# PBH formation from overdensities in delayed vacuum transitions

Kiyoharu Kawana<EMAIL_ADDRESS>School of Physics, Korea Institute for
Advanced Study, Seoul 02455, Korea  TaeHun Kim<EMAIL_ADDRESS>School of
Physics, Korea Institute for Advanced Study, Seoul 02455, Korea  Center for
Theoretical Physics, Department of Physics and Astronomy, Seoul National
University, Seoul 08826, Korea  Philip Lu<EMAIL_ADDRESS>Center for
Theoretical Physics, Department of Physics and Astronomy, Seoul National
University, Seoul 08826, Korea

###### Abstract

Primordial black hole (PBH) formation from first-order phase transitions
(FOPTs) combines two prevalent elements of beyond the Standard Model physics
with wide-ranging consequences. We elaborate on a recently proposed scenario
in which inhomogeneities in vacuum energy decay seed the overdensities that
collapse to PBHs. In this scenario, the PBH mass is determined by the Hubble
mass as in conventional formation scenarios, while its number density is
determined by the nucleation dynamics of the FOPT. We present a detailed study
of the formation probability including parameter dependencies. In addition, we
generate populations in the open mass window as well as for the HSC and OGLE
candidate microlensing events. This mechanism inevitably creates PBHs in
generic FOPTs, with significant populations produced in slow and moderately
strong phase transitions.

##  I Introduction

The existence of primordial black holes (PBHs) would have many interesting and
diverse consequences. In the asteroid mass window $10^{-16}M_{\odot}\lesssim
M_{\rm PBH}\lesssim 10^{-11}M_{\odot}$, the PBH mass fraction is free of
constraints, making this mass range a popular candidate for PBH dark matter
(DM) Carr and Kuhnel (2020); Carr  _et al._ (2021a, 2010). As a subdominant
component of DM, PBHs can contribute to the LIGO-VIRGO-KAGRA population of
detected binary black hole mergers Nakamura  _et al._ (1997); Bird  _et al._
(2016); Cotner and Kusenko (2016, 2017); Raidal  _et al._ (2017); Eroshenko
(2018); Sasaki  _et al._ (2016); Clesse and García-Bellido (2017); Cotner  _et
al._ (2018, 2019); Flores and Kusenko (2021a, b, c); Wang  _et al._ (2023) or
seed supermassive black holes Bean and Magueijo (2002); Kawasaki  _et al._
(2012); Clesse and García-Bellido (2015). Interestingly, two microlensing
experiments HSC Niikura  _et al._ (2019a) and OGLE Niikura  _et al._ (2019b)
have reported candidate events which may be consistent with a population of
PBHs at $M_{\rm PBH}\sim 10^{-9}M_{\odot}$ and $M_{\rm PBH}\sim
10^{-4}M_{\odot}$, respectively. PBHs are leading candidates for DM and ones
with abundant astrophysical consequences.

It is therefore unsurprising that PBH-production mechanisms have saturated the
literature in recent years. In addition to the classic formation from
primordial overdensities Carr and Hawking (1974); Carr (1975), models
involving first-order phase transitions (FOPTs) have become popular Crawford
and Schramm (1982); Hawking  _et al._ (1982); La and Steinhardt (1989); Moss
(1994); Konoplich  _et al._ (1998, 1999); Kodama  _et al._ (1982); Lewicki and
Vaskonen (2020); Kusenko  _et al._ (2020) (see also Animali and Vennin (2023)
for an inflationary scenario but with metastable vacuum). Generically, FOPTs
soften the equation of state of the relativistic plasma and stimulate the
growth of overdensities, leading to increased PBH production Jedamzik and
Niemeyer (1999); Musco and Miller (2013); Byrnes  _et al._ (2018); Jedamzik
(1998, 2021); Carr  _et al._ (2021b, c); Dolgov and Postnov (2020);
Franciolini  _et al._ (2022); Juan  _et al._ (2022); Lu  _et al._ (2023). In
models with additional particles, they can accumulate and form PBHs and other
compact objects Gross  _et al._ (2021); Baker  _et al._ (2021a, b); Kawana and
Xie (2022); Marfatia and Tseng (2022); Huang and Xie (2022); Davoudiasl  _et
al._ (2022); Jung and Okui (2021); Hashino  _et al._ (2022); Maeso  _et al._
(2021); Lu  _et al._ (2022a); Kawana (2022); Lu  _et al._ (2022b); Kawana  _et
al._ (2022).

In particular, Ref. Liu  _et al._ (2022) calculated the abundance of FOPT-
induced PBHs that result from the overdensities in the false vacuum (FV)
regions with delayed decay (see He  _et al._ (2022) for a specific FOPT model
realizing the same scenario). The essence of this mechanism is the stochastic
nature of the FOPT, which creates inhomogeneities in the subsequent radiation
energy distribution after the vacuum transition. These perturbations may then
grow and collapse into PBHs in the standard manner. We seek to elaborate and
present a more detailed investigation of this interesting scenario, providing
a quantitative understanding of the production probability. The PBH prediction
of this mechanism is sensitive to the parameters of the FOPT, so it is
possible to realize any mass and energy fraction given suitable models. In
this paper, we simply choose benchmark parameters of FOPTs that correspond to
the open mass window ($\sim 10^{-15}M_{\odot}$) and the two microlensing
events mentioned above, leaving concrete model buildings for future
investigation.

In this paper, we describe the formation of PBHs from vacuum energy decay in a
FOPT. In Sec. II, a general overview of FOPTs and the relevant equations are
given. In Sec. III, we construct the formalism to calculate the PBH formation
probability and abundance. In Sec. IV, the numerical process and results are
described. Finally, in Sec. V, we discuss the results and draw conclusions.

##  II First-Order Phase Transition

We summarize the progression of a FOPT and list out key equations. In the
following, we take a phenomenological approach for the FOPT parameters
$\alpha$ and $\beta$ defined below. Our arguments are largely independent of
the underlying model, provided that the phase transition has a strong vacuum
energy $\alpha\sim\mathcal{O}(0.1)$ and slow pace $\beta/H\sim\mathcal{O}(1)$,
where $H$ is the Hubble expansion rate.

In a FOPT, the Universe is initially in a stable FV state which becomes
unstable below the critical temperature $T_{\rm cri}$ (see Refs. Megevand and
Ramirez (2017); Kobakhidze  _et al._ (2017); Ellis  _et al._ (2019, 2020);
Wang  _et al._ (2020)) and corresponding time $t_{\rm cri}$. The nucleation
rate for true vacuum (TV) bubbles is given by

$\Gamma(t)=A(t)e^{-S(t)}~{},$ (1)

where $S(t)$ is the bounce action of the three- (from thermal fluctuations) or
four-dimensional (from quantum tunneling) instanton solution. $\Gamma(t)$ is
the number of bubble nucleations per unit time per unit physical volume in FV
regions.

The time dependence of the nucleation rate stems from the temperature
dependence of the finite-temperature effective potential. As the temperature
varies with time due to the expansion of the Universe and the progression of
the phase transition, the effective potential and the corresponding nucleation
rate $\Gamma(t)$ vary accordingly.

The qualitative behavior of the effective potential depends on whether the
potential barrier separating the two vacua disappears in a finite temperature
below $T_{\rm cri}$ or it persists even in the zero-temperature limit Megevand
and Ramirez (2017). As the Universe cools down, the nucleation rate $\Gamma$
continually increases for the former case. In the latter case, it peaks at
some intermediate temperature and subsequently decreases, where only the
quantum tunneling can nucleate new bubbles in the zero-temperature limit. The
statistics of the postphase transition universe and the completion of the
phase transition (having no FV regions) depend on the low-temperature
behavior. We discuss this consistency problem in Sec. III.1 and in more detail
in Appendix A, by stating the conditions that should be satisfied in order to
have a FOPT that successfully ends while producing PBHs by the scenario
considered here.

The temperature evolution is complicated by the moderate reheating from the
release of latent heat and the differing Hubble rate in a two-component
radiation-vacuum universe. The effects of reheating are minor in a weak FOPT
with low latent heat. In a strong detonationlike FOPT, the bubble walls move
at nearly the speed of light, faster than the sound speed, so the latent heat
remains in TV and the nucleation rate in the FV is unaffected. An
inhomogeneous Hubble rate arises from the constant energy density in the
vacuum-dominated FV regions and the redshifting radiation-dominated TV regions
(see Sec. III). However, we neglect this effect as the inhomogeneous expansion
rates are not directly relevant to the focus of this paper, and instead use
the average energy density to calculate the Hubble rate.

For analytical purposes, the exponential form

$\Gamma(t)\approx\Gamma_{0}e^{\beta(t-t_{0})}~{},~{}~{}\beta=-\frac{dS(t)}{dt}\bigg{|}_{t=t_{0}}~{}$
(2)

is a good approximation when $\beta/H\gg 1$, and the second derivative of $S$
is negligible with respect to $\beta$ Megevand and Ramirez (2017) at an
arbitrary fixed time $t_{0}$. Hereafter, we take $t_{0}=t_{\rm cri}$, the
critical time in which the two vacua become degenerate. The average size of
the FV bubbles is related to the bubble wall velocity $v_{w}$ and $\beta$ by
$R\sim 1.12v_{w}/\beta$ Kawana and Xie (2022); Lu  _et al._ (2022a), which
implies that $\beta/H\lesssim{\cal O}(1)$ is necessary to have enough horizon-
sized FV regions that can collapse into a detectable number of PBHs. For
numerical calculations, we consider values of $\beta/H\sim\mathcal{O}(1)$ to
marginally satisfy both these conditions. Another key parameter is the ratio
of the vacuum energy to the plasma energy density $\alpha\equiv\Delta
V/\rho_{\rm SM}(T_{\rm cri})$, which measures the strength of the FOPT. In
this paper, we make an approximation by ignoring the temperature evolution of
$\Delta V$ and let it be a constant after $T_{\rm cri}$. This corresponds to a
situation where only the height of the potential barrier varies, giving Eq.
(2), while the vacua energies remain the same. Also, we simply focus on
$\alpha\lesssim\mathcal{O}(1)$ to avoid a long-duration second inflationary
phase.

As the phase transition progresses, the volume fraction of the FV region
shrinks from $f_{\rm fv}=1$ at $T>T_{\rm cri}$ to $f_{\rm fv}\xrightarrow{}0$
at $T\ll T_{\rm cri}$, and can be computed by

$f_{\rm fv}(t)=\exp\left[-\frac{4\pi}{3}\int^{t}_{t_{\rm
cri}}dt^{\prime}\,a^{3}(t^{\prime})\,\Gamma(t^{\prime})\,r^{3}(t,t^{\prime})\right]~{},$
(3)

where

$r(t,t^{\prime})=v_{w}\int^{t}_{t^{\prime}}dt^{\prime\prime}\frac{1}{a(t^{\prime\prime})}~{}$
(4)

is the comoving size of a bubble nucleated at $t^{\prime}$. Under the constant
scale factor approximation or constant Hubble rate approximation during the
transition and using the exponential approximation (2), $f_{\rm fv}(t)$ can be
analytically solved to be Turner  _et al._ (1992a)

$\displaystyle f_{\rm fv}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-I_{\ast}e^{\beta(t-t_{\rm cri})}\right],$
$\displaystyle I_{\ast}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{8\pi v_{w}^{3}\Gamma_{\rm
cri}}{\beta^{4}}&\text{(const $a$),}\\\ \frac{8\pi v_{w}^{3}\Gamma_{\rm
cri}}{\beta(H+\beta)(2H+\beta)(3H+\beta)}&\text{(const $H$).}\end{cases}$ (5)

Note that we have approximated $t_{\rm cri}\rightarrow-\infty$ when $H=$
const. Here we see that there are competing effects for PBH production. As
detailed in Sec. III, the constant vacuum energy $\Delta V$ grows in strength
to the surrounding plasma at lower temperatures and later times. Conversely,
the FV fraction steadily decreases with increasing time, constraining the PBH
production to peak at intermediate times.

##  III PBH Formation Mechanism

In the classical analytic prescription for PBH formation, a PBH forms when
horizon-sized perturbations exceed a critical overdensity threshold
$\delta>\delta_{c}\sim 0.45$ Musco  _et al._ (2021). The spectrum of curvature
perturbations is usually generated during inflation and reenters the horizon
after reheating. Here, we consider the scenario outlined in Ref. Liu  _et al._
(2022) in which overdensities are generated stochastically during a FOPT.
However, we investigate the formation mechanism in more detail and comment on
the different methodologies in this paper and Ref. Liu  _et al._ (2022) in
Sec. III.3.

For a moderately strong FOPT $\alpha\sim\mathcal{O}(0.1\text{--}1)$, the
vacuum energy is a significant component of the overall energy density which
even temporarily keeps the Hubble rate $H(t)$ constant. This vacuum energy
will seed the fluctuations when converted to radiation by the phase
transition. Because of the conservation of entropy, the background plasma
radiation density redshifts with scale factor $a$ as $\propto a^{-4}$, but the
vacuum energy density $\Delta V$ stays constant. Therefore, regions of FV will
have proportionally larger energy densities with increasing time. If the
resultant density contrasts are large enough, they gravitationally collapse
and PBHs are formed (see Sec. III.1 for detailed PBH formation criterion),
whereas deficient overdensities will decay and disappear soon due to the
radiation pressure.

In the paper, we focus on the PBHs from horizon-sized overdensities only and
discard the subhorizon ones, since the formation criterion for these are not
well understood.

###  III.1 Formation criterion

Here the quantitative criterion for PBH formation is stated. First, we define
a time $t_{1.45}$ at which the local energy density of a FV point is
$1+\delta_{c}\simeq 1.45$ of the background average density,

$1+\delta:=\frac{\Delta V+\rho_{\rm SM}(t_{\rm 1.45})}{\bar{\rho}(t_{\rm
1.45})}=1.45~{},$ (6)

where $\rho_{\rm SM}(t)$ is the Standard Model (SM) plasma energy density that
was present before the transition, and $\bar{\rho}(t)$ is the average total
energy density of the Universe (see Sec. IV.1 for more concrete definitions).

Then we use a simple but justified criterion for PBH formation: Any Hubble
volume that is entirely covered by FV after time $t_{\rm 1.45}$ eventually
collapses to a PBH. There are a few subtleties excluded by this prescription.
First, the value of $\delta_{c}=0.45$ is applicable only to radiation-
dominated plasma and should be scaled according to its effective equation-of-
state parameter $w$ Musco and Miller (2013) which would be different due to
the presence of vacuum energy. Furthermore, the overdensity $\delta$ cannot
accumulate and collapse when the horizon is dominated by the FV until it
decays into radiation, delaying the growth phase of the perturbation past
$t_{1.45}$. However, this delayed growth is offset by the relative increase in
the FV energy $\Delta V$ to the surrounding plasma during this time. These
omitted effects tend to cancel and we leave a more in-depth investigation of
PBH formation involving numerical simulations to future work.

The horizon-sized FV regions will expand exponentially if they remain intact.
This resembles the old inflation scenario, which was abandoned due to the
prediction that inflation never ends because the vacuum transition cannot
overcome the exponential expansion of FV regions. However, there is one
important difference for FOPT: The nucleation rate is not restricted to be
much smaller than the Hubble scale and also allowed to increase exponentially
as in Eq. (2). So for our scenario to be consistent with the successful
completion of a FOPT, the nucleation rate should catch up with the volume
expansion rate, making the physical volume of FV $\propto f_{\rm
fv}(t)\,a^{3}(t)$ decrease Turner  _et al._ (1992b); Kawana (2022). More
detailed conditions to achieve this are explained in Appendix A with a
numerical demonstration. We are assuming one of the three cases with
exponential approximation (2) for the nucleation rate; see the Appendix for
more details. Any deviation from our assumption will result in different
quantitative results, but the physics of PBH formation is not altered.

###  III.2 Formation probability

Figure 1:  Wall cone diagram for PBH formation. A nucleation in either the
horizon volume (red) or the causative past wall cone (blue) would trigger the
phase transition. A nucleation in the green spacetime slice would induce the
phase transition between time $t$ and $t+dt$.

We calculate the time-dependent probability $p(t)$ of PBH formation per unit
time for any given Hubble volume. We then apply this in Sec. IV to find the
PBH mass spectrum and abundance.

In the following, we use the wall cone formalism developed in Ref. Lu  _et
al._ (2022a) with the comoving radius of a TV bubble given by Eq. (4). The
wall cone formalism describes the propagation of bubble walls as cones in
spacetime, corresponding to light cones only with light speed $c$ replaced by
wall velocity $v_{w}$. A future wall cone coming out from a nucleation point
represents the region of TV, while the past wall cone of a spacetime point
represents the region for nucleation to put that point in TV.

For a Hubble volume to be entirely in FV at time $t$, (i) there can be no
nucleation within the Hubble volume for time $t^{\prime}<t$ and (ii) no
nucleation within the past wall cone of the Hubble volume. A nucleation
outside the horizon at time $t^{\prime}<t$ can propagate to the Hubble volume
if it is less than a distance $r(t,t^{\prime})$ away. In Fig. 1, we display
these three regions in spacetime. The probability $p(t)dt$ represents the
transition probability of the Hubble volume between $t$ and $t+dt$. We now
derive this probability. To streamline the discussion, we use comoving
quantities, with the comoving nucleation rate defined as

$\Gamma_{\rm com}(t)=\Gamma(t)a^{3}(t).$ (7)

In an infinitesimal time interval $dt$, the probability of no nucleation in an
infinitesimal comoving volume $dV_{\rm com}$ is $1-V_{\rm com}\Gamma_{\rm
com}dt$. Summing infinitesimal volumes and taking the limit, the probability
of no nucleation within a given comoving volume until the time
$t_{*}(\vec{x})$ for each comoving point $\vec{x}\in V_{\rm com}$ is

$\displaystyle P_{\rm\not{n}}(V_{\rm com};t_{*})$ $\displaystyle=$
$\displaystyle\prod_{i}[1-V_{\rm com}\Gamma_{\rm com}(t_{i})\,dt]$ (8)
$\displaystyle=$ $\displaystyle\exp\left[-\int_{V_{\rm
com}}d^{3}\vec{x}\int^{t_{*}(\vec{x})}_{t_{\rm
cri}}a^{3}(t)\,\Gamma(t)\,dt\right].\quad\ $

We apply this formula to the transition probability $p(t)$ of a given Hubble
volume by using the comoving radius from its center

$r_{x}:=|\vec{x}|\qquad\text{and}\qquad r_{H}(t):=\frac{1}{a(t)H(t)}~{}.$ (9)

First, each point within the Hubble volume $r_{x}<r_{H}(t)$ should remain in
the FV, implying $t_{*}(\vec{x})=t$. This is shown by the red region in Fig.
1. On the other hand, outside the horizon, the past wall cone has ($t_{*}<t$)

$\displaystyle
r_{x}(t_{*})=r_{H}(t)+v_{w}\int^{t}_{t_{*}}\frac{1}{a(t)}dt:=r_{H}(t)+r(t,t_{*})$
(10)

where we assume a constant wall velocity $v_{w}\leq 1$, and the maximum
comoving radius is denoted by $r_{\rm max}:=r_{x}(t_{\rm
cri})=r_{H}(t)+r(t,t_{\rm cri})$. The corresponding outside region is
represented in blue in Fig. 1. If we denote the inverse of Eq. (10) as
$t_{*}(r_{x})$, each outside point $\vec{x}$ must remain in the FV until
$t_{*}(r_{x})$. Then, the survival probability of the Hubble volume, or the
probability of having no TV up to time $t>t_{\rm cri}$, is given by the
product of the probabilities of having no nucleation in the horizon itself or
in its past wall cone as

$\displaystyle P(t)=$ $\displaystyle\exp\left[-\frac{4\pi}{3}\int^{t}_{t_{\rm
cri}}\frac{a^{3}(t^{\prime})}{a^{3}(t)}H^{-3}(t)\,\Gamma(t^{\prime})\,dt^{\prime}\right]$
$\displaystyle\times\exp\left[-4\pi\int^{r_{\rm
max}}_{r_{H}(t)}dr_{x}\,r_{x}^{2}\int^{t_{*}(r_{x})}_{t_{\rm
cri}}a^{3}(t^{\prime})\,\Gamma(t^{\prime})\,dt^{\prime}\right].$ (11)

The complement $1-P(t)$ is simply the cumulative distribution function of
$p(t)$, so the desired transition probability is given by

$p(t)=-\frac{d}{dt}P(t)~{}.$ (12)

As previously mentioned, we limit the domain of $p(t)$ to $t>t_{1.45}$ so that
$p(t)$ can be interpreted as the probability density for the onset of PBH
collapse. The mass distribution of PBH is determined by the temporal
distribution of formation, as the horizon size and the overdensity $\delta$
vary with time. The mass of the PBH is related to the horizon mass at the
transition time by

$M(t)=\gamma M_{H}(t)=\gamma\frac{4\pi}{3}\bar{\rho}(t)H^{-3}(t)~{},$ (13)

where we choose the numerical prefactor $\gamma=1$, and $\bar{\rho}(t)$ is the
average energy density (combined radiation and vacuum). If another value of
$\gamma$ or a different PBH mass relation (e.g., critical collapse; see Sec.
IV.2) is used, the mass distribution is simply shifted with little consequence
for our purposes. $p(t)$ can be converted to the standard PBH formation
parameter $\beta(M)$, which is the fraction of PBH abundance in the Universe
at the formation time Carr and Kuhnel (2020); Carr  _et al._ (2021a, 2010).
The (physical) number density at the formation of new PBHs for a time interval
between $t$ and $t+dt$ is

$dn_{\rm PBH}=p(t)\left(\frac{4\pi}{3}\frac{1}{H^{3}(t)}\right)^{-1}dt$ (14)

giving

$\displaystyle\beta(M)$ $\displaystyle=$ $\displaystyle\frac{1}{\rho_{\rm
SM}(T)}\frac{d\rho_{\rm PBH}}{d\log
M}=\frac{4M}{3T}\frac{1}{s(T)}\frac{dn_{\rm PBH}}{d\log M}$ (15)
$\displaystyle=$ $\displaystyle\frac{MH^{3}(t)}{\pi
T}\frac{p(t)}{s(T)}\left(\frac{d\log M}{dt}\right)^{-1},$

where $s(T)$ is the entropy density of SM plasma, and both $t$ and $T$ on the
rhs are understood as functions of $M$ via Eq. (13). Finally, the PBH energy
fraction $f_{\rm PBH}(M)$ is given in terms of $\beta(M)$ De Luca  _et al._
(2021) as

$f_{\rm PBH}(M)=\frac{1}{\Omega_{\rm DM}}\left(\frac{M_{\rm
eq}}{M}\right)^{1/2}\beta(M)~{},$ (16)

with $\Omega_{\rm DM}=0.264$ and the horizon mass at matter-radiation equality
$M_{\rm eq}=2.8\times 10^{17}M_{\odot}$.

###  III.3 Comparison with 2106.05637

Figure 2:  The underlying spacetime diagram for Refs. Liu  _et al._ (2022); He
_et al._ (2022) with the missing wall cones, which should have been somewhere
between the solid and the dashed blue curves.

In Liu et al. 2021 Liu  _et al._ (2022), the authors proposed a PBH formation
scenario from vacuum energy during a FOPT and provided some estimates of the
formation probability and PBH statistics. We elaborate on this mechanism and
present improved calculations in this paper using the wall cone approach of
Ref. Lu  _et al._ (2022a). We find that Refs. Liu  _et al._ (2022); He  _et
al._ (2022) overestimate the PBH abundance. In contrast, although our methods
also use analytic simplifications and contain uncertainties, they are more
sophisticated and tend to conservatively estimate the formation probability,
leading to a lower bound on the PBH abundance. Here we also try to clarify
certain points that were implied in the qualitative exposition of Refs. Liu
_et al._ (2022); He  _et al._ (2022).

Although a direct comparison is difficult, the primary difference between
these two methods is the past wall cone region that we consider. Our
interpretation of the spacetime diagram for Refs. Liu  _et al._ (2022); He
_et al._ (2022) is shown in Fig. 2. Their condition for PBH production
requires only delayed nucleation in the Hubble horizon itself, treating each
Hubble volume as a separate universe and neglecting TV propagation from the
surrounding wall cone. We find numerically that the wall cone region
contributes an exponential suppression factor comparable to the contribution
from the Hubble volume itself. This would result in much lower production
probability and smaller $\beta(M)$, although production is also exponentially
sensitive to FOPT parameters $\alpha$ and $\beta$ which can be changed to
compensate for the extra suppression. Thus, we conclude that the production
mechanism proposed in Ref. Liu  _et al._ (2022) is valid, but we present a
modified version and provide some details that we found missing.

##  IV PBH Abundance Calculation

We numerically evolve the FOPT equations to find the formation probability
$p(t)$ and energy fraction $\beta(M)$ or $f_{\rm PBH}(M)$. We describe the
generic numerical procedure in Sec. IV.1 and show the results in Sec. IV.2.

###  IV.1 FOPT evolution

During a FOPT, the Universe has two energy components: the vacuum energy of FV
$\Delta V$ and the radiation energy $\bar{\rho}_{R}(t)$. The FOPT starts at
the critical temperature $T_{\rm cri}$. At this moment, there are only
homogeneous SM plasma with energy density $\frac{\pi^{2}}{30}g(T_{\rm
cri})T_{\rm cri}^{4}$ and the vacuum energy with density $\alpha$ times the
former. As the transition progresses, the vacuum energy is converted into
kinetic energy for the bubble walls and delivers latent heat into the plasma.
The average total energy density is then expressed by

$\bar{\rho}(t)=\bar{\rho}_{V}(t)+\bar{\rho}_{R}(t)~{},$ (17)

where

$\bar{\rho}_{V}(t)=\Delta V\times f_{\rm fv}(t)$ (18)

is the average vacuum energy density for the FV volume fraction given by Eqs.
(3) and (4), and

$\bar{\rho}_{R}(t)=\rho_{\rm SM}(t)+\bar{\rho}_{\rm heat}(t)+\bar{\rho}_{\rm
wall}(t)$ (19)

is the total average radiation energy density, the sum of the SM plasma from
the reheating after inflation, the latent heat, and the bubble wall energy.
Hereafter, the wall energy is included in $\bar{\rho}_{R}(t)$ as it is subject
to the same redshift dependence of $\propto a^{-4}$ Liu  _et al._ (2022) and
will release gravitational waves after collisions. $\bar{\rho}_{R}(t)$ evolves
as

$\frac{d\bar{\rho}_{R}}{dt}+4H\bar{\rho}_{R}=-\frac{d\bar{\rho}_{V}}{dt}~{},$
(20)

where the converted vacuum energy goes into the last two terms in Eq. (19),
and the Hubble rate $H(t)$ is determined by the Friedmann equation

$H^{2}(t)=\frac{\bar{\rho}(t)}{3M_{P}^{2}}~{}.$ (21)

To simplify our calculations, we assumed $H(t)$ to be homogeneous, regardless
of the local composition of energy contents hence neglecting the backreaction
of the inhomogeneous vacuum transition to the expansion rate. Also, we use the
exponential form of $\Gamma(t)$, Eq. (2), which is marginally valid for our
parameter choice of $\beta/H\gtrsim 1$. For even slower phase transitions,
which are conducive to forming horizon-sized bubbles, the Gaussian
approximation should be used Megevand and Ramirez (2017); however, this may
suffer from the consistency problem as explained in Appendix A.

For a given set of FOPT and cosmological parameters, we first perform a
numerical evolution of the FOPT to calculate $t_{1.45}$, i.e., Eq. (6). Then,
using Eqs. (11) and (12), the transition probability of PBH formation $p(t)$
for $t>t_{1.45}$ is calculated. The mass and energy fractions of the
corresponding PBH are estimated by Eqs. (13), (15), and (16).

###  IV.2 Numerical results

Model | $T_{\rm cri}$ | $\Gamma_{\rm cri}^{1/4}$ | $\beta/H$ | $\alpha$ | $v_{w}$
---|---|---|---|---|---
A |  $3.7\times 10^{6}$ GeV | $5.39\times 10^{-6}\,\text{GeV}$ | 2.5 | 1.0 | 1
B |  $3.7\times 10^{3}$ GeV | $5.47\times 10^{-12}\,\text{GeV}$ | 2.5 | 1.0 | 1
C |  $39$ GeV | $5.91\times 10^{-16}\,\text{GeV}$ | 2.5 | 1.4 | 1
Table 1: FOPT parameters for models A (PBH mass window), B (HSC), and C (OGLE)
in Fig. 5. $\Gamma_{\rm cri}$ and $\beta$ are for Eq. (2) with $t_{0}=t_{\rm
cri}$, and $\beta/H$ and $\alpha$ are evaluated at $t_{\rm cri}$. Model A is
also used in Figs. 3 and 4 to demonstrate generic features of the phase
transition and PBH formation.

As benchmarks, we chose three phenomenological FOPT models summarized in Table
1.111While the nucleation rate starts with $0$ at $T_{\rm cri}$ Megevand and
Ramirez (2017), we extend the exponential approximation (2) to this point for
numerical calculation and take $\Gamma_{\rm cri}/H_{\rm cri}^{4}\ll 1$. This
value has only little effect on the results. Model A assumes a high-
temperature phase transition resulting in light mass PBHs in the mass window
for DM. The other two happening at lower temperatures can produce heavier mass
PBHs, which can be responsible for the candidate events of the HSC and OGLE
experiments.

Figure 3:  The average radiation density $\bar{\rho}_{R}(t)$, average FV
energy density $\bar{\rho}_{V}$, and average total density $\bar{\rho}(t)$ as
functions of time. $\rho_{\rm fv}(t)$ is the local density of unperturbed FV
regions.

We first display the energy density evolution of model A in Fig. 3, whose
behavior is generic to all three models. The average radiation energy density
$\bar{\rho}_{R}(t)$ initially falls with increasing scale factor but is
subsequently heated up by the decaying FV.222Figure 3 is for the average
densities. Locally, FV regions are not reheated as discussed above Eq. (2).
The nucleation rate there thus can still be approximated by the exponential
form, except for the common deviation by the actual potential shape outlined
in Appendix A. The energy density in the FV $\rho_{\rm fv}$ becomes dominated
by $\Delta V$ as the radiation component redshifts away. During the phase
transition, the FV regions become significantly overdense $\delta>\delta_{c}$
with respect to the average density at $t_{1.45}$.

Figure 4:  Top: the transition probability $p(t)$ for a Hubble volume as a
function of time in model A. Transitions in the shaded region can only form
PBHs under favorable circumstances, whereas transitions that occur after
$t_{1.45}$ in the unshaded region are significantly overdense and highly
likely to form PBHs. Bottom: variations of $p(t)$ and $t_{1.45}$ for faster
(red) or stronger (blue) FOPT parameters.

If the onset of the phase transition in a Hubble volume occurs after this
time, then the overdensity there will tend to grow until the FV regions are
depleted so that PBH formation is almost certain. We show the PBH-producing
tail of the probability density distribution $p(t)$ in Fig. 4, and the
discarded region where PBH formation is unlikely. It is possible for some
Hubble volumes near the boundary that begin transitioning at $t\lesssim
t_{1.45}$ to have $\delta>\delta_{c}$ by the time the phase transition
culminates. However, we discard these “probabilistic” regions to provide a
conservative estimate on the PBH abundance.

In the bottom panel of Fig. 4, we show the effect of changing the FOPT
parameters $\beta$, which determines the timescale of the FOPT, and $\alpha$
or the vacuum energy density $\Delta V$ (with $\beta/H_{\rm cri}$ fixed).
Faster phase transitions (larger $\beta$) result in earlier TV nucleation and
exponentially suppressed PBH production. Alternatively, the average size of
the FV pockets is inversely proportional to $\beta$ Kawana and Xie (2022); Lu
_et al._ (2022a), so that there are fewer horizon-sized ones. Naturally,
stronger phase transitions (larger $\Delta V$) result in larger overdensities
between FV regions with energy density $\rho_{\rm fv}$ and the background
$\bar{\rho}$ facilitating PBH formation. All these effects exponentially
impact $p(t)$ and dominate the final PBH abundance.

Figure 5:  The PBH mass function $f_{\rm PBH}(M)$. The excluded regions by
existing bounds are shown by the shaded regions. $H_{0}=67.66\text{ km/s/Mpc}$
and $\Omega_{\rm DM}=0.264$ are used. The three colored dots represent the
abundance of PBHs in the Window (red), HSC (green), and OGLE (blue) models.
The extended mass functions are sharply peaked (see Fig. 4), so we represent
them as points.

We show the PBH distribution of the three models in Fig. 5. The survival
probability drops sharply after $t_{1.45}$, and the scaling $\ln
p(t)\sim-e^{\beta(t-t_{\rm cri})}$ can be obtained from Eq. (24) with
$t_{1.45}$ replaced by $t$ and Eq. (12). We have numerically confirmed this
behavior in Fig. 4. Combined with the mass relation Eq. (13), the resulting
PBH distributions are sharply peaked and resemble Dirac delta functions
$\sim\delta(M_{\rm PBH}-M(t_{1.45}))$. However, the actual mass function would
be slightly broadened to lower mass ranges if critical collapse is considered
Choptuik (1993); Koike  _et al._ (1995); Niemeyer and Jedamzik (1998),
modifying Eq. (13). The variation of $\delta$ of a Hubble volume after the
transition time depending on the internal progression of phase transition
would be responsible for it.

Figure 6:  Contours of the final PBH abundance as a function of the FOPT
parameters $\alpha$ and $\beta$, with $T_{\rm cri}$ and $\Gamma_{\rm
cri}/H_{\rm cri}^{4}$ of model A (red dot). $\beta/H_{\rm cri}$ is fixed when
$\alpha$ is varied. The red curve is where the PBHs from the FOPT constitute
the full DM.

We further explore the parameter space of $\alpha$ and $\beta$ and show the
PBH abundance in Fig. 6 using model A as a reference. As a whole, we see that
the abundance increases with $\alpha$ and decreases with $\beta$, as expected
from the behavior of $p(t)$ in Fig. 4 and also the physical considerations.
The dependence is stronger for $\beta$ than $\alpha$, and the $\alpha$
dependence is even more suppressed for $\alpha\gtrsim 10$. The $\alpha$
dependence becomes saturated by resulting in the same vacuum-dominated
universe, hence the same FOPT progression for fixed $\Gamma_{\rm cri}/H_{\rm
cri}^{4}$ and $\beta/H_{\rm cri}$; see also Gouttenoire and Volansky (2023).
These scaling behaviors correspond with our semianalytic analysis in Appendix
B.

Because of the exponential dependence of the PBH fraction on these input
parameters, only a very narrow parameter region near the red contour with
$f_{\rm PBH}=1$ is practically viable to produce DM. On the other hand, from a
phenomenological point of view, any reasonable PBH mass $M$ and fraction
$f_{\rm PBH}$ can be generated in this scenario by changing the model
parameters.

##  V Discussion

In this paper, we presented a thorough calculation of the PBH mass
distribution function arising from vacuum energy decay in a FOPT. We examined
the condition of Ref. Liu  _et al._ (2022) for a Hubble volume to be entirely
in the FV, and found that the surrounding wall cone was neglected. This
omitted factor was found numerically to be of similar magnitude to the primary
contribution of the Hubble volume itself and can therefore result in
significant suppression of the PBH abundance.

Through our numerical simulations, we generated PBH populations that make up
all of DM in the PBH mass window and that can originate the candidate HSC and
OGLE microlensing events. Although we included the additional term and have a
slightly different formalism, our results generally agreed with those of Ref.
Liu  _et al._ (2022). Because of the sensitive dependence of the PBH abundance
on the FOPT parameters, the extra suppression we found can be compensated by a
modest change in parameters.

The PBH formation probability and the resulting abundance are exponentially
sensitive to FOPT speed $\beta$ and strength $\alpha$. Among them, the
dependence on $\beta$ was shown to be exceedingly strong, while the dependence
on $\alpha$ was relatively mild, and further suppressed for $\alpha\gtrsim
10$. These are supported by semianalytic analysis, and they also align with
intuition. For horizon-sized perturbations to survive late into the FOPT,
small $\beta/H\lesssim\mathcal{O}(1)$ is needed as it determines the timescale
of the phase transition and is inversely proportional to the bubble size
Kawana and Xie (2022); Lu  _et al._ (2022a). Naturally, small values of the
vacuum energy suppress PBH formation and large values increase the formation
rate, but make no further difference once a fully vacuum-dominated universe is
reached.

Altogether, the PBH formation from horizon-sized vacuum decay is a viable and
versatile scenario and can apply to a generic FOPT in beyond the Standard
Model physics. Phenomenologically, setting the critical time/temperature
determines the PBH mass, and minor changes in the FOPT parameters can produce
the desired PBH mass function. We found that significant production of PBH
occurs in phase transitions that are slow and moderately strong.

## ACKNOWLEDGMENTS

The work of K. K. is supported by KIAS Individual Grants, Grant No. 090901.
The work of T. H. K. is supported by a KIAS Individual Grant No. PG095201 at
Korea Institute for Advanced Study and National Research Foundation of Korea
under Grant No. NRF-2019R1C1C1010050. The work of P. L. is supported by
National Research Foundation of Korea under Grant No. NRF-2019R1C1C1010050. We
thank Hyung Do Kim, Wan-Il Park, Dhong Yeon Cheong, Gansukh Tumurtushaa,
Masahide Yamaguchi, Chang Sub Shin, and Alexander Kusenko for constructive
comments which helped improve the paper.

##  Appendix A Consistency check for completion of FOPT

Here we show that our scenario is generically consistent with the completion
of FOPT. We first discuss several options for completing a FOPT with
(super)horizon-sized FV regions at $t_{\rm 1.45}$, and then present a
numerical demonstration under a simplified treatment.

The major concern is that the horizon-sized FV regions soon become vacuum
energy dominated and undergo a de Sitter expansion, starting a second
inflation that resembles the old inflation scenario. Since the latter was
abandoned due to the failure of the graceful exit, we must also check whether
these FV regions do not harm the completion of FOPT.

We first state that just having a de Sitter expanding FV region does not
necessarily mean a never-ending inflation. The precise reason that the old
inflation was unsuccessful is that the constant nucleation rate was restricted
to be much smaller than $H^{4}$ in order to solve the horizon and the flatness
problems. This turned out to be incompatible with the completion of the vacuum
transition and thermalization Guth and Weinberg (1983). Here in our scenario,
no such restriction applies and we can have much higher nucleation rates
during the phase transition. If the nucleation rate is high enough, the bubble
nucleation and growth can overcome the volume expansion of FV so that the
physical FV volume decreases to zero and the phase transition ends Turner  _et
al._ (1992b); Kawana (2022).

To clarify, assume a simplified situation of a de Sitter expanding background
with constant $H$ and a constant nucleation rate $\Gamma$. The physical FV
volume is $V^{\rm phys}_{\rm fv}\propto f_{\rm fv}(t)a^{3}(t)$, so its
decrement requires

$\displaystyle\frac{1}{V^{\rm phys}_{\rm fv}}\frac{dV^{\rm phys}_{\rm
fv}}{dt}<0$
$\displaystyle\quad=3H(t)-\frac{4\pi}{3}\frac{d}{dt}\int^{t}_{t_{\rm
cri}}dt^{\prime}\,a^{3}(t^{\prime})\,\Gamma(t^{\prime})\,r^{3}(t,t^{\prime})\qquad$
(22)

which is, for constant $H$ and $\Gamma$,

$\Gamma>\frac{9H^{4}}{4\pi}\approx 0.7H^{4}.$ (23)

Thus, even a quite slow FOPT can complete while having horizon-sized FV
regions if the nucleation rate in those regions is higher than about
$0.7H^{4}$.

Then the question comes down to how high the nucleation rates in the FV
regions can be after passing $T_{\rm cri}$. This is determined by the
temperature dependence of the effective potential as discussed in Sec. II. If
the barrier disappears at some finite temperature, the nucleation rate in FV
regions keeps increasing, and hence there is no problem in finishing the phase
transition; after the barrier disappears, the phase transition becomes second
order and ends through spinodal decomposition. On the other hand, if the
barrier persists in the zero-temperature limit, the nucleation rate by thermal
fluctuation decreases to zero. Then nucleation can only occur through quantum
tunneling, and it is likely that the phase transition never ends, resulting in
a second inflationary phase.

Typically, the latter case does not pose a serious problem because the
remaining FV regions are fragmented into subhorizon sizes before the
nucleation rate becomes too small. These regions are totally converted into TV
by the bubble walls approaching from outside, instead of relying on bubble
nucleations inside. However, as our scenario requires having horizon-sized FV
regions at least until $t_{\rm 1.45}$, we need a stronger condition to have a
consistent scenario.

In conclusion, there are generically three ways to have completion of FOPT
within our scenario. First, if the effective potential gives the bubble
nucleation rate that keeps increasing with decreasing temperature, the FOPT
will finish without any problem. The exponential approximation we take (2) in
our study corresponds to this case. Second, even if the nucleation rate peaks
and decreases it is still possible to have a high enough nucleation rate after
$t_{\rm 1.45}$ so that all the horizon-sized FV regions are dismantled into
subhorizon pieces before the nucleation rate becomes too small. The third
option is to have a sufficiently high quantum tunneling rate that can finish
the FOPT even at zero temperature.

The actual nucleation rates in these three cases will have deviations from the
simple analytic exponential approximation in Eq. (2), although more realistic
forms of the nucleation rate for the current scenario are left to future
studies.

In the rest of this appendix, we numerically demonstrate the decrease of
physical FV volume after $t_{\rm 1.45}$ for the first option in a simplified
setup. We regard the FV region at $t_{1.45}$ as a separate universe and assume
it undergoes a de Sitter expansion with the Hubble rate coming from the vacuum
energy density $\Delta V$. These are quite good approximations for horizon-
sized FV regions, as they are vacuum dominated and the bubble walls from
outside cannot entirely cover them. The exponential approximation (2) is used
for the nucleation rate.

Figure 7:  The normalized FV volume of a FV region at $t_{1.45}$, simplified
to be a separated de Sitter universe. Black: the FV volume fraction equivalent
to the normalized FV comoving volume. Red: the FV volume fraction times
$a^{3}$ equivalent to the normalized FV physical volume.

In Fig. 7, we plot $f_{\rm fv}(t)$ and $f_{\rm fv}(t)\,a^{3}(t)$ for the
separated universe corresponding to a FV region at $t_{1.45}$ with the same
parameters as in Fig. 3. We see that the physical volume of the FV initially
increases due to the volume expansion, but it soon rapidly decreases due to
the nucleation and growth of TV bubbles. The nucleation rate at $t_{1.45}$ is
already about $\Gamma(t_{1.45})\sim\mathcal{O}(10)\times H^{4}$ due to the
exponential growth from $t_{\rm cri}$, and even keeps increasing further.
Referring to Eq. (23), this nucleation rate is sufficient to convert all the
remaining FV regions soon after the nucleation begins.

We also note that this large nucleation rate at late times is one of the two
reasons why the survival probability [or $p(t)$ as its differential] is
suppressed as depicted in Fig. 4. The other reason is of course the wall cone
contribution in Fig. 1. This tiny probability is compensated by the
cosmological expansion during the radiation-dominated epoch, where the
radiation density redshifts $\propto 1/a^{4}$ while the PBH only undergoes a
number density dilution $\propto 1/a^{3}$. Because of this effect, the small
fraction of horizons that have delayed transitions in the early Universe can
contribute significantly to $\Omega_{\rm PBH}$ in the present day.

##  Appendix B Semianalytic analysis of PBH formation probability

We derive the qualitative dependence of the PBH formation probability on the
FOPT parameters $\alpha$ and $\beta$, and confirm the agreement to the
intuitive expectations while also supporting our numerical results in Sec.
IV.2. As the actual PBH formation in this model is exponentially sensitive to
the parameters, the quantitatively correct probabilities and resultant
abundances need to be numerically computed as in the main text.

Here we focus on the formation probability (Fig. 4), as this is the main point
of our calculation; analytic expressions of $\beta(M)$ and $f(M)$ follow
directly from Eqs. (12), (15), and (16). We approximate the integrated
formation probability $\sim\int_{t_{1.45}}p(t)dt$ as simply the survival
probability $P(t_{1.45})$, since the majority of the PBH production occurs at
$t_{1.45}$ [see the rapidly dropping curves of $p(t)$ in Fig. 4]. Furthermore,
it is sufficient to consider only the first exponent in Eq. (11) for our
purposes since the behavior of the second exponent is similar (see Fig. 1).

As can be seen in Fig. 3, the average energy density $\bar{\rho}$ stays
approximately constant before $t_{1.45}$. This is generically true in our
scenario, as the vacuum energy necessarily becomes dominant over the SM
plasma. Then as the FV energy is converted into latent heat, the average
density will decline to a value $\bar{\rho}=(\Delta V+\rho_{\rm
SM}(t_{1.45}))/1.45\approx 0.69\Delta V$ by $t=t_{1.45}$. Thus, the average
density and the Hubble rate [Eq. (21)] are approximately constant for $t_{\rm
cri}<t<t_{1.45}$ as $H_{*}^{2}\simeq\Delta V/3M_{P}^{2}$, and hence,
$a(t)\propto e^{H_{*}t}$.

Then in this constant Hubble rate approximation, the first exponent of Eq.
(11) is (we calibrate $t$ by setting $t_{\rm cri}=0$ hereafter)

$\displaystyle\ln P_{\rm HV}(t_{1.45})$ $\displaystyle\simeq$
$\displaystyle-\frac{4\pi}{3}\int^{t_{1.45}}_{0}e^{-3H_{*}(t_{1.45}-t)}\frac{\Gamma_{\rm
cri}}{H_{*}^{3}}\,e^{\beta t}\,dt$ (24) $\displaystyle\simeq$
$\displaystyle-\frac{4\pi}{3}\frac{\Gamma_{\rm
cri}}{(3H_{*}+\beta)H_{*}^{3}}\,e^{\beta\,t_{1.45}}$

where the subscript HV denotes the Hubble volume only (excluding the wall cone
contribution) and $e^{\beta t_{\rm 1.45}}$, $e^{3Ht_{\rm 1.45}}\gg 1$. Since
$H_{\ast}^{2}\propto\alpha$, only the behavior of $e^{\beta t_{1.45}}$ remains
to be figured out.

To see how it scales, we use Eqs. (6) and (17)-(20). First, integrating Eq.
(20) with Eqs. (18) and (19) gives

$\bar{\rho}_{R}(t)=\rho_{\rm SM}(0)e^{-4Ht}+\Delta
V\int^{t}_{0}\left(-\frac{df_{\rm
fv}}{dt^{\prime}}\right)e^{-4H_{*}(t-t^{\prime})}dt^{\prime}.$ (25)

Then substituting Eqs. (17) and (25) into Eq. (6) and eliminating $\Delta V$
by $\alpha\equiv\Delta V/\rho_{\rm SM}(T_{\rm cri})$ gives

$f_{\rm fv}(t_{1.45})+\frac{0.31}{\alpha}e^{-4H_{*}\,t_{1.45}}+(1-f_{\rm
fv}(t_{1.45}))X=0.69~{},$ (26)

where $X$ represents the average redshift of the transitioned vacuum energy
defined by

$(1-f_{\rm fv}(t_{1.45}))X\equiv\int^{t_{1.45}}_{t_{\rm
cri}=0}\left(-\frac{df_{\rm fv}}{dt}\right)e^{-4H_{*}(t_{\rm
1.45}-t)}\,dt~{}.$ (27)

From the constant Hubble rate approximation in Eq. (II),

$f_{\rm fv}(t_{1.45})\simeq\exp\left[-\frac{8\pi v_{w}^{3}\Gamma_{\rm
cri}\,e^{\beta\,t_{1.45}}}{\beta(H_{*}+\beta)(2H_{*}+\beta)(3H_{*}+\beta)}\right].$
(28)

From Eq. (26), we extract the variation of $f_{\rm fv}(t_{1.45})$ with
$\alpha$ and $\beta$. Then we use Eq. (28) to solve for $e^{\beta t_{1.45}}$
as a function of $f_{\rm fv}(t_{1.45})$ and determine the behavior of $P_{\rm
HV}$ in Eq. (24).

We first examine the dependence on $\beta$. Faster phase transitions with
larger $\beta$ have smaller $t_{1.45}$, which results in less redshifting and
greater $X$.333While not mathematically clear, it is physically reasonable,
and we also numerically checked for all three cases in Table 1, with varying
$\beta$ by $\mathcal{O}(1)$ times. The FV fraction at formation, $f_{\rm
fv}(t_{1.45})$, decreases to satisfy Eq. (26) and lower the background energy
density.444At linear order, the variation of the lhs in Eq. (26) gives
$(1-f_{\rm fv})\delta X+(1-X)\delta f_{\rm fv}$. Since $1-f_{\rm fv}\geq 0$,
$\delta X>0$ means $\delta f_{\rm fv}<0$ to maintain Eq. (26); note that the
second term containing $\alpha$ also increases. Equation (28) then demands
$e^{\beta t_{1.45}}$ to greatly increase to get over the denominator. Solving
for it gives the scaling of $e^{\beta t_{1.45}}\sim-\beta^{4}\ln f_{\rm fv}$.
Since $0<f_{\rm fv}<1$ itself decreases with increasing $\beta$, using this
relation in Eq. (24) shows that $P_{\rm HV}(t_{1.45})$ decreases rapidly,
faster than negative exponential of powers of $\beta$.

On the other hand, Eq. (26) shows that larger $\alpha$ results in larger
$f_{\rm fv}(t_{1.45})$,555Similar to the previous case, this is not
mathematically clear but physically reasonable since the critical density
contrast will be reached in the earlier stage of FOPT. We checked this
numerically too. but the effect is suppressed by the exponential factor
$e^{-4H_{*}t_{1.45}}$ and $\alpha$ itself. Furthermore, Eqs. (24) and (28)
show no explicit $\alpha$ dependence as long as $\Gamma_{\rm cri}/H_{*}^{4}$
and $\beta/H_{*}$ are fixed. Therefore, $P_{\rm HV}(t_{1.45})$ increases with
$\alpha$, but the effect is suppressed if $\alpha$ keeps growing.

These behaviors are depicted in the bottom panel of Fig. 4 and Fig. 6.
Increasing $\beta$ drastically lowers $p(t)$ and hence $f_{\rm PBH}$.
Increasing $\alpha$ gives larger $p(t)$ and $f_{\rm PBH}$ but relatively
mildly. Clearly, these agree with the rule of thumb expectations: The faster
the phase transition, the less time for the redshift to generate density
contrast, so PBH formation probability decreases with $\beta$; greater vacuum
energy (the source of density contrast) over the homogeneous SM radiation
always facilitates the PBH production, so the probability increases with
$\alpha$, but asymptotes at large values of $\alpha$ corresponding to a
vacuum-dominated universe.

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