# Primordial black holes from bubble collisions during a first-order phase
transition

Tae Hyun Jung<EMAIL_ADDRESS>Department of Physics, Florida State
University, Tallahassee, FL 32306, USA  Takemichi Okui<EMAIL_ADDRESS>Department of Physics, Florida State University, Tallahassee, FL 32306, USA
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan

###### Abstract

We show that collisions of true vacuum bubbles during a first-order phase
transition can produce a sizable abundance of primordial black holes (PBHs),
provided that bubble walls have a non-negligible thickness and the transition
lasts long enough. We predict an approximately monochromatic PBH mass
spectrum, and the PBHs can constitute the whole dark matter with their masses
in the currently allowed range of $10^{-15}$–$10^{-11}$ solar masses if the
phase transition occurs around $10^{5}$–$10^{7}\,\mathrm{GeV}$. Our
calculation combined with bounds on the PBH abundance can constrain models
with a first-order phase transition.

††preprint: KEK-TH-2350

Introduction: First-order phase transition in the early universe is a well
motivated field of study both theoretically and experimentally. A generic
quantum field theory has multiple phases and it is natural for the early
universe to undergo various stages of phase transitions, some of which may
well be at first order. A first-order phase transition necessarily leads to
nucleation of true vacuum bubbles and their subsequent expansions, and they
must eventually collide to fill the universe with the true vacuum. This can
result in many cosmological implications such as dark matter Chway:2019kft ;
Baker:2019ndr ; Ahmadvand:2021vxs ; Hong:2020est ; Azatov:2021ifm ;
Asadi:2021pwo  and baryogenesis Shaposhnikov:1987tw ; Cline:2018fuq ;
Hall:2019ank ; Fujikura:2021abj ; Arakawa:2021wgz ; Azatov:2021irb . Since
gravitational wave production from bubble collisions during a first-order
phase transition is generically expected Witten:1984rs ; Hogan:1986qda ;
Kosowsky:1991ua ; Kosowsky:1992vn ; Kamionkowski:1993fg ; Espinosa:2010hh ;
Caprini:2019egz ; Schmitz:2020syl , those ideas may be testable by near-future
gravitational wave experiments Harry:2010zz ; LIGOScientific:2014pky ;
VIRGO:2014yos ; LISA:2017pwj ; Baker:2019nia ; Seto:2001qf ; Kawamura:2011zz ;
Yagi:2011wg ; Isoyama:2018rjb ; Crowder:2005nr ; Corbin:2005ny ; Harry:2006fi
; Somiya:2011np ; Aso:2013eba ; KAGRA:2018plz ; KAGRA:2019htd ;
Michimura:2019cvl ; LIGOScientific:2016wof ; Reitze:2019iox ; Punturo:2010zz ;
Hild:2010id ; Sathyaprakash:2012jk ; Maggiore:2019uih ; Burke-Spolaor:2018bvk
; McLaughlin:2013ira ; NANOGRAV:2018hou ; Aggarwal:2018mgp ; Brazier:2019mmu ;
Manchester:2012za ; Shannon:2015ect ; Kramer:2013kea ; Lentati:2015qwp ;
Babak:2015lua ; Hobbs:2009yy ; Manchester:2013ndt ; Verbiest:2016vem ;
Hazboun:2018wpv ; Carilli:2004nx ; Janssen:2014dka ; Weltman:2018zrl  (see
Caprini:2019egz ; Schmitz:2020syl  for reviews).

Since every first-order phase transition must end with bubble collisions, it
is important to seek other significant outcomes of bubble collisions. In
particular, given many PBH bounds and searches Carr:2020gox ; Carr:2020xqk ;
Marani:1998sh ; Nemiroff:2001bp ; Barnacka:2012bm ; Katz:2018zrn ;
Griest:2013esa ; Griest:2013aaa ; Niikura:2017zjd ; Paczynski:1985jf ;
Hamadache:2006fw ; Wyrzykowski:2009ep ; CalchiNovati:2009kq ;
Wyrzykowski:2010bh ; Wyrzykowski:2010mh ; Wyrzykowski:2011tr ; Niikura:2019kqi
; Zumalacarregui:2017qqd ; Garcia-Bellido:2017imq ; Page:1976wx ; Carr:2009jm
; Carr:2016hva ; Boudaud:2018hqb ; DeRocco:2019fjq ; Laha:2019ssq ;
Dasgupta:2019cae ; Laha:2020ivk ; Chan:2020zry ; Belotsky:2014twa ;
Kim:2020ngi , we would like to know whether or not bubble collisions can
produce an observationally meaningful abundance of PBHs, a subject initiated
by Hawking:1982ga . One might be tempted to disregard this possibility and
favor other PBH formation mechanisms Sato:1981bf ; Maeda:1981gw ;
Kodama:1982sf ; Hall:1989hr ; Kusenko:2020pcg  Khlopov:1999ys ;
Lewicki:2019gmv  Gross:2021qgx ; Baker:2021nyl ; Kawana:2021tde ;
Baker:2021sno  as it seems highly improbable for a large amount of energy to
be compacted into a small volume by bubble collisions. Note, however, that the
number of PBHs per Hubble volume _at the time of their formation_ must be
extremely small anyway:

$\frac{n_{\rm PBH}}{H^{3}}\sim 10^{-15}f_{\rm
PBH}\\!\left(\frac{10^{-11}M_{\odot}}{M_{\rm
PBH}}\right)\\!\\!\left(\frac{10^{5}\,\rm
GeV}{T_{*}}\right)^{\\!\\!3}\\!\\!\left(\frac{100}{g_{*}}\right)^{\\!\\!\frac{1}{2}}\\!\\!,$
(1)

where $f_{\rm PBH}\equiv\Omega_{\mathrm{PBH}}/\Omega_{\mathrm{DM}}$, and
$T_{*}$ and $g_{*}$ are the reheating temperature and radiation degrees of
freedom after the phase transition. Therefore, it requires a careful
quantitative calculation to really see whether or not bubble collisions can be
a viable origin of PBHs.

In this Letter we report such a calculation with the following results: (i)
bubble collisions during a first-order phase transition can produce PBHs _only
if_ the bubble radii become near-horizon-sized and the bubble walls have a
non-negligible thickness when they collide; (ii) the PBH abundance can be
observationally significant if the phase transition lasts long enough to allow
many such large bubbles to grow before they collide; and (iii) the conditions
in (i) and (ii) seem compatible with sensible quantum field theories. Because
of the requirement of a finite wall thickness, our positive result does not
contradict with the negative conclusion of Moss:1994pi  for infinitely thin
walls.

We restrict our analysis below to first-order phase transitions that go into a
super-cooled phase, where the positive vacuum energy of the false vacuum
quickly dominates over other forms of energy such as radiation. This
simplifies the analysis in two ways. First, the spacetime geometry is either
flat (inside a bubble) or a de Sitter (outside bubbles). Second, since
friction from the plasma become negligible in this regime, the bubble walls
expand nearly at the speed of light.

The hoop conjecture and the definition of mass: As a criterion for black hole
formation, we employ _the hoop conjecture_ Thorne:1972ji  — a black hole forms
if and only if a mass $M$ gets compacted into a region whose largest
circumference ${\cal C}_{\rm max}$ is less than the critical circumference
${\cal C}_{\rm cr}\sim 2\pi\cdot 2GM=M/2$, where $8\pi G=1$.

The utility of the hoop conjecture crucially depends on what we mean by $M$.
Consider three volumes labelled as $\mathcal{V}_{1,2,3}$ in Fig. 1. In volume
$\mathcal{V}_{1}$ the two bubble walls are overlapping now, in volume
$\mathcal{V}_{2}-\mathcal{V}_{1}$ an overlap has occurred, and in volume
$\mathcal{V}_{3}-\mathcal{V}_{2}$ they have not overlapped yet. Since no part
of an expanding single bubble in isolation would never form a black hole,
region $\mathcal{V}_{3}-\mathcal{V}_{2}$ should not contribute to the $M$ for
the hoop criterion. A good definition of $M$ with this desirable property is
inspired by the Schoen-Yau theorem Schoen:1983aa . The theorem gives a
rigorous sufficient condition for a region to collapse into a black
hole,111See Flanagan:1991aa  for a nice summary and discussion of the the
theorem in the context of the hoop conjecture. although it is too strong for
our purpose (as it is chosen to be strong enough to allow a rigorous proof by
mathematicians’ standards). We only adopt their definition of mass density and
use it to define $M$ as follows. Consider a spacelike hypersurface ${\cal H}$
and a timelike vector field $n^{\mu}$ normal to ${\cal H}$ with $n^{2}=-1$. We
define the mass $M({\cal V})$ of a volume ${\cal V}\subset{\cal H}$ by

$\displaystyle M({\cal V})\equiv\int_{\cal V}\\!\mathrm{d}^{3}x\sqrt{{\rm
det}\,(g_{ij})}\left(\rho-\sqrt{J^{\mu}J_{\mu}}\right),$ (2)

where $\rho\equiv T^{\mu\nu}n_{\mu}n_{\nu}$ and $J^{\mu}\equiv
T^{\mu\nu}n_{\nu}+\rho n^{\mu}$ following Schoen:1983aa , and $g_{ij}$ denotes
the induced metric on $\mathcal{H}$. One can check that the $\rho$ and
$\sqrt{J^{\mu}J_{\mu}}$ terms cancel with each other for relativistic walls
outside $\mathcal{V}_{2}$ even if $\mathcal{V}$ is chosen to be larger than
$\mathcal{V}_{2}$ (see the Supplemental Material). Similarly, for region
$\mathcal{V}_{2}-\mathcal{V}_{1}$, we interpret the results of numerical
simulations in Johnson:2011wt ; Cutting:2019zws ; Jinno:2019bxw  as indicating
that the motion in this region is highly relativistic and hence assume that
region $\mathcal{V}_{2}-\mathcal{V}_{1}$ does not contribute to $M$. Inside
$\mathcal{V}_{1}$, the two bubbles’ contributions to $J_{\mu}$ tend to cancel
while those to $\rho$ tend to add up, thereby contributing to $M$. So, Fig. 1
should be understood as capturing the moment when this contribution to $M$
from $\mathcal{V}_{1}$ is maximal.

Figure 1: A schematic picture of collision of two bubbles.

Necessity for near-horizon-sized bubbles: Having established what to include
in $M$, let us use it to make a quick estimate to show that PBH formation
requires bubbles to be as large as $H^{-1}$ when they collide. Consider
collision of two bubbles with their radii and wall widths taken to be similar,
$R_{1}\simeq R_{2}\simeq R$ and $L_{1}\simeq L_{2}\simeq L$, for simplicity.

If $R\ll H^{-1}$, we approximately have $\rho\sim\Delta V\,R^{3}/(R^{2}L)\sim
R\Delta V/L$ and $J_{\mu}\sim 0$ in region $\mathcal{V}_{1}$, where $\Delta
V>0$ is the vacuum energy density difference between the false and true vacua.
Region $\mathcal{V}_{1}$ takes the shape of a biconvex lens with its radius
and volume given respectively by $\sim\sqrt{RL}$ and $\sim RL^{2}$, so the
hoop criterion becomes

$\displaystyle 1<\frac{{\cal C}_{\rm cr}}{{\cal C}_{\rm
max}}\sim\frac{M}{\sqrt{RL}}\sim\frac{RL^{2}\rho}{\sqrt{RL}}\sim
H^{2}R^{2}\sqrt{\frac{L}{R}}\,$ (3)

with $H^{2}=\Delta V/3$. Since $R\ll H^{-1}$ and $L\ll R$, the inequality (3)
is severely violated. It is not possible to satisfy the criterion by having
$N_{B}$ bubbles collide with $N_{B}\gg\gg 1$ as the probability for such an
event is given by an exponentially small bubble nucleation rate raised to the
$N_{B}$-th power multiplied by a tiny phase space in which $N_{B}$ bubbles
colliding at a point.222Hawking:1982ga  reached a relatively optimistic
estimate that $\sim 8$ bubbles are needed to collide at a point for PBH
formation based on a definition of $M$ that picks up significant contributions
also from region outside $\mathcal{V}_{1}$.

Figure 2: [Left]: A single bubble in $(u,\bar{r})$ coordinates. [Middle]: The
same bubble in $(t,r)$ coordinates, where the dashed vertical line shows the
horizon. [Right]: The collision of two bubbles, where a PBH may form in the
blue region with “?”.

We therefore conclude that we need to give up the assumption $R\ll H^{-1}$. We
also see that we cannot neglect the ratio $L/R$, but since this ratio must be
$<1$ we need “$R\gg H^{-1}$” to satisfy the hoop criterion. This is
meaningless, however, as we now must carefully take into account the de Sitter
space outside the bubbles and define what we mean by $R$. We must also pay
attention to the causal structure of the de Sitter space, in particular its
horizon. We now move on to setting up a full curved space analysis.

Vaidya-de Sitter metric for a single bubble: Consider a single bubble. As we
can approximate the trajectories of the bubble walls by null worldlines, let
us introduce an outgoing null coordinate $u$ such that the bubble wall
trajectories are described by constant values of $u$. As we increase $u$ and
move from outside the bubble ($u<u_{1}$) to inside ($u>u_{1}+L$), the energy
density drops from $\Delta V$ to $0$. (See Fig. 2 left.)

For simplicity, we assume that this situation can be approximately described
by the Vaidya-de Sitter metric Mallett:1985lsn :

$\displaystyle
ds^{2}=-\left(1-\frac{\bar{\rho}(u)}{3}\bar{r}^{2}\right)du^{2}-2dud\bar{r}+\bar{r}^{2}d\Omega^{2}\,,$
(4)

where $\bar{\rho}(u)$ is not to be confused with the $\rho$ in (2). The
corresponding $T_{\mu\nu}$ is given by

$\displaystyle
T_{\mu\nu}=G_{\mu\nu}=-\frac{1}{3}\bar{r}\bar{\rho}^{\prime}(u)l_{\mu}l_{\nu}-\bar{\rho}(u)g_{\mu\nu},$
(5)

where $l_{\mu}=\partial u/\partial x^{\mu}$ in an arbitrary coordinate system
$x^{\mu}$ and ′ denotes $d/du$. It is nontrivial to calculate $\bar{\rho}(u)$
in region $u_{1}<u<u_{1}+L$ by solving the scalar equation of motion and
Einstein equation simultaneously. For the purpose of our estimate, we simply
model $\bar{\rho}(u)$ in this region as a linear function that drops from
$\Delta V$ to $0$.

It is essential that the wall width parameter $L$ is defined as the thickness
in the profile of $\bar{\rho}$ as above and in Fig. 2 left. Since $\bar{\rho}$
appears in $T_{\mu\nu}$ as in Eq. (5), it parametrizes the _total_ energy
energy density, not just that of the scalar field. Therefore, this in
particular means that $L$ can be much different from the width of the scalar
field configuration alone.

To describe the collision of two bubbles taking place in the background of the
false vacuum, we find it more convenient to switch to a coordinate system in
which the de Sitter space between the bubbles is described by the standard
LFRW metric. So, we define new coordinates $(t,r)$ via

$\displaystyle u=t-\frac{1}{H}\log\left(1+e^{Ht}Hr\right),$ (6)
$\displaystyle\bar{r}=re^{Ht},$ (7)

where $H$ is a _constant_ defined to be equal to the Hubble rate outside
bubbles, $H^{2}\equiv\Delta V/3$. Then, the metric in $(t,r)$ becomes

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dt^{2}+e^{2Ht}(dr^{2}+r^{2}d\Omega^{2})$ (8) $\displaystyle-\frac{\Delta
H^{2}(u)\,r^{2}e^{2Ht}}{(1+Hre^{Ht})^{2}}(dt-e^{Ht}dr)^{2}\,,$

where $\Delta H^{2}(u)\equiv H^{2}-\bar{\rho}(u)/3\geq 0$. For $u<u_{1}$
(outside the bubble), we have $\Delta H^{2}(u)=0$ and this metric simply takes
a translationally invariant LFRW form. This will allow us to add more bubbles
easily. For $u>u_{1}+L$ (inside the bubble), we have a flat spacetime although
the metric (8) looks complicated.

Eq. (6) shows that the $r$ coordinate of any outgoing null geodesic reaches an
asymptotic value as $t$ increases. For example, the $r$ coordinate of the
outer surface of the bubble wall nucleated at $t=t_{1}\equiv u_{1}$ asymptotes
to $R_{\rm h1}=H^{-1}e^{-Ht_{1}}$ (see Fig. 2 middle). However, Eq. (7) shows
that the _physical_ radius of the bubble will keep growing as $e^{Ht}$. In
this way we can give a precise meaning to the condition “$R\gg H^{-1}$”
necessary to form a PBH from bubble collisions.

Similarly, the bubble wall width in the $r$ coordinate asymptotes to $HLR_{\rm
h1}$, that is, the ratio of the wall width to the radius is asymptotically
$HL$. Therefore, we treat $HL$ as one of the effective parameters
characterizing a given first-order phase transition.

Finally, we show in the Supplemental Material that $M(\mathcal{V})$ vanishes
on any $\mathcal{V}$ on any constant $t$ hypersurface in the $(t,r)$
coordinate system for a single bubble, as we expect physically.

Collision of two bubbles and the hoop criterion: Let us consider two bubbles
as described in the right panel of Fig. 2. As we pointed out earlier,
describing two bubbles is trivial in the $(t,r)$ coordinates as long as
$t<t_{c}$, where $t_{c}$ is the time when the outer surfaces of the two
bubbles meet. Since what happens after $t_{c}$ is highly complicated and model
dependent, we estimate $M(\mathcal{V})$ at $t=t_{c}$ by neglecting the
$\sqrt{J_{\mu}J^{\mu}}$ term by hand and carefully choosing $\mathcal{V}$ to
mimic the state of overlapping walls of region $\mathcal{V}_{1}$ in Fig. 1.

With these approximations, we find the following results (see the Supplemental
Material for details). For $t_{1}=t_{2}$, we obtain

$M({\cal V})\sim\frac{4\pi}{3}H^{-3}\Delta V\frac{HL}{(1-R_{1}/R_{\rm
h1})^{2}}\quad\text{for}\>1-\frac{R_{1}}{R_{\rm h1}}\ll 1,$ (9)

where $R_{1}$ is the bubble radius in the $r$ coordinate at $t=t_{c}$ (see
Fig. 2 right). Since $M\propto L$, our estimation does not contradict with
Ref. Moss:1994pi  in which it was argued that a two-bubble collision process
does not form a PBH in $L=0$ limit. We also show, in the Supplemental
Material, that $M({\cal V})$ in $R_{1}\ll R_{h1}$ limit recovers the
estimation (3) in the flat spacetime approximation. Using the mass (9), the
hoop criterion becomes

$1<\frac{C_{\rm cr}}{C_{\rm max}}\sim\frac{\sqrt{HL}}{1-R_{1}/R_{\rm
h1}}\>\>\Longrightarrow\>\>1-\frac{R_{1}}{R_{\rm h1}}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\sqrt{HL}\,.$ (10)

For $t_{1}\neq t_{2}$, the criterion becomes

$r_{12}>R_{\rm h1}+R_{\rm h2}-2f(t_{1},t_{2})\,R_{\rm h2}\sqrt{HL},$ (11)

where $t_{1}$ $(t_{2})$ is the nucleation time of the bigger (smaller) bubble,
$R_{{\rm h}i}=H^{-1}e^{-Ht_{i}}$ is the horizon size of bubble $i$, and
$r_{12}$ is the distance between the bubble centers in the $r$ coordinate and
is bounded from above by $R_{\rm h1}+R_{\rm h2}$. The function
$f(t_{1},t_{2})$ is the ratio of the volume of $\mathcal{V}_{1}$ to its
largest circumference normalized as $f(t_{1},t_{1})=1$:

$\displaystyle
f(t_{1},t_{2})=\sqrt{\frac{2R_{h1}}{R_{h1}+R_{h2}}\frac{R_{h1}}{R_{h2}}}\,.$
(12)

The PBH abundance and spectrum: Given a bubble nucleated at $t_{1}$, the
average number of PBHs formed can be calculated by

$N(t_{1})=\int_{t_{1}}^{\infty}\\!\\!\\!dt_{2}\int_{{\rm
Eq.}\,\eqref{eq:criterion_final}}d^{3}\vec{r}_{12}\,\sqrt{-g}\,\Gamma_{n}(t_{2})P_{f}(t_{c})$
(13)

where the range of $\vec{r}_{12}$ integration is restricted by the condition
(11) and $\Gamma_{n}$ is the bubble nucleation rate. $P_{f}(t_{c})$ is the
probability that the collision point at $R_{1}$ has remained in the false
vacuum until $t_{c}$, and it depends on $t_{1}$, $t_{2}$, $r_{12}$. The PBH
number density can then be estimated as

$\displaystyle n_{\rm
PBH}=\frac{\displaystyle{\int_{-\infty}^{\infty}\\!\\!dt_{1}\,e^{3Ht_{1}}\,N(t_{1})\,\Gamma_{n}(t_{1})}}{\displaystyle{\int_{-\infty}^{\infty}\\!\\!dt\,e^{3Ht}\,P_{f}(t)}}\,.$
(14)

See the Supplemental Material for detailed calculations.

The mass of a PBH at its formation depends on its $t_{c}$. However, since
$P_{f}(t_{c})$ rapidly drops as $t_{c}$ increases, the PBH mass is mostly
distributed around the threshold obtained from (9) and (10), $M_{\rm
PBH}\sim\frac{4\pi}{3}H^{-3}\Delta V$, so we predict an approximately
monochromatic PBH mass spectrum.

Figure 3: $f_{\rm PBH}$ vs $M_{\rm PBH}$ for $HL=10^{-3}$. The lines with
different colors correspond to different values of $\beta$, where $\beta=6$,
8, 10, 12 from top to bottom. The dots on each line indicate the values of
$T_{*}/{\rm GeV}$. Dots connected by the same vertical gray line have the same
$T_{*}$. The current observational constraints are depicted by gray shaded
regions. On the left of the dashed line, PBHs fully evaporate before today.

Results: For a quick estimation, we parameterize $\Gamma_{n}(t)$ as
Turner:1992tz ; Kamionkowski:1993fg

$\displaystyle\Gamma_{n}(t)=H^{4}e^{\beta H(t-t_{n})}$ (15)

where $\beta$ and $t_{n}$ are free parameters. $(\beta H)^{-1}$ parameterizes
the duration of the phase transition. We expect decreasing $\beta$ should
increase $n_{\rm PBH}$ as a long duration is necessary to grow near-horizon-
sized bubbles. $t_{n}$ is the time at which the nucleation rate per Hubble
volume becomes comparable to the expansion rate, although $n_{\rm PBH}$ is
independent of $t_{n}$.

In Fig. 3, we show the results for $HL=10^{-3}$. The lines with different
colors correspond to different values of $\beta$, where $\beta=6$, 8, 10, 12
from top to bottom. The dots represent different choices of $T_{*}$, the
reheating temperature after the phase transition, with black numbers
indicating the values of $T_{*}$ in GeV. Dots connected by the same vertical
gray line have the same $T_{*}$. We approximate $T_{*}$ by $\Delta
V\sim\frac{\pi^{2}}{30}g_{*}T_{*}^{4}$ and assume that $g_{*}\sim 100$ does
not change much.

Note that $f_{\rm PBH}=\Omega_{\rm PBH}/\Omega_{\rm DM}$ is evaluated at the
time of PBH formation, and does not take into account possible evaporation by
Hawking radiation Hawking:1974rv ; Hawking:1975vcx . If $M_{\rm PBH}\gg
5\times 10^{14}\,{\rm g}$, the evaporation effect is negligible Carr:2016hva ,
so the abundance at present is given by $f_{\rm PBH}$. In the opposite case,
PBHs have completely evaporated, so the current PBH abundance is zero. This
boundary is depicted by the dashed line in Fig. 3.

The constraints in Fig. 3 are taken from Carr:2020gox ; Carr:2020xqk  and
depicted by gray regions. These constraints can now be applied to restrict
cosmological phase transition parameters $T_{*}$, $\beta$ and $HL$. The
constraints from gravitational lensing Marani:1998sh ; Nemiroff:2001bp ;
Barnacka:2012bm ; Katz:2018zrn ; Griest:2013esa ; Griest:2013aaa ;
Niikura:2017zjd ; Paczynski:1985jf ; Hamadache:2006fw ; Wyrzykowski:2009ep ;
CalchiNovati:2009kq ; Wyrzykowski:2010bh ; Wyrzykowski:2010mh ;
Wyrzykowski:2011tr ; Niikura:2019kqi ; Zumalacarregui:2017qqd ; Garcia-
Bellido:2017imq  are relevant in the PBH mass range of
$10^{-10}M_{\odot}$–$10\,M_{\odot}$, corresponding to $100\,{\rm
MeV}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}T_{*}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}10\,\mathrm{TeV}$. $f_{\rm PBH}$ for
PBH mass below $10^{-16}\,M_{\odot}$ (roughly $T_{*}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}10^{7}\,\mathrm{GeV}$) is strongly contrained, since PBHs in
this mass range evaporate and can leave signals in various channels
Page:1976wx ; Carr:2009jm ; Carr:2016hva ; Boudaud:2018hqb ; DeRocco:2019fjq ;
Laha:2019ssq ; Dasgupta:2019cae ; Laha:2020ivk ; Chan:2020zry ;
Belotsky:2014twa ; Kim:2020ngi .

For $T_{*}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{12}\,\mathrm{GeV}$, PBHs
completely evaporate before BBN. Although these PBHs cannot leave an
observable signature, these ultra-light PBHs have several cosmological
implications such as baryogenesis and dark matter Fujita:2014hha ;
Allahverdi:2017sks ; Lennon:2017tqq ; Morrison:2018xla ; Hooper:2019gtx ;
Lunardini:2019zob ; Chaudhuri:2020wjo ; Masina:2020xhk ; Baldes:2020nuv ;
Gondolo:2020uqv ; Bernal:2020kse ; Bernal:2020ili ; Bernal:2020bjf ;
Auffinger:2020afu ; Kitabayashi:2021hox ; Barrow:1990he ; Baumann:2007yr ;
Hook:2014mla ; Hamada:2016jnq ; Hooper:2020otu ; Perez-Gonzalez:2020vnz ;
Aliferis:2020dxr ; Datta:2020bht ; Sandick:2021gew .

Conclusion: We have shown that collisions of true vacuum bubbles during a
first-order phase transition can produce a sizable abundance of PBHs with an
approximately monochromatic mass spectrum, including the possibility that the
PBHs constitute an $O(1)$ fraction of dark matter or even over-close the
universe, provided that bubble walls have a non-negligible thickness and the
transition lasts long enough. We have demonstrated this by a benchmark value
for the wall width parameter, $HL=10^{-3}$, with various choices for the value
of the inverse duration parameter $\beta$ around $\sim 10$.

Regarding $HL$, recall that $L$ is the width of the total energy density
profile $\bar{\rho}$, rather than the thickness of the scalar field
configuration. Hence, we expect that such non-negligible $HL$ may be realized
in a sensible quantum field theory by taking into account the plasma bulk
motion Kamionkowski:1993fg ; Espinosa:2010hh . It is, however, beyond the
scope of this paper to take into account complicated modifications to the
Vaidya-de Sitter metric (4) and the corresponding $T_{\mu\nu}$ (5) due to such
fluid dynamics.

The value of $\beta$ somewhat below 10 suggests that the underlying quantum
field theory should be at least moderately strongly coupled since $\beta\sim
4\pi/g^{2}$ where $g$ is a characteristic coupling constant in the scalar
sector responsible for the phase transition. See Schmitz:2020syl ;
Baldes:2021aph  for examples of candidate models with such $\beta$.

PBH lensing and evaporation bounds provide meaningful constraints on $\beta$
and $HL$ for $T_{*}$ in the range of
$[100\,\mathrm{MeV},10^{4}\,\mathrm{GeV}]$ and
$[10^{7}\,\mathrm{GeV},10^{11}\,\mathrm{GeV}]$, respectively. Since our
calculation is based on these phenomenological parameters, these constraints
are valid independently of the underlying particle physics model.

$T_{*}\in[10^{4}\,\mathrm{GeV},10^{7}\,\mathrm{GeV}]$ is currently a blind
spot in PBH searches, and therefore PBHs in this range can constitute the
whole component of dark matter. This region may be partly probed through
Hawking evaporation by future MeV gamma-ray telescopes Coogan:2020tuf ;
Ray:2021mxu ; Ghosh:2021gfa  such as AdEPT Hunter:2013wla , AMEGO
AMEGO:2019gny , eASTROGAM e-ASTROGAM:2017pxr , GECCO GECCO , MAST
Dzhatdoev:2019kay , PANGU Wu:2014tya , GRAMS Aramaki:2019bpi  and XGIS
Labanti:2021gji ; THESEUSConsortium:2021yvf , as well as a future extension of
the analysis of Ref. Niikura:2017zjd  with more data. This region may also be
covered by future gravitational wave experiments because the small $\beta$ can
enhance the gravitational wave production Caprini:2019egz ; Schmitz:2020syl .
Moreover, the small $\beta$ is expected to broaden the bubble radius
distribution, which may give a distinct feature in the gravitational wave
spectrum. We leave these interesting questions for future work.

Acknowledgement We thank Chang Sub Shin for helpful discussions regarding the
hoop conjecture. This work is supported by the US Department of Energy grant
DE-SC0010102 and the Japan Society for Promotion of Science (JSPS) grant
KAKENHI 21H01086.

## References

  * (1) D. Chway, T. H. Jung, and C. S. Shin, “Dark matter filtering-out effect during a first-order phase transition,” Phys. Rev. D 101 no. 9, (2020) 095019, arXiv:1912.04238 [hep-ph].
  * (2) M. J. Baker, J. Kopp, and A. J. Long, “Filtered Dark Matter at a First Order Phase Transition,” Phys. Rev. Lett. 125 no. 15, (2020) 151102, arXiv:1912.02830 [hep-ph].
  * (3) M. Ahmadvand, “Filtered asymmetric dark matter during the Peccei-Quinn phase transition,” arXiv:2108.00958 [hep-ph].
  * (4) J.-P. Hong, S. Jung, and K.-P. Xie, “Fermi-ball dark matter from a first-order phase transition,” Phys. Rev. D 102 no. 7, (2020) 075028, arXiv:2008.04430 [hep-ph].
  * (5) A. Azatov, M. Vanvlasselaer, and W. Yin, “Dark Matter production from relativistic bubble walls,” JHEP 03 (2021) 288, arXiv:2101.05721 [hep-ph].
  * (6) P. Asadi, E. D. Kramer, E. Kuflik, G. W. Ridgway, T. R. Slatyer, and J. Smirnov, “Thermal Squeezeout of Dark Matter,” arXiv:2103.09827 [hep-ph].
  * (7) M. E. Shaposhnikov, “Baryon Asymmetry of the Universe in Standard Electroweak Theory,” Nucl. Phys. B 287 (1987) 757–775.
  * (8) J. M. Cline, “TASI Lectures on Early Universe Cosmology: Inflation, Baryogenesis and Dark Matter,” PoS TASI2018 (2019) 001, arXiv:1807.08749 [hep-ph].
  * (9) E. Hall, T. Konstandin, R. McGehee, H. Murayama, and G. Servant, “Baryogenesis From a Dark First-Order Phase Transition,” JHEP 04 (2020) 042, arXiv:1910.08068 [hep-ph].
  * (10) K. Fujikura, K. Harigaya, Y. Nakai, and R. Wang, “Electroweak-like Baryogenesis with New Chiral Matter,” arXiv:2103.05005 [hep-ph].
  * (11) J. Arakawa, A. Rajaraman, and T. M. P. Tait, “Annihilogenesis,” arXiv:2109.13941 [hep-ph].
  * (12) A. Azatov, M. Vanvlasselaer, and W. Yin, “Baryogenesis via relativistic bubble walls,” JHEP 10 (2021) 043, arXiv:2106.14913 [hep-ph].
  * (13) E. Witten, “Cosmic Separation of Phases,” Phys. Rev. D 30 (1984) 272–285.
  * (14) C. J. Hogan, “Gravitational radiation from cosmological phase transitions,” Mon. Not. Roy. Astron. Soc. 218 (1986) 629–636.
  * (15) A. Kosowsky, M. S. Turner, and R. Watkins, “Gravitational radiation from colliding vacuum bubbles,” Phys. Rev. D 45 (1992) 4514–4535.
  * (16) A. Kosowsky and M. S. Turner, “Gravitational radiation from colliding vacuum bubbles: envelope approximation to many bubble collisions,” Phys. Rev. D 47 (1993) 4372–4391, arXiv:astro-ph/9211004.
  * (17) M. Kamionkowski, A. Kosowsky, and M. S. Turner, “Gravitational radiation from first order phase transitions,” Phys. Rev. D 49 (1994) 2837–2851, arXiv:astro-ph/9310044.
  * (18) J. R. Espinosa, T. Konstandin, J. M. No, and G. Servant, “Energy Budget of Cosmological First-order Phase Transitions,” JCAP 06 (2010) 028, arXiv:1004.4187 [hep-ph].
  * (19) C. Caprini et al., “Detecting gravitational waves from cosmological phase transitions with LISA: an update,” JCAP 03 (2020) 024, arXiv:1910.13125 [astro-ph.CO].
  * (20) K. Schmitz, “New Sensitivity Curves for Gravitational-Wave Signals from Cosmological Phase Transitions,” JHEP 01 (2021) 097, arXiv:2002.04615 [hep-ph].
  * (21) LIGO Scientific Collaboration, G. M. Harry, “Advanced LIGO: The next generation of gravitational wave detectors,” Class. Quant. Grav. 27 (2010) 084006.
  * (22) LIGO Scientific Collaboration, J. Aasi et al., “Advanced LIGO,” Class. Quant. Grav. 32 (2015) 074001, arXiv:1411.4547 [gr-qc].
  * (23) VIRGO Collaboration, F. Acernese et al., “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quant. Grav. 32 no. 2, (2015) 024001, arXiv:1408.3978 [gr-qc].
  * (24) LISA Collaboration, P. Amaro-Seoane et al., “Laser Interferometer Space Antenna,” arXiv:1702.00786 [astro-ph.IM].
  * (25) J. Baker et al., “The Laser Interferometer Space Antenna: Unveiling the Millihertz Gravitational Wave Sky,” arXiv:1907.06482 [astro-ph.IM].
  * (26) N. Seto, S. Kawamura, and T. Nakamura, “Possibility of direct measurement of the acceleration of the universe using 0.1-Hz band laser interferometer gravitational wave antenna in space,” Phys. Rev. Lett. 87 (2001) 221103, arXiv:astro-ph/0108011.
  * (27) S. Kawamura et al., “The Japanese space gravitational wave antenna: DECIGO,” Class. Quant. Grav. 28 (2011) 094011.
  * (28) K. Yagi and N. Seto, “Detector configuration of DECIGO/BBO and identification of cosmological neutron-star binaries,” Phys. Rev. D 83 (2011) 044011, arXiv:1101.3940 [astro-ph.CO]. [Erratum: Phys.Rev.D 95, 109901 (2017)].
  * (29) S. Isoyama, H. Nakano, and T. Nakamura, “Multiband Gravitational-Wave Astronomy: Observing binary inspirals with a decihertz detector, B-DECIGO,” PTEP 2018 no. 7, (2018) 073E01, arXiv:1802.06977 [gr-qc].
  * (30) J. Crowder and N. J. Cornish, “Beyond LISA: Exploring future gravitational wave missions,” Phys. Rev. D 72 (2005) 083005, arXiv:gr-qc/0506015.
  * (31) V. Corbin and N. J. Cornish, “Detecting the cosmic gravitational wave background with the big bang observer,” Class. Quant. Grav. 23 (2006) 2435–2446, arXiv:gr-qc/0512039.
  * (32) G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner, and E. S. Phinney, “Laser interferometry for the big bang observer,” Class. Quant. Grav. 23 (2006) 4887–4894. [Erratum: Class.Quant.Grav. 23, 7361 (2006)].
  * (33) KAGRA Collaboration, K. Somiya, “Detector configuration of KAGRA: The Japanese cryogenic gravitational-wave detector,” Class. Quant. Grav. 29 (2012) 124007, arXiv:1111.7185 [gr-qc].
  * (34) KAGRA Collaboration, Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Yamamoto, “Interferometer design of the KAGRA gravitational wave detector,” Phys. Rev. D 88 no. 4, (2013) 043007, arXiv:1306.6747 [gr-qc].
  * (35) KAGRA Collaboration, T. Akutsu et al., “KAGRA: 2.5 Generation Interferometric Gravitational Wave Detector,” Nature Astron. 3 no. 1, (2019) 35–40, arXiv:1811.08079 [gr-qc].
  * (36) KAGRA Collaboration, T. Akutsu et al., “First cryogenic test operation of underground km-scale gravitational-wave observatory KAGRA,” Class. Quant. Grav. 36 no. 16, (2019) 165008, arXiv:1901.03569 [astro-ph.IM].
  * (37) Y. Michimura et al., “Prospects for improving the sensitivity of KAGRA gravitational wave detector,” in 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories.  6, 2019.  arXiv:1906.02866 [gr-qc].
  * (38) LIGO Scientific Collaboration, B. P. Abbott et al., “Exploring the Sensitivity of Next Generation Gravitational Wave Detectors,” Class. Quant. Grav. 34 no. 4, (2017) 044001, arXiv:1607.08697 [astro-ph.IM].
  * (39) D. Reitze et al., “Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO,” Bull. Am. Astron. Soc. 51 no. 7, (2019) 035, arXiv:1907.04833 [astro-ph.IM].
  * (40) M. Punturo et al., “The Einstein Telescope: A third-generation gravitational wave observatory,” Class. Quant. Grav. 27 (2010) 194002.
  * (41) S. Hild et al., “Sensitivity Studies for Third-Generation Gravitational Wave Observatories,” Class. Quant. Grav. 28 (2011) 094013, arXiv:1012.0908 [gr-qc].
  * (42) B. Sathyaprakash et al., “Scientific Objectives of Einstein Telescope,” Class. Quant. Grav. 29 (2012) 124013, arXiv:1206.0331 [gr-qc]. [Erratum: Class.Quant.Grav. 30, 079501 (2013)].
  * (43) M. Maggiore et al., “Science Case for the Einstein Telescope,” JCAP 03 (2020) 050, arXiv:1912.02622 [astro-ph.CO].
  * (44) S. Burke-Spolaor et al., “The Astrophysics of Nanohertz Gravitational Waves,” Astron. Astrophys. Rev. 27 no. 1, (2019) 5, arXiv:1811.08826 [astro-ph.HE].
  * (45) M. A. McLaughlin, “The North American Nanohertz Observatory for Gravitational Waves,” Class. Quant. Grav. 30 (2013) 224008, arXiv:1310.0758 [astro-ph.IM].
  * (46) NANOGRAV Collaboration, Z. Arzoumanian et al., “The NANOGrav 11-year Data Set: Pulsar-timing Constraints On The Stochastic Gravitational-wave Background,” Astrophys. J. 859 no. 1, (2018) 47, arXiv:1801.02617 [astro-ph.HE].
  * (47) K. Aggarwal et al., “The NANOGrav 11-Year Data Set: Limits on Gravitational Waves from Individual Supermassive Black Hole Binaries,” Astrophys. J. 880 (2019) 2, arXiv:1812.11585 [astro-ph.GA].
  * (48) A. Brazier et al., “The NANOGrav Program for Gravitational Waves and Fundamental Physics,” arXiv:1908.05356 [astro-ph.IM].
  * (49) R. N. Manchester et al., “The Parkes Pulsar Timing Array Project,” Publ. Astron. Soc. Austral. 30 (2013) 17, arXiv:1210.6130 [astro-ph.IM].
  * (50) R. M. Shannon et al., “Gravitational waves from binary supermassive black holes missing in pulsar observations,” Science 349 no. 6255, (2015) 1522–1525, arXiv:1509.07320 [astro-ph.CO].
  * (51) M. Kramer and D. J. Champion, “The European Pulsar Timing Array and the Large European Array for Pulsars,” Class. Quant. Grav. 30 (2013) 224009.
  * (52) L. Lentati et al., “European Pulsar Timing Array Limits On An Isotropic Stochastic Gravitational-Wave Background,” Mon. Not. Roy. Astron. Soc. 453 no. 3, (2015) 2576–2598, arXiv:1504.03692 [astro-ph.CO].
  * (53) S. Babak et al., “European Pulsar Timing Array Limits on Continuous Gravitational Waves from Individual Supermassive Black Hole Binaries,” Mon. Not. Roy. Astron. Soc. 455 no. 2, (2016) 1665–1679, arXiv:1509.02165 [astro-ph.CO].
  * (54) G. Hobbs et al., “The international pulsar timing array project: using pulsars as a gravitational wave detector,” Class. Quant. Grav. 27 (2010) 084013, arXiv:0911.5206 [astro-ph.SR].
  * (55) R. N. Manchester, “The International Pulsar Timing Array,” Class. Quant. Grav. 30 (2013) 224010, arXiv:1309.7392 [astro-ph.IM].
  * (56) J. P. W. Verbiest et al., “The International Pulsar Timing Array: First Data Release,” Mon. Not. Roy. Astron. Soc. 458 no. 2, (2016) 1267–1288, arXiv:1602.03640 [astro-ph.IM].
  * (57) J. S. Hazboun, C. M. F. Mingarelli, and K. Lee, “The Second International Pulsar Timing Array Mock Data Challenge,” arXiv:1810.10527 [astro-ph.IM].
  * (58) C. L. Carilli and S. Rawlings, “Science with the Square Kilometer Array: Motivation, key science projects, standards and assumptions,” New Astron. Rev. 48 (2004) 979, arXiv:astro-ph/0409274.
  * (59) G. Janssen et al., “Gravitational wave astronomy with the SKA,” PoS AASKA14 (2015) 037, arXiv:1501.00127 [astro-ph.IM].
  * (60) A. Weltman et al., “Fundamental physics with the Square Kilometre Array,” Publ. Astron. Soc. Austral. 37 (2020) e002, arXiv:1810.02680 [astro-ph.CO].
  * (61) B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, “Constraints on Primordial Black Holes,” arXiv:2002.12778 [astro-ph.CO].
  * (62) B. Carr and F. Kuhnel, “Primordial Black Holes as Dark Matter: Recent Developments,” Ann. Rev. Nucl. Part. Sci. 70 (2020) 355–394, arXiv:2006.02838 [astro-ph.CO].
  * (63) G. F. Marani, R. J. Nemiroff, J. P. Norris, K. Hurley, and J. T. Bonnell, “Gravitationally lensed gamma-ray bursts as probes of dark compact objects,” Astrophys. J. Lett. 512 (1999) L13, arXiv:astro-ph/9810391.
  * (64) R. J. Nemiroff, G. F. Marani, J. P. Norris, and J. T. Bonnell, “Limits on the cosmological abundance of supermassive compact objects from a millilensing search in gamma-ray burst data,” Phys. Rev. Lett. 86 (2001) 580, arXiv:astro-ph/0101488.
  * (65) A. Barnacka, J. F. Glicenstein, and R. Moderski, “New constraints on primordial black holes abundance from femtolensing of gamma-ray bursts,” Phys. Rev. D 86 (2012) 043001, arXiv:1204.2056 [astro-ph.CO].
  * (66) A. Katz, J. Kopp, S. Sibiryakov, and W. Xue, “Femtolensing by Dark Matter Revisited,” JCAP 12 (2018) 005, arXiv:1807.11495 [astro-ph.CO].
  * (67) K. Griest, A. M. Cieplak, and M. J. Lehner, “New Limits on Primordial Black Hole Dark Matter from an Analysis of Kepler Source Microlensing Data,” Phys. Rev. Lett. 111 no. 18, (2013) 181302.
  * (68) K. Griest, A. M. Cieplak, and M. J. Lehner, “Experimental Limits on Primordial Black Hole Dark Matter from the First 2 yr of Kepler Data,” Astrophys. J. 786 no. 2, (2014) 158, arXiv:1307.5798 [astro-ph.CO].
  * (69) H. Niikura et al., “Microlensing constraints on primordial black holes with Subaru/HSC Andromeda observations,” Nature Astron. 3 no. 6, (2019) 524–534, arXiv:1701.02151 [astro-ph.CO].
  * (70) B. Paczynski, “Gravitational microlensing by the galactic halo,” Astrophys. J. 304 (1986) 1–5.
  * (71) C. Hamadache et al., “Galactic bulge microlensing optical depth from eros-2,” Astron. Astrophys. 454 (2006) 185–199, arXiv:astro-ph/0601510.
  * (72) L. Wyrzykowski et al., “The OGLE View of Microlensing towards the Magellanic Clouds. I. A Trickle of Events in the OGLE-II LMC data,” Mon. Not. Roy. Astron. Soc. 397 (2009) 1228–1242, arXiv:0905.2044 [astro-ph.GA].
  * (73) S. Calchi Novati, L. Mancini, G. Scarpetta, and L. Wyrzykowski, “LMC self lensing for OGLE-II microlensing observations,” Mon. Not. Roy. Astron. Soc. 400 (2009) 1625, arXiv:0908.3836 [astro-ph.GA].
  * (74) L. Wyrzykowski et al., “The OGLE View of Microlensing towards the Magellanic Clouds. II. OGLE-II SMC data,” Mon. Not. Roy. Astron. Soc. 407 (2010) 189–200, arXiv:1004.5247 [astro-ph.GA].
  * (75) L. Wyrzykowski et al., “The OGLE View of Microlensing towards the Magellanic Clouds. III. Ruling out sub-solar MACHOs with the OGLE-III LMC data,” Mon. Not. Roy. Astron. Soc. 413 (2011) 493, arXiv:1012.1154 [astro-ph.GA].
  * (76) L. Wyrzykowski et al., “The OGLE View of Microlensing towards the Magellanic Clouds. IV. OGLE-III SMC Data and Final Conclusions on MACHOs,” Mon. Not. Roy. Astron. Soc. 416 (2011) 2949, arXiv:1106.2925 [astro-ph.GA].
  * (77) H. Niikura, M. Takada, S. Yokoyama, T. Sumi, and S. Masaki, “Constraints on Earth-mass primordial black holes from OGLE 5-year microlensing events,” Phys. Rev. D 99 no. 8, (2019) 083503, arXiv:1901.07120 [astro-ph.CO].
  * (78) M. Zumalacarregui and U. Seljak, “Limits on stellar-mass compact objects as dark matter from gravitational lensing of type Ia supernovae,” Phys. Rev. Lett. 121 no. 14, (2018) 141101, arXiv:1712.02240 [astro-ph.CO].
  * (79) J. Garcia-Bellido, S. Clesse, and P. Fleury, “Primordial black holes survive SN lensing constraints,” Phys. Dark Univ. 20 (2018) 95–100, arXiv:1712.06574 [astro-ph.CO].
  * (80) D. N. Page and S. W. Hawking, “Gamma rays from primordial black holes,” Astrophys. J. 206 (1976) 1–7.
  * (81) B. J. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, “New cosmological constraints on primordial black holes,” Phys. Rev. D 81 (2010) 104019, arXiv:0912.5297 [astro-ph.CO].
  * (82) B. J. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, “Constraints on primordial black holes from the Galactic gamma-ray background,” Phys. Rev. D 94 no. 4, (2016) 044029, arXiv:1604.05349 [astro-ph.CO].
  * (83) M. Boudaud and M. Cirelli, “Voyager 1 $e^{\pm}$ Further Constrain Primordial Black Holes as Dark Matter,” Phys. Rev. Lett. 122 no. 4, (2019) 041104, arXiv:1807.03075 [astro-ph.HE].
  * (84) W. DeRocco and P. W. Graham, “Constraining Primordial Black Hole Abundance with the Galactic 511 keV Line,” Phys. Rev. Lett. 123 no. 25, (2019) 251102, arXiv:1906.07740 [astro-ph.CO].
  * (85) R. Laha, “Primordial Black Holes as a Dark Matter Candidate Are Severely Constrained by the Galactic Center 511 keV $\gamma$ -Ray Line,” Phys. Rev. Lett. 123 no. 25, (2019) 251101, arXiv:1906.09994 [astro-ph.HE].
  * (86) B. Dasgupta, R. Laha, and A. Ray, “Neutrino and positron constraints on spinning primordial black hole dark matter,” Phys. Rev. Lett. 125 no. 10, (2020) 101101, arXiv:1912.01014 [hep-ph].
  * (87) R. Laha, J. B. Muñoz, and T. R. Slatyer, “INTEGRAL constraints on primordial black holes and particle dark matter,” Phys. Rev. D 101 no. 12, (2020) 123514, arXiv:2004.00627 [astro-ph.CO].
  * (88) M. H. Chan and C. M. Lee, “Constraining Primordial Black Hole Fraction at the Galactic Centre using radio observational data,” Mon. Not. Roy. Astron. Soc. 497 no. 1, (2020) 1212–1216, arXiv:2007.05677 [astro-ph.HE].
  * (89) K. M. Belotsky and A. A. Kirillov, “Primordial black holes with mass $10^{16}-10^{17}$ g and reionization of the Universe,” JCAP 01 (2015) 041, arXiv:1409.8601 [astro-ph.CO].
  * (90) H. Kim, “A constraint on light primordial black holes from the interstellar medium temperature,” arXiv:2007.07739 [hep-ph].
  * (91) S. W. Hawking, I. G. Moss, and J. M. Stewart, “Bubble Collisions in the Very Early Universe,” Phys. Rev. D 26 (1982) 2681.
  * (92) K. Sato, M. Sasaki, H. Kodama, and K.-i. Maeda, “Creation of Wormholes by First Order Phase Transition of a Vacuum in the Early Universe,” Prog. Theor. Phys. 65 (1981) 1443.
  * (93) K.-i. Maeda, K. Sato, M. Sasaki, and H. Kodama, “Creation of De Sitter-schwarzschild Wormholes by a Cosmological First Order Phase Transition,” Phys. Lett. B 108 (1982) 98–102.
  * (94) H. Kodama, M. Sasaki, and K. Sato, “Abundance of Primordial Holes Produced by Cosmological First Order Phase Transition,” Prog. Theor. Phys. 68 (1982) 1979.
  * (95) L. J. Hall and S. Hsu, “Cosmological Production of Black Holes,” Phys. Rev. Lett. 64 (1990) 2848–2851.
  * (96) A. Kusenko, M. Sasaki, S. Sugiyama, M. Takada, V. Takhistov, and E. Vitagliano, “Exploring Primordial Black Holes from the Multiverse with Optical Telescopes,” Phys. Rev. Lett. 125 (2020) 181304, arXiv:2001.09160 [astro-ph.CO].
  * (97) M. Y. Khlopov, R. V. Konoplich, S. G. Rubin, and A. S. Sakharov, “First order phase transitions as a source of black holes in the early universe,” Grav. Cosmol. 2 (1999) S1, arXiv:hep-ph/9912422.
  * (98) M. Lewicki and V. Vaskonen, “On bubble collisions in strongly supercooled phase transitions,” Phys. Dark Univ. 30 (2020) 100672, arXiv:1912.00997 [astro-ph.CO].
  * (99) C. Gross, G. Landini, A. Strumia, and D. Teresi, “Dark Matter as dark dwarfs and other macroscopic objects: multiverse relics?,” JHEP 09 (2021) 033, arXiv:2105.02840 [hep-ph].
  * (100) M. J. Baker, M. Breitbach, J. Kopp, and L. Mittnacht, “Primordial Black Holes from First-Order Cosmological Phase Transitions,” arXiv:2105.07481 [astro-ph.CO].
  * (101) K. Kawana and K.-P. Xie, “Primordial black holes from a cosmic phase transition: The collapse of Fermi-balls,” arXiv:2106.00111 [astro-ph.CO].
  * (102) M. J. Baker, M. Breitbach, J. Kopp, and L. Mittnacht, “Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions,” arXiv:2110.00005 [astro-ph.CO].
  * (103) I. G. Moss, “Black hole formation from colliding bubbles,” Phys. Rev. D 50 (6, 1994) 676–681, arXiv:gr-qc/9405045.
  * (104) K. S. Thorne, “Nonspherical gravitational collapse: A short review,” in Magic Without Magic: John Archibald Wheeler, J. Klauder, ed., p. 231.  Freeman, San Francisco, 1972.
  * (105) R. Schoen and S. Yau, “The existence of a black hole due to condensation of matter,” Commun.Math. Phys. 90 (1983) 575–579.
  * (106) E. Flanagan, “Hoop conjecture for black-hole horizon formation,” Phys. Rev. D 44 (Oct, 1991) 2409–2420. https://link.aps.org/doi/10.1103/PhysRevD.44.2409.
  * (107) M. C. Johnson, H. V. Peiris, and L. Lehner, “Determining the outcome of cosmic bubble collisions in full General Relativity,” Phys. Rev. D 85 (2012) 083516, arXiv:1112.4487 [hep-th].
  * (108) D. Cutting, M. Hindmarsh, and D. J. Weir, “Vorticity, kinetic energy, and suppressed gravitational wave production in strong first order phase transitions,” Phys. Rev. Lett. 125 no. 2, (2020) 021302, arXiv:1906.00480 [hep-ph].
  * (109) R. Jinno, T. Konstandin, and M. Takimoto, “Relativistic bubble collisions—a closer look,” JCAP 09 (2019) 035, arXiv:1906.02588 [hep-ph].
  * (110) R. L. Mallett, “Radiating Vaidya metric imbedded in de Sitter space,” Phys. Rev. D 31 no. 2, (1985) 416–417.
  * (111) M. S. Turner, E. J. Weinberg, and L. M. Widrow, “Bubble nucleation in first order inflation and other cosmological phase transitions,” Phys. Rev. D 46 (1992) 2384–2403.
  * (112) S. W. Hawking, “Black hole explosions,” Nature 248 (1974) 30–31.
  * (113) S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975) 199–220. [Erratum: Commun.Math.Phys. 46, 206 (1976)].
  * (114) T. Fujita, M. Kawasaki, K. Harigaya, and R. Matsuda, “Baryon asymmetry, dark matter, and density perturbation from primordial black holes,” Phys. Rev. D 89 no. 10, (2014) 103501, arXiv:1401.1909 [astro-ph.CO].
  * (115) R. Allahverdi, J. Dent, and J. Osinski, “Nonthermal production of dark matter from primordial black holes,” Phys. Rev. D 97 no. 5, (2018) 055013, arXiv:1711.10511 [astro-ph.CO].
  * (116) O. Lennon, J. March-Russell, R. Petrossian-Byrne, and H. Tillim, “Black Hole Genesis of Dark Matter,” JCAP 04 (2018) 009, arXiv:1712.07664 [hep-ph].
  * (117) L. Morrison, S. Profumo, and Y. Yu, “Melanopogenesis: Dark Matter of (almost) any Mass and Baryonic Matter from the Evaporation of Primordial Black Holes weighing a Ton (or less),” JCAP 05 (2019) 005, arXiv:1812.10606 [astro-ph.CO].
  * (118) D. Hooper, G. Krnjaic, and S. D. McDermott, “Dark Radiation and Superheavy Dark Matter from Black Hole Domination,” JHEP 08 (2019) 001, arXiv:1905.01301 [hep-ph].
  * (119) C. Lunardini and Y. F. Perez-Gonzalez, “Dirac and Majorana neutrino signatures of primordial black holes,” JCAP 08 (2020) 014, arXiv:1910.07864 [hep-ph].
  * (120) A. Chaudhuri and A. Dolgov, “PBH evaporation, baryon asymmetry,and dark matter,” arXiv:2001.11219 [astro-ph.CO].
  * (121) I. Masina, “Dark matter and dark radiation from evaporating primordial black holes,” Eur. Phys. J. Plus 135 no. 7, (2020) 552, arXiv:2004.04740 [hep-ph].
  * (122) I. Baldes, Q. Decant, D. C. Hooper, and L. Lopez-Honorez, “Non-Cold Dark Matter from Primordial Black Hole Evaporation,” JCAP 08 (2020) 045, arXiv:2004.14773 [astro-ph.CO].
  * (123) P. Gondolo, P. Sandick, and B. Shams Es Haghi, “Effects of primordial black holes on dark matter models,” Phys. Rev. D 102 no. 9, (2020) 095018, arXiv:2009.02424 [hep-ph].
  * (124) N. Bernal and O. Zapata, “Self-interacting Dark Matter from Primordial Black Holes,” JCAP 03 (2021) 007, arXiv:2010.09725 [hep-ph].
  * (125) N. Bernal and O. Zapata, “Gravitational dark matter production: primordial black holes and UV freeze-in,” Phys. Lett. B 815 (2021) 136129, arXiv:2011.02510 [hep-ph].
  * (126) N. Bernal and O. Zapata, “Dark Matter in the Time of Primordial Black Holes,” JCAP 03 (2021) 015, arXiv:2011.12306 [astro-ph.CO].
  * (127) J. Auffinger, I. Masina, and G. Orlando, “Bounds on warm dark matter from Schwarzschild primordial black holes,” Eur. Phys. J. Plus 136 no. 2, (2021) 261, arXiv:2012.09867 [hep-ph].
  * (128) T. Kitabayashi, “Primordial black holes and scotogenic dark matter,” Int. J. Mod. Phys. A 36 (2021) 2150139, arXiv:2101.01921 [hep-ph].
  * (129) J. D. Barrow, E. J. Copeland, E. W. Kolb, and A. R. Liddle, “Baryogenesis in extended inflation. 2. Baryogenesis via primordial black holes,” Phys. Rev. D 43 (1991) 984–994.
  * (130) D. Baumann, P. J. Steinhardt, and N. Turok, “Primordial Black Hole Baryogenesis,” arXiv:hep-th/0703250.
  * (131) A. Hook, “Baryogenesis from Hawking Radiation,” Phys. Rev. D 90 no. 8, (2014) 083535, arXiv:1404.0113 [hep-ph].
  * (132) Y. Hamada and S. Iso, “Baryon asymmetry from primordial black holes,” PTEP 2017 no. 3, (2017) 033B02, arXiv:1610.02586 [hep-ph].
  * (133) D. Hooper and G. Krnjaic, “GUT Baryogenesis With Primordial Black Holes,” Phys. Rev. D 103 no. 4, (2021) 043504, arXiv:2010.01134 [hep-ph].
  * (134) Y. F. Perez-Gonzalez and J. Turner, “Assessing the tension between a black hole dominated early universe and leptogenesis,” arXiv:2010.03565 [hep-ph].
  * (135) G. Aliferis and V. Zarikas, “Electroweak baryogenesis by primordial black holes in Brans-Dicke modified gravity,” Phys. Rev. D 103 no. 2, (2021) 023509, arXiv:2006.13621 [gr-qc].
  * (136) S. Datta, A. Ghosal, and R. Samanta, “Baryogenesis from ultralight primordial black holes and strong gravitational waves from cosmic strings,” JCAP 08 (2021) 021, arXiv:2012.14981 [hep-ph].
  * (137) P. Sandick, B. S. Es Haghi, and K. Sinha, “Asymmetric reheating by primordial black holes,” Phys. Rev. D 104 no. 8, (2021) 083523, arXiv:2108.08329 [astro-ph.CO].
  * (138) I. Baldes, Y. Gouttenoire, F. Sala, and G. Servant, “Supercool Composite Dark Matter beyond 100 TeV,” arXiv:2110.13926 [hep-ph].
  * (139) A. Coogan, L. Morrison, and S. Profumo, “Direct Detection of Hawking Radiation from Asteroid-Mass Primordial Black Holes,” Phys. Rev. Lett. 126 no. 17, (2021) 171101, arXiv:2010.04797 [astro-ph.CO].
  * (140) A. Ray, R. Laha, J. B. Muñoz, and R. Caputo, “Near future MeV telescopes can discover asteroid-mass primordial black hole dark matter,” Phys. Rev. D 104 no. 2, (2021) 023516, arXiv:2102.06714 [astro-ph.CO].
  * (141) D. Ghosh, D. Sachdeva, and P. Singh, “Future Constraints on Primordial Black Holes from XGIS-THESEUS,” arXiv:2110.03333 [astro-ph.CO].
  * (142) S. D. Hunter et al., “A Pair Production Telescope for Medium-Energy Gamma-Ray Polarimetry,” Astropart. Phys. 59 (2014) 18–28, arXiv:1311.2059 [astro-ph.IM].
  * (143) AMEGO Collaboration, R. Caputo et al., “All-sky Medium Energy Gamma-ray Observatory: Exploring the Extreme Multimessenger Universe,” arXiv:1907.07558 [astro-ph.IM].
  * (144) e-ASTROGAM Collaboration, M. Tavani et al., “Science with e-ASTROGAM: A space mission for MeV–GeV gamma-ray astrophysics,” JHEAp 19 (2018) 1–106, arXiv:1711.01265 [astro-ph.HE].
  * (145) A. Moiseev, S. Profumo, A. Coogan, and L. Morrison, “Snowmass 2021 Letter of Interest: Searching for Dark Matter and New Physics with GECCO, 2020,”. https://www.snowmass21.org/docs/files/summaries/CF/SNOWMASS21-CF3_CF1_Stefano_Profumo-007.pdf.
  * (146) T. Dzhatdoev and E. Podlesnyi, “Massive Argon Space Telescope (MAST): A concept of heavy time projection chamber for $\gamma$-ray astronomy in the 100 MeV–1 TeV energy range,” Astropart. Phys. 112 (2019) 1–7, arXiv:1902.01491 [astro-ph.HE].
  * (147) X. Wu, M. Su, A. Bravar, J. Chang, Y. Fan, M. Pohl, and R. Walter, “PANGU: A High Resolution Gamma-ray Space Telescope,” Proc. SPIE Int. Soc. Opt. Eng. 9144 (2014) 91440F, arXiv:1407.0710 [astro-ph.IM].
  * (148) T. Aramaki, P. Hansson Adrian, G. Karagiorgi, and H. Odaka, “Dual MeV Gamma-Ray and Dark Matter Observatory - GRAMS Project,” Astropart. Phys. 114 (2020) 107–114, arXiv:1901.03430 [astro-ph.HE].
  * (149) C. Labanti et al., “The X/Gamma-ray Imaging Spectrometer (XGIS) on-board THESEUS: design, main characteristics, and concept of operation,” in SPIE Astronomical Telescopes + Instrumentation 2020.  2, 2021.  arXiv:2102.08701 [astro-ph.IM].
  * (150) THESEUS Consortium Collaboration, L. Amati et al., “The THESEUS space mission: science goals, requirements and mission concept,” arXiv:2104.09531 [astro-ph.IM].
  * (151) S. R. Coleman, “The Fate of the False Vacuum. 1. Semiclassical Theory,” Phys. Rev. D 15 (1977) 2929–2936. [Erratum: Phys.Rev.D 16, 1248 (1977)].
  * (152) C. L. Wainwright, “CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields,” Comput. Phys. Commun. 183 (2012) 2006–2013, arXiv:1109.4189 [hep-ph].

Supplemental Material

##  I 1\. The hoop criterion in Vaidya-de Sitter metric

###  I.1 1.1. A single bubble

Here, we show that the mass density function $\rho-\sqrt{J_{\mu}J^{\mu}}$ in a
single bubble expansion is zero (or a microscopic scale up to $\Delta V$) at
any point. Inside a bubble, the energy-momentum tensor is zero and therefore
$\rho=0$ and $J_{\mu}=0$, so the mass density is automatically zero. Outside a
bubble, $T_{\mu\nu}=-\Delta Vg_{\mu\nu}$, so $\rho=\Delta V$ and
$J^{\mu}=-\Delta Vn^{\mu}+\Delta Vn^{\mu}=0$. Therefore, the mass density
becomes $\Delta V$ which is a microscopic scale.

To discuss the bubble wall region $t_{1}<u<t_{1}+L$, let us start from the
metric and energy momentum tensor for a single bubble:

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-1-\frac{\Delta
H^{2}(u)\,r^{2}e^{2Ht}}{(1+Hre^{Ht})^{2}}&\frac{\Delta
H^{2}(u)r^{2}e^{2Ht}}{(1+Hre^{Ht})^{2}}&0&0\\\ \frac{\Delta
H^{2}(u)r^{2}e^{2Ht}}{(1+Hre^{Ht})^{2}}&e^{2Ht}\left(1-\frac{\Delta
H^{2}(u)\,r^{2}e^{2Ht}}{(1+Hre^{Ht})^{2}}\right)&0&0\\\ 0&0&r^{2}e^{2Ht}&0\\\
0&0&0&r^{2}e^{2Ht}\sin^{2\\!}\theta\end{array}\right),$ (S5) $\displaystyle
T_{\mu\nu}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{3}\frac{r}{L}e^{Ht}\Delta
V\times\left(\begin{array}[]{cccc}l_{t}^{2}&l_{t}l_{r}&0&0\\\
l_{t}l_{r}&l_{r}^{2}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right),$ (S10)

where we dropped the $g_{\mu\nu}$ part in $T_{\mu\nu}$ since $\bar{r}/L\gg 1$
and

$\displaystyle l_{t}=\frac{\partial u}{\partial
t}=\frac{1}{1+e^{Ht}Hr},~{}l_{r}=\frac{\partial u}{\partial
r}=-\frac{e^{Ht}}{1+e^{Ht}Hr}.$ (S11)

For an hypersurface $\cal H$ being $t=t_{c}$, $n^{\mu}$ is given by solving
$g_{\rm\mu\nu}n^{\mu}X^{\nu}=0$ and $n^{\mu}n_{\mu}=-1$ where
$X^{\mu}=(0,1,0,0)$, so we obtain

$\displaystyle n^{t}$ $\displaystyle=$ $\displaystyle\sqrt{1-\frac{\Delta
H^{2}(u)r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}}},$ (S12) $\displaystyle n^{r}$
$\displaystyle=$ $\displaystyle\frac{-\Delta
H^{2}r^{2}e^{Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}\sqrt{1-\frac{\Delta
H^{2}r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}}}},$ (S13) $\displaystyle
n^{\theta}$ $\displaystyle=$ $\displaystyle n^{\phi}=0.$ (S14)

At large $t_{c}$, $n^{t}\to 1$ and $n^{r}\to 0$, which can simplify the
equations further, but we do not use this approximation.

By using the definitions $\rho=T_{\mu\nu}n^{\mu}n^{\nu}$ and
$J_{\mu}=T_{\mu\nu}n^{\nu}+\rho n_{\mu}$, we obtain

$\displaystyle\rho_{s}$ $\displaystyle=$
$\displaystyle\sqrt{{J_{s}}_{\mu}J_{s}^{\mu}}=\frac{re^{Ht_{c}}\Delta
V}{3L}\frac{1}{(1+Hre^{Ht_{c}})^{2}(1-\frac{\Delta
H^{2}r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}})}$ (S15)

where the subscript $s$ denotes for the single bubble case. Therefore,
$\rho_{s}-\sqrt{{J_{s}}_{\mu}J_{s}^{\mu}}$ is also zero on the bubble wall
region.

###  I.2 1.2. Two bubbles

First, let us consider two bubbles are nucleated at the same time
$t_{1}=t_{2}$, and therefore the radii at the collision are same,
$R_{1}=R_{2}$. For the simplicity, we replace the collision time $t_{c}$ in
terms of $R_{1}$ as

$\displaystyle H(t_{c}-t_{1})=-\log(1-R_{1}/R_{h1}).$ (S16)

Remind that $R_{h1}=H^{-1}e^{-Ht_{1}}$ is the corresponding particle horizon
in $r$ coordinate.

In the following, we consider only the bubble wall region
$R_{1}-Le^{-Ht_{1}}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}r<R_{1}$ since the mass density
function is zero in other regions. Then, $\Delta H^{2}(u)$ is written as

$\displaystyle\Delta
H^{2}(u)=H^{2}-\bar{\rho}(u)/3=H^{2}(u-t_{1})/L=-\frac{H}{L}\log\left(1-\frac{R_{1}-r}{R_{h1}}\right)\simeq\frac{H}{L}\frac{R_{1}-r}{R_{h1}}$
(S17)

where we take $R_{1}-r\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}Le^{-Ht_{1}}\ll R_{h1}$ approximation
in the last step. By using Eq. (S17), we can rewrite $\rho_{s}$ as

$\displaystyle\rho_{s}\simeq\frac{\Delta
V}{3}\frac{(1-R_{1}/R_{h1})R_{1}/R_{h1}}{HL-(R_{1}-r)R_{1}^{2}/R_{h1}^{3}}.$
(S18)

Let us obtain the mass $M(\mathcal{V}_{1})$. As explained in the main text, we
take $t=t_{c}$, drop $\sqrt{J_{\mu}J^{\mu}}$ term, and restrict the
integration range to $\mathcal{V}_{s}$, where $\mathcal{V}_{s}$ is a region
that will evolve to become $\mathcal{V}_{1}$. That is,

$\displaystyle M({\cal V}_{1})\sim 2\int_{{\cal
V}_{s}}\\!\\!d^{3}\tilde{x}\sqrt{\det(\tilde{g}_{ij})}\,\rho_{s}.$ (S19)

where $\tilde{x}$ is defined by $\tilde{g}_{00}=-1$ and $\tilde{g}_{0i}=0$,
and the integral range of ${\cal V}_{s}$ is defined by
$R_{1}-Le^{-Ht_{1}}<r<R_{1}$, $0<\phi<2\pi$, and
$1-\frac{Le^{-Ht_{1}}}{2R_{1}}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}\cos\theta<1$. $\tilde{x}$ can be
obtained by $dx^{\mu}=A^{\mu}_{~{}\nu}d\tilde{x}^{\nu}$, where

$\displaystyle\left.A^{\mu}_{~{}~{}\nu}\right|_{t=t_{c}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}n^{t}&0\\\ n^{r}&1\end{array}\right).$
(S22)

$A^{\mu}_{~{}\nu}$ at $t\neq t_{c}$ will be more complicated to make the
transformation integrable, but we do not need it since we just need
$\tilde{g}_{ij}$ on $\cal H$. From
$\tilde{g}_{\mu\nu}=g_{\rho\sigma}A^{\rho}_{~{}\mu}A^{\sigma}_{~{}\nu}$, we
obtain

$\displaystyle\left.\tilde{g}_{\tilde{r}\tilde{r}}\right|_{t=t_{c}}=e^{2Ht_{c}}\left(1-\frac{\Delta
H^{2}r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}}\right).$ (S23)

Since $r=\tilde{r}$ on $\cal H$, the integral becomes

$\displaystyle
d^{3}\tilde{x}\sqrt{\det(\tilde{g}_{ij})}=dr\,d\Omega\,r^{2}e^{3Ht_{c}}\sqrt{1-\frac{\Delta
H^{2}r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}}}.$ (S24)

Finally, since $\Delta\Omega\sim\pi Le^{-Ht_{1}}/R_{1}$, the $M({\cal V}_{1})$
becomes

$\displaystyle M({\cal V}_{1})$ $\displaystyle\sim$ $\displaystyle
2\pi\frac{Le^{-Ht_{1}}}{R_{1}}\int_{R_{1}-Le^{-Ht_{1}}}^{R_{1}}dr\,r^{2}e^{3Ht_{c}}\sqrt{1-\frac{\Delta
H^{2}r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}}}\frac{re^{Ht_{c}}\Delta
V}{3L}\frac{1}{(1+Hre^{Ht_{c}})^{2}(1-\frac{\Delta
H^{2}r^{2}e^{2Ht_{c}}}{(1+Hre^{Ht_{c}})^{2}})}$ (S25) $\displaystyle\simeq$
$\displaystyle\frac{2\pi}{3}H^{-3}\Delta
V\int_{R-Le^{-Ht_{1}}}^{R}\frac{dr}{R_{h1}}\,\frac{(R_{1}/R_{h1})^{2}\sqrt{HL}}{(1-R_{1}/R_{h1})^{2}\sqrt{HL-(R_{1}-r)R_{1}^{2}/R_{h1}^{3}}}$
(S26) $\displaystyle\simeq$ $\displaystyle\frac{4\pi}{3}H^{-3}\Delta
V\,\frac{HL(1-\sqrt{1-R_{1}^{2}/R_{h1}^{2}})}{(1-R_{1}/R_{h1})^{2}}$ (S27)
$\displaystyle\simeq$
$\displaystyle\begin{cases}\frac{2\pi}{3}\left(\frac{R_{1}}{HR_{h1}}\right)^{2}L\Delta
V&\text{ for }R_{1}\ll R_{h1},\\\ \frac{4\pi}{3}H^{-3}\Delta
V\frac{HL}{(1-R_{1}/R_{h1})^{2}}&\text{ for }1-R_{1}/R_{h1}\ll 1.\end{cases}$
(S28)

This recovers the result in the flat space time approximation by taking
$R_{\rm flat}\sim R_{1}e^{Ht_{1}}=R_{1}/(HR_{h1})$ in the limit of $R_{1}\ll
R_{h1}$ where $R_{\rm flat}$ is the physical bubble radius in the flat
spacetime approximation.

The critical circumference of $\cal V$ is given by

$\displaystyle C_{\rm cr}=2\pi\frac{M({\cal V}_{1})}{4\pi}\sim 2\pi
H^{-1}\frac{HL}{(1-R_{1}/R_{h1})^{2}},$ (S29)

where $\Delta V=3H^{2}$ was used in the last step. Since the largest
circumference of the colliding region is given by ${\cal C}_{\rm max}\simeq
2\pi e^{Ht_{c}}\sqrt{(Le^{-Ht_{1}})R_{1}}$, the hoop criterion becomes

$\displaystyle\frac{C_{\rm cr}}{C_{\rm
max}}\sim\sqrt{\frac{R_{h1}}{R_{1}}}\frac{\sqrt{HL}}{1-R_{1}/R_{h}}\simeq\frac{\sqrt{HL}}{1-R_{1}/R_{h1}}>1.$
(S30)

By using this condition back in Eq. (S28), the black hole mass is roughly
bounded from below as

$\displaystyle M({\cal V}_{1})\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}\frac{4\pi}{3}H^{-3}\Delta V.$ (S31)

Now, let us consider two bubbles are nucleated in different time $t_{1}$ and
$t_{2}$ and take $t_{1}<t_{2}$ without loss of generality. At $t=t_{c}$ (see
the right panel of Fig. 2), their radii are given by

$\displaystyle R_{1}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}(r_{12}+R_{h1}-R_{h2})$ (S32) $\displaystyle R_{2}$
$\displaystyle\simeq$ $\displaystyle\frac{1}{2}(r_{12}-R_{h1}+R_{h2})$ (S33)

where $R_{hi}=H^{-1}e^{-Ht_{i}}$ for $i=1,2$ is the horizon size of each
bubble, and $r_{12}$ is the distance between two bubble centers in $r$
coordinate. When they collide, the integrand of (2) can be obtained with
$\rho\sim\rho_{1}+\rho_{2}$ and $J_{\mu}\sim J_{1\mu}+J_{2\mu}$, and it
becomes roughly $\rho-\sqrt{J_{\mu}J^{\mu}}\sim 2\rho_{2}$ even if $t_{1}\ll
t_{2}$. Therefore, the value of $\rho-\sqrt{J_{\mu}J^{\mu}}$ for an asymmetric
two-bubble collision with $R_{1}>R_{2}$ is numerically similar to that for a
symmetric collision with $R_{2}$. However, the volume and the circumference of
the overlapping region change according to the different bubble wall widths
and radii as Fig. 1. We take this geometric correction as $f(t_{1},t_{2})$ and
multiply it to the hoop criterion

$\displaystyle\frac{C_{c}}{C_{\rm
max}}=f(t_{1},t_{2})\frac{\sqrt{HL}}{1-R_{2}/R_{h2}}>1,$ (S34)

where

$\displaystyle f(t_{1},t_{2})$ $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{2R_{1}}{R_{1}+R_{2}}\frac{R_{h1}}{R_{h2}}}\simeq\sqrt{\left(1+\frac{1}{Hd}(e^{-Ht_{1}}-e^{-Ht_{2}})\right)e^{H(t_{2}-t_{1})}}$
(S35) $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{2e^{H(t_{2}-t_{1})}}{1+e^{H(t_{2}-t_{1})}}e^{H(t_{2}-t_{1})}}\,.$
(S36)

We used $r_{12}\simeq R_{h1}+R_{h2}$ in the last step since black holes form
only when radii are close to their horizon sizes.

In terms of $r_{12}$, Eq. (S34) can be rewritten as

$\displaystyle r_{12}>R_{h1}+R_{h2}-2f(t_{1},t_{2})R_{h2}\sqrt{HL}$ (S37)

by using Eq. (S33).

##  II 2\. Detailed analysis in the estimation of PBH number density

The bubble nucleation rate is defined as

$\displaystyle\Gamma_{n}=\lim_{V,\,\Delta t\to 0}\frac{\langle
N_{b}\rangle_{V}}{V\Delta t}$ (S38)

where $\langle N_{b}\rangle_{V}$ is the expected number of bubble nucleations
in a space-like false-vacuum volume $V$. $\Gamma_{n}$ can be calculated by
using the bounce action of the temperature dependent scalar potential for a
given particle physics model Coleman:1977py ; Wainwright:2011kj .

$\Gamma_{n}$ is useful for calculating probabilities of physical processes in
bubble dynamics. For example, directly from the definition, $\Gamma_{n}dV$ is
the probability that a bubble is nucleated in an infinitesimal spacetime
volume of $dV$ in the false vacuum state. By using it, the probability that a
spacetime point $(t,\vec{r})$ remains in the false vacuum state can be
estimated as

$\displaystyle P_{f}(t)=\lim_{\begin{subarray}{c}N\to\infty\\\ \Delta t\to
0\end{subarray}}\prod_{i=1}^{t/\Delta t}\left(1-\Delta
t\frac{V(t_{i})}{N}\Gamma_{n}(t_{i})\right)^{\\!\\!N}=\exp\left(-\frac{4\pi}{3}H^{-3}\int_{-\infty}^{t}dt^{\prime}(1-e^{-H(t-t^{\prime})})^{3}\Gamma_{n}(t^{\prime})\right),$
(S39)

where $t_{i}$ is the $i$-th time slice of the past light cone of the spacetime
point and $V(t_{i})=\frac{4\pi}{3}H^{-3}(1-e^{-H(t-t_{i})})^{3}$ is the
corresponding spacelike volume. Here, we assumed that the bubble wall velocity
is close to the speed of light. Otherwise, the past light cone integration
should be modified by the worldline corresponding to the bubble wall velocity.
In the left panel of Fig. S1, we show $P_{f}$ from $\Gamma_{n}$ of the form
(15) with different $\beta=6,\,8,\,10,\,12,\,14$. The physical duration of the
phase transition is given by $(\beta H)^{-1}$.

The expected number of bubbles nucleated in an arbitrary spacetime volume
$V_{4}$ can be estimated as $\int dV_{4}\Gamma_{n}P_{f}$. Note that it is not
just $\int dV_{4}\Gamma_{n}$ since $V_{4}$ may contain a nonzero portion of
the true-vacuum volume. Eq. (13) is written in the same way. For a bubble
nucleated at $t$ to form PBH by a collision with another bubble nucleated at
$t^{\prime}>t$, we need 1) a large separation between two bubbles satisfying
the condition (11), and 2) the would-be collision area to remain in the false
vacuum state until the collision happens. These conditions directly derive Eq.
(13) and we can rewrite it in terms of explicit coordinates as

$\displaystyle N(t)$ $\displaystyle=$
$\displaystyle\int_{t_{1}}^{\infty}dt_{2}\int_{R_{h1}+R_{h2}-2f(t_{1},t_{2})R_{h2}\sqrt{HL}}^{R_{h1}+R_{h2}}dr_{12}4\pi
r_{12}^{2}e^{3Ht_{2}}\Gamma_{n}(t_{2})P_{f}(t_{\rm c})$ (S40)
$\displaystyle\simeq$ $\displaystyle 8\pi
k\sqrt{HL}\int_{Ht_{1}}^{\infty}d(Ht_{2})\,(1+e^{H(t_{2}-t_{1})})^{2}\,f(t_{1},t_{2})\,\left(H^{-4}\Gamma_{n}(t_{2})\right)\int_{0}^{1}dxP_{f}(t_{c})$
(S41)

where we replace $r_{12}$ to anohter integral variable $x$ by using
$r_{12}\equiv R_{h1}+R_{h2}-2f(t_{1},t_{2})R_{h2}\sqrt{HL}\,x$. Here, $t_{c}$
is the collision time of bubbles nucleated at $t_{1}$ and $t_{2}$ with
$r_{12}$ separation,

$\displaystyle t_{c}$ $\displaystyle=$
$\displaystyle-H^{-1}\log\left(\frac{H(R_{h1}+R_{h2}-r_{12})}{2}\right)=t_{2}-H^{-1}\log\left(f(t_{1},t_{2})\sqrt{HL}\,x\right)$
(S42)

We avoid the double counting issue by setting $t_{1}<t_{2}$.

Figure S1: $P_{f}(t)$ from $\Gamma_{n}$ of the form (15) with different
$\beta=6,\,8,\,10,\,12,\,14$. The duration of the phase transition is given by
$(\beta H)^{-1}$.

Figure S2: $dn_{\rm PBH}/dt$ as a function of $t$ for
$\beta=6,\,8,\,10,\,12,\,14$ and $HL=10^{-3},\,10^{-4},\,10^{-5}$ and
$10^{-6}$.

Then, the PBH number density is given by

$\displaystyle n_{\rm
PBH}=\frac{1}{R_{f}^{3}}\int_{-\infty}^{\infty}dt_{1}e^{3Ht_{1}}N(t_{1})\Gamma_{n}(t_{1}).$
(S43)

We present $dn_{\rm PBH}/dt=R_{f}^{-3}e^{3H(t_{1})}N(t_{1})\Gamma_{n}(t_{1})$
in Fig. S2 for $\beta=6,\,8,\,10,\,12,\,14$ and
$HL=10^{-3},\,10^{-4},\,10^{-5}$ and $10^{-6}$. Reminding that $t_{1}$ is the
nucleation time of the larger bubble and the collision region should remain in
the false vacuum at $t=t_{c}$, we can intuitively understand why $dn_{\rm
PBH}/dt$ drops quickly before $t_{n}$; a bubble nucleated too late has no
chance to form a PBH since it collides with other bubbles before it becomes
sufficiently large. In order to avoid this huge suppression from $P_{f}$ at
$t=t_{c}$, bubbles should be nucleated before roughly
$t_{n}+H^{-1}\log\sqrt{HL}$ as $t_{c}$ is given by Eq. (S42) (taking $x\sim
f\sim 1$ for the estimation). This feature can be seen in the right panel of
Fig. S2.

Figure S3: $H^{-3}n_{\rm PBH}$ for $\beta=6,\,8,\,10,\,12,\,14$ and
$HL=10^{-3},\,10^{-4},\,10^{-5},\,10^{-6}$.

In Fig. S3, we show the numerical estimation of $H^{-3}n_{\rm PBH}$ for
$\beta=6,\,8,\,10,\,12,\,14$ and $HL=10^{-3},\,10^{-4},\,10^{-5},\,10^{-6}$.
Since $H^{-3}n_{\rm PBH}$ is independent of the size of $H$, one can read the
plot and estimate the PBH abundance for a given value of $\Delta V$ (or
equivalently $H=\sqrt{\Delta V/3}$) depending on models.

In the following, we describe how to find the corresponding point in the space
of $M_{\rm PBH}$ and $f_{\rm PBH}$ for a given value of $\Delta V$.

The PBH mass is approximately monochromatic since the phase space opens when
$1-R_{i}/R_{h}\sim k\sqrt{HL}$ and the probability quickly decreases as
$R_{i}/R_{h}$ becomes close to $1$ because of the suppression in
$P_{f}(t_{c})$ where $t_{c}\to\infty$ as $R_{i}/R_{h}\to 1$. Therefore, the
initially produced PBH mass is roughly

$\displaystyle M_{\rm PBH}\sim\frac{4\pi}{3}H^{-3}\Delta
V\sim\frac{12\sqrt{10}}{\sqrt{g_{*}}}\frac{M_{\rm Pl}^{3}}{T_{*}^{2}},$ (S44)

where $T_{*}$ is the temperature after the phase transition. We approximate
$\Delta V=3M_{\rm Pl}^{2}H^{2}\sim\frac{\pi^{2}}{30}g_{*}T_{*}^{4}$.

The PBH mass can be reduced from the initial value by the Hawking radiation,
and the current mass of PBH becomes

$\displaystyle M_{\rm PBH}\to M^{\rm(today)}_{\rm PBH}\simeq\left({M_{\rm
PBH}}^{3}-{M_{c}}^{3}\right)^{1/3},$ (S45)

where $M_{c}\sim 5\times 10^{14}\,{\rm g}$ Hawking:1974rv ; Hawking:1975vcx ;
Carr:2016hva . If $M_{\rm PBH}<M_{c}$, PBHs fully evaporate, and the current
PBH abundance becomes zero. For $M_{\rm PBH}\gg M_{c}$, we can ignore the
evaporation effect.

The PBH yield, $Y_{\rm PBH}=n_{\rm PBH}/s_{*}$, is frozen by the end of the
phase transition, where $s_{*}=\frac{2\pi^{2}}{45}g_{*}T_{*}^{3}$ is the
entropy density after the phase transition. One can rewrite $Y_{\rm PBH}$ as

$\displaystyle Y_{\rm PBH}=\frac{H^{3}}{s_{*}}(n_{\rm
PBH}H^{-3})\sim\frac{\pi\sqrt{g_{*}}}{12\sqrt{10}}\frac{T_{*}^{3}}{M_{\rm
Pl}^{3}}(n_{\rm PBH}H^{-3}),$ (S46)

and obtain $f_{\rm PBH}$ as

$\displaystyle f_{\rm PBH}$ $\displaystyle=$ $\displaystyle\frac{M_{\rm
PBH}Y_{\rm PBH}}{3.6\times 10^{-10}\,\mathrm{GeV}}\sim\frac{\pi T_{*}(n_{\rm
PBH}\,H^{-3})}{3.6\times 10^{-10}\,\mathrm{GeV}}$ (S47)

where $g_{*}$ dependence is canceled.

The PBH abundance at present should be estimated by

$\displaystyle f_{\rm PBH}^{\rm(today)}$ $\displaystyle=$
$\displaystyle\frac{\left({M_{\rm PBH}}^{3}-{M_{c}}^{3}\right)^{1/3}Y_{\rm
PBH}}{3.6\times 10^{-10}\,\mathrm{GeV}}.$ (S48)

However, PBH constraints from various evaporation signatures are given in
terms of the initial abundance since they are also testable by using signals
at high redshifts such as BBN, CMB and cosmic rays from extragalactic sources.
That is the reason why we present our result in terms of the initial abundance
in the main text.

In some literatures such as Ref. Carr:2020gox ; Carr:2009jm , the evaporation
constraints are given in terms of $\beta^{\prime}$ parameter defined as

$\displaystyle\beta^{\prime}=\frac{\rho_{\rm PBH}(t_{i})}{\rho_{\rm
tot}(t_{i})},$ (S49)

where $t_{i}$ is the time of the PBH formation. It can be related with $f_{\rm
PBH}$ as

$\displaystyle\beta^{\prime}=\frac{M_{\rm PBH}\,n_{\rm
PBH}}{\frac{3}{4}T_{*}s_{*}}=\frac{4}{3}\frac{M_{\rm PBH}}{T_{*}}Y_{\rm
PBH}=\frac{4}{3}\frac{3.6\times 10^{-10}\,\mathrm{GeV}}{T_{*}}f_{\rm PBH}.$
(S50)

In Fig. 3, we take constraints in $\beta^{\prime}$ parameter from the Ref.
Carr:2020gox  and translate into $f_{\rm PBH}$ by using

$\displaystyle f_{\rm PBH}=\frac{0.7\times 10^{26}}{g_{*}^{1/4}\sqrt{M_{\rm
PBH}/{\rm gram}}}\beta^{\prime},$ (S51)

where Eq. (S44) is used to replace $T_{*}$ by $M_{\rm PBH}$.