# Reflections on Bubble Walls

Isabel Garcia Garcia<EMAIL_ADDRESS>Center for Cosmology and Particle
Physics, Department of Physics, New York University, New York, NY 10003, USA
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540,
USA  Giacomo Koszegi<EMAIL_ADDRESS>Department of Physics,
University of California, Santa Barbara, CA 93106, USA  Rudin Petrossian-Byrne
<EMAIL_ADDRESS>Abdus Salam International Centre for Theoretical Physics,
Strada Costiera 11, 34151, Trieste, Italy

###### Abstract

Abstract

We discuss the dynamics of expanding bubble walls in the presence of massive
dark photons whose mass changes as they cross the wall. For sufficiently thin
walls, we show that there exists a transient kinematic regime characterized by
a constant reflection probability of longitudinal – but not transverse –
modes. This effect can have important implications for the dynamics of
expanding vacuum bubbles in the early Universe. Most notably, it leads to a
new source of pressure on the expanding interface, featuring a non-monotonic
dependence on the $\gamma$-factor of the bubble walls and reaching a peak at
intermediate $\gamma$-factors that we dub Maximum Dynamic Pressure. When this
pressure is large enough to halt the acceleration of the bubble walls, the
difference in vacuum energy densities goes into making a fraction of the dark
photons relativistic, turning them into dark radiation. If the dark radiation
remains relativistic until late times, an observable contribution to $\Delta
N_{\text{eff}}$ is possible for phase transitions with strength $\alpha\sim
10^{-2}-10^{-1}$.

(Dated: )

###### Contents

  1. I  Introduction
  2. II  Set-up
  3. III  Reflection and transmission probabilities
    1. III.1  Massive electromagnetism
    2. III.2  A step wall
    3. III.3  A smooth wall
  4. IV  Dark photon friction on bubble walls
    1. IV.1  Maximum Dynamic Pressure
    2. IV.2  Equilibrium $\gamma$-factor and energy budget
    3. IV.3  Fate of the dark radiation
  5. V  Conclusions
  6. A Beyond the effective theory
    1. A.1 Abelian Higgs UV-completion
    2. A.2 Interaction rates
  7. B Reflection probabilities – supplemental material
    1. B.1 Longitudinal reflection from Goldstone Equivalence
    2. B.2 Scattering on a $\delta^{\prime}$ potential
    3. B.3 The Born approximation
    4. B.4 Numerical methods

##  I  Introduction

Cosmological phase transitions in the early Universe that proceed via
nucleation of bubbles are a well-motivated possibility in minimal extensions
of the Standard Model, as well as in more general scenarios featuring hidden
sectors with their own dynamics. Such first order transitions result in the
emission of gravitational radiation Witten:1984rs ; Hogan:1986qda ;
Kosowsky:1991ua ; Kosowsky:1992rz ; Kamionkowski:1993fg  that current and
future observatories may be able to detect in the form of a stochastic
background of gravitational waves (see e.g. Caprini:2015zlo ; Caprini:2019egz
for reviews). An observation of this kind would provide unambiguous evidence
for the existence of degrees of freedom beyond the Standard Model. This has
spurred significant interest in the gravitational wave signatures of hidden
sectors in recent times Schwaller:2015tja ; Breitbach:2018ddu ;
Helmboldt:2019pan ; Jinno:2019jhi ; Azatov:2019png ; Nakai:2020oit ;
Halverson:2020xpg ; Huang:2021rrk ; Ertas:2021xeh ; Dent:2022bcd ;
Fairbairn:2019xog ; Jinno:2022fom .

Despite the abundance of particle physics models susceptible of undergoing an
out-of-equilibrium transition, our ability to make use of the resulting
gravitational wave signal to extract information about the relevant dynamics
is extremely limited. The most revealing features concern the frequency peak
of the stochastic background, as well as its spectral shape at high
frequencies. The former determines the epoch at which the transition takes
place, whereas the latter contains information about the dominant source of
gravitational radiation. For example, if most of the energy released during
the transition goes into accelerating the bubble walls (as in vacuum
Coleman:1977py ), these become relativistic and continue to expand at ever-
increasing velocities. In this case, collisions of these “run-away” bubbles
constitute the main source of gravitational waves, and the resulting signal
falls off as $f^{-1}$ at high frequencies Huber:2008hg . Alternatively,
pressure on the bubble walls due to particles in the thermal plasma may cause
the expanding walls to reach a terminal speed. In this case, most of the
latent heat gets damped instead into the thermal fluid, and it is its
subsequent motion that provides the dominant source of gravitational radiation
Espinosa:2010hh . The high-frequency fall-off of the stochastic background is
steeper, e.g. decreasing as $f^{-4}$ for radiation sourced by sound waves
Hindmarsh:2015qta ; Hindmarsh:2016lnk ; Hindmarsh:2017gnf ; Hindmarsh:2019phv
. On the other hand, assuming radiation-domination at the time of
gravitational wave production, causal propagation restricts the low frequency
shape of the spectrum to grow as $f^{3}$, independently of the dominant
production channel Caprini:2009fx . 111More generally, the low-frequency shape
of the stochastic background depends on the equation of state
Barenboim:2016mjm ; Hook:2020phx  and on the existence of free-streaming
particles Weinberg:2003ur ; Hook:2020phx , and could thus provide non-trivial
information about the Universe at early times.

Understanding the dynamics of expanding bubble walls in the early Universe is
clearly crucial to determine both quantitative and qualitative features of the
resulting gravitational wave signal. But given the intrinsic degeneracy
present in any stochastic background, it is equally important to explore
alternative probes of the relevant dynamics. For example, the upcoming LISA
experiment will be sensitive to phase transitions at electroweak to multi-TeV
scale temperatures, probing the nature of the electroweak phase transition and
potentially shedding light on the dynamics behind baryogenesis and electroweak
symmetry breaking. In this range of energies, the complementarity between LISA
and current and future colliders will no doubt be key in furthering our
understanding of physics at and around the weak scale (see e.g.
Morrissey:2012db ; Caprini:2015zlo ; Caprini:2019egz ; AlAli:2021let  for
reviews and references). Beyond (and below) the electroweak scale, phase
transitions within hidden sectors may occur at virtually any temperature, and
the corresponding stochastic background could fall anywhere from the low
frequency range of PTA observatories NANOGRAV:2018hou ; Lentati:2015qwp
($10^{-9}-10^{-7}\ \text{Hz}$), to the high frequencies probed by LIGO
($10-10^{3}\ \text{Hz}$), and beyond. However, vacuum bubbles nucleated in
hidden sector transitions may feature very different dynamics to those linked
to the weak scale, and the relevant degrees of freedom may be inaccessible at
laboratory experiments. Our work is motivated by the goal to more broadly
understand the potential behavior of expanding bubbles in the early Universe,
as well as to identify alternative predictions that may accompany an
observable stochastic background of gravitational waves.

In a first order phase transition, bubbles of true vacuum are nucleated at
rest, and begin to expand fueled by the difference in free energy densities at
either side of the interface. In vacuum, the velocity of the bubble walls
evolves according to Coleman:1977py

$\frac{d|\vec{v}|}{dt}=\frac{1}{\gamma^{3}R_{0}}\propto\frac{1}{\gamma^{3}}\frac{\Delta
V}{\sigma}\ ,$ (1)

with $\gamma=1/\sqrt{1-\vec{v}^{\,2}}$ the usual Lorentz $\gamma$-factor,
$R_{0}\propto\sigma/\Delta V$ the critical bubble radius, and where $\Delta V$
and $\sigma$ refer to the difference in vacuum energy densities and surface
tension of the bubble wall. In reality, bubbles do not expand against a sea of
false vacuum, but rather within a non-trivial environment that in the early
Universe must include the Standard Model plasma as well as, potentially, other
ingredients such as dark matter. As the bubble wall speed grows, friction from
the surrounding environment can exert a pressure on the interface that opposes
the expansion of the bubble walls. Their evolution can still be written as in
eq. 1, after the replacement Arnold:1993wc ; Moore:1995si :

$\Delta V\rightarrow\Delta V-\mathcal{P}\ ,$ (2)

where $\mathcal{P}$ refers to the frictional pressure resulting from the
interaction between the interface and the surrounding medium, and is in
general dependent on the speed of the bubble wall.

Two qualitatively different scenarios are thus possible. If $\Delta
V\gg\mathcal{P}$ during the entire evolution of the bubble walls, these
effectively behave as if they were in vacuum and most of the energy released
as the bubbles grow goes into accelerating the expanding interface. The walls
then “run away” – that is, they continue to expand with ever increasing
velocities. Alternatively, if $\mathcal{P}$ grows large enough so as to
neutralize the difference in vacuum energies, $\Delta V=\mathcal{P}$, the
bubble walls reach an equilibrium regime of constant speed. Once in
equilibrium, the fraction of the total energy that becomes localized on the
interface quickly becomes tiny. Instead, most of the energy gets damped into
whatever sector of the surrounding environment is responsible for halting the
acceleration of the bubble walls. Calculating the pressure experienced by
bubble walls as they expand is a classic problem Turok:1992jp ; Dine:1992wr ;
Liu:1992tn ; Arnold:1993wc ; Moore:1995ua ; Moore:1995si  that has received
renewed attention in recent times Bodeker:2009qy ; Espinosa:2010hh ;
Leitao:2015ola ; Bodeker:2017cim ; Azatov:2020ufh ; Ai:2021kak ; Mancha_2021 ;
Bea:2021zsu ; Gouttenoire:2021kjv ; Azatov:2022tii ; Laurent:2022jrs ;
DeCurtis:2022hlx .

Within a thermal plasma, particles with phase-dependent mass create a pressure
on the expanding walls that asymptotes to a constant
$\mathcal{P}_{\infty}\sim\Delta m^{2}T^{2}$ in the ultra-relativistic limit
222The ultra-relativistic limit refers to the kinematic regime where the
energy of the incident particles in the rest frame of the bubble wall is the
largest energy scale. Alternatively, this corresponds to the limit
$\gamma\rightarrow\infty$ for the $\gamma$-factor of the bubble wall.,
independently of the type of particle Bodeker:2009qy . An additional source of
friction may be present if the spectrum contains gauge bosons, of the form
$\mathcal{P}_{\infty}\sim\gamma g^{2}\Delta m_{v}T^{3}$, with $g$ the relevant
gauge coupling and $\Delta m_{v}$ the change in the mass of the gauge boson at
either side of the bubble wall Bodeker:2017cim ; Azatov:2020ufh ;
Gouttenoire:2021kjv . This effect has its origin in the transition radiation
emitted by charged particles as they cross the wall and its
$\gamma$-dependence can easily render it the most significant source of
friction on fast expanding bubbles. Indeed, if the electroweak phase
transition were first order, transition radiation would likely cause the
bubble walls to reach an equilibrium $\gamma$-factor as low as
$\gamma_{\text{eq}}=\mathcal{O}(10)$ Bodeker:2017cim .

A crucial aspect of all sources of friction known so far is that the
corresponding pressure is a monotonically increasing function of the wall
speed. As put forward in Bodeker:2009qy ; Bodeker:2017cim , this allows for a
simple criterion to determine whether bubble walls during a cosmological phase
transition become run-away, by comparing the pressure in the ultra-
relativistic limit, $\mathcal{P}_{\infty}$, to the difference in vacuum energy
density across the wall:

$\text{Run-away criterion:}\qquad\qquad\Delta
V>\mathcal{P}_{\infty}\qquad\qquad\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bodeker:2009qy,Bodeker:2017cim}{\@@citephrase{(}}{\@@citephrase{)}}}}\
.$ (3)

In this article, we discuss a new physical effect that can qualitatively alter
the dynamics of bubble walls during a cosmological phase transition. Namely,
the existence of a transient relativistic regime characterized by an
approximately constant reflection probability of longitudinal massive vectors
off an expanding interface. Effectively, the wall behaves temporarily like an
imperfect mirror that reflects a fraction of longitudinal – but not transverse
– modes. Two conditions need to be satisfied for this regime to be accessible:
(_i_) that the expansion of the bubble walls takes place against a population
of massive vectors whose mass changes across the interface; and (_ii_) that
the expanding walls are sufficiently “thin”. By thin, we mean that the wall
thickness (in the rest frame of the bubble wall) be much smaller than the
Compton wavelength of the massive vector, i.e.

$L\ll m^{-1}\ .$ (4)

In this case, the regime of constant longitudinal reflection corresponds to
Lorentz $\gamma$-factors in the range

$1\ll\gamma\ll(Lm)^{-1}\ ,$ (5)

ending when $\gamma$ is so large that the Lorentz-contracted Compton
wavelength of the dark photon becomes smaller than the wall thickness. We will
refer to this kinematic regime as the region of “inter-relativistic” motion.
Eq.(4) ensures that $(Lm)^{-1}\gg 1$, and that this regime is indeed
accessible during the evolution of the expanding bubbles. Once $\gamma$
becomes $\gg(Lm)^{-1}$, reflection probabilities for all polarizations die off
exponentially – a well-known feature of the ultra-relativistic limit.

Most notably, the effect described above leads to an additional source of
friction on expanding bubble walls. Unlike previously known cases, the
corresponding pressure features a characteristic non-monotonic dependence on
the relevant $\gamma$-factor, reaching a maximum at $\gamma\sim(Lm)^{-1}$
before turning-off at larger values. In (superficial) analogy with the
behavior of spacecraft shortly after launch, we will refer to this pressure
peak as Maximum Dynamic Pressure. Its existence can make it much harder for
bubble walls to become run-away than previously believed, and we will show
that eq. 3 can be qualitatively misleading in phase transitions where the
bubble walls expand against an existing population of phase-dependent massive
dark photons.

This article is dedicated to deriving the claims made in the previous two
paragraphs, as well as illustrating some of their phenomenological
consequences. We will refer to the temperature of the Standard Model plasma at
the epoch of the phase transition as $T_{*}$, and denote the phase transition
strength via the usual dimensionless quantity Caprini:2015zlo ;
Caprini:2019egz :

$\alpha\equiv\frac{\Delta V}{\rho_{\text{SM}}(T_{*})}=\frac{\Delta
V}{\frac{\pi^{2}}{30}g_{*}(T_{*})T_{*}^{4}}\ .$ (6)

We will focus on bubble walls that expand against a population of cold and
non-interacting dark photons. Despite its simplicity, this system will be
relevant in some physically interesting cases, such as when the dark photons
furnish the dark matter Hu:2000ke ; Hui:2016ltb ; Nelson:2011sf ; Arias:2012az
; Graham:2015rva ; Agrawal:2018vin  – a well-motivated benchmark that we often
refer to throughout this work. When the bubble walls reach an equilibrium
regime as a result of longitudinal reflections, the fraction of the total
energy that goes into accelerating the bubble walls becomes increasingly
small. Instead, most of the available energy goes into making the reflected
dark photons relativistic, turning them into dark radiation. If the dark
radiation remains relativistic until late times, an observable contribution to
$\Delta N_{\text{eff}}$ is possible. In particular, current bounds on $\Delta
N_{\text{eff}}$ could probe phase transitions with strength $\alpha\gtrsim
10^{-1}$, whereas CMB S-4 measurements could be sensitive to scenarios down to
$\alpha\sim 10^{-2}$ for all relevant frequencies.

Extensions of the Standard Model featuring massive vectors are popular both
because of their minimality as well as their potential to furnish the dark
matter, with a variety of production mechanisms spanning a wide mass range
Nelson:2011sf ; Arias:2012az ; Graham:2015rva ; Agrawal:2018vin . Before we
move on, let us summarize why the existence of _massive_ dark photons whose
mass changes in the course of our cosmological history is not only a well-
motivated possibility, but may be an unavoidable feature in a wide class of
models. At the renormalizable level, the physical system that we focus on is
described by a lagrangian of the form

$\mathcal{L}\supset\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}V_{\mu}V^{\mu}\
,$ (7)

where $\phi$ is a real scalar field and $V_{\mu}$ is the massive dark photon.
By assumption, the potential for $\phi$ features two non-degenerate vacua such
that the scalar sector undergoes a phase transition in the early Universe.
WLOG we take the false and the true vacuum to lie at $\langle\phi\rangle=0$
and $\langle\phi\rangle=v$. $V(\phi)$ may be a finite temperature effective
potential, or it may be a zero-temperature potential in which case the
transition proceeds via quantum tunneling. If the scalar sector is in thermal
equilibrium with the Standard Model then $T_{*}\sim v$, whereas $T_{*}\ll v$
is possible if the transition is ‘super-cooled’, or if $\phi$ belongs in a
hidden sector that is decoupled from the thermal plasma. At nucleation, the
thickness of the bubble walls is determined by the features of the scalar
potential, and typically $L\sim(\sqrt{\lambda}\,v)^{-1}$, with $\lambda$ the
typical quartic coupling in $V(\phi)$. The size of the dimensionless
combination $T_{*}L$ will have a quantitative effect on our results, as we
will discuss when relevant.

At the level of eq. 7, the scalar and dark photon sectors are fully decoupled.
Beyond the renormalizable terms of eq. 7, this need not remain true. For
example, the following operator

$\mathcal{L}\supset\frac{\kappa}{2}\phi^{2}V^{\mu}V_{\mu}\ ,$ (8)

leads to an additional contribution to the vector mass in the true vacuum, of
the form $\Delta m^{2}=\kappa v^{2}$. As the scalar vev ‘turns on’ in the
early Universe the dark photon mass will shift accordingly. Of course, the
scalar field may be complex instead of real, which can be trivially
accommodated by writing $|\phi|^{2}$ instead. Indeed, an effective interaction
$\propto|H|^{2}V^{\mu}V_{\mu}$, with $H$ the Standard Model Higgs doublet,
would lead to a shift in the dark photon mass before and after the electroweak
phase transition. We emphasize that eq. 8 cannot be forbidden on the basis of
symmetry, and its manifest lack of gauge redundancy is a moot point given that
the theory under consideration already contains a mass for $V^{\mu}$.
Legalistically, one might object to eq. 8 on the basis that it differs from a
Stückelberg mass, $\frac{1}{2}m^{2}V^{\mu}V_{\mu}$, in that the former is not
a renormalizable interaction and demands UV-completion. However, upholding the
laws of effective field theory, we have no choice but to overrule this
objection. Accepting the existence of a finite cutoff in our description of
nature, a theory with a massive dark photon and scalar degrees of freedom will
in general feature effective interactions as in eq. 8.

Indeed, a non-zero $\kappa$ in eq. 8 sets an upper bound on the scale of UV
completion. Provided that the term in eq. 8 only accounts for a subleading
contribution to the overall mass of the dark photon in the true vacuum
($\Delta m^{2}\ll m^{2}$), the upper bound on the UV cutoff can be
conveniently written as Lebedev:2011iq

$\Lambda\lesssim\frac{4\pi v}{\sqrt{\Delta m^{2}/m^{2}}}\ .$ (9)

Here, we indeed focus on cases where the change in the dark photon mass is
tiny. This allows for a separation between the scale of the phase transition,
$v$, and the UV cutoff, allowing us to neglect the effect of heavy degrees of
freedom on the dynamics of the phase transition and focus instead on the
consequences of non-zero $\Delta m^{2}$ on the evolution of the bubble walls.
As we will see, even when $\Delta m^{2}/m^{2}\ll 1$, the implications for the
evolution of cosmological vacuum bubbles can be significant. 333Although most
of our subsequent discussion will proceed within the effective theory defined
in eq. 7-8, we discuss in section A.1 how this effective description can arise
from a UV-complete model.

An important comment before we proceed. As made clear in the preceding
paragraph, the content of this paper is only relevant in the presence of
vector bosons that are _massive_ at either side of the bubble wall. Our
results therefore do not affect the pressure created by _massless_ gauge
bosons that gain a mass as they cross the wall, and so we have nothing to add
to e.g. the pressure created by $W$ and $Z$ bosons during a first order
electroweak phase transition.

The rest of this paper is organized as follows. We begin in section II with a
summary of the physical set-up that we will focus on in the remainder of this
manuscript. In section III, we present our calculation of the reflection
probability for phase-dependent massive vectors. Fig. 2 illustrates the main
result of this section: the existence of a transient relativistic regime
characterized by an approximately constant reflection probability of
longitudinally polarized dark photons. Section IV focuses on fleshing out the
consequences of our results for the evolution of bubble walls in the early
Universe, including the existence of a Maximum Dynamic Pressure in section
IV.1, as well as a self-consistent determination of the equilibrium
$\gamma$-factor when this pressure is large enough to halt the acceleration of
the bubble walls in section IV.2. The fate of the reflected dark photons
depends sensitively on a variety of considerations, most notably on whether
the sector undergoing the phase transition is hot or cold, as we discuss in
section IV.3. We summarize our conclusions in section V, and a number of
appendices supplement the discussion in the main text.

##  II  Set-up

The rest of this article is dedicated to calculating the pressure on an
expanding bubble wall due to an existing population of phase-dependent massive
dark photons and to discussing the implications of our results. With this
goal, we consider an expanding planar interface, representing a portion of a
sufficiently large bubble wall, moving with local velocity $\vec{v}$ and
corresponding $\gamma$-factor $\gamma\equiv 1/\sqrt{1-\vec{v}^{\,2}}$. The
wall is not expanding in vacuum, but rather against a population of cold and
non-interacting massive vector bosons with number density $n_{V}$. WLOG, we
take the velocity of the bubble wall in the rest frame of the dark photons to
be along the negative $z$-axis, $\vec{v}=-|\vec{v}|\hat{z}$. At leading order,
to compute the pressure on the wall we need the momentum transfer from
particles that either reflect or transmit across the interface. Our assumption
that the dark photon sector is non-interacting allows us to consider the
individual interactions of particles with the wall, although this approach is
more generally valid whenever the relevant mean free path exceeds the wall
thickness.

Like previous work focused on computing pressure on expanding bubbles, we find
it convenient to work in the rest frame of the interface. In this frame, the
scalar order parameter characterizing the transition varies only as a function
of the spatial coordinates, which in our convention will be along the
$z$-axis. A dark photon wind moving from the false to the true vacuum hits the
bubble wall with velocity $-\vec{v}=|\vec{v}|\hat{z}$. All particles hit the
interface at normal incidence and there are no particles traveling in the
opposite direction – both observations follow from our assumption that the
dark photon sector is cold 444Quantitatively, the assumption of normal
incidence requires that, in the wall frame, the normal component of the dark
photon momentum is much greater than the components along the plane of the
bubble wall, i.e. $|\vec{k}_{\perp}|\gg|\vec{k}_{\parallel}|$. For a gas of
dark photons at temperature $T_{\text{dp}}\ll m$ and a relativistic wall,
$|\vec{k}_{\perp}|\sim\gamma
m\gg|\vec{k}_{\parallel}|\sim\sqrt{T_{\text{dr}}m}$ provided
$\gamma\gg\sqrt{T_{\text{dp}}/m}$, which is satisfied trivially. It should be
be straightforward to generalize our results beyond these assumptions.. This
set-up is depicted in fig. 1.

Figure 1: In the rest frame of the bubble wall, a dark photon wind hits the
interface from the region of false vacuum. The energy of the incident dark
photons, $\omega=\gamma m$, remains conserved in the interaction with the
wall, whereas momentum along the $z$-direction changes resulting in a net
momentum transfer to the interface. $L$ refers to the wall thickness, i.e. the
typical length scale over which the order parameter varies. In this article,
we focus on the case where the vector mass changes by a small amount at either
side of the bubble wall, i.e. $\Delta m^{2}\ll m^{2}\simeq\tilde{m}^{2}$. Much
of our work will be concerned with the dynamics of the inter-relativistic
kinematic regime of eq. 5 where $m\ll\omega\ll L^{-1}$.

By assumption, the dark photon mass depends on the order parameter
characterizing the phase transition. As anticipated in the Introduction, and
further discussed in section A.1, realistic situations often feature a
dependence of the form $m^{2}_{V}(z)=m^{2}+\kappa v(z)^{2}$, with $v$ the
relevant order parameter and $\kappa$ a constant determined by the underlying
model. However, for most of our discussion, it will be sufficient to just
assume that the vector mass-squared parameter is $z$-dependent, with
asymptotic values $m^{2}$ and $\tilde{m}^{2}$ for large negative and positive
$z$, as indicated in fig. 1. We define the difference in squared masses as
$\Delta m^{2}\equiv\tilde{m}^{2}-m^{2}>0$ and we will always assume that
$\Delta m^{2}\ll m^{2}\simeq\tilde{m}^{2}$, in keeping with the discussion
around eq. 9.

Since the wall background is independent of $t$, the energy of the incident
particles, $\omega=\gamma m$, remains conserved in the interaction with the
wall, whereas momentum in the $z$-direction changes in a way that results in a
net momentum transfer to the interface. In total, the pressure exerted on the
wall as a result of this dark photon wind can be written as

$\mathcal{P}=\gamma|\vec{v}|n_{V}\,\times\frac{1}{3}\,\sum_{\lambda}\left(R_{\lambda}\,\Delta
k_{R}+T_{\lambda}\,\Delta k_{T}\right)\ .$ (10)

The factor $\gamma|\vec{v}|n_{V}$ corresponds to the flux of incoming massive
vectors hitting the wall (in the wall frame), whereas the quantity in
parentheses represents the average momentum transfer to the wall from an
incoming particle with fixed polarization $\lambda$, with $R_{\lambda}$ and
$T_{\lambda}$ the corresponding reflection and transmission probabilities. The
momentum transfer from reflected and transmitted particles is given by $\Delta
k_{R}=2k_{z}$ and $\Delta k_{T}=k_{z}-\tilde{k}_{z}$, with $k_{z}$ and
$\tilde{k}_{z}$ the asymptotic transverse momenta at either side of the bubble
wall, as defined in fig. 1. The sum over $\lambda$ includes the three physical
polarizations of a massive vector, and the lack of mixing between different
polarizations in eq. 10 follows as a result of normal incidence.

For example, in the regime of ultra-relativistic motion – when $\omega$ is the
largest energy scale in the problem, $\omega\gg m,L^{-1}$ – reflection and
transmission probabilities must asymptote to zero and unity respectively, and
eq. 10 takes the form,

$\mathcal{P}_{\infty}=\lim_{\gamma\rightarrow\infty}\mathcal{P}\simeq\gamma
n_{V}\Delta k_{T}=\rho_{V}\frac{\Delta m^{2}}{2m^{2}}\ ,$ (11)

where we have used $\Delta k_{T}\simeq\Delta m^{2}/(2\gamma m)$, and
$\rho_{V}=mn_{V}$ is the energy density of dark photons. If the massive vector
population outside of the bubble wall was instead described in terms of a
fully thermal distribution, with $T\gg m$, we would have $\omega\sim\gamma T$
and $n_{V}\sim T^{3}$, and therefore $\mathcal{P}_{\infty}\sim T^{2}\Delta
m^{2}$ as is well known Bodeker:2009qy . As seen in eq. 11, the frictional
pressure reaches a constant (independent of the wall velocity) in this limit.

A comment is in order before we move on. Comparing the asymptotic pressure in
eq. 11 to the difference in vacuum energy densities, one finds

$\frac{\mathcal{P}_{\infty}}{\Delta V}\ \simeq\ \frac{\Delta m^{2}}{2m^{2}}\
\frac{\rho_{V}}{\Delta V}\ \sim\ \alpha\ \frac{\Delta m^{2}}{m^{2}}\
\frac{\rho_{V}(T_{*})}{\rho_{\text{dm}}(T_{*})}\
\frac{\rho_{\text{dm}}(T_{*})}{\rho_{\text{SM}}(T_{*})}\ ,$ (12)

where $\alpha$ is defined in eq. 6 and in the last step we have made it
explicit that all the relevant energy densities are to be evaluated at the
phase transition epoch. The ratio $\rho_{V}/\rho_{\text{dm}}$ in eq. 12 is 1
if the dark photons account for all of the dark matter, whereas $\alpha$ and
$\Delta m^{2}/m^{2}$ are $\ll 1$ as discussed previously. The ratio of dark
matter to Standard Model radiation in the early Universe is
$\rho_{\text{dm}}/\rho_{\text{SM}}\lll 1$, making eq. 12 correspondingly tiny.
As per the run-away criterion of eq. 3, one would be tempted to conclude that
a small change in the dark matter mass would have a negligible effect on the
evolution of cosmological bubble walls. Our results will show that – in
general – this expectation can be mistaken.

To proceed, we must obtain reflection and transmission probabilities, as a
function of $\gamma$, for massive vectors interacting with the non-trivial
background of the bubble wall. This problem reduces to solving the equations
of motion for massive electromagnetism with a spatially varying photon mass,
as we discuss next.

##  III  Reflection and transmission probabilities

We begin in section III.1 with a brief discussion regarding massive
electromagnetism with a spatially varying vector mass. In III.2, we obtain
reflection and transmission probabilities in one the of the few non-trivial
cases where an analytic solution is accessible: a step function change in the
mass of the dark photon. This corresponds to the limit of vanishing wall
thickness, and it provides an accurate description of the system in the regime
$\omega\ll L^{-1}$. Considering a step wall allows us to illustrate one of our
main results: that the reflection probability for longitudinal modes
asymptotes to a constant in the inter-relativistic regime of eq. 5. In III.3
we turn to the realistic situation of finite width, and show that the
existence of a relativistic regime of near-constant longitudinal reflection is
a generic feature of thin walls with finite thickness.

###  III.1  Massive electromagnetism

In the absence of charged sources, Maxwell’s equations in the presence of a
varying vector mass read

$\partial_{\mu}F^{\mu\nu}+m^{2}_{V}(x)V^{\nu}=0\ ,$ (13)

which, in turn, imply the consistency condition

$\partial_{\mu}\left(m^{2}_{V}(x)V^{\mu}\right)=0\ .$ (14)

As is well known, when the vector mass is constant, $m_{V}^{2}(x)=m^{2}$, eq.
14 reduces to the familiar requirement that $\partial_{\mu}V^{\mu}=0$, and
Maxwell’s equations admit plane-wave solutions of the form

$V^{\mu}(x)=v^{\mu}e^{-ik\cdot x}\qquad\qquad\text{(constant vector mass)}\ ,$
(15)

for all $k^{\mu}$ such that $k^{2}=m^{2}$, and the $v^{\mu}$ are momentum-
dependent complex coefficients satisfying $k_{\mu}v^{\mu}=0$. 555The field
$V^{\mu}$ is of course real-valued. It is implicitly assumed that only the
real part of eq. 15 has physical significance. The same applies to all other
expressions of this form in the remainder of this paper. As usual, the
$v^{\mu}$ can be written as a sum over unit-normalized polarization vectors,
one for each of the three physical degrees of freedom of a massive spin-1
particle. For $\vec{k}=\pm|\vec{k}|\hat{z}$, a convenient basis is given by

$\varepsilon^{\mu}_{x}=(0,1,0,0)\
,\qquad\varepsilon^{\mu}_{y}=(0,0,1,0)\qquad\text{and}\qquad\varepsilon^{\mu}_{l}=\left(\frac{|\vec{k}|}{m},0,0,\pm\frac{\omega}{m}\right)\
.$ (16)

When the photon mass has a non-trivial profile, analytic solutions to eq.
13-(14) only exist in some special cases. Assuming the vector mass-squared
features no time dependence, as appropriate in the rest frame of the bubble
wall, eq. 14 can be written as

$\displaystyle\partial_{\mu}V^{\mu}=-\left(\vec{\nabla}\log
m_{V}^{2}(\vec{x})\right)\cdot\vec{V}\ .$ (17)

If $\vec{k}\parallel\vec{\nabla}m_{V}^{2}$, as in the case of normal
incidence, eq. 17 is non-vanishing for the longitudinal component, whereas
$\partial_{\mu}V^{\mu}=0$ for the transverse modes. As summarized around fig.
1, this is indeed the set-up that we focus on in this work.

###  III.2  A step wall

We will first consider a step-function change in the vector mass:

$m_{V}^{2}(z)=m^{2}+\Delta m_{V}^{2}(z)\qquad\text{with}\qquad\Delta
m_{V}^{2}(z)=\Delta m^{2}\,\Theta(z)\ ,$ (18)

where $\Delta m^{2}=\tilde{m}^{2}-m^{2}>0$ and $\Theta(z)$ is the Heaviside
step function. With this mass profile, the equations of motion feature plane-
wave solutions on either side of the wall localized at $z=0$. These can be
written as

$V^{\mu}_{\perp}(t,z)=e^{-i\omega
t}\begin{cases}(0,1,1,0)e^{ik_{z}z}+r_{\perp}(0,1,1,0)e^{-ik_{z}z}&\quad
z<0\\\ t_{\perp}(0,1,1,0)e^{i\tilde{k}_{z}z}&\quad z>0\end{cases}$ (19)

for the transverse modes, 666We have taken advantage of rotational invariance
in the $x-y$ plane to set $t_{x}=t_{y}\equiv t_{\perp}$ and $r_{x}=r_{y}\equiv
r_{\perp}$. and

$V^{\mu}_{l}(t,z)=e^{-i\omega
t}\begin{cases}\left(\frac{k_{z}}{m},0,0,\frac{\omega}{m}\right)e^{ik_{z}z}+r_{l}\left(\frac{k_{z}}{m},0,0,-\frac{\omega}{m}\right)e^{-ik_{z}z}&\quad
z<0\\\
t_{l}\left(\frac{\tilde{k}_{z}}{\tilde{m}},0,0,\frac{\omega}{\tilde{m}}\right)e^{i\tilde{k}_{z}z}&\quad
z>0\end{cases}$ (20)

for the longitudinal component, with $k_{z}=\sqrt{\omega^{2}-m^{2}}$ and
$\tilde{k}_{z}=\sqrt{\omega^{2}-\tilde{m}^{2}}$ as defined in fig. 1. The
overall normalization of eq. 19-(20) is arbitrary, and with this choice the
reflection and transmission probabilities are given by

$R_{\alpha}=|r_{\alpha}|^{2}\qquad\text{and}\qquad
T_{\alpha}=\frac{\tilde{k}_{z}}{k_{z}}|t_{\alpha}|^{2}\ ,$ (21)

with $\alpha=\perp,l$ for transverse and longitudinal modes respectively.

We can now obtain analytic solutions for both reflection and transmission
probabilities by integrating the equations of motion across the interface. Let
us discuss the transverse modes first. As discussed below eq. 17, the
transverse modes satisfy $\partial_{\mu}V^{\mu}_{\perp}=0$, and eq. 13 reads
$(\Box+m_{V}^{2}(z))V_{\perp}^{\mu}=0$. We then have:

$\lim_{\epsilon\rightarrow
0}\int_{-\epsilon}^{+\epsilon}dz\,\left(\Box+m_{V}^{2}(z)\right)V_{\perp}^{\mu}=0\qquad\Rightarrow\qquad\partial_{z}V^{\mu}_{\perp}\
\text{is continuous at}\ z=0\ .$ (22)

Combined with the requirement that $V^{\mu}_{\perp}$ itself remains
continuous, we can solve for $r_{\perp}$ and $t_{\perp}$. In particular, the
reflection probability is given by

$R_{\perp}=|r_{\perp}|^{2}=\left|\frac{k_{z}-\tilde{k}_{z}}{k_{z}+\tilde{k}_{z}}\right|^{2}\xrightarrow{\omega\gg
m,\tilde{m}}\left(\frac{\Delta
m^{2}}{4\omega^{2}}\right)^{2}=\gamma^{-4}\left(\frac{\Delta
m^{2}}{4m^{2}}\right)^{2}\ .$ (23)

Unsurprisingly, $R_{\perp}$ falls off rapidly in the regime of relativistic
motion.

The behavior of the longitudinal mode is starkly different. We can obtain a
first matching condition by integrating eq. 14 across the wall:

$\lim_{\epsilon\rightarrow
0}\int_{-\epsilon}^{+\epsilon}dz\,\partial_{\mu}\left(m_{V}^{2}(z)V^{\mu}_{l}\right)=0\qquad\Rightarrow\qquad
m_{V}^{2}(z)V^{3}_{l}\ \text{is continuous at}\ z=0\ .$ (24)

Integrating eq. 13 provides the second condition we need. Since
$\partial_{\mu}V_{l}^{\mu}$ no longer vanishes, expanding Eq.(13) we now have
$(\Box+m_{V}^{2}(z))V_{l}^{\mu}-\partial^{\mu}(\partial_{\nu}V_{l}^{\nu})=0$.
The matching condition arising from the $\mu=0$ equation is degenerate with
eq. 24. Instead, focusing on $\mu=3$, we find

$\lim_{\epsilon\rightarrow
0}\int_{-\epsilon}^{+\epsilon}dz\,\left((\Box+m_{V}^{2}(z))V_{l}^{3}+\partial_{z}(\partial_{\nu}V_{l}^{\nu})\right)=0\qquad\Rightarrow\qquad\partial_{t}V_{l}^{0}\
\ \text{is continuous at }z=0\ .$ (25)

Given that the time dependence of $V^{\mu}$ is of the form $e^{-i\omega t}$
for all $z$, the previous requirement is equivalent to demanding that
$V_{l}^{0}$ itself remains continuous. The corresponding reflection
probability now reads

$R_{l}=|r_{l}|^{2}=\left|\frac{\tilde{m}^{2}\,k_{z}-m^{2}\,\tilde{k}_{z}}{\tilde{m}^{2}\,k_{z}+m^{2}\,\tilde{k}_{z}}\right|^{2}\xrightarrow{\omega\gg
m,\tilde{m}}\left(\frac{\tilde{m}^{2}-m^{2}}{\tilde{m}^{2}+m^{2}}\right)^{2}\simeq\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}\ ,$ (26)

where the last step assumes $\Delta m^{2}\ll m^{2}\simeq\tilde{m}^{2}$. As
advertised in the Introduction, in the regime of relativistic motion the
longitudinal reflection probability approaches a constant, independent of
$\gamma$. 777At the level of a Stückelberg theory, a discontinuity in the
number of degrees of freedom (dof) occurs in the limit $m\rightarrow 0$, with
2 vs 3 physical polarizations at either side of the interface. The fact that
$R_{l}\rightarrow 1$ as $m\rightarrow 0$ in eq. 26 reflects the observation
that the longitudinal mode would be unphysical in the region $z<0$, and
therefore must not propagate into the region $z>0$. In an Abelian Higgs UV-
completion, the number of dof of course stays continuous, with the would-be
longitudinal accounted for by the appropriate linear combination of the two
real dof of the complex Higgs. The physical dof are two transverse modes and
the suitable combination of the real scalars, and therefore the treatment
presented in this section is not applicable in the massless regime. Scattering
with $m=0$ has been studied long ago, e.g. Farrar:1994vp . As emphasized in
the Introduction, and in section A.1, we instead focus on cases where (in the
Abelian Higgs language) the Higgs vev is ‘on’ on both sides of the wall.
888See section B.1 for an alternative derivation of eq. 26 making use the of
Goldstone nature of longitudinals at high energies.

###  III.3  A smooth wall

In a realistic situation where the wall thickness is finite and the vector
mass varies smoothly, the analytic results of the previous subsection only
provide an accurate approximation to the reflection and transmission
probabilities in the regime $\omega\ll L^{-1}$. In what follows, we perform a
more general analysis of the case of finite width. We parametrize the dark
photon mass as in eq. 18, except now

$\Delta m_{V}^{2}(z)=\Delta m^{2}\Theta_{L}(z)\ ,$ (27)

where $\Theta_{L}(z)$ is no longer a step function, but rather some smooth
function that approaches 0 and 1 for large negative and positive $z$
respectively, and with an appropriate step-function limit as $L\rightarrow 0$.
We will discuss a specific choice of mass profile at the end of this
subsection, but will otherwise keep things general.

Let us start by building some intuition behind the radically different
behavior exhibited by the transverse and longitudinal components discussed in
III.2 by taking a closer look at the corresponding field equations. It will be
helpful to factor out the time dependence of $V^{\mu}$, as follows

$V^{\mu}(t,z)=v^{\mu}(z)e^{-i\omega t}\ ,$ (28)

and instead focus on the behavior of the $v^{\mu}(z)$. As we discussed
previously, the equations of motion for the transverse modes read
$(\Box+m_{V}^{2}(z))V_{\perp}^{\mu}=0$. In terms of the $v^{\mu}$, this can be
written as

$\displaystyle\left(\partial_{z}^{2}+k_{z}^{2}\right)v^{\mu}_{\perp}(z)=\Delta
m^{2}_{V}(z)\,v^{\mu}_{\perp}(z)\ ,$ (29)

with $k_{z}^{2}=\omega^{2}-m^{2}$. The transverse components obey a
Schrödinger-like equation for a particle moving in one dimension with “energy”
$k_{z}^{2}$ in the presence of a potential $U_{\perp}(z)=\Delta m_{V}^{2}(z)$.
In the $L\rightarrow 0$ limit, our problem reduces to one-dimensional quantum
mechanical scattering on a step potential, with energies above the step.
Indeed, eq. 23 is just the reflection probability for this classic problem.

On the other hand, the Schrödinger equation governing the behavior of the
longitudinal component reads

$\displaystyle\left(\partial_{z}^{2}+k_{z}^{2}\right)\lambda(z)=U_{l}(z)\lambda(z)\qquad\text{with}\qquad
U_{l}(z)=\Delta
m_{V}^{2}(z)+\frac{3}{4}\left(\frac{\partial_{z}m_{V}^{2}(z)}{m_{V}^{2}(z)}\right)^{2}-\frac{1}{2}\frac{\partial^{2}_{z}m_{V}^{2}(z)}{m_{V}^{2}(z)}\
,$ (30)

where we have defined

$\lambda(z)\equiv\frac{m_{V}(z)}{\omega}v^{3}_{l}(z)\qquad\text{such
that}\qquad\lambda(z)\rightarrow\begin{cases}e^{ik_{z}z}+r_{l}e^{-ik_{z}z}&\quad\text{for}\qquad
z\ll-L\\\ t_{l}e^{i\tilde{k}_{z}z}&\quad\text{for}\qquad z\gg+L\end{cases}\ .$
(31)

The effective scattering potential for the longitudinal component, $U_{l}(z)$,
depends on the length scale over which the mass changes not just through the
choice of mass profile but through its derivatives – much unlike its
transverse counterparts. It is this crucial difference that leads to the
strikingly different behavior of longitudinal and transverse modes.

In the relativistic limit, $k_{z}\simeq\omega\gg m$, we might neglect the
first term in $U_{l}(z)$. The derivative terms in the effective potential are
localized in a region of thickness $\sim L$, and the second-derivative term is
dominant whenever $\Delta m^{2}/m^{2}$ is tiny. We then have

$U_{l}(z)\simeq-\frac{1}{2}\left(\frac{\partial^{2}_{z}m_{V}^{2}(z)}{m_{V}^{2}(z)}\right)\simeq-\frac{\Delta
m^{2}}{2m^{2}}\Theta_{L}^{\prime\prime}(z)\ .$ (32)

The step-function limit discussed in the previous subsection corresponds to
$\Theta_{L}^{\prime\prime}(z)\xrightarrow{L\rightarrow 0}\delta^{\prime}(z)$,
and eq. 30 reduces to the Schrödinger equation in a potential
$U_{l}(z)=-\frac{\Delta m^{2}}{2m^{2}}\delta^{\prime}(z)$. Solving this
equation subject to the appropriate boundary conditions, one indeed recovers
the result of eq. 26 to leading order in the ratio $\Delta m^{2}/m^{2}$, as we
summarize in appendix B.2.

Since the longitudinal reflection coefficient in the relativistic regime is
$R_{l}\ll 1$ whenever the change in the dark photon mass is tiny, we can go
further and leverage the one-dimensional Born approximation familiar from
quantum mechanics to obtain an analytic expression for the reflection
probability in the case of a smoothly varying mass. As we summarize in
appendix B.3, the Born approximation to the longitudinal reflection
coefficient can be written in closed form in terms of the effective scattering
potential in eq. 30 as

$R_{l\text{,\,Born}}=\frac{1}{4k_{z}^{2}}\left|\int_{-\infty}^{\infty}dz\
e^{2ik_{z}z}\ U_{l}(z)\right|^{2}\ .$ (33)

Plugging eq. 32 into eq. 33 and integrating by parts, we find

$\displaystyle R_{l\text{,\,Born}}$
$\displaystyle\simeq\frac{1}{4k_{z}^{2}}\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}\left|\Big{[}e^{2ik_{z}z}\
\Theta_{L}^{\prime}(z)\Big{]}_{-\infty}^{+\infty}-2ik_{z}\int_{-\infty}^{\infty}dz\
e^{2ik_{z}z}\ \Theta_{L}^{\prime}(z)\right|^{2}$ (34)
$\displaystyle\simeq\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}\left|\int_{-L}^{L}dz\
\left(1+2ik_{z}z+\mathcal{O}(k_{z}^{2}z^{2})\right)\
\Theta_{L}^{\prime}(z)\right|^{2}$ (35) $\displaystyle=\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}\left(1+\mathcal{O}(k_{z}^{2}L^{2})\right)\ .$ (36)

To get the second line, we used that $\Theta_{L}^{\prime}$ quickly vanishes
for $|z|\gtrsim L$, and we Taylor expanded the exponential for $|k_{z}z|\ll
1$. Since $\Theta_{L}$ interpolates between 0 and 1, $\Theta_{L}^{\prime}$ can
be treated as a probability density function. With this interpretation, the
integral in the second line is essentially an average of $1+2ik_{z}z$ over the
region $(-L,L)$, weighted by this probability density. The average of $k_{z}z$
is bounded in magnitude by $k_{z}L$, giving the final line. Eq. (36) coincides
with the step-function result of eq. 26 provided $k_{z}\simeq\omega\ll
L^{-1}$, as expected. Moreover, it highlights how the existence of a kinematic
regime where the longitudinal reflection coefficient stays nearly constant is
in fact a generic feature of any smooth mass profile, lasting all the way up
to $k_{z}\simeq\omega\sim L^{-1}$.

Given a specific profile, we can use eq. 33 to obtain an analytic
approximation to the reflection probability. For example, for a wall profile
of the familiar kink form

$\Theta_{L}(z)=\frac{1}{4}\left[1+\tanh(z/L)\right]^{2}\ ,$ (37)

one finds

$\displaystyle R_{l\text{,\,Born}}\simeq\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}\frac{\pi^{2}(k_{z}L)^{2}\left[1+(k_{z}L)^{2}\right]}{\sinh^{2}(\pi
k_{z}L)}\ .$ (38)

Eq. (38) reproduces the result of eq. 26 in the regime $\omega\simeq k_{z}\ll
L^{-1}$, up to corrections of $\mathcal{O}(k_{z}^{2}L^{2})$, whereas in the
ultra-relativistic limit it takes the form

$R_{l\text{,\,Born}}\simeq\left(\frac{\Delta m^{2}}{2m^{2}}\right)^{2}\times
4\pi^{2}\left(k_{z}L\right)^{4}e^{-2\pi k_{z}L}\qquad\text{for}\qquad
k_{z}\simeq\omega\gg L^{-1}\ .$ (39)

Indeed, the longitudinal reflection probability dies off exponentially fast –
as expected – in the regime of ultra-relativistic motion.

The results of this subsection are best summarized in fig. 2, where we show
the reflection probability for longitudinal and transverse vectors for the
mass profile in eq. 37, obtained by numerically solving the corresponding
equations of motion. As advertised, the longitudinal reflection coefficient
features a plateau for $\gamma$-factors in the regime
$1\ll\gamma\ll(Lm)^{-1}$, which is well-approximated by the step-function
result of eq. 26. The consequences of this behavior for the evolution of
bubble walls are the topic of the next section.

Figure 2: Reflection probability for longitudinal (blue) and transverse
(orange) massive vectors scattering off a planar wall in the limit of normal
incidence, as a function of the Lorentz $\gamma$-factor. The change in mass
across the interface is $1\%$ and $0.1\%$ for the left and right panels
respectively. Solid lines show numerical results for a smooth mass profile of
the form $m_{V}^{2}(z)=m^{2}+\Delta m^{2}\Theta_{L}(z)$, with $\Theta_{L}(z)$
as in eq. 37. Dashed lines correspond to the step function results of eq. 23
and (26). As anticipated, the step-function analysis provides an excellent
approximation up to $\gamma\sim(mL)^{-1}$. Details on how we obtain our
numerical results are summarized in appendix B.4.

##  IV  Dark photon friction on bubble walls

We now discuss the implications of the results presented in section III for
the dynamics of expanding vacuum bubbles in the early Universe. In section
IV.1, we compute the pressure on an expanding wall due to the presence of dark
photons, and argue that the requirement for run-away walls can be much
stronger than previously believed. Section IV.2 focuses on the dynamics of
bubble walls that reach an equilibrium regime as a result of longitudinal
friction, including a self-consistent determination of the equilibrium
$\gamma$-factor. Once in equilibrium, most of the energy released in the
transition goes into making a fraction of the dark photons relativistic. The
fate of this dark radiation depends sensitively on the size of self-
interactions among the dark photons, as well as between the dark photons and
particles in the thermal plasma, as we discuss in section IV.3. If interaction
rates are negligible – an assumption more likely to hold if the sector
undergoing vacuum decay is cold – we argue that the reflected dark photons
accumulate in a thin “shell” of dark radiation surrounding the expanding
bubbles. If they remain relativistic until late times, their contribution to
$\Delta N_{\text{eff}}$ could be observable for phase transitions with
strength $\alpha\sim 10^{-2}-10^{-1}$. Alternatively, if the scalar sector is
in equilibrium with the thermal plasma, interactions between the reflected
dark photons and $\phi$ particles can easily be efficient, in which case the
energy density in dark radiation gets transferred instead into the thermal
fluid.

###  IV.1  Maximum Dynamic Pressure

It is helpful to rearrange eq. 10 using the relationship
$T_{\lambda}=1-R_{\lambda}$, as follows:

$\mathcal{P}=\gamma|\vec{v}|n_{V}\left\\{\frac{1}{3}R_{l}(k_{z}+\tilde{k}_{z})+(k_{z}-\tilde{k}_{z})+\frac{2}{3}R_{\perp}(k_{z}+\tilde{k}_{z})\right\\}\
.$ (40)

In the relativistic limit, $k_{z}+\tilde{k}_{z}\simeq 2\gamma m$ and
$k_{z}-\tilde{k}_{z}\simeq\Delta m^{2}/(2\gamma m)$, and eq. 40 reads

$\mathcal{P}\simeq\frac{2}{3}\gamma^{2}\rho_{V}R_{l}+\rho_{V}\frac{\Delta
m^{2}}{2m^{2}}+\frac{4}{3}\gamma^{2}\rho_{V}R_{\perp}\qquad\text{for}\qquad\gamma\gg
1\ .$ (41)

The second term above is just the asymptotic contribution of eq. 11. The last
term corresponds to reflections of transverse modes. As discussed around eq.
23, $R_{\perp}\propto\gamma^{-4}$ already in the inter-relativistic regime,
and therefore this third term falls off as $\gamma^{-2}$ for large $\gamma$.
In contrast, the peculiar behavior of the longitudinal reflection probability,
$R_{l}$, which stays approximately constant in the region of inter-
relativistic motion, leads to a contribution to the overall pressure on the
expanding interface that grows $\propto\gamma^{2}$. In total, neglecting the
last term above and substituting the expression for $R_{l}$ appropriate in the
inter-relativistic regime (remember eq. 26), we have:

$\displaystyle\mathcal{P}\simeq\underbrace{\frac{2}{3}\gamma^{2}\rho_{V}\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}}_{\text{longitudinal reflections}}+\
\underbrace{\rho_{V}\frac{\Delta
m^{2}}{2m^{2}}}_{\mathcal{P}_{\infty}}\qquad\text{for}\qquad
1\ll\gamma\ll(mL)^{-1}\ .$ (42)

We emphasize that this $\gamma^{2}$-growing pressure is a transient phenomenon
that is only present for $\gamma$-factors within the range indicated in eq.
42. Once $\gamma\gg(mL)^{-1}$, the longitudinal reflection probability dies
off exponentially fast, leaving the second term above as the sole significant
contribution to the overall pressure.

A consequence of our previous discussion is that the pressure exerted on the
expanding interface will reach a maximum near the end of the inter-
relativistic regime, before dropping down to
$\mathcal{P}_{\infty}\simeq\rho_{V}\times\Delta m^{2}/2m^{2}$ in the ultra-
relativistic limit. As advertised in the Introduction, we will refer to this
pressure peak as Maximum Dynamic Pressure. Parametrically, it is given by

$\displaystyle\mathcal{P}_{\text{mdp}}\simeq\frac{2}{3}\gamma_{\text{max}}^{2}\rho_{V}R_{l}+\rho_{V}\frac{\Delta
m^{2}}{2m^{2}}\sim\frac{\rho_{V}}{(Lm)^{2}}\left(\frac{\Delta
m^{2}}{2m^{2}}\right)^{2}+\rho_{V}\frac{\Delta m^{2}}{2m^{2}}\ ,$ (43)

where $\gamma_{\text{max}}\sim(mL)^{-1}$ is the value of $\gamma$ at Maximum
Dynamic Pressure. As a result, for bubble walls to become run-away, the
following condition must be satisfied:

$\text{Run-away criterion:}\qquad\qquad\Delta
V>\mathcal{P}_{\text{mdp}}\qquad\qquad\text{(this work)}\ .$ (44)

Eq.(44) replaces eq. 3 as a diagnostic of run-away bubble walls in the
presence of phase-dependent massive dark photons in transitions with access to
the regime of inter-relativistic motion, and is a primary new result of this
work.

If the first term in eq. 43 is a small correction on top of the asymptotic
pressure, $\mathcal{P}_{\infty}$, the effect of longitudinal mode reflections
will be largely irrelevant to describe the dynamics of the expanding bubbles.
However, if the first term dominates then eq. 44 can be much stronger than the
requirement $\Delta V>\mathcal{P}_{\infty}$ quoted in eq. 3. Parametrically,
the effect of longitudinal reflections will dominate the overall pressure on
the expanding interface provided

$mL\ll\sqrt{\frac{\Delta m^{2}}{m^{2}}}\ .$ (45)

When $\Delta m^{2}/m^{2}\ll 1$, this is a stronger requirement than the
condition in eq. 5 for the inter-relativistic regime to be accessible,
although the main feature in both cases is that there must be a significant
hierarchy between the mass of the dark photon and the energy scale $L^{-1}$
characterizing the thickness of the bubble wall.

Our discussion thus far has only been concerned with the pressure due to a
population of cold dark photons. If the sector undergoing the transition is
cold, such that vacuum decay proceeds via quantum tunneling and there is no
thermal plasma that interacts with the bubble walls, this will be a good
approximation to the overall pressure. Alternatively, if the scalar sector is
in equilibrium with the thermal fluid, an additional source of pressure will
be present due to interactions between the plasma and the wall. As mentioned
in the Introduction, particles in the plasma that gain mass across the wall
contribute as $\mathcal{P}_{\infty}\sim\Delta m^{2}T^{2}$ in the relativistic
limit Bodeker:2009qy . In perturbative theories this source of friction can
easily fall below $\Delta V$, and therefore won’t be large enough to obstruct
the acceleration of the bubble walls. Potentially more significant are those
cases where the phase transition sector features gauge bosons that acquire a
mass as they cross the wall. As anticipated in section I, this can lead to an
additional source of pressure from the transition radiation emitted as charged
particles cross the interface, of the form $\mathcal{P}_{\infty}\sim\gamma
g^{2}\Delta m_{v}T^{3}$ Bodeker:2017cim ; Azatov:2020ufh ; Gouttenoire:2021kjv
. For this source of friction to be subdominant to that from longitudinal dark
photons, one would need

$\gamma_{\text{eq}}g^{2}\Delta m_{v}T_{*}^{3}\lesssim\Delta V\ .$ (46)

This condition can be interpreted as an upper bound on $T_{*}$ relative to the
typical energy scale of the phase transition. Parametrically, taking for
simplicity $\Delta V\sim v^{4}$ and $\Delta m_{v}\sim gv$, we find

$\displaystyle\frac{T_{*}}{v}$
$\displaystyle\lesssim\frac{1}{g\,\gamma_{\text{eq}}^{1/3}}$ (47)
$\displaystyle\sim 10^{-3}\,\left(\frac{1}{g}\right)\left(\frac{\Delta
m^{2}/m^{2}}{10^{-4}}\right)^{1/3}\left(\frac{10^{-2}}{\alpha}\right)^{1/6}\left(\frac{100\
\text{GeV}}{T_{*}}\right)^{1/6}\left(\frac{\rho_{\text{dm}}}{\rho_{V}}\right)^{1/6}\
.$ (48)

If the relevant gauge couplings are $g=\mathcal{O}(1)$ – as in the Standard
Model – then neglecting transition radiation would require $T_{*}\ll v$, i.e.
the transition needs to be significantly super-cooled. More generally, in
hidden sectors where the relevant gauge couplings are $g\ll 1$, the above
condition could be satisfied even within ‘standard’ thermal transitions where
$T_{*}\sim v$. A more comprehensive analysis of the class of thermal
transitions for which this assumption holds is an interesting direction for
future investigation.

Fig. 3 shows the pressure on an expanding interface due to massive dark
photons, relative to its asymptotic value in the limit
$\gamma\rightarrow\infty$, in cases where the inter-relativistic kinematic
regime identified in eq. 5 is accessible during the wall’s expansion. The
evolution of bubble walls that fail the run-away condition of eq. 44 as a
result of longitudinal friction is the topic to which we now turn.

Figure 3:  The pressure exerted by massive dark photons on an expanding bubble
wall can feature a non-monotonic dependence on the wall $\gamma$-factor due to
reflections of longitudinal modes, reaching a peak at $\gamma\sim(mL)^{-1}$
that is potentially much larger than its asymptotic value in the ultra-
relativistic limit. The criterion for run-away bubble walls can therefore be
far stronger than simply requiring $\Delta V>\mathcal{P}_{\infty}$, as
discussed around eq. 44. Left: increasing the relative change in the dark
photon mass raises the Maximum Dynamic Pressure without significantly altering
its position. Right: decreasing the wall thickness increases _both_ the height
and position of the MDP. This behavior is in qualitative agreement with the
discussion in and around eq. 43. Both plots have been obtained by evaluating
eq. 40 after numerically obtaining reflection coefficients corresponding to
the smooth wall profile of eq. 37.

###  IV.2  Equilibrium $\gamma$-factor and energy budget

If the Maximum Dynamic Pressure is dominated by the reflection of longitudinal
modes, and the run-away criterion of eq. 44 is not satisfied, then bubble
walls will reach an equilibrium regime once
$\mathcal{P}(\gamma_{\text{eq}})\simeq\Delta V$. Parametrically, the
equilibrium $\gamma$-factor is given by

$\displaystyle\gamma_{\text{eq}}$ $\displaystyle\simeq\left(\frac{3\Delta
V}{2\rho_{V}R_{l}}\right)^{1/2}$ (49) $\displaystyle\sim
10^{9}\left(\frac{10^{-4}}{\Delta
m^{2}/m^{2}}\right)\left(\frac{\alpha}{10^{-2}}\right)^{1/2}\left(\frac{T_{*}}{100\
\text{GeV}}\right)^{1/2}\left(\frac{\rho_{\text{dm}}}{\rho_{V}}\right)^{1/2}\
.$ (50)

How easy is it for bubble walls to reach an equilibrium regime as a result of
longitudinal reflections? An obvious self-consistency condition on our
determination of $\gamma_{\text{eq}}$ in eq. 49 is that it lies below the
$\gamma$-factor at Maximum Dynamic Pressure, i.e.

$\gamma_{\text{eq}}\lesssim\frac{1}{mL}\ .$ (51)

Additionally, a “kinematic” condition is that the expanding walls reach
equilibrium _before_ the bubble walls collide and the phase transition ends.
The bubble radius at collision is set by the Hubble scale at the epoch of the
phase transition. Demanding that the size of the expanding bubbles at the
onset of the equilibrium regime is a fraction $x<1$ of the Hubble radius, we
find

$R_{\text{eq}}\simeq\gamma_{\text{eq}}R_{0}\lesssim
xH(T_{*})^{-1}\qquad\Rightarrow\qquad\gamma_{\text{eq}}\lesssim\frac{x}{R_{0}H(T_{*})}\
,$ (52)

where $R_{0}$ is the critical bubble radius.

Eq.(51) and (52) can be interpreted in various ways, but perhaps the most
relevant for us is to regard them as _lower bounds_ on the fractional change
of the dark photon mass for the effect of longitudinal pressure to be large
enough to stop the acceleration of the bubble walls, as follows:

$\frac{\Delta m^{2}}{m^{2}}\gtrsim\left(\frac{\Delta
m^{2}}{m^{2}}\right)_{\text{min}}\equiv 2\left(\frac{3\Delta
V}{2\rho_{V}}\right)^{1/2}\,\times\,\text{Max}\left\\{mL,\,x^{-1}R_{0}H(T_{*})\right\\}\
.$ (53)

This is illustrated in fig. 4, where we show contours of $\sqrt{(\Delta
m^{2}/m^{2})_{\text{min}}}$ for various choices of the underlying parameters,
as described in the caption. As can be seen in fig. 4, even extremely small
changes in the mass of the dark photon across the interface can cause enough
friction to halt the acceleration of the bubble walls.

Figure 4: Contour lines indicating the lower bound on the fractional change of
the dark photon mass for longitudinal reflections to create enough friction to
halt the acceleration of the bubble walls. These contours saturate the
inequality in eq. 53, with the choice $x=10^{-1}$ for illustration. The energy
density in cold dark photons is taken to be that of the dark matter, and the
difference in energy densities, $\Delta V$, is taken to be $1\%$ of that in
the Standard Model bath (i.e. $\alpha=10^{-2}$ in eq. 6). The blue shaded area
corresponds to dark photon masses such that the corresponding Compton
wavelength exceeds the Hubble radius, $m^{-1}>H(T_{*})^{-1}$ – a regime that
lies outside the validity of our discussion. Left: The wall thickness is
related to the temperature of the thermal plasma at the phase transition epoch
as $L=T_{*}^{-1}$, as expected in a ‘standard’ thermal transition. Right:
$L=(10^{4}\,T_{*})^{-1}\ll T_{*}^{-1}$, as could be the case if the phase
transition took place within a cold hidden sector with a characteristic energy
scale $\gg T_{*}$, or for a thermal transition featuring significant super-
cooling (recall discussion below eq. 7).

After the bubble walls reach an equilibrium speed, they carry a decreasing
fraction of the total energy available in the transition:

$\frac{E_{\text{wall}}}{E_{\text{total}}}\simeq\frac{4\pi
R(t)^{2}\gamma_{\text{eq}}\sigma}{\frac{4\pi}{3}R(t)^{3}\Delta
V}\sim\frac{\gamma_{\text{eq}}\sigma}{\Delta V\,R(t)}\ ,$ (54)

where $R(t)$ is the bubble radius at time $t$. As $R(t)$ grows,
$E_{\text{wall}}/E_{\text{total}}$ quickly becomes tiny. By comparison, the
above ratio is identically 1 for transitions in vacuum, where all the
available energy goes into accelerating the bubble walls Coleman:1977py . In
thermal transitions where the walls reach an equilibrium speed due to friction
from the thermal plasma, most of the available energy goes instead into
producing motion in the form of sound waves or hydrodynamic turbulence, as
mentioned in the Introduction and summarized in Caprini:2015zlo ;
Caprini:2019egz . In the case at hand, as we will now show, the energy
released in the transition goes instead into accelerating a fraction of the
dark photons, turning them into dark radiation.

In the rest frame of the bubble wall, reflected longitudinal modes propagate
away from the interface with Lorentz-factor $\gamma_{\text{eq}}$, while in the
rest frame of the dark photons far away from the wall – loosely, the rest
frame of the dark matter – the $\gamma$-factor of the reflected dark photons
is given by

$\gamma_{\text{dr}}=\frac{1+|\vec{v}_{\text{eq}}|^{2}}{1-|\vec{v}_{\text{eq}}|^{2}}\simeq
2\gamma_{\text{eq}}^{2}\gg\gamma_{\text{eq}}\ .$ (55)

Within a Hubble volume, the average number density of dark photons that become
relativistic as the bubble walls sweep across the dark matter from the onset
of the equilibrium regime until the bubbles collide can be approximated by

$\langle
n_{\text{dr}}\rangle\simeq\frac{1}{3}R_{l}n_{V}\left(1-\frac{R_{\text{eq}}^{3}}{R_{\text{coll}}^{3}}\right)\
.$ (56)

The factor in parenthesis takes care of the fact that we are only interested
in keeping track of the fraction of dark matter that gets converted into dark
radiation during equilibrium (before, most of the energy released as the
bubbles grow goes into accelerating the bubble walls). Obviously, if the
bubble walls only reach equilibrium right before they collide,
$R_{\text{eq}}\approx R_{\text{coll}}$, then $\langle n_{\text{dr}}\rangle$
will be correspondingly tiny. However, provided the onset of equilibrium takes
place well before collision, then $\langle
n_{\text{dr}}\rangle\simeq\frac{1}{3}R_{l}n_{V}$. For example, taking
$R_{\text{eq}}\simeq\gamma_{\text{eq}}R_{0}$ and $R_{\text{eq}}\simeq
xH(T_{*})^{-1}$, we find $R_{\text{eq}}^{3}/R_{\text{coll}}^{3}\sim
10^{-15}\ll 1$, where we have evaluated all parameters as in eq. 50, and for
illustration we have taken $x=10^{-2}$ as well as $R_{0}^{-1}=T_{*}=100\
\text{GeV}$. More generally, notice that having $R_{\text{eq}}\lesssim
R_{\text{coll}}$ is a requirement for dark photon reflections to affect
significantly the evolution of the bubble walls (as per our discussion around
eq. 50), and so we proceed under this assumption.

Right after bubble walls collide, the volume-averaged energy density in dark
radiation is therefore

$\rho_{\text{dr}}=\omega_{\text{dr}}\,\langle
n_{\text{dr}}\rangle\simeq\frac{2}{3}\gamma_{\text{eq}}^{2}\,R_{l}\,\rho_{V}\qquad\text{at}\qquad
T\sim T_{*}\ ,$ (57)

where $\omega_{\text{dr}}=\gamma_{\text{dr}}m$ and $\gamma_{\text{dr}}$ is
given in eq. 55. Notice the right-hand-side above is equal to $\Delta V$ by
eq. 49 – consistent with energy conservation in the rest frame of the dark
matter. Thus, in phase transitions where bubble walls reach an equilibrium
regime as a result of friction from the dark photons, most of the difference
in vacuum energy densities goes into turning a fraction of the (cold) dark
photons into dark radiation.

###  IV.3  Fate of the dark radiation

The fate of the reflected dark photons depends sensitively on the values of
the various underlying parameters. As a result, general statements about the
later evolution of the dark radiation are not possible, and a comprehensive
study spanning all available parameter space is well beyond the scope of this
work. Instead, we will focus on highlighting the possible outcomes given
certain assumptions.

Although sweeping statements are not possible, there are two qualitatively
different cases that merit separate consideration, depending on whether the
sector undergoing vacuum decay is part of the thermal plasma or belongs in a
cold hidden sector such that the transition proceeds via quantum tunneling. We
will focus on the cold scenario first. In this case, the absence of a thermal
population of particles that interact with the wall background leaves the dark
photons are the sole source of pressure on the expanding interface, and the
evolution of the dark radiation as the bubbles grow depends primarily on the
strength of the self-interactions among the dark photons. As discussed in
section A.1, quartic dark photon self-interactions are generally part of the
low energy theory, and in the context of an Abelian Higgs UV-completion the
relevant quartic coupling is as given in eq. 68. Interactions among the dark
radiation and the cold dark photons occur at a center-of-mass energy
$\sqrt{s}/2\simeq\gamma_{\text{eq}}m$, which always fall below the cutoff
scale of the effective theory. In this kinematic regime, the relevant cross
section is as given in eq. 72, and one finds that the corresponding
interaction rate can be tiny compared to $H(T_{*})$ across significant
portions of parameter space.

When interactions between the dark matter and the reflected dark photons are
negligible, the following picture emerges. Going back to eq. 55, notice that
(in the dark matter frame) the speed of the dark radiation is ever-so-slightly
larger than the speed of the bubble wall:

$\displaystyle\Delta v_{\text{dr}}$
$\displaystyle\equiv|\vec{v}_{\text{dr}}|-|\vec{v}_{\text{eq}}|=|\vec{v}_{\text{eq}}|\frac{1-|\vec{v}_{\text{eq}}|^{2}}{1+|\vec{v}_{\text{eq}}|^{2}}\simeq\frac{|\vec{v}_{\text{eq}}|}{2\gamma_{\text{eq}}^{2}}\ll
1\ .$ (58)

As a result, from the beginning of the equilibrium regime until the moment of
bubble wall collisions, reflected dark photons become uniformly distributed on
a shell of thickness $\Delta L$ in front of the interface, with

$\Delta L\sim\Delta v_{\text{dr}}\times\Delta
t\simeq\frac{|\vec{v}_{\text{eq}}|\Delta
t}{2\gamma_{\text{eq}}^{2}}\lesssim\frac{R_{\text{coll}}}{2\gamma_{\text{eq}}^{2}}\ll
R_{\text{coll}}\ ,$ (59)

where $\Delta t$ refers to the time interval between when equilibrium is
reached and collision of the bubble walls, and we have used
$|\vec{v}_{\text{eq}}|\Delta t\lesssim R_{\text{coll}}$. Thus, as the bubbles
grow bigger, a shell of dark radiation forms, moving ever-so-slightly in front
of the bubble walls. Although the walls move at constant speed, most of the
energy density remains localized in a thin layer close to the surface of the
expanding bubbles.

The fate of these shells of radiation depends on the interactions between
relativistic dark photons as the bubble walls meet. Now, the relevant center-
of-mass-energy is $\sqrt{s}/2\sim\omega_{\text{dr}}\simeq
2m\gamma_{\text{eq}}^{2}$, and interactions are often well-described within
the high energy Goldstone regime of eq. 74. To estimate the interaction rate,
notice that although the average number density of dark radiation is well-
approximated by $\langle n_{\text{dr}}\rangle\simeq\frac{1}{3}R_{l}n_{V}$, the
distribution of relativistic dark photons is highly inhomogeneous, with
$n_{\text{dr}}\sim\langle n_{\text{dr}}\rangle H^{-1}/\Delta L\gg\langle
n_{\text{dr}}\rangle$ in a thin shell surrounding the bubble walls and zero
elsewhere. This cross section defines a mean-free-path
$\lambda_{\text{mfp}}=(\sigma n_{\text{dr}})^{-1}$. Comparing this length
scale to the typical thickness of one of these shells we find, e.g.

$\displaystyle\frac{\lambda_{\text{mfp}}}{\Delta
L}\sim\frac{10^{8}}{\lambda_{\Phi}^{2}}\left(\frac{m}{1\
\text{eV}}\right)^{3}\left(\frac{10^{-4}}{\Delta
m^{2}/m^{2}}\right)^{6}\left(\frac{T_{*}}{100\
\text{GeV}}\right)\left(\frac{\alpha}{10^{-2}}\right)^{2}\left(\frac{\rho_{\text{dm}}}{\rho_{V}}\right)^{3}\
.$ (60)

The above expression has a strong dependence on a number of parameters,
especially the fractional change in the mass of the dark photon. But
interestingly it can remain $\gg 1$ in a large region of the relevant
parameter space where longitudinal reflections are relevant to the evolution
of the bubble walls. In this case, the shells of dark radiation will pass each
other without significant dissipation, a process that could last for much
longer than the usual duration of a cosmological phase transitions. The long-
lasting motion of these shells could greatly enhance the strength of the
gravitational wave signal, a possibility that motivates more careful
exploration of this potential new source of gravitational waves. We will
return to this topic in future work.

Moreover, it is interesting to consider the limiting possibility that the
reflected dark photons remain relativistic long after the phase transition is
over, leading to a dark radiation signal. Making the optimistic assumption
that significant losses to gravitational radiation and other “inelastic”
processes can be ignored, the requirement that the reflected dark photons
remain relativistic at temperatures $T\leq T_{*}$ can be written as:

$\displaystyle\gamma_{\text{dr}}(T\leq T_{*})\simeq
2\gamma_{\text{eq}}^{2}\frac{a(T_{*})}{a(T)}\approx 2\times
10^{13}\left(\frac{10^{-4}}{\Delta
m^{2}/m^{2}}\right)^{2}\left(\frac{\alpha}{10^{-2}}\right)\left(\frac{\rho_{\text{dm}}}{\rho_{V}}\right)\left(\frac{T}{1\
\text{MeV}}\right)\gtrsim 1\ .$ (61)

Of particular interest are the temperatures of Big Bang Nucleusynthesis
($T_{\text{BBN}}\sim 1\ \text{MeV}$) and recombination ($T_{\text{rec}}\sim 1\
\text{eV}$), when $\Delta N_{\text{eff}}$ bounds exist. As can be inferred
from eq. 61, this is easily the case provided $\alpha$ is not too tiny. When
this is the case, the corresponding contribution to $\Delta N_{\text{eff}}$
can be written as

$\displaystyle\Delta N_{\rm eff}$
$\displaystyle=\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\left(\frac{\rho_{\text{dr}}(T_{*})}{\rho_{\gamma,0}}\right)\left(\frac{a(T_{*})}{a(T_{0})}\right)^{4}$
(62) $\displaystyle\simeq
1.4\left(\frac{\alpha}{0.1}\right)\frac{g_{*}(T_{*})}{g_{*,s}(T_{*})^{4/3}}$
(63) $\displaystyle\simeq
0.3\left(\frac{\alpha}{0.1}\right)\left(\frac{g_{*}(100\ \rm
GeV)}{g_{*}(T_{*})}\right)^{1/3}\ ,$ (64)

where $\rho_{\gamma,0}$ refers to the current energy density in (Standard
Model) photons. In the first step above we have used
$\rho_{\text{dr}}(T_{*})\simeq\Delta V$, as discussed around eq. 56-(57), and
in the final step we have ignored the difference between $g_{*,s}(T_{*})$ and
$g_{*}(T_{*})$. Interestingly, phase transitions with $\alpha\sim 10^{-1}$
would already be probed by current $\Delta N_{\text{eff}}$ constraints –
though we emphasize that this is under the assumption that the highly
relativistic dark photons suffer no significant energy loss since the time of
the phase transition other than redshift due to Hubble expansion. Under this
assumption, fig. 5 shows the prediction for $\Delta N_{\text{eff}}$ as a
function of the characteristic temperature of the phase transition, for
various values of $\alpha$. Strong phase transitions with $\alpha>0.1$ would
be probed by current measurements, while weaker transitions with
$\alpha\gtrsim 10^{-2}$ could be probed by CMB-Stage 4 observations CMBstage4
.

Figure 5:  In phase transitions where bubble walls reach an equilibrium regime
as a result of dark photon reflections, most of the difference in vacuum
energy densities goes into turning a fraction of the (cold) dark photons into
dark radiation. If the dark radiation remains relativistic until late times,
an observable contribution to $\Delta N_{\text{eff}}$ is possible. Plotted are
the maximum possible contributions to $\Delta N_{\text{eff}}$, as discussed in
the main text, as a function of the thermal plasma at the epoch of the
transition, $T_{*}$. Solid gray lines correspond to current bounds
Aghanim:2018eyx , and dashed gray lines show the projected sensitivities of
upcoming observatories CMBstage4 ; SimonsObservatory:2018koc .

Because bubble acceleration is stalled by the reflection of particles, this
effect is a particularly efficient manner of converting the latent heat of the
false vacuum to relativistic matter, thus leading to a significant
contribution to $\Delta N_{\rm eff}$ if the reflected particles remain
relativistic until late times. When instead bubbles of the new phase are run-
away or reach a terminal velocity by dispersing energy into the thermal
plasma, only the resulting stochastic background of gravitational waves
contributes towards dark radiation, and $\Delta N_{\text{eff}}$ is a less
important observable.

The second scenario highlighted early on in this subsection corresponds to the
case where the phase transition sector is in equilibrium with the thermal
plasma. In this case, the reflected dark photons will also interact with the
population of $\phi$’s in the thermal fluid via eq. 8 (or its UV-complete
version, as described in section A.1). Interactions take place at center-of-
mass energies
$\sqrt{s}/2\sim\sqrt{2\,T_{*}\,\omega_{\text{dr}}}/2\simeq\gamma_{\text{eq}}\sqrt{m\,T_{*}}$,
and the corresponding cross section is well-approximated by eq. 75 or eq. 76
depending on whether $\sqrt{s}$ falls above or below the cutoff scale of the
effective theory. As discussed in section A.2, the corresponding interaction
rate can be well above $H(T_{*})$, although once more this depends sensitively
on the values of the underlying parameters. At any rate, interactions between
the reflected dark photons and the thermal plasma provide an additional
obstacle for the dark radiation to propagate undisturbed. It is therefore more
likely that the energy density in the dark radiation gets distributed within
the primordial plasma, in which case the main source of gravitational
radiation would come from motion within the thermal fluid, similar to the case
of phase transitions that reach an equilibrium regime as a result of thermal
pressure.

Determining more precisely under what circumstances the reflected dark photons
remain relativistic until late times, and therefore contribute to $\Delta
N_{\text{eff}}$, is an important question worthy of further attention.

##  V  Conclusions

In this work, we have discussed a new physical effect that can affect the
evolution of cosmological vacuum bubbles expanding against a population of
phase-dependent massive dark photons, with a special focus on the case where
the dark photons furnish the dark matter. Namely, the existence of a transient
relativistic regime, for sufficiently thin walls, characterized by a constant
reflection probability of longitudinal dark photons. The reflection of
longitudinal modes creates a pressure on the expanding interface that features
a characteristic non-monotonic dependence on the $\gamma$-factor of the bubble
wall, reaching a peak at intermediate values of $\gamma$ that we have dubbed
Maximum Dynamic Pressure. The existence of a MDP that exceeds the asymptotic
value of the pressure in the ultra-relativistic limit can make it much harder
for bubble walls to become run-away, even in the absence of a thermal plasma
that interacts with the wall background.

Our work opens a number of avenues for future exploration. In phase
transitions where bubble walls reach an equilibrium regime as a result of this
effect, the later evolution of the reflected dark photons could modify the
features of the resulting gravitational wave signal and, in some cases, lead
to an observable contribution to $\Delta N_{\text{eff}}$ if the reflected dark
photons remain relativistic until late times. As discussed in section IV
(specially IV.3), the extent to which this happens depends on a variety of
considerations, most importantly on whether the sector undergoing vacuum decay
is cold or, instead, is in thermal equilibrium with the primordial plasma
surrounding the expanding bubbles. Understanding more generally the possible
implications of the dark radiation for these two broad classes of models is
clearly an important topic that deserves further attention.

Moreover, although we have focused exclusively on the case of phase-dependent
massive dark photons, it is possible that the phenomenon of MDP on expanding
bubble walls could be realized in scenarios beyond this example. Given the
impact this can have on the dynamics of bubble walls, which in turn largely
determine the features of the resulting gravitational wave signal, this is an
important question worthy of further investigation.

##  Acknowledgments

We thank Alex Azatov, Nathaniel Craig, Yann Gouttenoire, John March-Russell,
Mehrdad Mirbabayi, Ken Van Tilburg and Govanni Villadoro for helpful
discussions. IGG gratefully acknowledges support from NSF Grant PHY-2207584,
and from the James Arthur Postdoctoral Fellowship at NYU. RPB is grateful for
the support and hospitality of the KITP through its graduate fellowship
program, which resulted in the present collaboration. At the KITP, this
research was supported in part by the National Science Foundation under Grant
No. NSF PHY-1748958. The research of GK was supported by the Len DeBenedictis
Graduate Fellowship and DOE grant number DE-SC0011702.

##  Appendix A Beyond the effective theory

###  A.1 Abelian Higgs UV-completion

In this appendix, we briefly discuss how the effective Lagrangian of eq. 7-(8)
may arise from a more complete framework. Our goal is not to be exhaustive,
but rather to identify the necessary features of underlying models giving rise
to our effective theory. The simplest UV-completion of eq. 7-(8) can be
written in terms of an Abelian Higgs model:

$\mathcal{L}_{\text{UV}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_{\mu}\Phi|^{2}-V_{\Phi}(|\Phi|)+\frac{1}{2}(\partial_{\mu}\phi)^{2}-V_{\phi}(\phi)\,+\,\frac{\eta}{2}\phi^{2}|\Phi|^{2}\
,$ (65)

where $D_{\mu}\Phi\equiv(\partial_{\mu}-ig^{\prime}V_{\mu})\Phi$. Assuming a
Higgs potential of the usual form:
$V_{\Phi}(|\Phi|)=\lambda_{\Phi}/2(|\Phi^{2}|-v^{\prime 2}/2)^{2}$, the
corresponding radial mode gets a mass
$m_{\rho}=\sqrt{\lambda_{\Phi}}v^{\prime}$, whereas $m=g^{\prime}v^{\prime}$
is the mass of the dark photon. Integrating out the radial mode, one obtains a
low energy effective theory featuring additional interactions among the light
degrees of freedom. In particular, a term such as eq. 8 – repeated here for
convenience – is generated at tree-level:

$\mathcal{L}\supset\frac{\kappa}{2}\phi^{2}V^{\mu}V_{\mu}\qquad\quad\text{with}\qquad\quad\kappa=\frac{\eta\,m^{2}}{m_{\rho}^{2}}=\frac{\eta
g^{\prime 2}}{\lambda_{\Phi}}\ .$ (66)

Notice in particular that the ratio $\Delta m^{2}/m^{2}$ is independent of
$g^{\prime}$, and it is therefore controlled by a different set of parameters
of the underlying model compared to the overall mass of the dark photon. In
this context, the scale of UV-completion is $\Lambda=m_{\rho}$, which can be
written as

$m_{\rho}=\frac{\sqrt{\eta}v}{\sqrt{\Delta m^{2}/m^{2}}}\lesssim\frac{4\pi
v}{\sqrt{\Delta m^{2}/m^{2}}}\ .$ (67)

This reproduces our EFT expectation for the upper bound on the cutoff scale in
eq. 9 subject to the perturbativity requirement $\eta\lesssim 16\pi^{2}$.

Of course, the low energy effective theory contains interactions among the
light degrees of freedom beyond eq. 8. Of particular interest are self-
interactions among the dark photons. Indeed, a quartic interaction is
generated at tree-level, of the form

$\mathcal{L}\supset\lambda_{V}(V_{\mu}V^{\mu})^{2}\qquad\quad\text{with}\qquad\quad\lambda_{V}\sim\frac{g^{\prime
4}}{\lambda_{\Phi}}=\lambda_{\Phi}\frac{m^{4}}{m_{\rho}^{4}}\lll 1\ .$ (68)

Interactions among the dark photons, as well as between the dark photons and a
potential population of $\phi$ particles in the thermal plasma, are relevant
(_i_) to assess the extent to which our assumptions that the dark matter is
cold and non-interacting are self-consistent, and (_ii_) to determine the
evolution of the reflected dark photons, which is the topic of section IV.3.
For convenience, we summarize the relevant interaction rates in the remainder
of this appendix.

###  A.2 Interaction rates

At low momenta, eq. 68 leads to a self-interaction among the dark photons of
the form

$\sigma_{\scriptstyle VV}\sim\frac{\lambda_{V}^{2}}{8\pi
m^{2}}=\frac{\lambda_{\Phi}^{2}}{8\pi
m_{\rho}^{2}}\left(\frac{m}{m_{\rho}}\right)^{6}\qquad\text{for}\qquad|\vec{k}|\ll
m\ .$ (69)

The corresponding dark matter self-interaction rate is always tiny compared to
the Hubble rate in the early Universe in the region of parameter space
relevant to this work. Potentially more relevant are interactions between the
dark photons and $\phi$ particles in the thermal fluid. In particular,
processes of the form $V\phi\rightarrow V\phi$ could be efficient at
transferring energy from the plasma to the dark matter, unless the
corresponding interaction rates are tiny. The relevant cross section is of the
form, parametrically:

$\sigma_{\scriptstyle V\phi}\sim\frac{\kappa^{2}}{8\pi
T_{*}^{2}}\sim\frac{1}{8\pi T_{*}^{2}}\left(\frac{\Delta
m^{2}}{m^{2}}\right)^{2}\frac{m^{4}}{T_{*}^{4}}\ ,$ (70)

where $\kappa\simeq\Delta m^{2}/v^{2}$, as discussed below eq. 8, and we’ve
taken $m_{\phi}\sim v\sim T_{*}$, which should be a good approximation at the
epoch of the phase transition (barring a tiny quartic coupling for $\phi$).
Assuming a thermal population of $\phi$ particles, with number density
$n_{\phi}\sim T_{*}^{3}$, we find

$\frac{\Gamma_{\scriptstyle V\phi}}{H(T_{*})}\sim 10^{-39}\left(\frac{\Delta
m^{2}/m^{2}}{10^{-4}}\right)^{2}\left(\frac{m}{1\
\text{eV}}\right)^{4}\left(\frac{100\ \text{GeV}}{T_{*}}\right)^{5}\ .$ (71)

Clearly, this ratio can be very small in much of the parameter space of
interest (see left panel in fig. 4).

Let us know discuss interactions involving the (highly relativistic) reflected
dark photons. These will be relevant in section IV.3 when we discuss the fate
of the dark radiation, and for convenience we summarize the relevant results
here. At large center-of-mass energies, i.e. $\sqrt{s}\gg m$, the two-to-two
scattering cross section among the dark photons is dominated by the scattering
of longitudinally-polarized vectors, and so it will be sufficient for our
purposes to just consider the process $V_{l}V_{l}\rightarrow V_{l}V_{l}$. Two
limiting kinematic regimes are of interest. In the region $m\ll\sqrt{s}\ll
m_{\rho}$, the relevant cross section is of the form

$\sigma_{\scriptstyle ll}\sim\frac{\lambda_{V}^{2}}{8\pi
s}\left(\frac{\sqrt{s}}{m}\right)^{8}\sim\frac{\lambda_{\Phi}^{2}}{8\pi
m_{\rho}^{2}}\left(\frac{\sqrt{s}}{m_{\rho}}\right)^{6}\qquad\text{for}\qquad
m\ll\sqrt{s}\ll m_{\rho}\ .$ (72)

This will be relevant for scattering between the dark radiation and the cold
dark photons, where the relevant center-mass-energy is given by
$\sqrt{s}/2\simeq\gamma_{\text{eq}}m$. Comparing the relevant interaction rate
to $H(T_{*})$, we find

$\frac{\Gamma_{\scriptstyle ll}}{H(T_{*})}\sim
10^{-12}\,\frac{\lambda_{\Phi}^{2}}{\eta^{4}}\left(\frac{m}{1\
\text{eV}}\right)^{5}\left(\frac{10^{-4}}{\Delta
m^{2}/m^{2}}\right)^{2}\left(\frac{T_{*}}{v}\right)^{8}\left(\frac{100\
\text{GeV}}{T_{*}}\right)^{4}\left(\frac{\alpha}{10^{-2}}\right)^{3}\left(\frac{\rho_{\text{dm}}}{\rho_{V}}\right)^{2}\
,$ (73)

where we have evaluated the cross section at $\sqrt{s}/2=\gamma_{\text{eq}}m$,
and substituted $m_{\rho}$ as given in eq. 67. Notice that the ratio $T_{*}/v$
could be $\lll 1$ if the phase transition sector is cold and decoupled from
the thermal plasma, as discussed below eq. 7, which would further suppress the
above ratio. If instead $\sqrt{s}\gg m_{\rho}$ then the cross section must be
computed in the complete theory of eq. 65, and we find

$\sigma_{\scriptstyle ll}\sim\frac{\lambda_{\Phi}^{2}}{8\pi
s}\qquad\text{for}\qquad\sqrt{s}\gg m_{\rho}\ ,$ (74)

as expected by virtue of Goldstone Equivalence. This is the form of the cross
section used in deriving eq. 60.

Another important class of interactions are those between the dark radiation
and the $\phi$ particles in the plasma when the phase transition sector is
thermal. In particular, if the process $V_{l}\phi\rightarrow V\phi$ happens
efficiently, the energy density stored in the dark radiation would quickly
become distributed among the thermal fluid, as discussed in section IV.3. The
typical $s$-parameter is of the form $s\sim 2T_{*}\omega_{\text{dr}}$. In the
regime $\sqrt{s}\ll m_{\rho}$,

$\sigma_{\scriptstyle l\phi}\sim\frac{\kappa^{2}}{8\pi
s}\left(\frac{\sqrt{s}}{m}\right)^{4}\qquad\text{for}\qquad m\ll\sqrt{s}\ll
m_{\rho}\ ,$ (75)

whereas in the high-energy limit $\sqrt{s}\gg m_{\rho}$,

$\displaystyle\sigma_{\scriptstyle l\phi}$
$\displaystyle\sim\frac{\eta^{2}}{8\pi s}\qquad\text{for}\qquad\sqrt{s}\gg
m_{\rho}\ .$ (76)

As an example, if the there’s a full thermal distribution of $\phi$ particles
with $n_{\phi}\sim T_{*}^{3}$, then the relevant interaction rate, relative to
$H(T_{*})$ is given by

$\frac{\Gamma_{\scriptstyle l\phi}}{H(T_{*})}\sim
10^{8}\,\eta^{2}\left(\frac{\Delta m^{2}}{m^{2}}\right)^{2}\left(\frac{1\
\text{eV}}{m}\right)^{2}\left(\frac{100\
\text{GeV}}{T_{*}}\right)^{2}\left(\frac{10^{-2}}{\alpha}\right)^{2}\left(\frac{\rho_{V}}{\rho_{\text{dm}}}\right)^{2}\
,$ (77)

where we have used eq. 76 in obtaining $\Gamma_{\scriptstyle l\phi}$, as
appropriate given our choice of parameters on the right-hand-side. Clearly,
interactions between the dark radiation and the thermal plasma can be very
efficient in parts of the relevant parameter space but – again – this
conclusion depends strongly on the choices of the various scales and
couplings.

##  Appendix B Reflection probabilities – supplemental material

This appendix contains additional details supplementing the discussion of III.

###  B.1 Longitudinal reflection from Goldstone Equivalence

In the main text we described the theory of a massive vector in eqs. 7 and 13
in terms of the Proca Lagrangian, featuring three degrees of freedom for the
three physical polarizations of a massive spin-1 particle. However, this is
often understood as a particular choice of gauge, after gauge redundancy is
restored via $V_{\mu}\longrightarrow V_{\mu}-\partial_{\mu}\theta$. Although
$\theta=0$ (unitary gauge) is naively simplest, often it is convenient to keep
the ‘Goldstone’ $\theta$. The Goldstone Equivalence Theorem states that in the
relativistic limit $\omega\gg m$ correct amplitudes can be obtained by
identifying the physical longitudinal vector with the scalar $\theta$, with an
error of order $\mathcal{O}\left(m^{2}/\omega^{2}\right)$. Using the usual
gauge-fixing parameter $\xi$, the theory is described by

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}m_{V}^{2}(z)(V_{\mu}-\partial_{\mu}\theta)^{2}-\frac{1}{2\xi}\left(\partial_{\mu}V^{\mu}+m^{2}\xi\theta\right)^{2}\
.$ (78)

Proceeding with the equation of motion for $\theta$:

$\partial_{\mu}\left(m_{V}^{2}(z)\partial^{\mu}\theta\right)+\xi
m^{4}\theta=0~{},$ (79)

where we have neglected mixing with $V^{\mu}$ in order to test the spirit of
the Goldstone Equivalent in our context. Working in the step function limit,
the scattering solution is simple for $z\neq 0$:

$\theta=e^{-i\omega t}\left\\{\begin{matrix}&e^{ik_{z}}+r\
e^{-ik_{z}},\quad&z<0\\\ &t\ e^{i\tilde{k}_{z}},\quad&z>0\end{matrix}\right.,$
(80) $\quad\quad k_{z}=\sqrt{\omega^{2}-\xi
m^{2}}~{},\quad\tilde{k}_{z}=\sqrt{\omega^{2}-\xi m^{4}/\tilde{m}^{2}}~{},$
(81)

and need only be supplemented by the matching conditions:

$m_{V}^{2}(z)\partial_{z}\theta\qquad\text{and}\qquad\theta\qquad\text{are
continuous at }z=0\ ,$ (82)

derived by integrating eq. 79 once and twice respectively. A little algebra
gives the reflection probability as

$R=|r|^{2}=\left(\frac{\tilde{k}_{z}\tilde{m}^{2}-k_{z}m^{2}}{\tilde{k}_{z}\tilde{m}^{2}+k_{z}m^{2}}\right)^{2}\xrightarrow{\omega\gg
m,\tilde{m}}\left(\frac{\tilde{m}^{2}-m^{2}}{\tilde{m}^{2}+m^{2}}\right)^{2}+\mathcal{O}\left(\tilde{m}^{2}/\omega^{2}\right)\,$
(83)

which is $\xi$ (gauge) independent and matches the leading order result
derived in the main text eq. 26 consistent with the spirit of the GET. This
result suggests that the enhanced reflection of the inter-relativistic limit
described in this work might fundamentally be a property of Nambu-Goldstone
bosons more generally, a topic that we will return to in future work.

###  B.2 Scattering on a $\delta^{\prime}$ potential

As anticipated in III.3, the effective scattering potential for the
longitudinal component in the limit of vanishing wall thickness takes the form

$U_{l}(z)\xrightarrow{L\rightarrow
0}\kappa\,\delta^{\prime}(z)\qquad\text{with}\qquad\kappa\equiv-\frac{\Delta
m^{2}}{2m^{2}}\ ,$ (84)

and eq. 30 reads

$(\partial_{z}^{2}+k_{z}^{2})\lambda(z)=\kappa\,\delta^{\prime}(z)\lambda(z)\
.$ (85)

We can obtain matching conditions for $\lambda$ and $\lambda^{\prime}$ by
integrating (twice) over this equation. First, integrating eq. 85 from
$z_{0}<0$ to $z$, we find

$\displaystyle\int_{z_{0}}^{z}d\hat{z}\,\left[(\partial_{\hat{z}}^{2}+k_{z}^{2})\lambda(\hat{z})\right]=\int_{z_{0}}^{z}d\hat{z}\,\kappa\,\delta^{\prime}(\hat{z})\lambda(\hat{z})$
(86)
$\displaystyle\Rightarrow\qquad\lambda^{\prime}(z)-\lambda^{\prime}(z_{0})+k_{z}^{2}\int_{z_{0}}^{z}d\hat{z}\,\lambda(\hat{z})=\kappa\,\lambda(z)\delta(z)-\frac{\kappa}{2}\left[\lambda^{\prime}(0^{+})+\lambda^{\prime}(0^{-})\right]\Theta(z)\
.$ (87)

Integrating eq. 87 from $z=-\epsilon$ to $z=+\epsilon$ and taking the limit
$\epsilon\rightarrow 0$, we find:

$\lambda(0^{+})-\lambda(0^{-})=-\frac{\kappa}{2}\left[\lambda(0^{+})+\lambda(0^{-})\right]\
.$ (88)

Similarly, taking $z_{0}=-\epsilon$ and $z=+\epsilon$ in eq. 87, with
$\epsilon\rightarrow 0$, one finds:

$\lambda^{\prime}(0^{+})-\lambda^{\prime}(0^{-})\simeq\frac{\kappa}{2}\left[\lambda^{\prime}(0^{+})+\lambda^{\prime}(0^{-})\right]\
,$ (89)

where in the right-hand-side we have only kept terms of $\mathcal{O}(\kappa)$.
From the last two equations, one finds $r_{l}\simeq\kappa$, and therefore

$R_{l}\simeq\kappa^{2}=\left(\frac{\Delta m^{2}}{2m^{2}}\right)^{2}\ ,$ (90)

in agreement with eq. 26 at leading order in $\kappa\ll 1$.

###  B.3 The Born approximation

In section III.3, we found it convenient to recast the equations of motion for
both transverse and longitudinal components as a one-dimensional Schrödinger
equation, of the form

$(\partial_{z}^{2}+k_{z}^{2})\psi(z)=U(z)\psi(z)\ ,$ (91)

with $U(z)\rightarrow U_{l}(z)$ and $\psi\rightarrow\xi$ for the longitudinal
component (c.f. eq. 30), and $U(z)\rightarrow U_{\perp}(z)$ and
$\psi\rightarrow v^{\mu}_{\perp}$ for the transverse modes (c.f. eq. 29). It
is an important feature that the effective scattering potential vanishses in
the limit $z\rightarrow-\infty$, as will become clear in due time.

As is well-known, it is possible to recast eq. 91 as an integral equation, as
follows

$\psi(z)=\psi_{0}(z)+\int_{-\infty}^{\infty}dz^{\prime}\
G(z-z^{\prime})U(z^{\prime})\psi(z^{\prime})\ ,$ (92)

where $\psi_{0}$ is any function satisfying the free particle equation,
$\big{(}\partial_{z}^{2}+k_{z}^{2}\big{)}\psi_{0}=0$, and $G(z)$ is a Green’s
function for the differential operator on the right-hand-side of eq. 91, i.e.

$\big{(}\partial_{z}^{2}+k_{z}^{2}\big{)}G(z)=\delta(z)\ .$ (93)

It is straightforward to check that eq. 92 is equivalent to eq. 91 by applying
$(\partial_{z}^{2}+k_{z}^{2})$ to both sides and using eq. 93. In order to
solve for $G(z)$, note that for $z\neq 0$ eq. 93 is just the free particle
equation. The matching conditions at $z=0$ are that $G$ must be continuous and
$G^{\prime}$ must have unit jump. Putting this together, the solutions are

$G(z)=\pm\frac{1}{2ik_{z}}e^{\pm ik_{z}|z|}\ ,$ (94)

up to addition of functions that satisfy the free particle equation. It turns
out that we only need one of these two solutions: the one with the $+$ sign.

To represent an incoming plane wave, we take $\psi_{0}(z)=e^{ik_{z}z}$. Then
our integral equation is

$\psi(z)=e^{ik_{z}z}+\frac{1}{2ik_{z}}\int_{-\infty}^{\infty}dz^{\prime}\
e^{ik_{z}|z-z^{\prime}|}\ U(z^{\prime})\psi(z^{\prime})\ .$ (95)

So far, we have made no approximations. However, if the correction to the
incoming wavefunction $\psi_{0}$ is small (meaning the reflection coefficient
is tiny), then we can plug this equation for $\psi$ into itself in the
integral and truncate higher order terms, leaving

$\displaystyle\psi(z)$ $\displaystyle\simeq
e^{ik_{z}z}+\frac{1}{2ik_{z}}\int_{-\infty}^{\infty}dz^{\prime}\
e^{ik_{z}|z-z^{\prime}|}\ U(z^{\prime})e^{ik_{z}z^{\prime}}$ (96)
$\displaystyle=e^{ik_{z}z}+\frac{e^{ik_{z}z}}{2ik_{z}}\int_{-\infty}^{z}dz^{\prime}\
U(z^{\prime})+\frac{e^{-ik_{z}z}}{2ik_{z}}\int_{z}^{\infty}dz^{\prime}\
e^{2ik_{z}z^{\prime}}\ U(z^{\prime})\ .$ (97)

We wish to extract a reflection probability from this ‘first Born
approximation’. To this end, consider the limit $z\to-\infty$. The second term
above clearly vanishes, given that $U(z^{\prime})\to 0$ as
$z^{\prime}\to-\infty$ as emphasized below eq. 91. Furthermore, the third term
can be identified as the reflected plane wave piece, with associated
probability

$R_{l\text{,\,Born}}=\frac{1}{4k_{z}^{2}}\left|\int_{-\infty}^{\infty}dz\
e^{2ik_{z}z}\ U(z)\right|^{2}\ .$ (98)

Keep in mind that $R_{l}\simeq R_{l\text{,\,Born}}$ only when $R_{l}\ll 1$.

###  B.4 Numerical methods

Finding the reflection and transmission coefficients for the longitudinal mode
reduces to solving eq. 30 with boundary conditions corresponding to plane-wave
behavior far away from the bubble wall. i.e.

$\lambda(z)=\begin{cases}e^{ik_{z}z}+r_{l}e^{-ik_{z}z}&\text{as}\quad
z\rightarrow-\infty\\\ t_{l}e^{i\tilde{k}_{z}z}&\text{as}\quad
z\rightarrow+\infty\end{cases}$ (99)

It will be convenient to rescale $\lambda$ by an overall complex constant, and
define $f(z)\equiv t_{l}^{-1}\lambda(z)$. The equation of motion for $f(z)$
remains as in Eq.(30), and the boundary conditions can now be written as
$f(z_{0})=1$ and $\partial_{z}f(z_{0})=i\tilde{k}_{z}$ for some $z_{0}>0$ far
away to the right of the wall. Our numerical solution will lie in the window
$z\in[-z_{0},z_{0}]$, where $z_{0}$ is set to be much larger than the incident
particle wavelength (constraining for low energies) and the thickness of the
bubble wall (constraining for high energies).

Lastly, we need a prescription for extracting the reflection probability from
the numerical solution. Consider evaluating the asymptotic expression for
$f(z)$ at the following points, where $n\in\mathbb{N}$, $n\to\infty$:

$\displaystyle f\left(-\frac{n\pi}{k_{z}}\right)$
$\displaystyle\longrightarrow\left(t_{l}^{-1}+t_{l}^{-1}\,r_{l}\right)(-1)^{n}$
(100) $\displaystyle f\left(-\frac{(n-\frac{1}{2})\pi}{k_{z}}\right)$
$\displaystyle\longrightarrow\left(t_{l}^{-1}-t_{l}^{-1}\,r_{l}\right)(-1)^{n}(i)$
(101)

These equations can then be easily inverted to obtain the scattering
coefficients $r_{l}$ and $t_{l}$, and ultimately the reflection and
transmission probabilities.

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