bubbles in (3.25). If a reference temperature is required for baryogenesis or
gravitational wave production, a full analysis of the transition is essential
for strongly supercooled phase transitions. We have attempted to treat the
transition analysis as rigorously as our analytic methods allow. Although
reheating from non-detonations could affect the accuracy of these conditions
for percolation and completion (as discussed in Appendix D), it does not
invalidate the analysis.

We have investigated two scenarios: unit nucleation without percolation, and
percolation without unit nucleation. We showed that both scenarios are
possible by using the analytic treatment in Section 3 and a full numerical
transition analysis in a toy model and the real scalar singlet model in
Section 4. Next-to-leading-order corrections in the nucleation rate would
change the number of bubbles nucleated throughout the transition, and gauge
and renormalisation scale dependence of M2 could also significantly affect the
transition. Nonetheless, the benchmarks along with the analytic treatment in
Section 3 provide an important proof of principle that the scenarios of unit
nucleation without percolation, and percolation without unit nucleation, can
occur in realistic models.

An interesting feature we did not capture in our analysis is how concrete
particle physics models can realise deflagrations or hybrids during strongly
supercooled transitions. Additionally, our analysis applies to transitions
that remain possible at arbitrarily low temperatures. Even within M2 there are
possible phase histories involving transitions between phases that do not
persist at zero temperature. However, our analytic treatment in Section 3 can
easily be adapted to transitions that are halted at some finite temperature,
which may arise for example when there is no tree-level barrier in M2.

We also performed a detailed examination of the conditions for a phase
transition to complete. The condition that the false vacuum fraction becomes
negligible is necessary but not sufficient for a phase transition to
successfully convert the entire Universe to the true vacuum. Even if the false
vacuum fraction vanishes asymptotically, the physical volume of the false
vacuum, $\mathcal{V}_{\text{phys}}$, may not decrease. To address this issue,
we provide correction factors $\alpha_{\text{CC}i}$ (see (3.26) and
surrounding text) to the bubble wall velocity bounds for completion. These
correction factors guarantee that $\mathcal{V}_{\text{phys}}$ decreases for
all benchmarks considered. Yet even a successful transition with $N(0)\lesssim
1$ may be at odds with the observed isotropy and homogeneity of the cosmic
microwave background [207, 208, 144]. If only a few large bubbles were
responsible for converting the observable Universe into the electroweak
symmetry-broken vacuum, it is not clear that the latent heat would be
guaranteed to thermalise. This issue is left for future work.

Despite this caveat, our completion criteria provide an important test of
extensions to the Standard Model of particle physics. For example, when a
first-order phase transition to our current vacuum is essential (e.g. for
baryogenesis or gravitational waves),222222However, gravitational wave
production merely requires that bubbles and their sound shells collide, not
that the transition completes. Thus, there may be scenarios where a phase
transition does not complete but leaves a detectable gravitational wave
signal. To our knowledge, such a possibility has not been considered before in
the literature. the phase transition must complete for that new physics
scenario to be realised in nature. The most promising parameter points in
terms of gravitational wave detectability typically involve strongly
supercooled transitions. Therefore, any regions of parameter space that are
constrained using less stringent completion criteria (such as merely requiring
unit nucleation) may be excluded when using our completion criteria. Plus,
particularly promising parameter points potentially providing phase
transitions that are strongly supercooled might be too hastily discarded, due
to the assumption that completion requires unit nucleation. This may lead to
incorrect bounds being placed on models, and potentially missing a scenario
that is actually realised in nature. This necessitates an increase in the
rigour in which transition analyses are performed. We hope this paper helps
advance that rigour.

## Acknowledgments

The work of P.A. in this paper has been supported by the Australian Research
Council Future Fellowship grant FT160100274 and the National Natural Science
Foundation of China (NNSFC) Research Fund for International Excellent Young
Scientists, grant No. 12150610460. The work of P.A. and C.B. was also
supported by the Australian Research Council Discovery Project grants
DP180102209 and DP210101636. The work of L.M. was supported by an Australian
Government Research Training Program (RTP) Scholarship and a Monash Graduate
Excellence Scholarship (MGES). L.M. would like to thank Ariel Mégevand for
clarifying notation for the energy density used throughout the literature, and
illuminating the difference in assumptions for $\rho_{H}$ used in Refs. [57,
122].

##  Appendix A Treatment of the real scalar singlet model

###  A.1 Symmetry breaking conditions at one-loop

We impose two constraints on the zero-temperature, one-loop potential: there
is a stationary point at $\boldsymbol{v}$, and the eigenvalues of the
(squared) mass matrix are the (squared) pole masses $m_{h}^{2}$ and
$m_{s}^{2}$. The first constraint reads

$\displaystyle\left.\partialderivative{V}{\phi_{h}}\right|_{\boldsymbol{v}}$
$\displaystyle=0,$ (A.1)
$\displaystyle\left.\partialderivative{V}{\phi_{s}}\right|_{\boldsymbol{v}}$
$\displaystyle=0,$ (A.2)

which respectively allow for the elimination of $\mu_{h}^{2}$ and
$\mu_{s}^{2}$ from the list of free parameters.

For the latter constraint we introduce a mixing angle, $\theta$, to
diagonalise the mass matrix, and we have

$\displaystyle\left.\partialderivative[2]{V}{\phi_{h}}\right|_{\boldsymbol{v}}$
$\displaystyle=m_{h}^{2}\cos[2](\theta)+m_{s}^{2}\sin[2](\theta),$ (A.3)
$\displaystyle\left.\partialderivative[2]{V}{\phi_{s}}\right|_{\boldsymbol{v}}$
$\displaystyle=m_{h}^{2}\sin[2](\theta)+m_{s}^{2}\cos[2](\theta),$ (A.4)
$\displaystyle\left.\frac{\partial^{2}V}{\partial\phi_{h}\partial\phi_{s}}\right|_{\boldsymbol{v}}$
$\displaystyle=\frac{1}{2}\\!\left(m_{h}^{2}-m_{s}^{2}\right)\sin(2\theta).$
(A.5)

This allows the elimination of $\lambda_{h}$, $\lambda_{s}$ and
$\lambda_{hs}$, leaving $\kappa_{hhs},\kappa_{sss},\theta,v_{s}$ and $m_{s}$
as free parameters, with $m_{h}$ and $v_{h}$ fixed by experimental
observations as described in Section A.2. We employ an iterative method to
numerically extract the $\mu^{2}_{i}$ and $\lambda_{i}$ Lagrangian parameters
from the system of implicit equations described by (A.1)-(A.5).

###  A.2 Mass spectrum, couplings and renormalisation scale

We use tree-level masses within the Coleman-Weinberg potential. We start with
$\alpha=1/137.036$, $m_{W}=80.379\,\text{GeV}$, $m_{Z}=91.1876\,\text{GeV}$
and $m_{t}=162.5\,\text{GeV}$, where the $W$ and $Z$ pole masses and the top
running mass have been used [209]. From this we calculate the Weinberg angle
to be $\theta_{W}=0.4918$, and $g=0.6412$, $g^{\prime}=0.3435$ and
$y_{t}=0.9167$. We take the Higgs mass to be $m_{h}=125.1\,\text{GeV}$ and set
the renormalisation scale to $\mu_{R}=m_{Z}$ (i.e. the scale at which the
couplings have been extracted). These couplings and the renormalisation scale
are held fixed, even though the masses are field- and temperature-dependent.
The Higgs VEV is then $v_{h}\approx 250.7\,\text{GeV}$.

The masses included in $V_{CW}$ and $V_{T}$ are $m_{t}$, $m_{W^{\pm}}$,
$m_{Z}$, $m_{\gamma}$, $m_{GB}$, $m_{h}$ and $m_{s}$. The scalar masses
$m_{h}$, $m_{s}$ and $m_{GB}$ are extracted at tree-level, hence using
$\mu_{h,0}^{2}$ and $\mu_{s,0}^{2}$ obtained through (A.1) and (A.2) with the
replacement $V\rightarrow V_{0}$. The degrees of freedom are $n_{t}=12$,
$n_{W^{\pm}}=6$, $n_{Z}=n_{\gamma}=n_{GB}=3$ and $n_{h}=n_{s}=1$. The thermal
mass corrections as well as the explicit forms for these masses (in a
$\mathbb{Z}_{2}$-symmetric variant of this model) can be found in Appendix C
of Ref. [162].

Lastly, we note that an infrared divergence appears in $V^{\prime\prime}$ due
to terms in the Coleman-Weinberg potential of the form

$\left.\frac{\partial^{2}V_{CW}}{\partial\phi_{j}\partial\phi_{k}}\right|_{\boldsymbol{v}}\supset\left.\partialderivative{m_{i}^{2}}{\phi_{j}}\right|_{\boldsymbol{v}}\left.\partialderivative{m_{i}^{2}}{\phi_{k}}\right|_{\boldsymbol{v}}\ln(\frac{m_{i}^{2}(\boldsymbol{v})}{\mu_{R}^{2}}).$
(A.6)

A vanishing mass in the logarithm causes a divergence if the derivative of the
mass at the VEV is non-zero. There are two particles in our mass spectrum that
vanish at the VEV at zero temperature; the photon and the Goldstone boson. The
photon term in (A.6) vanishes because the photon is field-independent at zero
temperature. However, the Goldstone boson is field-dependent, so the Goldstone
term in (A.6) diverges.

We adopt the following prescription to avoid this IR divergence [210, 211]. We
shift the Goldstone mass by its one-loop self-energy evaluated at $p^{2}=0$;
that is, $m_{GB}^{2}(\boldsymbol{\phi},T)\rightarrow
m_{GB}^{2}(\boldsymbol{\phi},T)+\kappa\Pi_{GB}(\boldsymbol{\phi})$ where
$\kappa=1/(16\pi^{2})$. This self-energy is calculated as

$\kappa\Pi_{GB}(\boldsymbol{\phi})=\frac{1}{h}\partialderivative{V^{\prime}_{CW}}{h},$
(A.7)

where the prime denotes the removal of the Goldstone boson contribution to the
Coleman-Weinberg potential. Thus,

$\Pi_{GB}(\boldsymbol{\phi})=\frac{1}{2h}\sum_{i\neq
GB}(-1)^{2s_{i}}n_{i}\partialderivative{m_{i}^{2}}{h}m_{i}^{2}(\boldsymbol{\phi})\\!\left[\log(\frac{m_{i}^{2}(\boldsymbol{\phi})}{\mu_{R}^{2}})-c_{i}+\frac{1}{2}\right].$
(A.8)

This mass shift prevents the IR divergence in the second derivative of the
Coleman-Weinberg potential, because the Goldstone mass no longer vanishes in
the VEV.

##  Appendix B Time-temperature relation

Under the assumption that the Universe expands adiabatically during the phase
transition,232323Reheating does not occur in the false vacuum if bubbles grow
as detonations, so reheating does not affect the dynamics of the phase
transition. However, temperature-dependent relics such as gravitational waves
are affected by this reheating. we have

$\derivative{t}\\!\left(s(t)a^{3}(t)\right)=0,$ (B.1)

where $s$ is the entropy density of the plasma and $a$ is the scale factor of
the Universe. This can be expressed as

$\derivative{s}{t}=-3H(t)s(t).$ (B.2)

The entropy density is given by

$s(T)=\partialderivative{p}{T}=-\partialderivative{V}{T}.$ (B.3)

Writing (B.2) in terms of temperature and using (B.3), we have

$\derivative{T}{t}\partialderivative[2]{V}{T}=-3H(T)\partialderivative{V}{T},$
(B.4)

from which we can extract

$\derivative{T}{t}=-3H(T)\frac{\partial_{T}V}{\partial_{TT}V}.$ (B.5)

Here we have used the notation $\partial_{T}V=\partialderivative{V}{T}$ and
$\partial_{TT}V=\partialderivative[2]{V}{T}$. Because we assume the energy
density is approximately homogeneous even in the presence of bubbles due to
energy conservation, we evaluate the derivatives of the potential at the false
vacuum. That is,

$\derivative{T}{t}=-3H(T)\frac{\partial_{T}V(\boldsymbol{\phi}_{f}(T),T)}{\partial_{TT}V(\boldsymbol{\phi}_{f}(T),T)}.$
(B.6)

The typical next step is to assume an equation of state to simplify (B.6) and
the transition analysis in general. A common choice is the MIT bag equation of
state [136], where the potential in the false vacuum is modelled as

$V(T)=aT^{4}+b,$ (B.7)

with $a$ and $b$ constant in temperature. Under this approximation, (B.6)
reduces to the familiar form of

$\derivative{T}{t}=-TH(T).$ (B.8)

This simple form is essential for analytic comparison of unit nucleation and
percolation in Section 3. We have numerically compared (B.6) and (B.8) and
found very good agreement in each of our benchmarks. The most significant
deviation occurs for $T<10\,\text{GeV}$, where sub-quartic thermal terms in
the potential become significant. That is, where the terms proportional to
$T$, $T^{2}$ and $T^{3}$ are no longer dominated by the terms proportional to
$T^{4}$. The former are neglected in the high-temperature approximation used
in the bag equation of state.

##  Appendix C Ratio of scale factors

The ratio of scale factors at different times and at different temperatures
respectively appear in the false vacuum fraction in (2.6) and (2.9). Here we
show how to evaluate these ratios. A particularly simple form arises when
using temperature.

The definition of the Hubble parameter,

$H(t)=\frac{\dot{a}(t)}{a(t)}=\frac{1}{a(t)}\derivative{a}{t},$ (C.1)

is a separable ordinary differential equation. Integrating, we have

$\int_{t_{0}}^{t}\\!dt^{\prime}\,H(t^{\prime})=\int_{a(t_{0})}^{a(t)}\\!da\,\frac{1}{a(t)},$
(C.2)

where $t_{0}<t$ is some reference time. We will see that the reference time is
not important here since it cancels when taking the ratio of scale factors.
Solving the integral equation for $a(t)$, we find

$a(t)=a(t_{0})\exp(\int_{t_{0}}^{t}\\!dt^{\prime}\,H(t^{\prime})).$ (C.3)

The ratio of scale factors at times $t_{1}$ and $t_{2}$ is then

$\frac{a(t_{1})}{a(t_{2})}=\exp(\int_{t_{2}}^{t_{1}}\\!dt^{\prime}\,H(t^{\prime})),$
(C.4)

where all dependence on the reference time $t_{0}$ has disappeared.

Now consider the ratio of scale factors at temperatures $T_{1}$ and $T_{2}$.
Using the adiabatic time-temperature relation (B.8) derived in Appendix B, the
ratio in (C.4) can be expressed in terms of temperature as

$\frac{a(T_{1})}{a(T_{2})}=\exp(\int_{T_{1}}^{T_{2}}\\!dT^{\prime}\,\frac{1}{T^{\prime}})=\frac{T_{2}}{T_{1}}.$
(C.5)

This simple form is exact if the Universe expands adiabatically and the MIT
bag equation of state holds.

##  Appendix D The effects of reheating

Reheating occurs when the energy density accumulated in and near bubble walls
is redistributed to the surrounding plasma [145, 74]. This has the effect of
increasing the temperature of the surrounding plasma. For bubbles expanding as
detonations, this reheating occurs inside the bubble, while for deflagrations
it occurs outside the bubble. For a hybrid expansion mode, reheating occurs on
both sides of the bubble wall. See Ref. [212] for a description of the three
bubble expansion modes. If a bubble propagates as a deflagration or hybrid,
then the reheating slows the expansion of the bubble and suppresses the bubble
nucleation rate nearby.242424The suppression of bubble nucleation due to
reheating assumes that the nucleation rate is lower at higher temperatures.
However, if $T<T_{\Gamma}$, where $T_{\Gamma}$ maximises the nucleation rate,
then reheating towards $T_{\Gamma}$ may bring about another era of significant
bubble nucleation. Then it would appear that our treatment of the true vacuum
volume is an upper bound on the correct result that incorporates reheating. We
now explain that this does not invalidate our demonstration of the existence
of Scenarios 1 and 2 (see Section 3) when the bubbles expand as deflagrations
or hybrids. Reheating for bubbles expanding as detonations is not expected to
affect bubble expansion or the nucleation rate.

First, consider Scenario 1: unit nucleation without percolation. We
potentially slightly overestimate bubble nucleation, encoded in $X(T)$ (see
(3.6)). However, $X(T)$ should be largely unaffected because the fraction of
the Universe in the reheated shell surrounding true vacuum bubbles must be
small. This is because $P_{f}(T)>0.71$ for all temperatures in Scenario 1.
Regardless, $X(T)$ equally affects the number of bubbles and the true vacuum
volume and thus cannot affect their comparison. This realisation follows from
the simplification made below (3.16). The bubble wall velocity will be reduced
to an unknown extent. A lower bubble wall velocity would reduce the true
vacuum volume, hence further delaying — perhaps inhibiting — the onset of
percolation. Hence, Scenario 1 only becomes easier to realise with the
incorporation of reheating effects. The upper bound on bubble wall velocity
for which Scenario 1 is possible then becomes an overly strong constraint. It
is possible for Scenario 1 to be realised for bubble wall velocities higher
than the predicted upper bound.

Next, consider Scenario 2: percolation without unit nucleation. Again, we
overestimate $X(T)$, but perhaps substantially this time as a significant
fraction of the Universe may be contained in the reheated shells surrounding
bubbles. The case of no unit nucleation is then easier to realise. However,
for this scenario we still need percolation to occur. With a slowing of the
bubble propagation and less bubbles nucleated, percolation is delayed just as
discussed for Scenario 1. The lower bound on bubble wall velocity for which
Scenario 2 is possible then becomes an insufficient constraint. It is possible
for Scenario 2 to not be realised for bubble wall velocities higher than the
predicted lower bound. However, by our arguments, at the very least
detonations still allow for the possibility of Scenario 2.

From these general principles, we then expect Scenario 1 to be easier to
realise than predicted, and Scenario 2 to still be possible because reheating
from detonations occurs only inside the bubbles and should not affect the
transition progress.

##  Appendix E Comparing bubble and Hubble volumes

Here we prove that bubbles can be larger than the Hubble volume provided
sufficient supercooling occurs. We need to show that

$r(T^{\prime},T)>r_{H}(T),$ (E.1)

where $r(T^{\prime},T)$ is the comoving radius of a bubble that nucleated at
temperature $T^{\prime}$ and has grown until temperature $T$, and $r_{H}(T)$
is the Hubble volume at temperature $T$, given by

$r_{H}(T)=\frac{1}{a(T)H(T)}.$ (E.2)

The largest bubble possible at any given temperature is one that nucleated
very close to the critical temperature. The initial comoving bubble radius,
$r_{0}$, at the critical temperature is not well-defined. Instead we consider
the bubble to have nucleated slightly below this temperature and ignore
$r_{0}$, with the expectation that $r(T_{c},T)\gg r_{0}$. Further assuming a
constant wall velocity, as justified in Section 2, the comoving bubble radius
(in terms of time) is

$\displaystyle
r(t^{\prime},t)=v_{w}\int_{t^{\prime}}^{t}\\!dt^{\prime\prime}\frac{1}{a(t^{\prime\prime})}.$
(E.3)

Using the time-temperature relation derived in Appendix B, (E.3) can be
expressed in terms of temperature as

$r(T^{\prime},T)=v_{w}\int_{T}^{T^{\prime}}\\!dT^{\prime\prime}\frac{1}{T^{\prime\prime}H(T^{\prime\prime})a(T^{\prime\prime})}.$
(E.4)

Multiplying by the scale factor at an arbitrary temperature $T_{*}>0$ and
using the ratio of scale factors derived in Appendix C, (E.4) simplifies to

$a(T_{*})r(T^{\prime},T)=\frac{v_{w}}{T_{*}}\int_{T}^{T^{\prime}}\\!dT^{\prime\prime}\frac{1}{H(T^{\prime\prime})}.$
(E.5)

Thus, we have

$a(T_{*})\left(r(T_{c},T)-r_{H}(T)\right)=\frac{v_{w}}{T_{*}}\int_{T}^{T_{c}}\\!dT^{\prime}\frac{1}{H(T^{\prime})}-\frac{T}{T_{*}H(T)},$
(E.6)

which we require to be positive to satisfy (E.1).

Figure 9: The bubble wall velocity bound for which the volume of bubbles
nucleated at the critical temperature can exceed the Hubble volume. The dashed
black line corresponds to the contour where a bubble wall velocity of
$v_{w}=1$ is required. The bubble volume can only exceed the Hubble volume
below this contour (in the coloured region). The dashed blue line corresponds
to the vacuum-dominated limit: $T<T_{c}/2$. The horizontal dotted green line
corresponds to $T=T_{\text{eq}}$, marking the boundary between the radiation-
and vacuum-dominated eras.

We proceed by using the model-independent approximation for the Hubble
parameter defined in (3.20). Pulling out a positive factor $1/(T_{*}H(T))$, we
can obtain a bound on the bubble wall velocity for which (E.1) is satisfied,

$v_{w}>\left(\sqrt{\left(\frac{T}{T_{\text{eq}}}\right)^{\\!\\!4}+1}\left[\frac{T_{c}}{T}\,{}_{2}F_{1}\\!\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\\!\left(\frac{T_{c}}{T_{\text{eq}}}\right)^{\\!\\!4}\right)-{}_{2}F_{1}\\!\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\\!\left(\frac{T}{T_{\text{eq}}}\right)^{\\!\\!4}\right)\right]\right)^{-1},$
(E.7)

independent of the choice of $T_{*}$. In the vacuum-dominated limit $T_{c}\ll
T_{\text{eq}}$, the hypergeometric functions ${}_{2}F_{1}$ approach unity,
leaving the bound

$T<T_{c}\\!\left(1+\frac{1}{v_{w}}\right)^{\\!\\!-1}.$ (E.8)

This bound reduces to $T<T_{c}/2$ when $v_{w}=1$. One can also obtain (E.8) by
using a constant Hubble parameter in (E.6).

When the right-hand side of (E.7) is larger than unity, even the largest
bubbles possible are smaller than the Hubble volume. The result (E.7) is shown
in Figure 9, along with the $(T,T_{c})$ contour for which the right-hand side
of (E.7) is equal to unity. We plot the bound $T<T_{c}/2$ obtained in the
vacuum-dominated limit, where subluminal solutions exist. In Figure 9, we see
that the largest bubble volume is smaller than the Hubble volume at high
temperature but may exceed the Hubble volume if permitted to grow for
sufficiently long. One could replace $T_{c}$ with $T_{\Gamma}$ in (E.7) and
Figure 9 to compare the volume of the most common bubbles to the Hubble
volume. One could replace $T$ with $T_{f}$ to determine whether bubbles can
outgrow the Hubble volume before the phase transition ends. For sufficiently
large $T_{c}/T_{\text{eq}}$ (specifically $T_{c}\gtrsim 4.55T_{\text{eq}}$),
we find that bubbles can be larger than the Hubble volume even during the
radiation-dominated era. However, the ratio $T/T_{c}$ for which (E.7) is
satisfied decreases with $T_{c}$. This suggests that a phase transition that
first becomes possible at a high energy scale compared to $T_{\text{eq}}$
requires more supercooling before bubbles can outgrow the Hubble volume.

This investigation demonstrates that it is possible for the volume of a single
bubble to exceed that of the Hubble volume if the phase transition is
sufficiently supercooled. Then the expectation of having one bubble per Hubble
volume is overly restrictive. However, even with bubbles smaller than the
Hubble volume, it is possible for these bubbles to percolate without having
one in every Hubble volume (assuming some form of Hubble sphere packing). In
our simulations with $N(0)\lesssim 1$ (e.g. the benchmarks listed in Tables 1
and 2) we found that the average bubble radius was below $R_{H}$ at the
percolation temperature and above $R_{H}$ at the completion temperature, where
$R_{H}(T)=1/H(T)$ is the proper Hubble radius.

##  Appendix F Numerical uncertainty of the bounce action

Although tools exist to determine the action, the underlying problem remains
computationally difficult and expensive. There is naturally a trade-off
between running time and accuracy which becomes particularly relevant for
studies that require some combination of parameter space scanning, precise
predictions, and strong supercooling. We used CosmoTransitions (version 2.0.6)
[168] to compute the action. We found it necessary to make a few key changes
to improve the accuracy and precision of our results. The following is a list
of changes we made:

  * •

We tightened the tolerances xtol and phitol input to tunneling1D.findProfile,
from $10^{-4}$ to $10^{-6}$. Further tightening was not found to consistently
improve precision, yet incurred substantial penalties to running time.

  * •

We resolved many underflow and overflow errors that commonly appear in thin-
walled cases and sometimes elsewhere.

  * •

We introduced a ThinWallError exception class to differentiate general failure
of the path deformation algorithm from failures that occur specifically for
thin-walled cases.

  * •

Phases were traced using PhaseTracer (version 1.1.0) [167] rather than with
CosmoTransitions.

  * •

Transitions were analysed using TransitionSolver rather than with
CosmoTransitions.

Together these changes prevent the otherwise frequent occurrence of crashes
and exceptions (outside of expected ThinWallErrors), and suppress sawtooth
patterns in the action curve arising from loose tolerances. ThinWallErrors are
raised typically for $T\approx T_{c}$ where nucleation is negligible, and
importantly provide a distinction between this expected case of failure and
more general failure due to unhandled exceptions. We find the ThinWallError to
be consistently raised above a threshold temperature near $T_{c}$, with all
action evaluations successful below this threshold temperature. Due to fixing
underflow and overflow errors, the more general failure of the action
evaluation no longer occurs.

Despite these improvements, we find an $\mathcal{O}(0.1\%)$ discontinuity at
the minimum of the action (i.e. at $T_{S_{\text{min}}}$), particularly in M2.
Additionally, the numerically determined action curve oscillates slightly near
$T_{S_{\text{min}}}$ due to negligible perturbations of the Lagrangian
parameters. The discontinuity and oscillations lead to noticeable uncertainty
in quantities directly related to $S_{\text{min}}$, as seen in Figure 8, and
ultimately limits our ability to resolve $\delta_{sc,n}\rightarrow 1$. This
uncertainty is slightly amplified by the discrete sampling of the action.
Nevertheless, our changes alleviate issues presented in the appendix of Ref.
[213] (and possibly the issues that motivated Appendix A of Ref. [214]) which
we also experienced. We suspect issues associated with evaluating the action
are more prevalent than the lack of such reports would suggest.

## References

  * [1] L.A. Dolan and R.W. Jackiw, _Symmetry Behavior at Finite Temperature_ , _Phys. Rev. D_ 9 (1974) 3320.
  * [2] S. Weinberg, _Gauge and Global Symmetries at High Temperature_ , _Phys. Rev. D_ 9 (1974) 3357.
  * [3] D.A. Kirzhnits and A.D. Linde, _A Relativistic phase transition_ , _Zh. Eksp. Teor. Fiz._ 67 (1974) 1263.
  * [4] D.A. Kirzhnits and A.D. Linde, _Symmetry Behavior in Gauge Theories_ , _Annals Phys._ 101 (1976) 195.
  * [5] M. Dine, R.G. Leigh, P.Y. Huet, A.D. Linde and D.A. Linde, _Towards the theory of the electroweak phase transition_ , _Phys. Rev. D_ 46 (1992) 550 [hep-ph/9203203].
  * [6] Y. Aoki, Z. Fodor, S.D. Katz and K.K. Szabó, _The QCD transition temperature: Results with physical masses in the continuum limit_ , _Phys. Lett. B_ 643 (2006) 46 [hep-lat/0609068].
  * [7] Y. Aoki, G. Endrődi, Z. Fodor, S.D. Katz and K.K. Szabó, _The Order of the quantum chromodynamics transition predicted by the standard model of particle physics_ , _Nature_ 443 (2006) 675 [hep-lat/0611014].
  * [8] S.R. Coleman, _The Fate of the False Vacuum. 1. Semiclassical Theory_ , _Phys. Rev. D_ 15 (1977) 2929.
  * [9] C.G. Callan, Jr. and S.R. Coleman, _The Fate of the False Vacuum. 2. First Quantum Corrections_ , _Phys. Rev. D_ 16 (1977) 1762.
  * [10] A.D. Linde, _Fate of the False Vacuum at Finite Temperature: Theory and Applications_ , _Phys. Lett. B_ 100 (1981) 37.
  * [11] A.D. Linde, _Decay of the False Vacuum at Finite Temperature_ , _Nucl. Phys. B_ 216 (1983) 421.
  * [12] M. Trodden, _Electroweak baryogenesis_ , _Rev. Mod. Phys._ 71 (1999) 1463 [hep-ph/9803479].
  * [13] J.M. Cline, _Baryogenesis_ , in _Les Houches Summer School - Session 86: Particle Physics and Cosmology: The Fabric of Spacetime_ , 9, 2006 [hep-ph/0609145].
  * [14] D.E. Morrissey and M.J. Ramsey-Musolf, _Electroweak baryogenesis_ , _New J. Phys._ 14 (2012) 125003 [1206.2942].
  * [15] White, Graham Albert, _A Pedagogical Introduction to Electroweak Baryogenesis_ , 2053-2571, Morgan & Claypool Publishers (2016), 10.1088/978-1-6817-4457-5.
  * [16] G. Bertone et al., _Gravitational wave probes of dark matter: challenges and opportunities_ , _SciPost Phys. Core_ 3 (2020) 007 [1907.10610].
  * [17] A. Vilenkin, _Gravitational radiation from cosmic strings_ , _Phys. Lett. B_ 107 (1981) 47.
  * [18] A. Vilenkin, _Cosmic Strings and Domain Walls_ , _Phys. Rept._ 121 (1985) 263.
  * [19] M. Kawasaki and K. Saikawa, _Study of gravitational radiation from cosmic domain walls_ , _JCAP_ 09 (2011) 008 [1102.5628].
  * [20] D.G. Figueroa, M. Hindmarsh and J. Urrestilla, _Exact Scale-Invariant Background of Gravitational Waves from Cosmic Defects_ , _Phys. Rev. Lett._ 110 (2013) 101302 [1212.5458].
  * [21] E. Witten, _Cosmic Separation of Phases_ , _Phys. Rev. D_ 30 (1984) 272.
  * [22] C.J. Hogan, _Gravitational radiation from cosmological phase transitions_ , _Mon. Not. Roy. Astron. Soc._ 218 (1986) 629.
  * [23] A. Kosowsky, M.S. Turner and R. Watkins, _Gravitational radiation from colliding vacuum bubbles_ , _Phys. Rev. D_ 45 (1992) 4514.
  * [24] A. Kosowsky, M.S. Turner and R. Watkins, _Gravitational waves from first order cosmological phase transitions_ , _Phys. Rev. Lett._ 69 (1992) 2026.
  * [25] K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, _Is there a hot electroweak phase transition at $m_{H}\gtrsim m_{W}$?_, _Phys. Rev. Lett._ 77 (1996) 2887 [hep-ph/9605288].
  * [26] K. Rummukainen, M. Tsypin, K. Kajantie, M. Laine and M.E. Shaposhnikov, _The Universality class of the electroweak theory_ , _Nucl. Phys. B_ 532 (1998) 283 [hep-lat/9805013].
  * [27] F. Csikor, Z. Fodor and J. Heitger, _Endpoint of the hot electroweak phase transition_ , _Phys. Rev. Lett._ 82 (1999) 21 [hep-ph/9809291].
  * [28] A. Beniwal, M. Lewicki, J.D. Wells, M. White and A.G. Williams, _Gravitational wave, collider and dark matter signals from a scalar singlet electroweak baryogenesis_ , _JHEP_ 08 (2017) 108 [1702.06124].
  * [29] G. Kurup and M. Perelstein, _Dynamics of Electroweak Phase Transition In Singlet-Scalar Extension of the Standard Model_ , _Phys. Rev. D_ 96 (2017) 015036 [1704.03381].
  * [30] C.-W. Chiang, M.J. Ramsey-Musolf and E. Senaha, _Standard Model with a Complex Scalar Singlet: Cosmological Implications and Theoretical Considerations_ , _Phys. Rev. D_ 97 (2018) 015005 [1707.09960].
  * [31] J. Bernon, L. Bian and Y. Jiang, _A new insight into the phase transition in the early Universe with two Higgs doublets_ , _JHEP_ 05 (2018) 151 [1712.08430].
  * [32] A. Alves, T. Ghosh, H.-K. Guo, K. Sinha and D. Vagie, _Collider and Gravitational Wave Complementarity in Exploring the Singlet Extension of the Standard Model_ , _JHEP_ 04 (2019) 052 [1812.09333].
  * [33] H.-L. Li, M.J. Ramsey-Musolf and S. Willocq, _Probing a scalar singlet-catalyzed electroweak phase transition with resonant di-Higgs boson production in the $4b$ channel_, _Phys. Rev. D_ 100 (2019) 075035 [1906.05289].
  * [34] P. Athron, C. Balázs, A. Fowlie, G. Pozzo, G. White and Y. Zhang, _Strong first-order phase transitions in the NMSSM — a comprehensive survey_ , _JHEP_ 11 (2019) 151 [1908.11847].
  * [35] X. Wang, F.P. Huang and X. Zhang, _Gravitational wave and collider signals in complex two-Higgs doublet model with dynamical CP-violation at finite temperature_ , _Phys. Rev. D_ 101 (2020) 015015 [1909.02978].
  * [36] S.V. Demidov, D.S. Gorbunov and E. Kriukova, _Gravitational waves from first-order electroweak phase transition in a model with light sgoldstinos_ , _JHEP_ 07 (2022) 061 [2112.06083].
  * [37] S. Schettler, T. Boeckel and J. Schaffner-Bielich, _Imprints of the QCD Phase Transition on the Spectrum of Gravitational Waves_ , _Phys. Rev. D_ 83 (2011) 064030 [1010.4857].
  * [38] S. Iso, P.D. Serpico and K. Shimada, _QCD-Electroweak First-Order Phase Transition in a Supercooled Universe_ , _Phys. Rev. Lett._ 119 (2017) 141301 [1704.04955].
  * [39] S. Rezapour, K. Bitaghsir Fadafan and M. Ahmadvand, _Gravitational waves of a first-order QCD phase transition at finite coupling from holography_ , _Annals Phys._ 437 (2022) 168731 [2006.04265].
  * [40] M. Daniel and C.E. Vayonakis, _The Phase Transition in the SU(5) Model at High Temperatures_ , _Nucl. Phys. B_ 180 (1981) 301.
  * [41] P.S.B. Dev and A. Mazumdar, _Probing the Scale of New Physics by Advanced LIGO/VIRGO_ , _Phys. Rev. D_ 93 (2016) 104001 [1602.04203].
  * [42] C. Balázs, A. Fowlie, A. Mazumdar and G. White, _Gravitational waves at aLIGO and vacuum stability with a scalar singlet extension of the Standard Model_ , _Phys. Rev. D_ 95 (2017) 043505 [1611.01617].
  * [43] D. Croon, T.E. Gonzalo and G. White, _Gravitational Waves from a Pati-Salam Phase Transition_ , _JHEP_ 02 (2019) 083 [1812.02747].
  * [44] W.-C. Huang, F. Sannino and Z.-W. Wang, _Gravitational Waves from Pati-Salam Dynamics_ , _Phys. Rev. D_ 102 (2020) 095025 [2004.02332].
  * [45] N. Okada, O. Seto and H. Uchida, _Gravitational waves from breaking of an extra $U(1)$ in $SO(10)$ grand unification_, _PTEP_ 2021 (2021) 033B01 [2006.01406].
  * [46] C. Caprini et al., _Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions_ , _JCAP_ 04 (2016) 001 [1512.06239].
  * [47] C. Caprini et al., _Detecting gravitational waves from cosmological phase transitions with LISA: an update_ , _JCAP_ 03 (2020) 024 [1910.13125].
  * [48] LISA collaboration, _Laser Interferometer Space Antenna_ , 1702.00786.
  * [49] C. Grojean and G. Servant, _Gravitational Waves from Phase Transitions at the Electroweak Scale and Beyond_ , _Phys. Rev. D_ 75 (2007) 043507 [hep-ph/0607107].
  * [50] S.J. Huber and T. Konstandin, _Gravitational Wave Production by Collisions: More Bubbles_ , _JCAP_ 09 (2008) 022 [0806.1828].
  * [51] R. Jinno and M. Takimoto, _Probing a classically conformal B-L model with gravitational waves_ , _Phys. Rev. D_ 95 (2017) 015020 [1604.05035].
  * [52] L. Marzola, A. Racioppi and V. Vaskonen, _Phase transition and gravitational wave phenomenology of scalar conformal extensions of the Standard Model_ , _Eur. Phys. J. C_ 77 (2017) 484 [1704.01034].
  * [53] J. Ellis, M. Lewicki, J.M. No and V. Vaskonen, _Gravitational wave energy budget in strongly supercooled phase transitions_ , _JCAP_ 06 (2019) 024 [1903.09642].
  * [54] J. Ellis, M. Lewicki and V. Vaskonen, _Updated predictions for gravitational waves produced in a strongly supercooled phase transition_ , _JCAP_ 11 (2020) 020 [2007.15586].
  * [55] M. Kierkla, A. Karam and B. Swiezewska, _Conformal model for gravitational waves and dark matter: A status update_ , 2210.07075.
  * [56] S.J. Huber, T. Konstandin, G. Nardini and I. Rues, _Detectable Gravitational Waves from Very Strong Phase Transitions in the General NMSSM_ , _JCAP_ 03 (2016) 036 [1512.06357].
  * [57] L. Leitao and A. Mégevand, _Gravitational waves from a very strong electroweak phase transition_ , _JCAP_ 05 (2016) 037 [1512.08962].
  * [58] A. Mégevand and S. Ramírez, _Bubble nucleation and growth in very strong cosmological phase transitions_ , _Nucl. Phys. B_ 919 (2017) 74 [1611.05853].
  * [59] A. Kobakhidze, C. Lagger, A. Manning and J. Yue, _Gravitational waves from a supercooled electroweak phase transition and their detection with pulsar timing arrays_ , _Eur. Phys. J. C_ 77 (2017) 570 [1703.06552].
  * [60] R.-G. Cai, M. Sasaki and S.-J. Wang, _The gravitational waves from the first-order phase transition with a dimension-six operator_ , _JCAP_ 08 (2017) 004 [1707.03001].
  * [61] J. Ellis, M. Lewicki and J.M. No, _On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal_ , _JCAP_ 04 (2019) 003 [1809.08242].
  * [62] X. Wang, F.P. Huang and X. Zhang, _Phase transition dynamics and gravitational wave spectra of strong first-order phase transition in supercooled universe_ , _JCAP_ 05 (2020) 045 [2003.08892].
  * [63] K. Freese and M.W. Winkler, _Have pulsar timing arrays detected the hot big bang: Gravitational waves from strong first order phase transitions in the early Universe_ , _Phys. Rev. D_ 106 (2022) 103523 [2208.03330].
  * [64] M. Lewicki and V. Vaskonen, _Gravitational waves from bubble collisions and fluid motion in strongly supercooled phase transitions_ , 2208.11697.
  * [65] C. Badger et al., _Probing Early Universe Supercooled Phase Transitions with Gravitational Wave Data_ , 2209.14707.
  * [66] LIGO Scientific, Virgo collaboration, _Observation of Gravitational Waves from a Binary Black Hole Merger_ , _Phys. Rev. Lett._ 116 (2016) 061102 [1602.03837].
  * [67] D. Cutting, M. Hindmarsh and D.J. Weir, _Vorticity, kinetic energy, and suppressed gravitational wave production in strong first order phase transitions_ , _Phys. Rev. Lett._ 125 (2020) 021302 [1906.00480].
  * [68] D.J. Weir, _Revisiting the envelope approximation: gravitational waves from bubble collisions_ , _Phys. Rev. D_ 93 (2016) 124037 [1604.08429].
  * [69] R. Jinno and M. Takimoto, _Gravitational waves from bubble collisions: An analytic derivation_ , _Phys. Rev. D_ 95 (2017) 024009 [1605.01403].
  * [70] M. Hindmarsh, _Sound shell model for acoustic gravitational wave production at a first-order phase transition in the early Universe_ , _Phys. Rev. Lett._ 120 (2018) 071301 [1608.04735].
  * [71] D. Bodeker and G.D. Moore, _Electroweak Bubble Wall Speed Limit_ , _JCAP_ 05 (2017) 025 [1703.08215].
  * [72] M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, _Shape of the acoustic gravitational wave power spectrum from a first order phase transition_ , _Phys. Rev. D_ 96 (2017) 103520 [1704.05871].
  * [73] R. Jinno and M. Takimoto, _Gravitational waves from bubble dynamics: Beyond the Envelope_ , _JCAP_ 01 (2019) 060 [1707.03111].
  * [74] A. Mégevand and S. Ramírez, _Bubble nucleation and growth in slow cosmological phase transitions_ , _Nucl. Phys. B_ 928 (2018) 38 [1710.06279].
  * [75] T. Konstandin, _Gravitational radiation from a bulk flow model_ , _JCAP_ 03 (2018) 047 [1712.06869].
  * [76] D. Cutting, M. Hindmarsh and D.J. Weir, _Gravitational waves from vacuum first-order phase transitions: from the envelope to the lattice_ , _Phys. Rev. D_ 97 (2018) 123513 [1802.05712].
  * [77] P. Niksa, M. Schlederer and G. Sigl, _Gravitational Waves produced by Compressible MHD Turbulence from Cosmological Phase Transitions_ , _Class. Quant. Grav._ 35 (2018) 144001 [1803.02271].
  * [78] A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili and A. Kosowsky, _Numerical simulations of gravitational waves from early-universe turbulence_ , _Phys. Rev. D_ 102 (2020) 083512 [1903.08585].
  * [79] R. Jinno, H. Seong, M. Takimoto and C.M. Um, _Gravitational waves from first-order phase transitions: Ultra-supercooled transitions and the fate of relativistic shocks_ , _JCAP_ 10 (2019) 033 [1905.00899].
  * [80] R. Jinno, T. Konstandin and M. Takimoto, _Relativistic bubble collisions—a closer look_ , _JCAP_ 09 (2019) 035 [1906.02588].
  * [81] M. Hindmarsh and M. Hijazi, _Gravitational waves from first order cosmological phase transitions in the Sound Shell Model_ , _JCAP_ 12 (2019) 062 [1909.10040].
  * [82] T. Alanne, T. Hugle, M. Platscher and K. Schmitz, _A fresh look at the gravitational-wave signal from cosmological phase transitions_ , _JHEP_ 03 (2020) 004 [1909.11356].
  * [83] M. Lewicki and V. Vaskonen, _On bubble collisions in strongly supercooled phase transitions_ , _Phys. Dark Univ._ 30 (2020) 100672 [1912.00997].
  * [84] J. Ellis, M. Lewicki and J.M. No, _Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source_ , _JCAP_ 07 (2020) 050 [2003.07360].
  * [85] F. Giese, T. Konstandin and J. van de Vis, _Model-independent energy budget of cosmological first-order phase transitions—A sound argument to go beyond the bag model_ , _JCAP_ 07 (2020) 057 [2004.06995].
  * [86] M. Barroso Mancha, T. Prokopec and B. Swiezewska, _Field-theoretic derivation of bubble-wall force_ , _JHEP_ 01 (2021) 070 [2005.10875].
  * [87] D. Cutting, E.G. Escartin, M. Hindmarsh and D.J. Weir, _Gravitational waves from vacuum first order phase transitions II: from thin to thick walls_ , _Phys. Rev. D_ 103 (2021) 023531 [2005.13537].
  * [88] M. Lewicki and V. Vaskonen, _Gravitational wave spectra from strongly supercooled phase transitions_ , _Eur. Phys. J. C_ 80 (2020) 1003 [2007.04967].
  * [89] H.-K. Guo, K. Sinha, D. Vagie and G. White, _Phase Transitions in an Expanding Universe: Stochastic Gravitational Waves in Standard and Non-Standard Histories_ , _JCAP_ 01 (2021) 001 [2007.08537].
  * [90] S. Höche, J. Kozaczuk, A.J. Long, J. Turner and Y. Wang, _Towards an all-orders calculation of the electroweak bubble wall velocity_ , _JCAP_ 03 (2021) 009 [2007.10343].
  * [91] A. Mégevand and F.A. Membiela, _Bubble wall correlations in cosmological phase transitions_ , _Phys. Rev. D_ 102 (2020) 103514 [2008.01873].
  * [92] D. Croon, O. Gould, P. Schicho, T.V.I. Tenkanen and G. White, _Theoretical uncertainties for cosmological first-order phase transitions_ , _JHEP_ 04 (2021) 055 [2009.10080].
  * [93] A. Friedlander, I. Banta, J.M. Cline and D. Tucker-Smith, _Wall speed and shape in singlet-assisted strong electroweak phase transitions_ , _Phys. Rev. D_ 103 (2021) 055020 [2009.14295].
  * [94] R. Jinno, T. Konstandin and H. Rubira, _A hybrid simulation of gravitational wave production in first-order phase transitions_ , _JCAP_ 04 (2021) 014 [2010.00971].
  * [95] A. Azatov and M. Vanvlasselaer, _Bubble wall velocity: heavy physics effects_ , _JCAP_ 01 (2021) 058 [2010.02590].
  * [96] S. Balaji, M. Spannowsky and C. Tamarit, _Cosmological bubble friction in local equilibrium_ , _JCAP_ 03 (2021) 051 [2010.08013].
  * [97] F. Giese, T. Konstandin, K. Schmitz and J. Van De Vis, _Model-independent energy budget for LISA_ , _JCAP_ 01 (2021) 072 [2010.09744].
  * [98] R.-G. Cai and S.-J. Wang, _Effective picture of bubble expansion_ , _JCAP_ 03 (2021) 096 [2011.11451].
  * [99] H.-K. Guo, K. Sinha, D. Vagie and G. White, _The benefits of diligence: how precise are predicted gravitational wave spectra in models with phase transitions?_ , _JHEP_ 06 (2021) 164 [2103.06933].
  * [100] A. Ekstedt, _Higher-order corrections to the bubble-nucleation rate at finite temperature_ , _Eur. Phys. J. C_ 82 (2022) 173 [2104.11804].
  * [101] O. Gould and J. Hirvonen, _Effective field theory approach to thermal bubble nucleation_ , _Phys. Rev. D_ 104 (2021) 096015 [2108.04377].
  * [102] A. Megévand and F.A. Membiela, _Gravitational waves from bubble walls_ , _JCAP_ 10 (2021) 073 [2108.05510].
  * [103] A. Mégevand and F.A. Membiela, _Model-independent features of gravitational waves from bubble collisions_ , _Phys. Rev. D_ 104 (2021) 123532 [2108.07034].
  * [104] W.-Y. Ai, B. Garbrecht and C. Tamarit, _Bubble wall velocities in local equilibrium_ , 2109.13710.
  * [105] Y. Gouttenoire, R. Jinno and F. Sala, _Friction pressure on relativistic bubble walls_ , 2112.07686.
  * [106] J. Hirvonen, J. Löfgren, M.J. Ramsey-Musolf, P. Schicho and T.V.I. Tenkanen, _Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory_ , _JHEP_ 07 (2022) 135 [2112.08912].
  * [107] J. Dahl, M. Hindmarsh, K. Rummukainen and D.J. Weir, _Decay of acoustic turbulence in two dimensions and implications for cosmological gravitational waves_ , _Phys. Rev. D_ 106 (2022) 063511 [2112.12013].
  * [108] G.C. Dorsch, S.J. Huber and T. Konstandin, _A sonic boom in bubble wall friction_ , _JCAP_ 04 (2022) 010 [2112.12548].
  * [109] X. Wang, F.P. Huang and Y. Li, _Sound velocity effects on the phase transition gravitational wave spectrum in the sound shell model_ , _Phys. Rev. D_ 105 (2022) 103513 [2112.14650].
  * [110] S. De Curtis, L.D. Rose, A. Guiggiani, A.G. Muyor and G. Panico, _Bubble wall dynamics at the electroweak phase transition_ , _JHEP_ 03 (2022) 163 [2201.08220].
  * [111] R.-G. Cai, K. Hashino, S.-J. Wang and J.-H. Yu, _Gravitational waves from patterns of electroweak symmetry breaking: an effective perspective_ , 2202.08295.
  * [112] D. Cutting, E. Vilhonen and D.J. Weir, _Droplet collapse during strongly supercooled transitions_ , _Phys. Rev. D_ 106 (2022) 103524 [2204.03396].
  * [113] P. Auclair, C. Caprini, D. Cutting, M. Hindmarsh, K. Rummukainen, D.A. Steer et al., _Generation of gravitational waves from freely decaying turbulence_ , _JCAP_ 09 (2022) 029 [2205.02588].
  * [114] M.A. Ajmi and M. Hindmarsh, _Thermal suppression of bubble nucleation at first-order phase transitions in the early Universe_ , _Phys. Rev. D_ 106 (2022) 023505 [2205.04097].
  * [115] S.-J. Wang and Z.-Y. Yuwen, _The energy budget of cosmological first-order phase transitions beyond the bag equation of state_ , _JCAP_ 10 (2022) 047 [2206.01148].
  * [116] T.V.I. Tenkanen and J. van de Vis, _Speed of sound in cosmological phase transitions and effect on gravitational waves_ , _JHEP_ 08 (2022) 302 [2206.01130].
  * [117] M. Hijazi, _Numerical Estimations of the Distribution of the Lifetime of Bubbles Emerging from First Order Cosmological Phase Transitions_ , 2208.10636.
  * [118] M. Lewicki, V. Vaskonen and H. Veermäe, _Bubble dynamics in fluids with N-body simulations_ , _Phys. Rev. D_ 106 (2022) 103501 [2205.05667].
  * [119] T.M. Davis and C.H. Lineweaver, _Expanding confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe_ , _Publ. Astron. Soc. Austral._ 21 (2004) 97 [astro-ph/0310808].
  * [120] G.F.R. Ellis and J.-P. Uzan, _Causal structures in inflation_ , _Comptes Rendus Physique_ 16 (2015) 928 [1612.01084].
  * [121] W. Chao, W.-F. Cui, H.-K. Guo and J. Shu, _Gravitational wave imprint of new symmetry breaking_ , _Chin. Phys. C_ 44 (2020) 123102 [1707.09759].
  * [122] A. Mégevand and A.D. Sánchez, _Supercooling and phase coexistence in cosmological phase transitions_ , _Phys. Rev. D_ 77 (2008) 063519 [0712.1031].
  * [123] L. Darmé, J. Jaeckel and M. Lewicki, _Towards the fate of the oscillating false vacuum_ , _Phys. Rev. D_ 96 (2017) 056001 [1704.06445].
  * [124] M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, _Numerical simulations of acoustically generated gravitational waves at a first order phase transition_ , _Phys. Rev. D_ 92 (2015) 123009 [1504.03291].
  * [125] A.H. Guth and S.-H.H. Tye, _Phase Transitions and Magnetic Monopole Production in the Very Early Universe_ , _Phys. Rev. Lett._ 44 (1980) 631.
  * [126] A.H. Guth and E.J. Weinberg, _Cosmological Consequences of a First Order Phase Transition in the SU(5) Grand Unified Model_ , _Phys. Rev. D_ 23 (1981) 876.
  * [127] W.A. Johnson, _Reaction kinetics in processes of nucleation and growth_ , _Am. Inst. Min. Metal. Petro. Eng._ 135 (1939) 416.
  * [128] M. Avrami, _Kinetics of Phase Change. I General Theory_ , _The Journal of Chemical Physics_ 7 (1939) 1103.
  * [129] M. Avrami, _Kinetics of Phase Change. II Transformation-Time Relations for Random Distribution of Nuclei_ , _The Journal of Chemical Physics_ 8 (1940) 212.
  * [130] M. Avrami, _Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III_ , _The Journal of Chemical Physics_ 9 (1941) 177.
  * [131] A.N. Kolmogorov, _On the statistical theory of metal crystallization_ , _Izv. Akad. Nauk SSSR, Ser. Math_ 1 (1937) 335.
  * [132] P. Athron, C. Balázs, A. Fowlie, L. Morris and L. Wu, _Cosmological Phase Transitions: from perturbative particle physics to gravitational waves_ , 2023.
  * [133] M. Fanfoni and M. Tomellini, _The Johnson-Mehl-Avrami-Kolmogorov model: a brief review_ , _Il Nuovo Cimento D_ 20 (1998) 1171.
  * [134] A.A. Burbelko, E. Fraś and W. Kapturkiewicz, _About Kolmogorov’s statistical theory of phase transformation_ , _Materials Science and Engineering: A_ 413-414 (2005) 429.
  * [135] N.V. Alekseechkin, _Extension of the Kolmogorov–Johnson–Mehl–Avrami theory to growth laws of diffusion type_ , _Journal of Non-Crystalline Solids_ 357 (2011) 3159.
  * [136] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, _New Extended Model of Hadrons_ , _Phys. Rev. D_ 9 (1974) 3471.
  * [137] J. Wu, Q. Li, J. Liu, C. Xue, S. Yang, C. Shao et al., _Progress in Precise Measurements of the Gravitational Constant_ , _Annalen Phys._ 531 (2019) 1900013.
  * [138] K. Enqvist, J. Ignatius, K. Kajantie and K. Rummukainen, _Nucleation and bubble growth in a first order cosmological electroweak phase transition_ , _Phys. Rev. D_ 45 (1992) 3415.
  * [139] C.D. Lorenz and R.M. Ziff, _Precise determination of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algorithm_ , _The Journal of Chemical Physics_ 114 (2001) 3659 [https://doi.org/10.1063/1.1338506].
  * [140] J. Lin and H. Chen, _Continuum percolation of porous media via random packing of overlapping cube-like particles_ , _Theoretical and Applied Mechanics Letters_ 8 (2018) 299.
  * [141] M. Li, H. Chen and J. Lin, _Numerical study for the percolation threshold and transport properties of porous composites comprising non-centrosymmetrical superovoidal pores_ , _Computer Methods in Applied Mechanics and Engineering_ 361 (2020) 112815.
  * [142] V.K.S. Shante and S. Kirkpatrick, _An introduction to percolation theory_ , _Advances in Physics_ 20 (1971) 325 [https://doi.org/10.1080/00018737100101261].
  * [143] A. Hunt, R. Ewing and B. Ghanbarian, _Percolation theory for flow in porous media_ , vol. 880, Springer (2014).
  * [144] M.S. Turner, E.J. Weinberg and L.M. Widrow, _Bubble nucleation in first order inflation and other cosmological phase transitions_ , _Phys. Rev. D_ 46 (1992) 2384.
  * [145] A.F. Heckler, _The Effects of electroweak phase transition dynamics on baryogenesis and primordial nucleosynthesis_ , _Phys. Rev. D_ 51 (1995) 405 [astro-ph/9407064].
  * [146] A. Kosowsky and M.S. Turner, _Gravitational radiation from colliding vacuum bubbles: envelope approximation to many bubble collisions_ , _Phys. Rev. D_ 47 (1993) 4372 [astro-ph/9211004].
  * [147] M. Kamionkowski, A. Kosowsky and M.S. Turner, _Gravitational radiation from first order phase transitions_ , _Phys. Rev. D_ 49 (1994) 2837 [astro-ph/9310044].
  * [148] A. Mégevand and F.A. Membiela, _Stability of cosmological deflagration fronts_ , _Phys. Rev. D_ 89 (2014) 103507 [1311.2453].
  * [149] A. Mégevand, F.A. Membiela and A.D. Sánchez, _Lower bound on the electroweak wall velocity from hydrodynamic instability_ , _JCAP_ 03 (2015) 051 [1412.8064].
  * [150] A. Ashoorioon and T. Konstandin, _Strong electroweak phase transitions without collider traces_ , _JHEP_ 07 (2009) 086 [0904.0353].
  * [151] P. Huang, A.J. Long and L.-T. Wang, _Probing the Electroweak Phase Transition with Higgs Factories and Gravitational Waves_ , _Phys. Rev. D_ 94 (2016) 075008 [1608.06619].
  * [152] K. Hashino, M. Kakizaki, S. Kanemura, P. Ko and T. Matsui, _Gravitational waves and Higgs boson couplings for exploring first order phase transition in the model with a singlet scalar field_ , _Phys. Lett. B_ 766 (2017) 49 [1609.00297].
  * [153] T. Huang, J.M. No, L. Pernié, M.J. Ramsey-Musolf, A.N. Safonov, M. Spannowsky et al., _Resonant di-Higgs boson production in the $b{\bar{b}}WW$ channel: Probing the electroweak phase transition at the LHC_, _Phys. Rev. D_ 96 (2017) 035007 [1701.04442].
  * [154] A. Alves, T. Ghosh, H.-K. Guo and K. Sinha, _Resonant Di-Higgs Production at Gravitational Wave Benchmarks: A Collider Study using Machine Learning_ , _JHEP_ 12 (2018) 070 [1808.08974].
  * [155] O. Gould, J. Kozaczuk, L. Niemi, M.J. Ramsey-Musolf, T.V.I. Tenkanen and D.J. Weir, _Nonperturbative analysis of the gravitational waves from a first-order electroweak phase transition_ , _Phys. Rev. D_ 100 (2019) 115024 [1903.11604].
  * [156] L. Bian, H.-K. Guo, Y. Wu and R. Zhou, _Gravitational wave and collider searches for electroweak symmetry breaking patterns_ , _Phys. Rev. D_ 101 (2020) 035011 [1906.11664].
  * [157] R. Zhou, L. Bian and H.-K. Guo, _Connecting the electroweak sphaleron with gravitational waves_ , _Phys. Rev. D_ 101 (2020) 091903 [1910.00234].
  * [158] A. Alves, D. Gonçalves, T. Ghosh, H.-K. Guo and K. Sinha, _Di-Higgs Blind Spots in Gravitational Wave Signals_ , _Phys. Lett. B_ 818 (2021) 136377 [2007.15654].
  * [159] W. Liu and K.-P. Xie, _Probing electroweak phase transition with multi-TeV muon colliders and gravitational waves_ , _JHEP_ 04 (2021) 015 [2101.10469].
  * [160] J. Ellis, M. Lewicki, M. Merchand, J.M. No and M. Zych, _The Scalar Singlet Extension of the Standard Model: Gravitational Waves versus Baryogenesis_ , 2210.16305.
  * [161] C.-W. Chiang, Y.-T. Li and E. Senaha, _Revisiting electroweak phase transition in the standard model with a real singlet scalar_ , _Phys. Lett. B_ 789 (2019) 154 [1808.01098].
  * [162] P. Athron, C. Balázs, A. Fowlie, L. Morris, G. White and Y. Zhang, _How arbitrary are perturbative calculations of the electroweak phase transition?_ , 2208.01319.
  * [163] L. Niemi, P. Schicho and T.V.I. Tenkanen, _Singlet-assisted electroweak phase transition at two loops_ , _Phys. Rev. D_ 103 (2021) 115035 [2103.07467].
  * [164] G.W. Anderson and L.J. Hall, _The Electroweak phase transition and baryogenesis_ , _Phys. Rev. D_ 45 (1992) 2685.
  * [165] A. Megévand, _First order cosmological phase transitions in the radiation dominated era_ , _Phys. Rev. D_ 69 (2004) 103521 [hep-ph/0312305].
  * [166] S. Profumo, M.J. Ramsey-Musolf and G. Shaughnessy, _Singlet Higgs phenomenology and the electroweak phase transition_ , _JHEP_ 08 (2007) 010 [0705.2425].
  * [167] P. Athron, C. Balázs, A. Fowlie and Y. Zhang, _PhaseTracer: tracing cosmological phases and calculating transition properties_ , _Eur. Phys. J. C_ 80 (2020) 567 [2003.02859].
  * [168] C.L. Wainwright, _CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields_ , _Comput. Phys. Commun._ 183 (2012) 2006 [1109.4189].
  * [169] S.W. Ham, Y.S. Jeong and S.K. Oh, _Electroweak phase transition in an extension of the standard model with a real Higgs singlet_ , _J. Phys. G_ 31 (2005) 857 [hep-ph/0411352].
  * [170] D. O’Connell, M.J. Ramsey-Musolf and M.B. Wise, _Minimal Extension of the Standard Model Scalar Sector_ , _Phys. Rev. D_ 75 (2007) 037701 [hep-ph/0611014].
  * [171] A. Ahriche, _What is the criterion for a strong first order electroweak phase transition in singlet models?_ , _Phys. Rev. D_ 75 (2007) 083522 [hep-ph/0701192].
  * [172] V. Barger, P. Langacker, M. McCaskey, M.J. Ramsey-Musolf and G. Shaughnessy, _LHC Phenomenology of an Extended Standard Model with a Real Scalar Singlet_ , _Phys. Rev. D_ 77 (2008) 035005 [0706.4311].
  * [173] J.R. Espinosa, T. Konstandin and F. Riva, _Strong Electroweak Phase Transitions in the Standard Model with a Singlet_ , _Nucl. Phys. B_ 854 (2012) 592 [1107.5441].
  * [174] J.M. No and M.J. Ramsey-Musolf, _Probing the Higgs Portal at the LHC Through Resonant di-Higgs Production_ , _Phys. Rev. D_ 89 (2014) 095031 [1310.6035].
  * [175] K. Fuyuto and E. Senaha, _Improved sphaleron decoupling condition and the Higgs coupling constants in the real singlet-extended standard model_ , _Phys. Rev. D_ 90 (2014) 015015 [1406.0433].
  * [176] S. Profumo, M.J. Ramsey-Musolf, C.L. Wainwright and P. Winslow, _Singlet-catalyzed electroweak phase transitions and precision Higgs boson studies_ , _Phys. Rev. D_ 91 (2015) 035018 [1407.5342].
  * [177] C.-Y. Chen, S. Dawson and I.M. Lewis, _Exploring resonant di-Higgs boson production in the Higgs singlet model_ , _Phys. Rev. D_ 91 (2015) 035015 [1410.5488].
  * [178] F. Sannino and J. Virkajärvi, _First Order Electroweak Phase Transition from (Non)Conformal Extensions of the Standard Model_ , _Phys. Rev. D_ 92 (2015) 045015 [1505.05872].
  * [179] J. Kozaczuk, _Bubble Expansion and the Viability of Singlet-Driven Electroweak Baryogenesis_ , _JHEP_ 10 (2015) 135 [1506.04741].
  * [180] S. Ghosh, A. Kundu and S. Ray, _Potential of a singlet scalar enhanced Standard Model_ , _Phys. Rev. D_ 93 (2016) 115034 [1512.05786].
  * [181] A.V. Kotwal, M.J. Ramsey-Musolf, J.M. No and P. Winslow, _Singlet-catalyzed electroweak phase transitions in the 100 TeV frontier_ , _Phys. Rev. D_ 94 (2016) 035022 [1605.06123].
  * [182] S. Kanemura, M. Kikuchi and K. Yagyu, _One-loop corrections to the Higgs self-couplings in the singlet extension_ , _Nucl. Phys. B_ 917 (2017) 154 [1608.01582].
  * [183] I.M. Lewis and M. Sullivan, _Benchmarks for Double Higgs Production in the Singlet Extended Standard Model at the LHC_ , _Phys. Rev. D_ 96 (2017) 035037 [1701.08774].
  * [184] C.-Y. Chen, J. Kozaczuk and I.M. Lewis, _Non-resonant Collider Signatures of a Singlet-Driven Electroweak Phase Transition_ , _JHEP_ 08 (2017) 096 [1704.05844].
  * [185] P. Ghorbani, _Vacuum stability vs. positivity in real singlet scalar extension of the standard model_ , _Nucl. Phys. B_ 971 (2021) 115533 [2104.09542].
  * [186] P. Huang and K.-P. Xie, _Primordial black holes from an electroweak phase transition_ , _Phys. Rev. D_ 105 (2022) 115033 [2201.07243].
  * [187] S.R. Coleman and E.J. Weinberg, _Radiative Corrections as the Origin of Spontaneous Symmetry Breaking_ , _Phys. Rev. D_ 7 (1973) 1888.
  * [188] M. Quiros, _Finite temperature field theory and phase transitions_ , in _ICTP Summer School in High-Energy Physics and Cosmology_ , pp. 187–259, 1, 1999 [hep-ph/9901312].
  * [189] R.R. Parwani, _Resummation in a hot scalar field theory_ , _Phys. Rev. D_ 45 (1992) 4695 [hep-ph/9204216].
  * [190] R.N. Lerner and J. McDonald, _Gauge singlet scalar as inflaton and thermal relic dark matter_ , _Phys. Rev. D_ 80 (2009) 123507 [0909.0520].
  * [191] ATLAS, CMS collaboration, _Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at $\sqrt{s}=7$ and 8 TeV_, _JHEP_ 08 (2016) 045 [1606.02266].
  * [192] M. Carena, Z. Liu and M. Riembau, _Probing the electroweak phase transition via enhanced di-Higgs boson production_ , _Phys. Rev. D_ 97 (2018) 095032 [1801.00794].
  * [193] CMS collaboration, _Combined Higgs boson production and decay measurements with up to 137 fb -1 of proton-proton collision data at $\sqrt{s}$ = 13 TeV_, Tech. Rep. CERN, Geneva (2020).
  * [194] ATLAS collaboration, _A detailed map of Higgs boson interactions by the ATLAS experiment ten years after the discovery_ , _Nature_ 607 (2022) 52 [2207.00092].
  * [195] D. Croon and G. White, _Exotic Gravitational Wave Signatures from Simultaneous Phase Transitions_ , _JHEP_ 05 (2018) 210 [1803.05438].
  * [196] A.P. Morais, R. Pasechnik and T. Vieu, _Multi-peaked signatures of primordial gravitational waves from multi-step electroweak phase transition_ , _PoS_ EPS-HEP2019 (2020) 054 [1802.10109].
  * [197] A.P. Morais and R. Pasechnik, _Probing multi-step electroweak phase transition with multi-peaked primordial gravitational waves spectra_ , _JCAP_ 04 (2020) 036 [1910.00717].
  * [198] A. Angelescu and P. Huang, _Multistep Strongly First Order Phase Transitions from New Fermions at the TeV Scale_ , _Phys. Rev. D_ 99 (2019) 055023 [1812.08293].
  * [199] Z. Zhao, Y. Di, L. Bian and R.-G. Cai, _Probing the electroweak symmetry breaking history with Gravitational waves_ , 2204.04427.
  * [200] D.J. Weir, _Gravitational waves from a first order electroweak phase transition: a brief review_ , _Phil. Trans. Roy. Soc. Lond. A_ 376 (2018) 20170126 [1705.01783].
  * [201] A. Masoumi, K.D. Olum and B. Shlaer, _Efficient numerical solution to vacuum decay with many fields_ , _JCAP_ 01 (2017) 051 [1610.06594].
  * [202] P. Athron, C. Balázs, M. Bardsley, A. Fowlie, D. Harries and G. White, _BubbleProfiler: finding the field profile and action for cosmological phase transitions_ , _Comput. Phys. Commun._ 244 (2019) 448 [1901.03714].
  * [203] R. Sato, _SimpleBounce : a simple package for the false vacuum decay_ , _Comput. Phys. Commun._ 258 (2021) 107566 [1908.10868].
  * [204] V. Guada, M. Nemevšek and M. Pintar, _FindBounce: Package for multi-field bounce actions_ , _Comput. Phys. Commun._ 256 (2020) 107480 [2002.00881].
  * [205] J.M. No, _Aspects of Phenomenology and Cosmology in Hidden sector extensions of the Standard Model_ , Ph.D. thesis, Universidad Autónoma de Madrid, 06, 2009.
  * [206] S.J. Huber and T. Konstandin, _Production of gravitational waves in the nMSSM_ , _JCAP_ 05 (2008) 017 [0709.2091].
  * [207] A.H. Guth and E.J. Weinberg, _Could the Universe Have Recovered from a Slow First Order Phase Transition?_ , _Nucl. Phys. B_ 212 (1983) 321.
  * [208] S.W. Hawking, I.G. Moss and J.M. Stewart, _Bubble Collisions in the Very Early Universe_ , _Phys. Rev. D_ 26 (1982) 2681.
  * [209] Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * [210] J. Elias-Miró, J.R. Espinosa and T. Konstandin, _Taming Infrared Divergences in the Effective Potential_ , _JHEP_ 08 (2014) 034 [1406.2652].
  * [211] S.P. Martin, _Taming the Goldstone contributions to the effective potential_ , _Phys. Rev. D_ 90 (2014) 016013 [1406.2355].
  * [212] J.R. Espinosa, T. Konstandin, J.M. No and G. Servant, _Energy Budget of Cosmological First-order Phase Transitions_ , _JCAP_ 06 (2010) 028 [1004.4187].
  * [213] F.F. Freitas, G. Lourenço, A.P. Morais, A. Nunes, J.a. Olívia, R. Pasechnik et al., _Impact of SM parameters and of the vacua of the Higgs potential in gravitational waves detection_ , 2108.12810.
  * [214] A. Azatov, G. Barni, S. Chakraborty, M. Vanvlasselaer and W. Yin, _Ultra-relativistic bubbles from the simplest Higgs portal and their cosmological consequences_ , _JHEP_ 10 (2022) 017 [2207.02230].


m}\|\nabla_{\text{b}}^{k}(v^{\prime}w^{\prime\prime})\nabla_{\text{b}}^{m-k}(m_{p}\varphi))\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle C_{m}\sum_{0\leq k\leq
m}\|\nabla_{\text{b}}^{k}(v^{\prime}w^{\prime\prime})\|_{L^{2,2}(S^{1})}\|\nabla_{\text{b}}^{m-k}(m_{p}\varphi)\|_{H^{2}_{\text{b}}}$
$\displaystyle\leq$ $\displaystyle C_{m}\sum_{0\leq k\leq
m}\|v^{\prime}w^{\prime\prime}\|^{1-\tfrac{k}{m}}_{L^{2,2}(S^{1})}\
\|v^{\prime}w^{\prime\prime}\|^{\tfrac{k}{m}}_{L^{m+2,2}(S^{1})}\
\|m_{p}\varphi\|^{1-\tfrac{k}{m}}_{H^{2}_{\text{b}}}\
\|m_{p}\varphi\|^{\tfrac{k}{m}}_{H^{m+2}_{\text{b}}}$

where we have used the Sobolev embedding $C^{0}\hookrightarrow H^{2}(S^{1})$
and then the interpolation inequalities with $m_{2}=m+2,m_{1}=2$. By Young’s
inequality with exponents $p=\tfrac{m}{k}$ and $q=\tfrac{m}{m-k}$, one finds
the above is bounded by

$\displaystyle\leq$ $\displaystyle C_{m}\left(\
\|v^{\prime}w^{\prime\prime}\|_{L^{m+2,2}(S^{1})}\|m_{p}\varphi\|_{L^{2,2}(S^{1})}\
+\
\|v^{\prime}w^{\prime\prime}\|_{L^{2,2}(S^{1})}\|m_{p}\varphi\|_{H^{m+2}_{\text{b}}}\right)$
$\displaystyle\leq$ $\displaystyle C_{m}\Big{(}\
\|v^{\prime}\|_{L^{m+4,2}(S^{1})}\|w^{\prime\prime}\|_{L^{4,2}(S^{1})}\ +\
\|v^{\prime}\|_{L^{4,2}(S^{1})}\|w^{\prime\prime}\|_{L^{m+4,2}(S^{1})}\ $
$\displaystyle+\
\|v^{\prime}\|_{L^{4,2}(S^{1})}\|w^{\prime\prime}\|_{L^{4,2}(S^{1})}\left(\|m_{p}\|_{H^{m+4}_{\text{b}}}\|\varphi\|_{H^{4}_{\text{b}}}+\|m_{p}\|_{H^{4}_{\text{b}}}\|\varphi\|_{H^{m+4}_{\text{b}}}\right)\Big{)}$
$\displaystyle\leq$ $\displaystyle C_{m}\Big{(}\
\|v\|_{L^{m+5,2}}\|w\|_{L^{6,2}}\ +\ \|v\|_{L^{5,2}}\|w\|_{L^{m+6,2}}\ +\
\|v\|_{L^{5,2}}\|w^{\prime\prime}\|_{L^{6,2}}\ \cdot\
\|(p,\eta,\varphi)\|_{m+6}\Big{)}\ $

where we have repeated the interpolation and Young’s steps from above with
$m_{2}=m+4$ and $m_{1}=4$ on both products, and then used the fact that $6\leq
m_{0}$ so that
$\|m_{p}\|_{H^{4}_{\text{b}}}+\|\varphi\|_{H^{4}_{\text{b}}}\leq C$. This
shows the desired estimate for $r,r^{\prime}=6$. The same steps apply to the
$r^{\nu}H^{m-1}_{\text{b}}$ term in the norm using the commutation relations
from Lemma 8.15. The other terms are similar, with the constant term in
$(1+\|(p,\eta,\varphi)\|_{m+r^{\prime}})$ on the right hand side arising from
the terms not involving $(p,\varphi,\eta)$ such as $\mathcal{B}_{\psi}(v)$. ∎

###  8.6 Proofs of Theorem 1.4 and Corollary 1.5

In this subsection, we invoke the Nash-Moser Implicit Function Theorem 7.4 to
conclude the proofs of Theorem 1.4 and Corollary 1.5, beginning with the
latter.

###### Proof of Corollary 1.5.

Lemmas 8.17, 8.19, and 8.20 verify hypotheses (I), (II), and (III) from
Section Hypothesis (I) respectively on
$V_{0}\times(U_{0}\cap\mathbb{P}_{\mathcal{X}})$. The formula (8.11) and Lemma
8.8 (which extends easily to the spaces $\mathcal{H}^{m,0}$) show that
$f_{p}\in\mathbb{P}_{\mathcal{Y}}$ and $\|f_{p}\|_{m}\leq C\|p\|_{m+s}$.

It remains to show that the property (P) of being polyhomogeneous is
propagated by the iteration in the sense of Definition 7.3. Lemma 8.12 and its
proof in Appendix B show that the smoothing operators
$S_{\varepsilon},S_{\varepsilon}^{\text{b}}$ preserve polyhomogeneity. It is
evident from the definition of the Dirac operator $\not{D}_{p}$ that for any
metric (including pullback metrics $p_{\eta}$)

$\displaystyle\varphi\in\mathbb{P}_{\mathcal{X}}\ \ \Rightarrow\ \
\overline{\not{\mathbb{D}}}_{p}(\eta,\varphi)\in\mathbb{P}_{\mathcal{Y}}$

preserves polyhomogeneity. To show polyhomogeneity (P) is propagated, we
therefore verify that

$\displaystyle\phantom{\varphi\in\mathbb{P}_{\mathcal{X}}}g\in\mathbb{P}_{\mathcal{Y}}\
\ \Rightarrow\ \
(\text{d}_{(\eta,\varphi)}\overline{\not{\mathbb{D}}}_{p})^{-1}g\in\mathbb{P}_{\mathcal{X}}.$

Let $u=(\text{d}_{(\eta,\varphi)}\overline{\not{\mathbb{D}}}_{p})^{-1}g$ be
the solution, and suppose that $g$ is polyhomogeneous with index set
$\mathbb{Z}^{+}-\tfrac{1}{2}$. By the block diagonal decomposition (8.32) from
Lemma 8.17, one has $\overline{T}_{p_{\eta}}(v)=\Pi_{p_{\eta}}(g)$ in the case
that the upper right entry $*$ vanishes. In this case, since $g\in Y$ is
smooth, then $\Pi_{0}(g)\in\text{\bf Ob}\cap H^{m}_{\text{b}}$ for every
$m\geq 0$ and admits a weak expansion as in Remark 3.10. Lemma 8.14 applies
identically in the case of index set $\mathbb{Z}^{+}-\tfrac{1}{2}$ to show
that the coefficients are smooth, hence $\Pi_{0}(g)$ is polyhomogeneous with
index set $\mathbb{Z}^{+}-\tfrac{1}{2}$. Since the latter is a vector space,
the right-hand side of the second equation

$\not{D}_{p_{\eta}}\phi=(1-\Pi_{0})(g)$

is polyhomogeneous with index set $\mathbb{Z}^{+}-\tfrac{1}{2}$. Since the
coefficients of $\not{D}_{p_{\eta}}$ are smooth, this left-hand side must also
be polyhomogeneous with index set $\mathbb{Z}^{+}+\tfrac{1}{2}$ (see [45],
Proposition 7.17) with the caveat of possibly having logarithm terms appear
with the $r^{1/2}$ coefficient. The case where $*$ is non-zero is the same as
$\Phi_{p_{\eta}}$ is polyhomogeneous.

The final point is to rule out the appearance of logarithm terms
$r^{1/2}\log(r)$. This is a consequence of the restrictions on the
$\theta_{p}$-Fourier modes in Fermi coordinates that appear with the $r^{1/2}$
coefficient and follows from formally solving the first term at each stage.
Logarithm terms $e^{ik\theta}r^{1/2}\log(r)$ would arise from the right-hand
side having terms $r^{-1/2}e^{\pm 3i\theta/2}$, but since the right-hand side
is always a combination of $\Pi_{0}(g)$ which has only modes $r^{-1/2}e^{\pm
i\theta/2}$ and the derivatives $\nabla_{z},\nabla_{\overline{z}}$ of a
polyhomogeneous spinor from the previous stage such terms never arise. This
property is preserved by the smoothing operators $S_{\varepsilon}^{\text{b}}$
by construction (see Appendix B). We conclude that the property (P) of being
polyhomogeneous in the sense of having expansions of the form (8.1)-(8.2) is
propagated.

By the Nash-Moser Implicit Function Theorem 7.4, there is an open neighborhood
$U\subset\mathcal{P}$ of smooth parameters such that for $p\in U$ there exists
a unique solution $(\mathcal{Z}_{p},\Phi_{p},\Lambda_{p})$ to the equation

${\not{\mathbb{D}}_{p}(\mathcal{Z}_{p},\Phi_{p})}=\Lambda_{p}\Phi_{p}$ (8.43)

and the triples $(\mathcal{Z}_{p},\Phi_{p},\Lambda_{p})$ define a smooth tame
graph of $U\subset U\times\mathcal{X}$. This completes the proof of Corollary
1.5 in the presence of Assumptions 4* and 5*. To eliminate the first is
trivial as the deformations of $\mathcal{Z}_{0}$ are local along each
component. In the absence of Assumption 5*, the standard Kuranishi picture
(see, e.g. Section 3.3 of [7]) applies to show that the set of parameters for
which (8.43) holds is described by the zero set of a smooth tame map

$\kappa_{p}:U\times\mathbb{R}^{n}\to\mathbb{R}^{n}$

where $n=\dim(\ker(T_{\Phi_{0}}))$ is the dimension of the kernel of the index
0 map from Section 6.3. ∎

###### Proof of Theorem 1.4.

The projection $\pi(\mathscr{M}_{\mathbb{Z}_{2}})\subseteq U\cap\mathcal{P}$
of the universal moduli space of $\mathbb{Z}_{2}$-harmonic spinors to the
parameter space is defined by the zero-set

$\pi(\mathscr{M}_{\mathbb{Z}_{2}})=\Lambda^{-1}(0)\cap U$

of the eigenvalue $\Lambda:U\to\mathbb{R}$ of Corollary 1.5, and there is
locally a unique $\mathbb{Z}_{2}$-harmonic spinor $(\mathcal{Z}_{p},\Phi_{p})$
up to normalization and sign for each $p\in\Lambda^{-1}(0)$, hence the
projection $\pi$ is a local homeomorphism.

In the presence of Assumption 5*, the map $\Lambda:U\to\mathbb{R}$ is
transverse to 0. To see this, let $p(s)$ be a path of parameters which we will
choose momentarily. By Corollary 1.5, such a path implicitly defines triples
$(\mathcal{Z}_{s},\Phi_{s},\Lambda_{s})$ for $s$ sufficiently small.
Differentiating (8.43) at $s=0$ yields the relation that

$\not{D}_{\dot{\mathcal{Z}}}\Phi_{0}+\not{D}_{\dot{p}}\Phi_{0}+\not{D}_{0}\dot{\Phi}=\dot{\Lambda}\Phi_{0}$
(8.44)

where $\cdot$ denotes the $s$-derivative. We now choose $p(s)=(g(s),B(s))$ so
that the derivative $(\dot{g},\dot{B})$ has the following properties. Let
$\dot{B}$ be a smooth perturbation supported on a neighborhood disjoint from
$N_{r_{0}}(\mathcal{Z}_{0})$ such that
$\langle\gamma(\dot{B})\Phi_{0},\Phi_{0}\rangle\neq 0$. Given this, we define
$\dot{g}$ in terms of $\dot{B}$ as follows. By Assumption 4*, we know that
$T_{\Phi_{0}}:L^{2,2}(\mathcal{Z}_{0};N\mathcal{Z}_{0})\to\text{ker}(\not{D}_{0}|_{L^{2}})$
is injective with closed range and 1-dimensional cokernel. Let
$\Phi_{1}\subseteq\ker(\not{D}|_{L^{2}})$ denote the orthogonal complement of
its range. Since $\overline{T}_{\Phi_{0}}$ is an isomorphism, we must have
have $\langle\Phi_{1},\Phi_{0}\rangle\neq 0$. Decompose
$\Pi_{0}(\gamma(\dot{B})\Phi_{0})=(c\Phi_{1},\xi)$, and set
$\dot{g}=\dot{g}_{\eta}$ where $T_{\Phi_{0}}(\eta)=-\xi$. Taking the inner
product of (8.45) with $\Phi_{1}$ then yields

$\cancel{\langle\not{D}_{\dot{\mathcal{Z}}}\Phi_{0},\Phi_{1}\rangle}+\langle-c\Phi_{1},\Phi_{1}\rangle+\cancel{\langle\not{D}_{0}\dot{\Phi},\Phi_{1}\rangle}=\dot{\Lambda}\langle\Phi_{0},\Phi_{1}\rangle$
(8.45)

wherein the first term vanishes by definition of $\Phi_{1}$ and the third via
integration by parts. We conclude that $\dot{\Lambda}\neq 0$ and that
$\Lambda:V_{0}\to\mathbb{R}$ is transverse to 0.

When Assumption 5* fails, the Kuranishi picture applies again to show that the
$\pi(\mathscr{M}_{\mathbb{Z}_{2}})$ is given locally by the zero-set of a
smooth tame map

$(\kappa_{p},\Lambda):U\times\mathbb{R}^{n}\to\mathbb{R}^{n}\times\mathbb{R}$

where $n=\dim(\ker(T_{\Phi_{0}}))$ is the dimension of the kernel of the index
0 map from Section 6.3. This concludes the proof of Theorem 1.4. ∎

##  Appendix A Appendix I: Exponential Decay

The goal of this appendix is to prove Lemma 4.7. To recall the notation,
$\not{D}_{0}$ here denotes the Dirac operator on
$(S^{1}\times\mathbb{R}^{2},dt^{2}+dx^{2}+dy^{2})$. Given a solution

$\not{D}_{0}\not{D}_{0}u_{\ell}=f_{\ell}$ (A.1)

of the second order equation where $f_{\ell}$ satisfies the assumptions of
4.7, we prove that the solution $u_{\ell}$ satisfies a discretized exponential
decay as in the conclusion of the lemma. The proof relies on a discretized
version of the maximum principle. Similar arguments may be found in [31] (see
Appendix A.2.1).

###### Proof.

The arguments of Proposition 2.4 and 2.7 are equally valid in the case of the
non-compact $Y_{0}$ (provided the scalar curvature $s$ and perturbation $B$
are compactly supported) since the boundary term at infinity vanishes when
integrating by parts. In fact, in this case $s=B=0$ hence the Weitzenböck
formula implies that $rH^{1}_{e}$-kernel is empty. It follows that

$\not{D}_{0}\not{D}_{0}:rH^{1}_{e}\longrightarrow rH^{-1}_{e}$

is an isomorphism and that (A.1) admits a unique solution. Since $\not{D}_{0}$
preserves Fourier modes, $u_{\ell}$ has Fourier modes in the range 4.10 as
well.

Denote by $\chi_{n}$ a cutoff-function equal to $1$ on $A_{n_{\ell}}$ such
that

$\text{supp}(d\chi_{n})\subseteq A_{(n-1)\ell}\cup A_{(n+1)\ell}\hskip
56.9055pt|d\chi_{n}|\leq\frac{c|\ell|}{R_{0}}.$

Taking the inner product of A.1 with $\chi_{n}^{2}u_{\ell}$, integrating by
parts, and using Young’s inequality yields the following where
$B_{n\ell}=A_{n-1,\ell}\cup A_{n,\ell}\cup A_{n+1,\ell}$ as before:

$\displaystyle\int_{B_{n\ell}}\chi_{n}^{2}|\not{D}_{0}{u_{\ell}}|^{2}\ dV_{0}$
$\displaystyle=$ $\displaystyle-\int_{B_{n\ell}}\langle
2\chi_{n}d\chi_{n}.u_{\ell},\not{D}_{0}u_{\ell}\rangle\
dV_{0}+\int_{B_{n\ell}}\langle\chi_{n}^{2}u_{\ell},-g_{\ell}\rangle\ dV_{0}$
(A.2) $\displaystyle\leq$ $\displaystyle
2c^{2}\frac{|\ell|^{2}}{R_{0}^{2}}\|u_{\ell}\|_{L^{2}(B_{n\ell})}+\frac{1}{2}\|\chi_{n}\not{D}_{0}{u_{\ell}}\|^{2}_{L^{2}(B_{n\ell})}$
(A.4)
$\displaystyle+\frac{1}{2c_{1}}\|g_{\ell}\|^{2}_{rH^{-1}_{e}(B_{n\ell})}+\frac{c_{1}}{2}\|\chi_{n}^{2}u_{\ell}\|^{2}_{rH^{1}_{e}(B_{n\ell})}$

where the inner product is that defined using the product metric
$g^{\text{Euc}}$ and likewise for the volume form $dV_{0}$.

The $\|\not{D}_{0}u_{\ell}\|^{2}$ and $\|\chi_{n}^{2}u_{\ell}\|^{2}$ terms on
the right side can be absorbed into the left once $c_{1}$ is chosen
sufficiently small. The first is clear from the factor of $1/2$, and for the
second apply the elliptic estimate
$\not{D}_{0}:rH^{1}_{e}\to\text{Range}(\not{D}_{0})$ to $\chi_{n}^{2}u_{\ell}$
to obtain

$\|\chi_{n}^{2}u_{\ell}\|^{2}_{rH^{1}_{e}(B_{n\ell})}\leq
C\|\not{D}_{0}(\chi_{n}^{2}u_{\ell})\|_{L^{2}(Y_{0})}^{2}\ \leq C\left(\
\|\chi_{n}^{2}\not{D}_{0}u_{\ell}\|_{L^{2}(B_{n\ell})}^{2}+\|d\chi_{n}.u_{\ell}\|^{2}_{L^{2}(B_{n\ell})}\right).$
(A.5)

By choosing $c_{1}$ sufficiently small (and using that
$|\chi_{n}^{2}|\leq|\chi_{n}|$), the $\|\chi_{n}^{2}\not{D}_{0}u_{\ell}\|^{2}$
term can be absorbed on the left hand side of A.2, and the
$\|d\chi_{n}.u_{\ell}\|^{2}$ term can be absorbed into A.4 by increasing
$c^{2}$.

Substituting A.5 again and increasing the constants yields

$\displaystyle\|u_{\ell}\|^{2}_{rH^{1}_{e}(A_{n\ell})}\leq\|\chi_{n}^{2}u_{\ell}\|^{2}_{rH^{1}_{e}(A_{n\ell})}$
$\displaystyle\leq$ $\displaystyle
C_{1}\frac{|\ell|^{2}}{R_{0}^{2}}\|u_{\ell}\|_{L^{2}(B_{n\ell})}+\frac{1}{2c_{1}}\|g_{\ell}\|_{rH^{-1}_{e}(B_{n\ell})}$

which shows, invoking the assumption on $g_{\ell}$, that

$\displaystyle\|u_{\ell}\|^{2}_{rH^{1}_{e}(A_{n\ell})}$ $\displaystyle\leq$
$\displaystyle
C_{1}\frac{|\ell|^{2}}{R_{0}^{2}}\|u_{\ell}\|_{rH^{1}_{e}(B_{n\ell})}+\frac{C^{\prime}_{m}}{|\ell|^{2+2m}}e^{-2n/c_{m}}.$
(A.6)

Then, because of the restriction of Fourier modes on $\eta_{\ell}$,

$C_{1}\frac{|\ell|^{2}}{R_{0}^{2}}\int_{B_{n\ell}}|u_{\ell}|^{2}\
dV\leq\frac{4C_{1}}{R_{0}^{2}}\left(\int_{A_{(n-1)\ell}}|\nabla u_{\ell}|^{2}\
dV+\int_{A_{n\ell}}|\nabla u_{\ell}|^{2}\ dV+\int_{A_{(n+1)\ell}}|\nabla
u_{\ell}|^{2}\ dV\right).$

Now set $\mathfrak{a}_{n}=\|u_{\ell}\|^{2}_{L^{1,2}(A_{n\ell})}$, and choose
$R_{0}$ so that $\tfrac{4C_{1}}{R_{0}}^{2}<\tfrac{1}{200}$. Equation A.6
implies the discrete differential inequality

$\mathfrak{a}_{n}-\tfrac{1}{100}(\mathfrak{a}_{n-1}+\mathfrak{a}_{n+1})\leq\frac{C^{\prime}_{m}}{|\ell|^{2+2m}}e^{-2n/c_{m}}.$

To conclude, we apply a discrete version of the maximum principle: let
$\mathfrak{s}_{n}=\tfrac{2C_{m}}{|\ell|^{2+2m}}e^{-2n/c_{m}}$. Possibly by
increasing $c_{m}$ to $c_{m}^{\prime}$, this rather trivially satisfies

$\mathfrak{s}_{n}-\tfrac{1}{100}(\mathfrak{s}_{n-1}+\mathfrak{s}_{n+1})\leq\frac{C^{\prime}_{m}}{|\ell|^{2+2m}}e^{-2n/c_{m}^{\prime}}.$

Hence the difference $\mathfrak{r}_{n}=\mathfrak{a}_{n}-\mathfrak{s}_{n}$
satisfies

$\mathfrak{r}_{n}-\tfrac{1}{100}(\mathfrak{r}_{n-1}+\mathfrak{r}_{n+1})\leq 0$
(A.7)

and, possibly increasing $C^{\prime}_{m}$, $\mathfrak{r}_{0}\leq 0$ and
$r_{n}\to 0$ as $n\to\infty$ (the latter requirement is simply from
integrability). The “maximum principle” then implies $\mathfrak{r}_{n}\leq 0$
for all $n$, since an interior maximum with
$\mathfrak{r}_{n}\geq\mathfrak{r}_{n-1},\mathfrak{r}_{n+1}$ would violate A.7.
We conclude that $u_{\ell}$ satisfies

$\|u_{\ell}\|^{2}_{L^{1,2}(A_{n\ell})}\leq\frac{C^{\prime}_{m}}{|\ell|^{2+2m}}\text{Exp}\left(-\frac{2n}{c_{m}^{\prime}}\right)$
(A.8)

which completes the lemma. ∎

##  Appendix B Appendix II: Boundary and Edge Regularity

This appendix gives proofs of two facts about regularity in the boundary and
edge Sobolev spaces, namely Lemma 8.12 and Lemma 8.14.

Recall the Fréchet spaces defined above in Lemma 8.12. To restate the
assertion of Lemma 8.12 succinctly:

######  Lemma B.1.

For $0<\varepsilon\leq 1$ there exist smoothing operators

$S_{\varepsilon}:C^{\infty}(\mathcal{Z}_{0};N\mathcal{Z}_{0})\to
C^{\infty}(\mathcal{Z}_{0};N\mathcal{Z}_{0})\hskip
56.9055ptS_{\varepsilon}^{\text{b}}:\bigcap_{m\geq
0}X_{m}^{\prime\prime}\to\bigcap_{m\geq 0}X_{m}^{\prime\prime}\hskip
56.9055ptS_{\varepsilon}^{\text{b}}:\mathcal{Y}\to\mathcal{Y}$

satisfying properties (i)-(iii) of Definition 7.1 and preserving the property
(P) of polyhomogeneity as in defined by (8.1)–(8.2). Additionally, in Fermi
coordinates around $\mathcal{Z}_{0}$, $S_{\varepsilon}^{\text{b}}$ does not
introduce new Fourier modes in $\theta$.

###### Proof.

On $\mathcal{X}_{0}=C^{\infty}(\mathcal{Z}_{0};N\mathcal{Z}_{0})$ the operator
$S_{\varepsilon}$ may be defined straightforwardly as the truncation of the
Fourier series at $|\ell|\sim\tfrac{1}{\varepsilon}$ in a global
trivialization. It is instructive for what follows, however, to construct
$S_{\varepsilon}$ using Schwartz kernels.

Alternatively, $S_{\varepsilon}$ may be defined as a convolution operator
using a Schwartz kernel that smoothly approximates the $\delta$-distribution
along the diagonal in
$(\mathcal{Z}_{0})^{2}=\mathcal{Z}_{0}\times\mathcal{Z}_{0}$. More precisely,
let $\chi(r)$ be a cut-off function equal to 1 near $r=0$ and vanishing for
$r>1$. Fix a collection $U_{j}\times\mathbb{C}$ for $j=1,..,n$ of
trivializations of $N\mathcal{Z}_{0}$ on contractible open sets, and for each
$j$, choose nested cut-off functions $\xi_{j},\beta_{j}$ such that
$\text{supp}(\beta_{j})\Subset\\{\xi_{j}=1\\}$. Then define

$S_{\varepsilon}(\eta)(t):=\frac{1}{\varepsilon}\sum_{j=1}^{n}\xi_{j}(t)\int_{\mathcal{Z}_{0}}\chi\left(\tfrac{|t-t^{\prime}|}{\varepsilon}\right)\beta_{j}(t^{\prime})\eta(t^{\prime})dt^{\prime}.$
(B.1)

where the constant $\tfrac{1}{\varepsilon}$ serves to normalize $\chi$ in
$L^{2}$. Properties (i)-(iii) now follow easily.

The construction of $S_{\varepsilon}^{\text{b}}$ is analogous, but now we de-
singularize the $\delta$-distribution on the diagonal in the blown-up product
defined as follows. Let $B=\mathcal{Z}_{0}\times\mathcal{Z}_{0}\subset Y\times
Y$, and let $D(B)$ denote a disk bundle of finite radius in the normal bundle.
Define

$Y^{2}_{\text{b}}:=(Y-(N_{r_{0}}\mathcal{Z}_{0}))^{2}\cup D(B).$

This blow-up is a compact 6-manifold with corners, having three boundary
strata of codimension 1 consisting of the interiors of
$\partial(N_{r_{0}}\mathcal{Z}_{0}\times Y),\partial(Y\times
N_{r_{0}}\mathcal{Z}_{0}),\partial(D(B))$ which intersect along codimension 2
corners. This space can be given local coordinates
$(s,\rho,\theta,\theta^{\prime},t,t^{\prime})$ in a neighborhood of the
diagonal, where $s=[r,r^{\prime}]$ is a projective coordinate along the blow-
up boundary, and $\rho=r^{\prime}$.

Away from these strata, $S_{\varepsilon}^{\text{b}}$ can be defined
analogously to (B.1); near the boundary strata it is defined as a product

$S_{\varepsilon}^{\text{b}}:=S_{\varepsilon}^{\theta}\circ
S^{\prime}_{\varepsilon}$

where $S_{\varepsilon}^{\theta}$ is defined by truncation of the
$\theta$-Fourier modes in a local trivialization, and
$S_{\varepsilon}^{\prime}$ is given in Fermi coordinates
$(s,\rho,t,t^{\prime})$ by

$S^{\prime}_{\varepsilon}(\psi)(r,t,\theta):=\frac{1}{\varepsilon^{2}}\chi\int_{Y-\mathcal{Z}_{0}}\chi\left(\tfrac{|s-1|}{\varepsilon}\right)\chi\left(\tfrac{|t-t^{\prime}|}{\varepsilon}\right)\frac{1}{r^{\prime}}(\beta\psi)dt^{\prime}dr^{\prime}$
(B.2)

where the factor of $1/r^{\prime}$ appears because $|r-r^{\prime}|\sim
r^{\prime}s$ and the $\delta$-distribution is homogeneous of order $-1$.

The properties (i)-(iii) for the spaces $H^{m}_{\text{b}}$ follow analogously
to the compact case. That $S_{\varepsilon}^{\text{b}}$ introduces no new
Fourier modes in $\theta$ is manifest from the definition, and the fact that
polyhomogeneity is preserved is a consequence of the pushforward theorem or of
direct inspection of the integral (B.2) (see [17] Section 3.1). Since the
ratio $r/r^{\prime}$ is uniformly bounded where $\chi\neq 0$, the commutators
$[\nabla^{e},S_{\varepsilon}^{\text{b}}]$ and
$[r^{\alpha},S_{\varepsilon}^{\text{b}}]$ are uniformly bounded, properties
(i)-(iii) for the space $\bigcap_{m\geq 0}H^{m,1}_{\text{b},e}$ follows from
the equivalent description of the norm (2.13). The same applies for the terms
$(r\partial_{r}\pm\tfrac{1}{2})\psi$ and therefore for the spaces
$r\mathcal{H}^{m,1}$ and $\mathcal{H}^{m,0}$.

∎

What remains is to prove Lemma 8.14, which requires several steps.

######  Lemma B.2.

If $\varphi\in r^{\alpha}H^{3}_{\text{b}}$ for $\alpha>1$ then the $\varphi$
satisfies the pointwise bound

$|\varphi(x)|\leq C\|\varphi\|_{rH^{3}_{\text{b}}}.$ (B.3)

###### Proof.

We first prove the lemma in the 1-dimensional case.

Consider $\mathbb{R}^{+}=(0,\infty)$ with the measure $rdr$ and suppose that
$\varphi\in rH^{1}_{b}(rdr)$ with $\text{supp}(\varphi)\subseteq(0,1]$. Then
we claim that there is a constant $C$ so that

$|\varphi(x)|\leq
C\|\varphi\|_{rH^{1}_{\text{b}}(rdr)}=C\left(\int_{\mathbb{R}^{+}}\frac{|\varphi|^{2}}{r^{2}}+|\nabla\varphi|^{2}\
rdr\right)^{1/2}.$ (B.4)

This follows from a dyadic decomposition. Since $r$ is uniformly bounded on
$[1/2,2]$ and $L^{1,2}[1/2,2]\hookrightarrow C^{0}[1/2,2]$ by the standard
Sobolev embedding, we have $|\varphi(1)|^{2}\leq
c\int|\varphi|^{2}+|\nabla\varphi|^{2}dr\leq
c\|\varphi\|_{rH^{1}_{\text{b}}}^{2}$. Then, by the Fundamental Theorem of
Calculus,

$\displaystyle|\varphi(1/2)|$ $\displaystyle\leq$
$\displaystyle|\varphi(1)|+\int_{1/2}^{1}|\varphi^{\prime}(\rho)|d\rho\leq|\varphi(1)|+\left(\int_{1/2}^{1}\rho|\varphi^{\prime}(\rho)|^{2}d\rho\right)^{1/2}\left(\int_{1/2}^{2}\frac{1}{\rho}d\rho\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle|\varphi(1)|+(\log
2)^{1/2}\|\varphi\|_{rH^{1}_{\text{b}}([1/2,1],rdr)}$

Similarly, $|\varphi(1/4)|\leq|\varphi(1/2)|+(\log
2)^{1/2}\|\varphi\|_{rH^{1}_{\text{b}}([1/4,1/2],rdr)}\leq|\varphi(1)|+(\log
2)^{1/2}\|\varphi\|_{rH^{1}_{\text{b}}([1/4,1],rdr)}$ where the second
inequality follows from substituting the above. In general, using the estimate
on $|\varphi(1)|$ we conclude that

$|\varphi{(2^{-k})}|\leq C\|\varphi\|_{rH^{1}_{\text{b}}(rdr)}.$

(B.4) then follows from applying the Fundamental Theorem of calculus again for
$x\in[2^{-k},2^{-k+1}]$.

In general, for $\varphi\in H^{m}_{\text{b}}(Y-\mathcal{Z}_{0};S_{0})$, the
lemma follows from the above by applying (B.4) to rays of constant
$(t,\theta)$ and after using the Sobolev restriction theorem (which increases
the number of derivatives needed).

∎

Next, we have the following fundamental fact about ODEs. For it, we use the
1-dimensional b-spaces $r^{\alpha}L^{1,2}_{\text{b}}([0,1],rdr)$ and
$r^{\alpha}L^{2}([0,1],rdr)$ defined by the norms

$\|u\|_{r^{\alpha}L^{1,2}_{\text{b}}}=\left(\int_{0}^{1}(|r\partial_{r}u|^{2}+|u|^{2})r^{-2\alpha}rdr\right)^{1/2}\hskip
56.9055pt\|u\|_{r^{\alpha}L^{2}_{\text{b}}}=\left(\int_{0}^{1}|u|^{2}r^{-2\alpha}rdr\right)^{1/2}.$

######  Lemma B.3.

Provided $\alpha>3/2$ then

$(r\partial_{r}-\tfrac{1}{2}):r^{\alpha}L^{1,2}_{\text{b}}(0,1]\to
r^{\alpha}L^{2}_{\text{b}}(0,1]$

is an isomorphism and there holds

$\|u\|_{r^{\alpha}L^{1,2}_{\text{b}}}\leq\|(r\partial_{r}-\tfrac{1}{2})u\|_{r^{\alpha}L^{2}_{\text{b}}}$
(B.5)

###### Proof.

Setting $r=e^{s}$ for $s\in(-\infty,0]$ the problem is equivalent to the
analogous statement for

$\partial_{s}-\tfrac{1}{2}:e^{(1-\alpha)s}L^{1,2}((-\infty,0],ds)\longrightarrow
e^{(1-\alpha)s}L^{2}((-\infty,0],ds)$

which is conjugate to

$\tfrac{1}{e^{(\alpha-1)s}}(\partial_{s}-\tfrac{1}{2})e^{(\alpha-1)s}=(\partial_{s}+\alpha-\tfrac{3}{2}):L^{1,2}((-\infty,0],ds)\to
L^{2}((-\infty,0],ds).$

The claim then follows directly from integrating by parts since the boundary
term $(\alpha-\tfrac{3}{2})|u(0)|^{2}>0$ is strictly positive. ∎

We now conclude the proof of Lemma 8.14.

###### Proof of Lemma 8.14.

If $\varphi$ is compactly supported in the region
$Y-N_{r_{0}/2}(\mathcal{Z}_{0})$ where $r$ is uniformly bounded below, the
lemma is immediate from the standard Sobolev Embedding Theorem. We may
therefore assume that $\varphi$ is supported in a tubular neighborhood of
$\mathcal{Z}_{0}$. Since $\varphi\in
r\mathcal{H}^{m,1}\cap\mathbb{P}_{\mathcal{X}}$ by assumption, we may write

$\varphi=A(t,\theta)r^{1/2}+B(t,\theta,r)$

in local coordinates, after which it suffices to show the bound for each term
individually.

Applying (B.5) to derivatives and integrating over the $t,\theta$ variables
leads to

$\|r\partial_{r}u\|_{r^{\alpha}H^{m}_{\text{b}}}+\|u\|_{r^{\alpha}H^{m}_{\text{b}}}\leq
C\|(r\partial_{r}-\tfrac{1}{2})u\|_{r^{\alpha}H^{m}_{\text{b}}}$

for $\alpha>3/2$ and in particular for $\alpha=1+\nu$. Applying this to
$B(t,\theta,r)$ and discarding the first term shows that

$\|B(r,t,\theta)\|_{r^{1+\nu}H^{\text{m}}_{\text{b}}}\leq
C\|(r\partial_{r}-\tfrac{1}{2})B\|_{r^{1+\nu}H^{m}_{\text{b}}}=C\|(r\partial_{r}-\tfrac{1}{2})\varphi\|_{r^{1+\nu}H^{m}_{\text{b}}}\leq
C\|\varphi\|_{r\mathcal{H}^{m+1,1}}$ (B.6)

since $(r\partial_{r}-\tfrac{1}{2})$ annihilates $A(t,\theta)r^{1/2}$. Then,
applying Lemma B.3 to $B(t,\theta,r)r^{-1/2}$ shows that

$|B(t,\theta,r)r^{-1/2}|\leq\|B(r,t,\theta)\|_{r^{3/2}H^{\text{m}}_{\text{b}}}\leq\|B(r,t,\theta)\|_{r^{1+\nu}H^{\text{m}}_{\text{b}}}\leq
C\|\varphi\|_{r\mathcal{H}^{m+1,1}}$

and the result for $B(t,\theta,r)$ follows after multiplying by $r^{1/2}$.

For the first term, the triangle inequality and (B.6) shows that

$\|A(t,\theta)r^{1/2}\|_{rH^{m}_{\text{b}}}=\|A(t,\theta)r^{1/2}+B\|_{rH^{m}_{\text{b}}}-\|B(r,t,\theta)\|_{rH^{\text{m}}_{\text{b}}}\leq
C\|\varphi\|_{r\mathcal{H}^{m+2,1}}.$

Finally, since
$\|A(t,\theta)r^{1/2}\|_{rH^{m}_{\text{b}}}\sim\|A(t,\theta)\|_{L^{m,2}(T^{2})}$,
the bound for the first term follows from the Sobolev embedding on $T^{2}$
after increasing $m+2$ to $m+4$. ∎

## References

  * [1] Serge Alinhac and Patrick Gérard, __Pseudo-differential Operators and the Nash-Moser Theorem__ , vol. 82, American Mathematical Soc., 2007.
  * [2] Christian Bär, Paul Gauduchon, and Andrei Moroianu, __Generalized Cylinders in semi-Riemannian and Spin Geometry__ , Mathematische Zeitschrift 249 (2005), no. 3, 545–580.
  * [3] Jean-Pierre Bourguignon and Paul Gauduchon, __Spineurs, Opérateurs de Dirac et Variations de Métriques__ , Communications in Mathematical Physics 144 (1992), no. 3, 581–599.
  * [4] Aleksander Doan, __Monopoles and Fueter Sections on Three-Manifolds__ , Ph.D. thesis, State University of New York at Stony Brook, 2019.
  * [5] Aleksander Doan and Semon Rezchikov, __Holomorphic Floer Theory and the Fueter Equation__ , arXiv Preprint: (2022), arXiv: 2210.12047.
  * [6] Aleksander Doan and Thomas Walpuski, __On Counting Associative Submanifolds and Seiberg-Witten Monopoles__ , Pure and Applied Mathematics Quarterly 15 (2019), no. 4, 1047–1133.
  * [7] , __Deformation Theory of the Blown-up Seiberg–Witten Equation in Dimension Three__ , Selecta Mathematica 26 (2020), no. 3, 1–48.
  * [8] , __On the Existence of Harmonic $\mathbb{Z}_{2}$-Spinors__, Journal of Differential Geometry 117 (2021), no. 3, 395–449.
  * [9] Simon Donaldson, __Deformations of Multivalued Harmonic Functions__ , The Quarterly Journal of Mathematics 72 (2021), no. 1-2, 199–235.
  * [10] Simon Donaldson and Ed Segal, __Gauge Theory in Higher Dimensions, II__ , Surveys in differential geometry Volume XVI. Geometry of special holonomy and related topics. 16 (2011), 1–41.
  * [11] Saman Habibi Esfahani, __Monopoles, Singularities, and Hyperkähler Geometry__ , Ph.D. thesis, Stony Brook University, 2022.
  * [12] , __Monopoles Fueter Floer Homoloy__ , (Forthcoming manuscript).
  * [13] Lawrence C Evans, __Partial Differential Equations__ , vol. 19, American Mathematical Soc., 2010.
  * [14] Alberto Fiorenza, Maria Rosaria Formica, Tomás G Roskovec, and Filip Soudskỳ, __Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks__ , Zeitschrift für Analysis und ihre Anwendungen 40 (2021), no. 2, 217–236.
  * [15] Laura Fredrickson, __Generic Ends of the Moduli Space of $SL(n,\mathbb{C})$ Higgs Bundles__, arXiv Preprint (2018), arXiv 1810.01556.
  * [16] Laura Fredrickson, Rafe Mazzeo, Jan Swoboda, and Hartmut Weiss, __Asymptotic Geometry of the Moduli Space of Parabolic $\text{SL}(2,\mathbb{C})$-Higgs Bundles__, Journal of the London Mathematical Society-Second Series (2022).
  * [17] Daniel Grieser, __Basics of the b-Calculus__ , Approaches to singular analysis, Springer, 2001, pp. 30–84.
  * [18] Richard S Hamilton, __The Inverse Function Theorem of Nash and Moser__ , Bulletin (New Series) of the American Mathematical Society 7 (1982), no. 1, 65–222.
  * [19] Andriy Haydys, __Gauge theory, Calibrated Geometry and Harmonic Spinors__ , Journal of the London Mathematical Society 86 (2012), no. 2, 482–498.
  * [20] , __Fukaya-Seidel Category and Gauge Theory__ , J. Symplectic Geom. 13 (2015), no. 1, 151–207.
  * [21] , __G2 Instantons and the Seiberg-Witten Monopoles__ , (2017), arXiv: 1703.06329.
  * [22] , __The Infinitesimal Multiplicities and Orientations of the Blow-up set of the Seiberg-Witten Equation with Multiple Spinors__ , Advances in Mathematics 5 (2019), 193–218.
  * [23] Andriy Haydys, Rafe Mazzeo, and Ryosuke Takahashi, forthcoming manuscript.
  * [24] Andriy Haydys and Thomas Walpuski, __A Compactness Theorem for the Seiberg–Witten Equation with Multiple Spinors in Dimension Three__ , Geometric and Functional Analysis 25 (2015), no. 6, 1799–1821.
  * [25] Siqi He, __Existence of Nondegenerate $\mathbb{Z}_{2}$-Harmonic 1-forms via $\mathbb{Z}_{3}$ Symmetry__, arXiv Preprint (2022), arXiv:2202.12283.
  * [26] , __The Branched Deformations of the Special Lagrangian Submanifolds__ , arXiv Preprint (2022), arXiv 2202.12282.
  * [27] Nigel Hitchin, __Harmonic Spinors__ , Advances in Mathematics 14 (1974), no. 1, 1–55.
  * [28] Sonja Hohloch, Gregor Noetzel, and Dietmar A Salamon, __Hypercontact Structures and Floer Homology__ , Geometry & Topology 13 (2009), no. 5, 2543–2617.
  * [29] Lars Hormander, __The Boundary Problems of Physical Geodesy__ , Archive for Rational Mechanics and Analysis 62 (1976), no. 1, 1–52.
  * [30] Peter B Kronheimer and Tomasz Mrowka, __Monopoles and Three-Manifolds__ , vol. 10, Cambridge University Press Cambridge, 2007.
  * [31] Cagatay Kutluhan, Steven Sivek, and C Taubes, __Sutured ECH is a Natural Invariant__ , Mem. Amer. Math. Soc. 275 (2022).
  * [32] H Blaine Lawson and Marie-Louise Michelsohn, __Spin Geometry__ , vol. 84, Princeton university press, 2016.
  * [33] Rafe Mazzeo, Jan Swoboda, Hartmut Weiss, and Frederik Witt, __Ends of the Moduli space of Higgs Bundles__ , Duke Mathematical Journal 165 (2016), no. 12, 2227–2271.
  * [34] Rafe Mazzeo and Boris Vertman, __Elliptic Theory of Differential Edge Operators, II: Boundary Value Problems__ , Indiana University Mathematics Journal (2014), 1911–1955.
  * [35] Rafe Mazzeo and Edward Witten, __The Nahm Pole Boundary Condition__ , The influence of Solomon Lefschetz in geometry and topology. Contemporary Mathematics (2013), no. 621, 171–226.
  * [36] , __The KW Equations and the Nahm Pole Boundary Condition with Knots__ , Communications in Analysis and Geometry 28 (2020), no. 4, 871–942.
  * [37] Richard Melrose, __The Atiyah-Patodi-Singer Index Theorem__ , AK Peters/CRC Press, 1993.
  * [38] John W Morgan, __The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-manifolds__ , vol. 44, Princeton University Press, 1996.
  * [39] Olaf Müller and Nikolai Nowaczyk, __A Universal Spinor Bundle and the Einstein–Dirac–Maxwell Equation as a Variational Theory__ , Letters in Mathematical Physics 107 (2017), no. 5, 933–961.
  * [40] Liviu I Nicolaescu, __Lectures on the Geometry of Manifolds__ , World Scientific, 2020.
  * [41] Gregory J. Parker, __Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations on 3-Manifolds__ , arXiv Preprint (2022), arXiv:2210.08148.
  * [42] , __Gluing $\mathbb{Z}_{2}$-Harmonic Spinors on 3-Manifolds__, Forthcoming manuscript.
  * [43] Victor Yaroslavovich Pidstrigach, __Hyperkähler Manifolds and Seiberg–Witten Equations__ , Trudy Matematicheskogo Instituta imeni VA Steklova 246 (2004), 263–276.
  * [44] Daniel Platt, __$G_{2}$ -Instantons on Resolutions of $G_{2}$-Orbifolds__, (2022), arXiv: 2208.10903.
  * [45] Mazzeo Rafe, __Elliptic Theory of Differential Edge Operators I__ , Communications in Partial Differential Equations 16 (1991), no. 10, 1615–1664.
  * [46] Xavier Saint Raymond, __A Simple Nash-Moser Implicit Function Theorem__ , L’Enseignement Mathématique 35 (1989), 217–226.
  * [47] Dietmar Salamon, __The Three-dimensional Fueter equation and Divergence-free Frames__ , Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 83, Springer, 2013, pp. 1–28.
  * [48] Paolo Secchi, __On the Nash-Moser Iteration Technique__ , Recent Developments of Mathematical Fluid Mechanics, Springer, 2016, pp. 443–457.
  * [49] Ryosuke Takahashi, __The Moduli Space of $S^{1}$-type Zero Loci for $\mathbb{Z}/2$-Harmonic Spinors in Dimension 3__, arXiv Preprint (2015), arXiv 1503.00767.
  * [50] Yuuji Tanaka and Richard Thomas, __Vafa-Witten Invariant for Projective Surfaces II: Semi-stable Case__ , Pure and Applied Math. Quarterly 13 (2017), no. 3, 517–562.
  * [51] , __Vafa-Witten Invariant for Projective Surfaces I: Stable Case__ , J. Algebraic Geom. 29 (2019), 603–668.
  * [52] Terence Tao and Gang Tian, __A Singularity Removal Theorem for Yang-Mills Fields in Higher Dimensions__ , Journal of the American Mathematical Society 17 (2004), no. 3, 557–593.
  * [53] Clifford Henry Taubes, __Nonlinear Generalizations of a 3-Manifold’s Dirac Operator__ , AMS IP Studies in Advanced Mathematics 13 (1999), 475–486.
  * [54] , __Compactness Theorems for $SL(2,{\mathbb{C}})$ Generalizations of the 4-dimensional Anti-Self Dual Equations__, arXiv Preprint (2013), arXiv:1307.6447.
  * [55] , __$\text{PSL}(2;\mathbb{C})$ Connections on 3-manifolds with $L^{2}$ Bounds on Curvature__, Cambridge Journal of Mathematics 1 (2013), 239–397.
  * [56] , __The Zero Loci of $\mathbb{Z}/2$-Harmonic Spinors in Dimension 2, 3 and 4__, arXiv Preprint (2014), arXiv:1407.6206.
  * [57] , __On the Behavior of Sequences of Solutions to $U(1)$-Seiberg-Witten Systems in Dimension 4__, arXiv Preprint (2016), arXiv:1610.07163.
  * [58] , __The Behavior of Sequences of Solutions to the Vafa-Witten Equations__ , arXiv Preprint (2017), arXiv:1702.04610.
  * [59] , __Sequences of Nahm pole Solutions to the $SU(2)$ Kapustin-Witten Equations__, arXiv Preprint (2018), arXiv:1805.02773.
  * [60] Clifford Henry Taubes and Yingying Wu, __Examples of Singularity Models of $\mathbb{Z}/2$-harmonic 1-forms and Spinors in Dimension 3__, arXiv Preprint (2020), arXiv 2001.00227.
  * [61] , __Topological Aspects of $\mathbb{Z}/2\mathbb{Z}$ Eigenfunctions for the Laplacian on $S^{2}$__, arXiv Preprint (2021), arXiv 2108.05017.
  * [62] Gang Tian, __Gauge Theory and Calibrated Geometry, I__ , Annals of Mathematics 151 (2000), no. 1, 193–268.
  * [63] Cumrun Vafa and Edward Witten, __A Strong Coupling Test of S-Duality__ , Nuclear Physics B 431 (1994), no. 1-2, 3–77.
  * [64] Thomas Walpuski, __A Compactness Theorem for Fueter sections__ , Commentarii Mathematici Helvetici 92 (2017), no. 4, 751–776.
  * [65] , __$G_{2}$ -Instantons, Associative Submanifolds and Fueter Sections__, Communications in Analysis and Geometry 25 (2017), no. 4, 847–893.
  * [66] , __Spin (7)-Instantons, Cayley Submanifolds and Fueter Sections__ , Communications in Mathematical Physics 352 (2017), no. 1, 1–36.
  * [67] Thomas Walpuski and Boyu Zhang, __On the Compactness Problem for a Family of Generalized Seiberg-Witten Equations in Dimension 3__ , Duke Mathematical Journal 170(17) (2021), 239–397.
  * [68] Edward Witten, __Fivebranes and Knots__ , Quantum Topology 3 (2011), no. 1, 1–137.
  * [69] , __Khovanov Homology and Gauge Theory__ , Proceedings of the Freedman Fest 18 (2012), 291–308.
  * [70] Fangyun Yang, __Dirac Operators and Monopoles with Singularities__ , Ph.D. thesis, Massachusetts Institute of Technology, 2007.
  * [71] Boyu Zhang, __Rectifiability and Minkowski Bounds for the Zero Loci of $\mathbb{Z}_{2}$-Harmonic Spinors in Dimension 4__, arXiv Preprint (to appear in Comm. Analysis and Geom.) (2017), arXiv 2202.12282.