^{2}},$ (88)

and the equation of motion on the flat background,

$\derivative{t}(\frac{r\dot{r}}{\sqrt{1-\dot{r}^{2}}})=-\sqrt{1-\dot{r}^{2}},$
(89)

can be solved by

$r(t)=r(0)\cos(\frac{t}{r(0)}),$ (90)

demonstrating again that the wall oscillates at a characteristic frequency
$k\sim r(0)^{-1}$.

##  Appendix B Details for the Computation of the Gravitational Wave Spectrum

In this appendix, we will provide some more details on how the gravitational-
wave spectrum was obtained in sections 3 and 4.

When the wall is still in the scaling regime ($t\lesssim H_{\text{re}}^{-1}$)
as discussed in section 3.3.1, we may solve the Boltzmann equation at
$t=H_{\text{re}}^{-1}$ as

$\evaluated{\partialderivative{\rho_{\text{GW, scaling}}}{\ln
k}}_{t=H_{\text{re}}^{-1}}=\int_{M_{\text{Pl}}v_{2}^{-2}}^{H_{\text{re}}^{-1}}\differential
t\;\quantity(\frac{a(t)}{a(H_{\text{re}}^{-1})})^{4}n(t)\partialderivative{P(t)}{\ln
k}.$ (91)

For simplicity, we will assume that the second phase transition happens
sufficiently early compared to the string re-entry time so that the lower
bound of the integral can be effectively taken to be zero. Numerically, as
long as the two timescales are separated by more than one order of magnitude,
the result will not change much as the redshift factor $\propto a^{4}$ peaks
at later time $t\sim H_{\text{re}}^{-1}$. Then, by assuming that
$n(t)\sim\rho_{\text{wall}}/E_{\text{wall}}\sim H^{-3}$, one obtains

$\begin{multlined}\evaluated{\partialderivative{\rho_{\text{GW, scaling}}}{\ln
k}}_{t=H_{\text{re}}^{-1}}\approx\frac{\pi\sigma^{2}C}{M_{\text{Pl}}^{2}}\int_{0}^{H_{\text{re}}^{-1}}\differential
t\;\frac{\quantity(H_{\text{re}}t)^{2}}{t}\frac{\quantity(k\sqrt{t/H_{\text{re}}})^{3}}{1+\quantity(k\sqrt{t/H_{\text{re}}})^{4}}\\\
=\frac{2\pi\sigma^{2}C}{7M_{\text{Pl}}^{2}}\quantity(\frac{k}{H_{\text{re}}})^{3}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{1\mskip
8.0mu7/4}{11/4};-\frac{k^{4}}{H_{\text{re}}^{4}}\right]}.\end{multlined}\evaluated{\partialderivative{\rho_{\text{GW,
scaling}}}{\ln
k}}_{t=H_{\text{re}}^{-1}}\approx\frac{\pi\sigma^{2}C}{M_{\text{Pl}}^{2}}\int_{0}^{H_{\text{re}}^{-1}}\differential
t\;\frac{\quantity(H_{\text{re}}t)^{2}}{t}\frac{\quantity(k\sqrt{t/H_{\text{re}}})^{3}}{1+\quantity(k\sqrt{t/H_{\text{re}}})^{4}}\\\
=\frac{2\pi\sigma^{2}C}{7M_{\text{Pl}}^{2}}\quantity(\frac{k}{H_{\text{re}}})^{3}{}_{2}F_{1}{\left[\genfrac{.}{.}{0.0pt}{}{1\mskip
8.0mu7/4}{11/4};-\frac{k^{4}}{H_{\text{re}}^{4}}\right]}.$ (92)

Here, the essential power law can be captured by expanding in small and large
$k$ limits, which we find to be

$\begin{gathered}\lim_{k\to 0}\evaluated{\partialderivative{\rho_{\text{GW,
scaling}}}{\ln
k}}_{t=H_{\text{re}}^{-1}}=\frac{2\pi\sigma^{2}C}{7M_{\text{Pl}}^{2}}\quantity(\frac{k}{H_{\text{re}}})^{3},\\\
\lim_{k\to\infty}\evaluated{\partialderivative{\rho_{\text{GW, scaling}}}{\ln
k}}_{t=H_{\text{re}}^{-1}}=\frac{2\pi\sigma^{2}C}{3M_{\text{Pl}}^{2}}\quantity(\frac{k}{H_{\text{re}}})^{-1}.\end{gathered}$
(93)

As the IR spectrum is more universal, we will focus on reproducing the IR
spectrum while making a power law approximation for this spectrum as

$\evaluated{\partialderivative{\rho_{\text{GW, scaling}}}{\ln
k}}_{t=H_{\text{re}}^{-1}}\approx\frac{2\pi\sigma^{2}C}{7M_{\text{Pl}}^{2}}\begin{dcases}\quantity(\frac{k}{H_{\text{re}}})^{3},&k\leq
H_{\text{re}},\\\
\quantity(\frac{H_{\text{re}}}{k}),&k>H_{\text{re}}.\end{dcases}$ (94)

After $t\approx H_{\text{re}}^{-1}$, the wall network becomes a string-wall
network and deviates from the scaling regime. This contribution to the
gravitational-wave energy density redshifts with radiation. The fractional
density at $t=\Gamma_{\text{tot}}^{-1}$ is

$\evaluated{\partialderivative{\Omega_{\text{GW, scaling}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}\approx\frac{2\pi\sigma^{2}C}{21M_{\text{Pl}}^{4}H_{\text{re}}^{2}}\begin{dcases}\quantity(\frac{k}{\sqrt{H_{\text{re}}\Gamma_{\text{tot}}}})^{3},&k\leq\sqrt{H_{\text{re}}\Gamma_{\text{tot}}},\\\
\quantity(\frac{\sqrt{H_{\text{re}}\Gamma_{\text{tot}}}}{k}),&k>\sqrt{H_{\text{re}}\Gamma_{\text{tot}}},\end{dcases}$
(95)

in which we take into account the redshift in frequency.

Now, we turn our attention to cosmic disks. After the cosmic string re-enters
the horizon, the network no longer grows but only oscillates for
$H_{\text{re}}^{-1}\lesssim t\lesssim\Gamma^{-1}$ (see section 3.3.2 for more
details). Here, we obtain the gravitational-wave density by integrating

$\evaluated{\partialderivative{\rho_{\text{GW, osc.}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}=\int_{H_{\text{re}}^{-1}}^{\Gamma_{\text{tot}}^{-1}}\differential
t\;\quantity(\frac{a(t)}{a(\Gamma_{\text{tot}}^{-1})})^{4}n(t)\partialderivative{P(t)}{\ln
k},$ (96)

with the number density $n$ given by

$n(t)\sim
H_{\text{re}}^{3}\quantity(\frac{a(H_{\text{re}}^{-1})}{a(t)})^{3}=\quantity(\Gamma_{\text{tot}}H_{\text{re}})^{3/2}\quantity(\frac{a(\Gamma_{\text{tot}}^{-1})}{a(t)})^{3},$
(97)

and power spectrum $\partialderivative*{P}{\ln k}$ given by eq. 17. We shall
also approximate the power spectrum as if all the gravitational-wave energy is
dumped into the fundamental mode, which has a frequency comparable to the
oscillation frequency of the wall. This means that the power spectrum will
roughly have a Dirac delta peak around this frequency, and integrating this
over a finite range of cosmic time $t$ leads to a Heaviside step function that
marks the UV and IR cutoff of the spectrum during this period, i.e.

$\evaluated{\partialderivative{\rho_{\text{GW, osc.}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}=\frac{2\pi\sigma^{2}}{M_{\text{Pl}}^{2}}\sqrt{\frac{\Gamma_{\text{tot}}}{H_{\text{re}}}}\quantity(\frac{k}{H_{\text{re}}})^{3}\trigbraces{\Theta}(H_{\text{re}}-k)\trigbraces{\Theta}(k-\sqrt{H_{\text{re}}\Gamma_{\text{tot}}})$
(98)

agreeing parametrically with eq. 18 from simple estimate.

In the last stage of the evolution ($t\gtrsim\Gamma_{\text{tot}}^{-1}$), the
string-wall network collapses rapidly and dumps most of all of its remaining
energy into gravitational waves as presented in section 3.3.2. The collapse of
the disk is reflected in its average radius that exponentially decreases its
size as

$\bar{r}(t)\approx
H_{\text{re}}^{-1}\exp(-\frac{\Gamma_{\text{tot}}}{2}\quantity(t-\Gamma_{\text{tot}}^{-1})).$
(99)

Then, the energy density of the gravitational wave can be evaluated as

$\evaluated{\partialderivative{\rho_{\text{GW, col.}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}=\int_{\Gamma_{\text{tot}}^{-1}}^{\infty}\differential
t\;n(\Gamma_{\text{tot}}^{-1})\partialderivative{P(t)}{\ln
k}=\frac{2\pi\sigma^{2}}{M_{\text{Pl}}^{2}}\sqrt{\frac{\Gamma_{\text{tot}}}{H_{\text{re}}}}\quantity(\frac{H_{\text{re}}}{k})^{2}\trigbraces{\Theta}(k-H_{\text{re}}),$
(100)

in which the number density is still estimated by eq. 97 similar to the
oscillating regime while the power spectrum is given by eq. 21 taking into
account the rapid shrinking of the network as described by eq. 99. After
normalizing with respect to the critical density, this leads to eq. 22.

For the evolution of cosmic belts (see section 3.3.3), we assumed that the
energy density of these objects follows the scaling law as shown in eq. 23.
Then, for $t=t_{\text{prod}}\lesssim\Gamma_{\text{tot}}^{-1}$, the power
spectrum roughly follows

$\partialderivative{P(t_{\text{prod}})}{\ln k}\sim\frac{\pi
w\ell}{M_{\text{Pl}}^{2}}\frac{a(t_{\text{obs}})k}{a(t_{\text{prod}})}\trigbraces{\delta}(\frac{a(t_{\text{obs}})k}{a(t_{\text{prod}})}-\frac{1}{w}),$
(101)

with $w\sim H_{\text{re}}^{-1}$ and $\ell\sim H^{-1}$. The scaling regime also
enforces that one observes $\sim\order{1}$ belt per Hubble volume so that the
number density can be estimated as $n\sim H^{3}$. Therefore, using eq. 96, we
obtain that

$\evaluated{\partialderivative{\rho_{\text{GW, osc., belt}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}=\frac{2\pi\sigma^{2}}{M_{\text{Pl}}^{2}}\frac{\Gamma_{\text{tot}}}{H_{\text{re}}}\quantity(\frac{k}{H_{\text{re}}})^{2}\trigbraces{\Theta}(H_{\text{re}}-k)\trigbraces{\Theta}(k-\sqrt{H_{\text{re}}\Gamma_{\text{tot}}}),$
(102)

similar to the estimate provided in eq. 25. Here, one may take
$t_{\text{prod}}\approx H^{-1}$ while letting
$t_{\text{obs}}\approx\Gamma_{\text{tot}}^{-1}$. Note that at the IR boundary,
$k=\sqrt{\Gamma_{\text{tot}}H_{\text{re}}}$, this density matches that of a
disk since this part of the gravitational wave is generated by cosmic strings
that first re-enter the horizon. Here, both belts and disks appear of similar
sizes. However, as the belt energy density scales with Hubble following
$\propto a^{-4}$, the disks’ contribution becomes more important. For
$t_{\text{prod}}\gtrsim\Gamma_{\text{tot}}^{-1}$, the network rapidly decays
into gravitational waves whose power spectrum is

$\partialderivative{P}{\ln
k}\approx\frac{\pi\sigma^{2}w(t)\ell}{M_{\text{Pl}}^{2}}k\trigbraces{\delta}(k-\frac{1}{w(t)}),$
(103)

in which $w(t)$ follows eq. 99 and $\ell\sim\Gamma_{\text{tot}}^{-1}$ is the
Hubble length when belts collapses. Computing an integral similar to eq. 96,
we find

$\evaluated{\partialderivative{\rho_{\text{GW, col., belt}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}=\frac{2\pi\Gamma_{\text{tot}}\sigma^{2}}{M_{\text{Pl}}^{2}H_{\text{re}}}\quantity(\frac{H_{\text{re}}}{k})\trigbraces{\Theta}(k-H_{\text{re}}),$
(104)

which agrees with eq. 26.

Lastly, when we consider walls bounded by global strings in section 4, we
argued that the UV spectrum is suppressed by an extra factor of $\sim k^{-2}$
due to NGB radiation. This can also be understood in the formalism discussed
in this appendix. We will illustrate this point by considering circular walls
bounded by a global string loop in its final collapse stage
$t\gtrsim\Gamma^{\text{NGB}}_{\text{wall}}(w_{i})^{-1}$. We will also assume
that $H_{\text{re}}=k_{\text{NGB}}$ so that the NGB radiation becomes
important immediately after the network collapses. This assumption is not
necessary for using this formalism, but it keeps the following computation
concise and highlights the salient physics. When
$H_{\text{re}}=k_{\text{NGB}}$, the evolution of the typical radius of the
wall during collapse follows approximately

$\derivative{(\sigma\bar{r}^{2})}{t}\approx-\gamma_{a}v_{2}^{2}\implies\bar{r}(t)\approx\sqrt{r_{0}^{2}-\frac{\gamma_{a}v_{2}^{2}}{\sigma}\quantity(t-t_{0})}\approx\frac{1}{H_{\text{re}}}\sqrt{2-\Gamma^{\text{NGB}}_{\text{wall}}(w_{i})t},$
(105)

in which we used
$\Gamma_{\text{tot}}\approx\Gamma^{\text{NGB}}_{\text{wall}}(w_{i})\approx
t_{0}^{-1}$. Then, using eqs. 100 and 21, we find that

$\evaluated{\partialderivative{\rho_{\text{GW, col., global}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}=\frac{2\pi\sigma^{2}}{M_{\text{Pl}}^{2}}\sqrt{\frac{\Gamma_{\text{tot}}}{H_{\text{re}}}}\quantity(\frac{H_{\text{re}}}{k})^{4}\trigbraces{\Theta}(k-H_{\text{re}}).$
(106)

The extra $\quantity(H_{\text{re}}/k)^{2}$ factors show how the more rapid
decay of the network due to NGB radiation appears in the gravitational-wave
spectrum. To relax our assumption $H_{\text{re}}=k_{\text{NGB}}$, one simply
needs to modify $t_{0}$ in eq. 105 to the appropriate timescale at which the
NGB radiation becomes efficient. Changing $t_{0}$ shifts where the extra $\sim
k^{-2}$ suppression appears on the spectrum, but the extra UV suppression
persists as it is essentially the same computation as done above.

##  Appendix C Reproducing GW Spectrum from Gauge Strings

In addition to the string-bounded walls, there is a stable gauge string for
the model in eq. 1. We will leave the following discussion general enough so
that not only the well-studied gauge string GW spectrum is reproduced, but one
can also apply it to the benchmark model considered in this work. We start by
estimating the gravitational-wave signal from individual string loops. Because
the gauge string loops tend to decay much later than Hubble time when they are
produced, they tend to emit more gravitational radiation at a particular
frequency than their long-string counterparts Allen:1991bk ; Vilenkin:2000jqa
. To find the spectrum of an individual string loop of an initial radius
$\ell\sim H_{p}^{-1}$, where $H_{p}$ is the Hubble scale when the loop is
produced, we partition its evolution into an oscillating stage and a decay
stage again. To connect with our previous estimates, it is convenient to
anchor our spectrum as that observed at the wall decay time
$t=\Gamma_{\text{tot}}^{-1}$. Here, the decay rate to string loop of size
$H_{p}^{-1}$ can be estimated by considering the following damping of gauge
string due to gravitational interactions

$\derivative{(\mu\ell)}{t}\approx
P_{\text{GW}}\sim-\frac{\mu^{2}}{M_{\text{Pl}}^{2}}\implies\Gamma_{\text{str}}\approx\frac{\mu
H_{p}}{M_{\text{Pl}}^{2}},$ (107)

in which $\Gamma_{\text{str}}$ is the characteristic decay rate at which the
string of size $H_{p}^{-1}$ decays efficiently into gravitational waves. Then,
in the oscillating stage, the string loop produces radiations with a frequency
$k\sim H_{p}$, and its fractional energy density at
$t=\Gamma_{\text{tot}}^{-1}$ can be estimated by

$\evaluated{\partialderivative{\Omega_{\text{GW, loop, osc.}}}{\ln k}{\ln
H_{p}}}_{t=\Gamma_{\text{tot}}^{-1}}\approx\frac{1}{T^{4}}\quantity(\mu
H_{p}^{2})\quantity(\frac{T}{T_{p}})^{3}\frac{\Gamma_{\text{str}}}{H}=\sqrt{\frac{\mu}{M_{\text{Pl}}^{2}}}\quantity(\frac{a(\Gamma_{\text{tot}}^{-1})}{a(\Gamma_{\text{str}})}\frac{k}{H_{p}})^{3}.$
(108)

in which we used the adiabatic invariant $k/T_{\Gamma_{\text{tot}}}=H_{p}/T$
and
$\mu^{1/2}H_{p}^{1/2}/\quantity(M_{\text{Pl}}\Gamma_{\text{tot}}^{1/2})=a(\Gamma_{\text{tot}}^{-1})/a(\Gamma_{\text{str}}^{-1})$
during radiation domination.191919 Similar to the cosmic disk case, we
technically need to impose the $k\gtrsim\sqrt{H_{p}\Gamma_{\text{tot}}}$ for
the oscillation contribution. However, on a large scale, these string loop
with length $\ell\sim H_{p}^{-1}$ provides a white-noise-like fluctuation. One
can see this by evaluating the GW generated by large field fluctuation
$\Omega_{\text{GW, fluc.}}\sim\quantity(\trigbraces{\delta}\rho/\rho_{c})^{2}$
around string re-entry and noticing the parametric agreement between the GW
abundance from the oscillating stage and the fluctuating stage. These
fluctuations will continue the $\sim k^{3}$ power-law dependence, analogous to
the scaling solution of domain walls. Therefore, the $\sim k^{3}$ spectral
shape continues to the deep IR.  In the decay stage, the loop energy, which is
proportionate to $\ell\propto k^{-1}$, is quickly dumped into gravitational
waves, resulting in a UV spectrum of the form

$\evaluated{\partialderivative{\Omega_{\text{GW, loop, col.}}}{\ln k}{\ln
H_{p}}}_{t=\Gamma_{\text{tot}}^{-1}}\approx\frac{1}{T_{\Gamma_{\text{str}}}^{4}}\quantity(\mu
H_{p}^{2})\quantity(\frac{T_{\Gamma_{\text{str}}}}{T_{p}})^{3}\quantity(\ell
H_{p})=\sqrt{\frac{\mu}{M_{\text{Pl}}^{2}}}\quantity(\frac{a(\Gamma_{\text{tot}}^{-1})}{a(\Gamma_{\text{str}}^{-1})}\frac{k}{H_{p}})^{-1}.$
(109)

Here, we included the appropriate redshift factor $\ell\approx
a(\Gamma_{\text{str}}^{-1})/\quantity(ka(\Gamma_{\text{tot}}^{-1}))$ because
we chose to evaluate the GW spectrum around the wall decay time
$t=\Gamma_{\text{tot}}^{-1}$ instead of the string decay time
$t=\Gamma_{\text{str}}^{-1}$. From this computation, we noticed that the
individual string loop has a GW spectrum that peaks at
$a(\Gamma_{\text{str}}^{-1})H_{p}/a(\Gamma_{\text{tot}}^{-1})$ and has a
maximal abundance of $\sqrt{\mu/M_{\text{Pl}}^{2}}$. The important feature is
that the abundance is mostly independent of $H_{p}$ while the peak frequency
changes with $H_{p}$.

Figure 11: Decomposition of gauge string spectrum: gravitational-wave spectrum
from individual string loops of size $\ell\approx H_{p}^{-1}$ are shown as
blue lines. A lighter color indicates a lower $H_{p}$ or later production of
string loop from string reconnection. The envelope (green line) of the
individual spectra gives rise to the total GW spectrum of the string network.
Some long string loops produced later do not have time to oscillate fully and
generate a large enough GW signal around matter-radiation equality. In other
words, their decay timescale is longer than that of equality. Thus, their GW
spectrum is prematurely terminated around equality, resulting in the $\sim
k^{3/2}$ spectrum. Had they oscillated fully, they would generate a GW
spectrum with a higher amplitude (dashed line).

Similar to section 3.4.2, we will now sum over individual string loop spectrum
by finding its envelope as shown in fig. 11. At the smallest $H_{p}$, the IR
spectrum looks like $\sim\sqrt{\mu}M_{\text{Pl}}^{-1}(k/k_{\text{min}})^{3}$;
at the largest $H_{p}$, the UV spectrum looks like
$\sim\sqrt{\mu}M_{\text{Pl}}^{-1}M_{\text{Pl}}(k_{\text{max}}/k)$. In between,
the envelope mainly traces out the peak amplitude, which is mostly flat.
However, when $H_{p}$ becomes more IR, the string may not have time to fully
oscillate to its maximum amplitude. Choosing our IR cutoff $H_{p,\text{min}}$
to be the Hubble scale around matter-radiation equality $H_{\text{eq}}$, the
string of size $\ell=H_{p}^{-1}$ cannot fully oscillate to reach its decay
time if

$\evaluated{\Gamma_{\text{str}}}_{H_{p}}<H_{\text{eq}}\implies
H_{p}<\frac{M_{\text{Pl}}^{2}}{\mu}H_{\text{eq}}.$ (110)

When the oscillation is incomplete, the $\sim k^{3}$ spectrum is prematurely
terminated at $k=H_{\text{eq}}$. The maximum gravitational-wave abundance
generated by these long loops is also suppressed by

$\Omega_{\text{GW}}\sim\frac{1}{T_{\text{eq}}^{4}}\quantity(\mu
H_{p}^{2})\quantity(\frac{T_{\text{eq}}}{T_{p}})^{3}\frac{\Gamma_{\text{str}}}{H_{\text{eq}}}=\frac{\sqrt{\mu}}{M_{\text{Pl}}}\quantity(\frac{\mu}{M_{\text{Pl}}^{2}}\frac{H_{p}}{H_{\text{eq}}})^{3/2}.$
(111)

More precisely, the spectrum for long loops still follows $\sim k^{3}$ power
law and has the form

$\evaluated{\partialderivative{\Omega_{\text{GW}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}\approx\sqrt{\frac{\mu}{M_{\text{Pl}}^{2}}}\quantity(\frac{\mu}{M_{\text{Pl}}^{2}}\frac{H_{p}}{H_{\text{eq}}})^{3/2}\quantity(\frac{a(\Gamma_{\text{tot}}^{-1})}{a(t_{\text{eq}})}\frac{k}{H_{p}})^{3},\quad
k<\frac{a(t_{\text{eq}})}{a(\Gamma_{\text{tot}}^{-1})}H_{p}.$ (112)

This is illustrated as the change in power-law dependence of the envelope
(green line) and missing part of the GW spectrum (dashed line) in fig. 11.
Therefore, integrating over these long loops produces a GW spectrum of the
form

$\evaluated{\partialderivative{\Omega_{\text{GW, str}}}{\ln
k}}_{t=\Gamma_{\text{tot}}^{-1}}\approx\frac{\sqrt{\mu}}{M_{\text{Pl}}}\begin{dcases}\quantity(\frac{k}{k_{\text{min}}})^{3}\quantity(\frac{k_{\text{min}}}{k_{\text{med}}})^{3/2},&k\lesssim
k_{\text{min}},\\\
\quantity(\frac{k}{k_{\text{med}}})^{3/2},&k_{\text{min}}\lesssim k\lesssim
k_{\text{med}},\\\ 1,&k_{\text{med}}\lesssim k\lesssim k_{\text{max}},\\\
\quantity(\frac{k_{\text{max}}}{k}),&k\gtrsim k_{\text{max}},\end{dcases}$
(113)

in which we defined

$k_{\text{min}}\equiv\sqrt{\Gamma_{\text{tot}}H_{\text{eq}}},\quad
k_{\text{med}}\equiv\frac{M_{\text{Pl}}^{2}\Gamma_{\text{tot}}^{1/2}}{\mu}H_{\text{eq}}^{1/2},\quad
k_{\text{max}}\equiv\frac{M_{\text{Pl}}\Gamma_{\text{tot}}^{1/2}}{\mu^{1/2}}H_{p,\text{max}}^{1/2}.$
(114)

Usually, the $\sim k^{3}$ spectral shape is hidden in the deep IR and much
below the experimental sensitivity. The remaining $\sim k^{3/2}\to\sim
k^{0}\to\sim k^{-1}$ power-law dependence agrees with previous studies on GW
signal generated by cosmic strings during radiation domination Blanco-
Pillado:2017oxo ; Cui:2018rwi ; Sousa:2020sxs .202020 Here, our naïve
estimate, assuming that cosmic string predominantly radiates in its
fundamental mode, produces $\sim k^{-1}$ GW spectrum for strings generated
during inflation. This claim agrees with some previous studies Guedes:2018afo
; Gouttenoire:2019kij . However, it is currently under debate whether higher-
order harmonics of the string may produce a shallower spectrum due to the
presence of cusps and kinks Cui:2019kkd .  For the usual gauge string
scenario, $H_{p,\text{max}}$ is usually set by either the Hubble scale of the
phase transition that produces cosmic strings or the Hubble scale when
friction on strings is subdominant and string’s scaling regime is reached. For
the inflated string-bounded wall discussed in this work, the choice of
$H_{p,\text{max}}$ should be the Hubble scale around the wall decay scale
$\sim\Gamma_{\text{tot}}$ because thin strings emerge from reconnecting rings
only after all the domain walls have decayed.212121The stable string has a
higher winding number in $\phi_{2}$; therefore, its production during the
first phase transition is suppressed. When one includes both the thin-string
mode (cf. section 3.4) and the stable string contribution, one may take
$H_{p,\text{max}}=H_{\text{re}}$.

##  Appendix D Remarks on Instantaneous Reheating Assumption

In our discussion in section 6, we assumed that the reheating is instantaneous
such that a radiation-dominated epoch follows immediately after the inflation
that pushes cosmic strings away. In this appendix, we discuss how non-
instantaneous reheating can minimally modify our results and evaluate if the
instantaneous reheating for thermal inflation is a reasonable assumption.

Regardless of the details of inflation, the dominant impact of a non-
instantaneous reheating is that it introduces a brief period of a matter-
domination epoch, which delays the string re-entry. This leads to a smaller
re-entry Hubble scale

$H_{\text{re}}\approx
e^{-2N_{\text{inf}}}H_{i}\quantity(\frac{T_{\text{R}}^{4}}{\rho_{\text{inf}}})^{1/6},$
(115)

in which $T_{\text{R}}$ denotes the reheating temperature, which should be
smaller than the fourth root of the inflaton energy density
$\rho_{\text{inf}}$. As long as the string re-entry happens after the end of
reheating, our estimate in section 3 should hold. This would mean that while
keeping the domain wall tension fixed, the gravitational wave will peak at a
slightly lower frequency while having a large maximal abundance as shown in
eqs. 38 and 39. This is because a delayed re-entry leads to a larger string
radius. This both gives a larger defect network, resulting in a larger
gravitational-wave signal and a lower frequency for the normal mode
oscillating on the domain wall in its oscillating stage
($H_{\text{re}}^{-1}<t<\Gamma_{\text{tot}}^{-1}$). More specifically, for a
second thermal inflation, a non-instantaneous reheating of the thermal
inflation can potentially impact our estimate in section 6. In this case, the
re-entry Hubble size is modified to

$H_{\text{re}}\approx\frac{m_{1}^{2}}{M_{\text{Pl}}}\quantity(\frac{T_{\text{R}}^{2}}{m_{1}v_{1}})^{1/3}$
(116)

so that the peak frequency and maximal abundance have a different power-law
dependence on model parameters

$\evaluated{f_{\text{peak}}}_{T_{0}}\propto
m_{1}^{5/3}\sigma^{-1/2}T_{\text{R}}^{2/3}v_{1}^{-1/3},\quad\evaluated{\Omega_{\text{GW}}h^{2}}_{T_{0},f_{\text{peak}}}\propto
m_{1}^{-5/6}\sigma^{1/2}T_{\text{R}}^{-1/3}v_{1}^{1/6}.$ (117)

If string re-entry happens during matter domination, the power-law dependence
of the frequency for its oscillating phase may be altered due to different
redshift factors and could lead to other interesting features on the
gravitational-wave spectrum. Nonetheless, as this complication is somewhat
tangential to our pursuit in this paper, we leave further discussions on this
possibility to a future study.

We now discuss if instantaneous reheating is possible for thermal inflation.
Let us consider a coupling of $\phi_{1}$ to chiral multiplets $\chi$ and
$\bar{\chi}$ with superpotential $W=y_{\chi}\phi_{1}\chi\bar{\chi}$. Via a
Yukawa-type interaction, $\phi_{1}$ decays into $\chi\bar{\chi}$ if
$y_{\chi}v_{1}<m_{1}$ with a decay rate
$y_{\chi}^{2}m_{1}/(8\pi)<m_{1}^{3}/(8\pi v_{1}^{2})$. Even if
$y_{\chi}v_{1}>m_{1}$, $\phi_{1}$ can still decay into light particles that
couple to $\chi\bar{\chi}$ with a decay rate
$(g_{\chi}^{2}/16\pi^{2})^{2}m_{1}^{3}/(8\pi v_{1}^{2})$, where $g_{\chi}$ is
the coupling of $\chi$ and $\bar{\chi}$ to the light particles. Thus, we may
generically parameterize the decay rate as

$\displaystyle\Gamma_{\phi_{1}}=\frac{\kappa^{2}}{8\pi}\frac{m_{1}^{3}}{v_{1}^{2}},~{}~{}\kappa\lesssim
1.$ (118)

The reheating temperature is

$\frac{\kappa^{2}}{8\pi}\frac{m_{1}^{3}}{v_{1}^{2}}\approx\frac{T_{\text{R}}^{2}}{M_{\text{Pl}}}\implies
T_{\text{R}}\approx\frac{\kappa}{\quantity(8\pi)^{1/2}}\frac{m_{1}^{3/2}M_{\text{Pl}}^{1/2}}{v_{1}}.$
(119)

An instantaneous reheating requires $T_{\text{R}}\gtrsim\sqrt{m_{1}v_{1}}$,
which translates to an upper bound on the wall symmetry breaking scale
$v_{1}$,

$v_{1}\lesssim${10}^{7}\text{\,}\mathrm{GeV}$\quantity(\frac{m_{1}}{$1\text{\,}\mathrm{TeV}$})^{2/3}\quantity(\frac{\kappa}{0.1})^{2/3}.$
(120)

Recall that the trilinear coupling $\lambda$ is at most $m_{1}/v_{2}$ so that
the domain wall tension is at most

$\sigma\sim\lambda^{1/2}m_{1}^{1/2}v_{2}^{1/2}v_{1}^{2}\lesssim
m_{1}v_{1}^{2};$ (121)

thus, to be consistent with an instantaneous reheating, the wall tension scale
should be

${\sigma^{1/3}}\lesssim$4.6\text{\times}{10}^{5}\text{\,}\mathrm{GeV}$\;\quantity(\frac{m_{1}}{$1\text{\,}\mathrm{TeV}$})^{5/9}\quantity(\frac{\kappa}{0.1})^{2/9}.$
(122)

However, decay of $\phi_{1}$ is not the most efficient way to terminate this
reheating. One may consider dissipation of the inflaton by the scattering with
particles in the thermal plasma. In this case, the dissipation rate is given
by

$\Gamma=\frac{\kappa^{2}}{8\pi}\frac{T^{3}}{v_{1}^{2}},~{}~{}\kappa\lesssim
1.$ (123)

The energy dumped into the radiation bath due to scattering of $\phi_{1}$
within a Hubble time after the end of thermal inflation is given by

$\Delta\rho_{\text{rad}}\approx\Delta T^{4}\approx\Gamma\rho_{\phi}\Delta
t\approx\frac{\kappa^{2}}{8\pi}\frac{T^{3}}{v_{1}^{2}}\quantity(m_{1}^{2}v_{1}^{2})\frac{M_{\text{Pl}}}{m_{1}v_{1}}\implies\Delta
T\sim\frac{\kappa^{2}}{32\pi}\frac{m_{1}M_{\text{Pl}}}{v_{1}}$ (124)

Demanding this process depletes all energy in the inflaton field into
radiation implies $\Delta T\gtrsim\sqrt{m_{1}v_{1}}$, and this provides an
upper bound on $v_{1}$

$v_{1}\lesssim${10}^{11}\text{\,}\mathrm{GeV}$\;\quantity(\frac{m_{1}}{$1\text{\,}\mathrm{TeV}$})^{1/3}\quantity(\frac{\kappa}{0.1})^{4/3}.$
(125)

This translates to an upper bound on ${\sigma^{1/3}}$,

${\sigma^{1/3}}\lesssim${10}^{8}\text{\,}\mathrm{GeV}$\;\quantity(\frac{m_{1}}{$1\text{\,}\mathrm{TeV}$})^{5/9}\quantity(\frac{\kappa}{0.1})^{8/9}.$
(126)

These constraints are generally satisfied for the parameter space considered
in fig. 10. Thus, at least for the parameter space of interest, instantaneous
reheating is a valid assumption for a second thermal inflation. Also, it is in
principle possible to enhance the dissipation rate by the cancellation of the
mass of daughter particles between $\phi_{1}$-dependent contribution and other
contributions Co:2020jtv , or by adding many fields.

##  Appendix E Suppressed Coupling between NGB and Domain Wall

In this appendix, we provide a quick parametric estimate for the coupling
between the NGB and the domain wall considered in this work in support of our
claim made in section 4.1 that oscillating walls almost do not radiate NGB
efficiently. In this section, we will assume, similar to the main text, that
$v_{2}\gg v_{1}$. Recall that the Lagrangian considered in our model contains

$\displaystyle\mathcal{L}_{\text{}}\supset$
$\displaystyle\absolutevalue{\partial_{\mu}\phi_{1}}^{2}+\quantity(\mu_{m}\phi_{1}^{2}\phi_{2}^{*}+\text{h.c.})-V(\absolutevalue{\phi_{1}},\absolutevalue{\phi_{2}})$
(127) $\displaystyle\to$
$\displaystyle\quantity(1+\frac{r_{1}}{v_{1}})^{2}\frac{1}{2}\quantity(\partial_{\mu}a_{1})^{2}+\frac{\mu_{m}v_{1}^{2}v_{2}}{\sqrt{2}}\quantity(1+\frac{r_{1}}{v_{1}})^{2}\quantity(1+\frac{r_{2}}{v_{2}})\cos(\frac{2a_{1}}{v_{1}}-\frac{a_{2}}{v_{2}})$
$\displaystyle\sim$
$\displaystyle\quantity(1+\frac{2r_{1}}{v_{1}})\frac{1}{2}\quantity[\partial_{\mu}\quantity(a_{h}+\frac{v_{1}}{2v_{2}}a_{l}+\order{v_{2}^{-2}})]^{2}+\frac{m_{a}^{2}v_{1}^{2}}{4}\quantity(1+\frac{2r_{1}}{v_{1}})\cos(\frac{2a_{h}}{v_{1}}+\order{v_{2}^{-2}}),$

in which we used $\phi_{i}=\quantity(v_{i}+r_{i})\exp(ia_{i}/v_{i})/\sqrt{2}$
and performed a field redefinition

$\begin{dcases}a_{h}=a_{1}+\frac{v_{1}}{2v_{2}}a_{2}+\order{v_{2}^{-2}},\\\
a_{l}=a_{2}-\frac{v_{1}}{2v_{2}}a_{1}+\order{v_{2}^{-2}}.\end{dcases}$ (128)

Here, the field redefinition allows us to identify the heavy angular direction
$a_{h}$ that forms the domain walls and separate it from the massless NGB mode
$a_{l}$. This leads to two interaction terms among $a_{l}$, $a_{h}$, and the
lightest radial mode $r_{1}$,

$\mathcal{L}_{\text{}}\supset\frac{1}{v_{2}}r_{1}\partial_{\mu}a_{h}\partial^{\mu}a_{l}+\frac{m_{a}^{2}v_{1}}{2}r_{1}\cos(\frac{2a_{h}}{v_{1}}).$
(129)

Here, the radial direction $r_{1}$ typically has a mass $m_{1}$ that is larger
than the angular mode. Then, by integrating out the heavy $r_{1}$, we obtain
an effective Lagrangian that includes interaction between the NGB mode $a_{l}$
and the heavy domain wall field $a_{h}$

$\mathcal{L}_{\text{eff}}\sim\frac{m_{a}^{2}v_{1}}{2m_{1}^{2}v_{2}}\quantity(\partial^{\mu}a_{l})\quantity(\partial_{\mu}a_{h})\cos(\frac{2a_{h}}{v_{1}})=\frac{2}{m_{1}^{2}v_{1}v_{2}}\quantity(\partial^{\mu}a_{l})\quantity(\partial_{\mu}a_{h})V(a_{h}).$
(130)

Note that away from the domain wall, both $V(a_{h})$ and $\partial_{\mu}a_{h}$
terms vanish. Only on domain walls does the heavy angular field $a_{h}$ have
fluctuation that can be coupled to the light axion field $a_{l}$. Integrating
along the direction transverse to the domain wall, we can find that

$\int\differential
z\;\partial_{z}a_{h}(z)V(a_{h}(z))=\frac{m_{a}^{2}v_{1}^{3}}{8}=\frac{\sigma^{2}}{32v_{1}}.$
(131)

The light NGB can then couple to the domain wall by dualizing
$\partial_{\mu}a_{l}$ to a 3-form field strength
$B_{\mu\nu\rho}\propto\epsilon_{\mu\nu\rho\sigma}\partial^{\sigma}a_{l}$ that
couples to the domain wall worldvolume. Thus, parametrically, we expect the
effective action between the domain wall at the light axion is

$S_{\text{wall}}\sim\frac{m_{a}^{2}v_{1}^{2}}{m_{1}^{2}v_{2}}\int\differential\Sigma^{\mu\nu\rho}B_{\mu\nu\rho}.$
(132)

This should be compared to the coupling between the cosmic string and NGB mode

$S_{\text{str}}\sim 2\pi v_{1}\int\differential\sigma^{\mu\nu}B_{\mu\nu},$
(133)

in which $B_{\mu\nu}$ denotes the 2-form potential of the 3-form field
strength $B_{\mu\nu\rho}$. Then, one may estimate the power loss due to NGB
radiation on a $H_{\text{re}}^{-1}\times H_{\text{re}}^{-1}$-sized wall
oscillating with $k\approx H_{\text{re}}$ as

$P_{\text{wall}}\approx\frac{m_{a}^{4}v_{1}^{4}}{m_{1}^{4}v_{2}^{2}}\approx\frac{m_{a}^{4}}{v_{2}^{2}},$
(134)

in which we assumed that $m_{1}\approx v_{1}$ in the 2nd equality. This should
be contrasted with the power radiated by the boundary string of length
$\ell\approx H_{\text{re}}^{-1}$ and frequency $k\approx H_{\text{re}}$, which
is

$P_{\text{NGB}}\approx\gamma_{a}v_{2}^{2},$ (135)

according to eq. 44. Indeed, as $m_{a}\ll v_{2}$, the power radiated by the
wall is highly suppressed. If we take $m_{1}\ll v_{1}$ instead so that thermal
inflation can be realized, there is an upper bound on $m_{a}\sim\sqrt{\lambda
m_{1}v_{2}}\lesssim m_{1}$ so that a large trilinear coupling does not spoil
thermal inflation. In this case, the power loss into NGB from the bulk defect
is

$P_{\text{wall}}\lesssim\frac{v_{1}^{4}}{v_{2}^{2}},$ (136)

which is suppressed compared to the boundary defect. This suppression of
direct NGB emission from walls is not too counterintuitive because (1) the
wall-producing field is almost orthogonal to the light NGB direction whereas
the string consists mainly of the NGB-producing field, and (2) despite the
large overall energy of the wall, the wall is locally always lighter than
cosmic strings due to $v_{1}\ll v_{2}$.

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