# Charting the Landscape in Our Neighborhood from the PBHs Mass Distribution
and GWs

Amjad Ashoorioon  Abasalt Rostami  Javad T. Firouzjaee

###### Abstract

We explicitly construct a double field inflationary model, which satisfies the
latest PLANCK constraints at the CMB scales and produces the whole dark matter
energy density as primordial black holes (PBHs), in the mass range
$10^{-17}~{}M_{\odot}\lesssim M_{{}_{\rm PBH}}\lesssim 10^{-13}~{}M_{\odot}$.
The PBHs are produced after the end of slow-roll inflation from the bubbles of
true vacuum that nucleate during the course of inflation. Obtaining PBHs in
this mass range enforces the scale of inflation to be extremely low,
$10^{-7}~{}{\rm GeV}\lesssim H\lesssim 10^{-3}~{}{\rm GeV}$, which makes the
efforts to observe gravitational waves at the CMB scales futile. However, it
is high enough to allow for a successful Big Bang Nucleosynthesis (BBN).
Instead, exit from the rolling metastable direction to the true vacuum of the
potential may happen with a first-order phase transition (PT) that leaves
behind a stochastic gravitational wave background (SGWB), which is potentially
observable by LISA. We show that PBHs produced during such a first-order PT
from the collision of bubble walls contribute negligibly to the PBH mass
spectrum. Examining the mass distribution of PBHs and possible SGWB from the
end of inflation, we may be able to gain invaluable information about the
topography of the landscape in our neighborhood.

##  1 Introduction

In the absence of detection of weakly interacting massive particles (WIMPs) at
the LHC and following the detection of gravitational waves by LIGO from the
merging of the binary of black holes with masses of the order of a couple of
decades heavier than sun [1, 2], the possibility of black holes being
candidates for dark matter has received an enormous amount of attention [3,
4]. Although it is quite acceptable that these black holes are astrophysical,
it will also be intriguing if they are primordial, namely, if they have been
produced during the early universe phases from large overdensities of
radiation or matter. In particular, a recent LIGO/VIRGO merging event,
GW190521 [5]), reveals component masses which cannot be easily explained by
the pair-instability in the standard stellar structure theory (c.f. [6] for
some variations of the standard model that can fill this caveat. Also, see [7]
for the accretion condition that needs to be satisfied, for the black holes to
be primordial). This may further raise the stakes for PBHs, being more than
just figments of imagination.

In fact, since PBHs lighter than $10^{15}$ g could have evaporated by today,
they have already been suggested to play a significant role in a variety of
astrophysical phenomena, such as galactic [8] and extragalactic [7]
$\gamma$-ray background radiation or short-period bursts [9], and the
reionization of pregalactic medium [10]. On the other hand, heavier PBHs that
are formed during radiation dominated era are not subject to the constraints
that baryonic matter is contingent on from BBN, about their contribution to
the energy budget of the Universe and therefore, they act as non-baryonic cold
dark matter (CDM), please see [11] and [12] for a comprehensive review.

Still, the contribution of the PBHs are very much constrained observationally
and theoretically in various mass ranges [13, 14, 15, 16]. The only current
open windows are in the sublunar mass range $10^{-17}-10^{-13}~{}M_{\odot}$,
in which the PBHs can constitute the whole dark matter energy density [11].
The other open window is $10$-${\rm few}\times 100~{}M_{\odot}$ PBHs in which
the PBHs can make up to $10\%$ of CDM energy density. The interesting point
about the first window is that astrophysical black holes cannot occupy this
mass range, as any star mass has to be at least about three solar masses [17]
to collapse into a black hole. Thus any evidence for black holes in this mass
region provides a massive boost to the idea that PBHs may have contributed
significantly to the CDM energy density.

Since 1971 that Hawking raised the possibility that highly overdense
inhomogeneities during the radiation dominated era can gravitationally
collapse to a black hole [18], a host of scenarios has been put forward to
explain the origin of PBHs in various mass ranges. Hawking himself suggested
the possibility that PBHs may form from the collapse of cosmic string loops
[19, 20]. Such PBHs emit $\gamma$-rays, which their lack of observation puts
an upper bound $G\mu\lesssim 10^{-6}$, which is only an order of magnitude
larger than the one obtained from SGWBs [21] from the European Pulsar Timing
Array and the PLANCK 2013 data [22]. Along with Moss and Stewart [23], he also
showed how the collision of enough number of bubble walls during a first-order
PT can lead to the formation of PBHs. This is one of the mechanisms that we
will check to see if PBHs are formed in our scenario.

Inflation is a prime candidate for generating primordial fluctuations in the
early universe cosmology. These fluctuations later reenter during the
radiation or matter-dominated phases and lead to the structures we observe
today. In fact, the collapse of large density fluctuations is another
mechanism of the formation of PBHs [24]. In this scenario, horizon reentry of
a mode with large fluctuating amplitude during the ensuing radiation
domination era leads to domination of gravity overpressure in the over-dense
region and its collapse to the black hole. The amplitude of scalar
fluctuations in the CMB scales is relatively small, $\mathcal{P}_{{}_{S}}\sim
2\times 10^{-9}$ [25], and if the power spectrum is almost scale-invariant up
to the scales relevant for PBH formation, their probability of formation is
extremely small, $\sim\exp(-10^{8})$. Therefore it is necessary that one would
enhance the spectrum to at least $\mathcal{P}_{{}_{S}}\sim 10^{-1}$, at the
relevant scales. Relating the mass of the PBHs to inflation provides us with
exceptional information about the primordial power spectrum and inflationary
potential, and other properties. Since with lensing, pulsar timing, and other
astrophysical methods [26, 27, 28, 29], we can substantially constrain not
only the initial energy density of PBHs $\beta(M_{{}_{\rm PBH}})=\rho_{{}_{\rm
PBH}}/\rho$ in various mass ranges, but also the shape of the primordial power
spectrum at the scales the CMB probes do not have any access (although, see
[30] for the shortcomings). One other advantage of this approach to the other
ones is that it can produce PBHs in the sub- or supersolar mass region,
depending on the size of the horizon when the wavelength of the mode with
large amplitude reenters.

Several mechanisms have been suggested along with this direction. One is
flattening the potential and using the ultra slow-roll inflation to enhance
the primordial power spectrum at the scales of interest for PBH formation [31,
32]. Although this mechanism has been used, for example, in the context of
Large Volume Scenario (LVS) inflation [33] to explain the origin of PBHs in
the mass range $10^{-17}-10^{-13}~{}M_{\odot}$, there are claims in the
literature [34] that it is difficult to go beyond $\mathcal{P}_{{}_{S}}\simeq
10^{-4}$ for the scales relevant within this mass range. This is not suitable
for PBH formation during inflation [35], which even in the case of broad
enhancement requires $\mathcal{P}_{{}_{S}}\simeq 10^{-2}$.

In [36], we proposed a mechanism of enhancement of the primordial power
spectrum generated during inflation, which is based on a sixth-order
polynomial dispersion relation, motivated in the context of Extended EFT of
Inflation, (EEFToI) [37, 38]. The mechanism was based on this observation that
with the sixth order polynomial dispersion relation if the coefficient of the
quartic term is negative and the ratio of quartic coefficient to the sixth
order one squared is smaller than a threshold, the power spectrum of
primordial fluctuations gets amplified. Since in the (Extended) EFToI
formalism, it is legitimate to assume that the couplings in the unitary gauge
action are time-dependent, one can envisage the parameters such that the
quartic coefficient becomes small and positive at the CMB scales but turns
negative around the scales relevant for PBH formation. The effect would
roughly take around two to three e-folds to show up in the power spectrum for
the parameters we chose. It was possible to enhance the power spectrum to the
threshold needed for PBH formation with the width, which could be controlled
by how long this period of negative quartic coupling lasts. The UV cutoff of
the theory can be adjusted to be well beyond the Hubble parameter, and the
strong coupling can be avoided if the coefficients of the interacting
operators are simply set to zero [39].

Nucleating cosmological defects such as cosmic strings and domain walls during
inflation has been considered as one of the mechanisms of producing seeds for
PBH formation [40]. Domain walls, whose size is smaller than a critical radius
and fall within the cosmological horizon early on, collapse due to their
tension, forming PBHs in the post-inflationary phase. The ones which are
larger than that exponentially expand under their gravitational repulsion, and
wormholes connecting the surrounding FRW Universe with the baby universes
inside are created, which finally closes up in the light-crossing time scale.
This will lead to the creation of two black holes at the two mouths of
wormholes [41, 42]. The resulting PBH mass spectrum has a wide distribution of
the masses; however, the heavier ones are those created much earlier in the
course of inflation. The mass function at such massive PBHs will get
suppressed due to the exponential expansion during inflation and peaks for the
ones whose mass is about the horizon mass at the end of inflation.

In this work, using the mechanism of [41, 42, 43], we try to construct an
inflationary scenario, compatible with the latest Planck data at the CMB
scales [25], such that it produces the whole dark matter energy density today
through this mechanism. The mass fraction of PBHs, even though broad, is
dominated by the ones that nucleate toward the end of inflation, whose mass is
given by the horizon mass at the time. Suppose we want to account for the
whole (or sizable) amount of dark matter energy density. In that case, we are
left with two windows of sublunar, $10^{-17}-10^{-13}~{}M_{\odot}$, and
intermediate-mass black holes, ${\rm few}-10^{5}~{}M_{\odot}$ [44, 45]. These
two mass scales correspond to extremely small Hubble parameter,
$10^{-7}~{}{\rm GeV}\lesssim H\lesssim 10^{-3}~{}{\rm GeV}$ in the former
case, and $10^{-20}~{}{\rm GeV}\lesssim H\lesssim 10^{-25}~{}{\rm GeV}$.
Assuming instant reheating at the end of inflation, one can estimate the
reheating temperature, respectively, $10^{6}~{}{\rm GeV}\lesssim T\lesssim
10^{8}~{}{\rm GeV}$ and $10^{-1}~{}{\rm GeV}\lesssim T\lesssim 10^{-4}~{}{\rm
GeV}$. In the first case, the reheating temperatures are large enough to allow
for successful BBN. In the latter, if the reheating temperature is larger than
few MeV, one can still construct a successful inflationary scenario.
Nonetheless, this will require that reheating be extremely efficient. In this
work, we focus mainly on constructing an inflationary model with Hubble
parameter that may give us sublunar PBHs, although the latter is also possible
to realize within our setup. In principle, our model is flexible to realize
inflation at any scale from the GUT scale down to $\gtrsim(0.01~{}{\rm
GeV})^{4}$.

If the string theory landscape [46] exists, rolling and nucleating both seem
to be quite commonplace in such a landscape (see [47] for obstructions in
realizing such a scenario). In fact it might be possible that it inflation has
happened through chains of nucleations in the landscape [48, 49, 50] (again
see [51] for the roadblocks along the path). It is also possible that
inflation has happened through a couple of rollings and nucleations. In this
work, we consider a double field inflationary scenario [52, 53] where
inflation happens by slow-rolling. Exit from inflation and settling to the
true vacuum, nonetheless, can happen through slow-roll violation, tachyonic
waterfall instability, first order PT, and sometimes a combination of the
aforementioned possibilities. Namely, inflation may end via the slow-roll
phase, but settling to the true vacuum happens afterward via a first or second
order PT. The scenario was analyzed in [54] using the setup of extended hybrid
inflation [55]. The model though is in conflict with the latest Planck bounds
on the tensor-to-scalar ratio, $r\lesssim 0.065$, [25] 111One could use
excited initial states for scalar fluctuations, to lower the predicted $r$,
[56].. In [54] it was anticipated that this is just the characteristic of the
potential chosen along the rolling direction, which was an $m^{2}\phi^{2}$
model. Here, we replace the quadratic rolling direction with a near-
inflection-point inflation [57, 58, 59], which has the benefit of having a
free energy scale, while simultaneously can convince the small tensor-to-
scalar ratio and scalar spectral index compatible with the latest Planck
results. The other benefit of the model is that field excursion in the model
remains sub-Planckian, and one would not run into difficulties with super-
Planckian displacements [60].

As rolling along the direction with near-inflection-point inflation happens,
another vacuum along with the other direction forms. As soon as these two
vacua become degenerate, bubbles of true vacuum begin to form. These bubbles
will act as seeds for PBHs once they collapse in the post-inflationary phase.
The tuning to the true vacuum has to be much less than one in order for
inflation to last long enough. It is also necessary to exclude the
overproduction of PBHs whose energy density exceeds the dark matter’s current
energy density. As mentioned, there are different possibilities for the
termination of inflation, and they affect the mass distribution of PBHs that
can come out of the process. What really matters for bubbles to turn to PBHs,
is that we have to exit inflation in a way to allow the bubbles to lose their
kinetic energy, come to rest with respect to the parent FRW universe, and then
collapse in the post-inflationary decelerating phase. A waterfall second order
PT will provide the bubbles (including the ones whose sizes are larger than
the Hubble radius at the true vacuum) with ample time to go through all these
effects. On the other hand, concluding inflation with an abrupt first order PT
to settle to the true vacuum will destroy these bubbles as the first order PT
only completes by the collision of the bubble walls formed in the sea of false
vacuum. There is an intermediate scenario in which, however, inflation
finishes by the violation of the slow-roll, the inflaton still rolls along the
false vacuum direction, but the expansion is no longer accelerating. Finally,
the field reaches a point where the nucleation rate per unit four-volume to
the true vacuum reaches the critical value at which the first order PT to the
true vacuum completes [61, 62]. The time elapsed from the end of inflation to
the first order PT’s termination allows for the collapse of most of these
bubbles. In addition, the first order PT allows a SGWBs [63, 64, 65, 66] to be
produced. For reasonable nucleation time, the peak frequency of the produced
gravitational waves falls in the LISA sensitivity band [67], which correlates
a high-frequency gravitational wave signal, with specific characteristics from
a first-order PT [64, 65, 66], with the PBHs in the sublunar mass range. We
also show that a negligible number of PBHs is formed from the collision of the
bubble walls and the first order PT.

The paper is structured as follows: In section 2, we introduce the setup of
double field inflation with near-inflection-point inflation as the rolling
direction, exemplifying the scenarios discussed above. In section 4, we
estimate the mass fraction of the PBHs produced from the bubbles collapse and
the ones created from bubble collisions, showing although the former can be
arranged to produce the whole or substantial part of CDM, the latter’s
contribution to PBHs mass function is negligible with a PT at such energy
scales. In section 3, in the cases where settling to the true vacuum happens
through a first order PT, we compute the SGWB resulting from the collision of
bubbles. We show that in particular for the one where PT occurs after the end
of the slow-roll violation, the model is capable of producing SGWB in the
reach of LISA or other high-frequency gravitational wave probes that scan that
region of frequency. We finally conclude the paper in the last section.
$M_{{}_{\rm P}}=(8\pi G)^{-1/2}=1$ in this paper, except for the places where
it is explicitly written.

##  2 Double field inflation

In this section, we will go through the basics of an inflationary double field
model, in which termination of inflation and settling to the true vacuum can
happen through either a second order tachyonic rolling or a first-order PT
[61, 62]. First order PT can either mark the end of inflation or happen
sometime after the accelerating phase’s termination with the slow-roll
violation. We argue that in the case where settling to the true vacuum happens
via tachyonic second order rolling, or when the first order PT happens after
the termination of inflation, one can produce large enough PBHs to account for
the dark matter energy density of the Universe. In the latter case, where PT
happens after the end of inflation, there will be a chance to produce an SGWB
from the collision of bubbles.

With the collision of true vacuum bubble walls and their natural conversion to
radiation, inflation termination becomes efficient. That is why in the old
inflationary scenario, Guth tried to invoke this mechanism with a trapped
metastable vacuum to produce the accelerating phase needed to solve the
problems of standard big bang cosmology (SBB) [68]. However, it was soon
realized that in order for inflation to last long enough to solve the SBB
problems, the Universe might never recover from the accelerated phase if the
rate of decay was as much as it was necessary for a graceful exit from the
accelerated expansion [69]. The ”new” inflationary scenario was proposed, in
which the scalar field was rolling slowly on a relatively flat potential,
which would become steeper toward the end of inflation [70, 71]. The coupling
of the rolling field to the other species would either perturbatively [72, 73]
or non perturbatively [74, 75] reheat the Universe.

Figure 1: $V(\phi,\psi)$ with rolling inflaton $\phi$ and the trigger field
$\psi$ that facilitates the PT or waterfall transition to the true vacuum.

Double-field inflation is a mixture of old and new inflationary scenarios [52,
53], in the sense that one of the fields rolls during inflation, as in the
slow-roll inflation, and the second field is initially trapped in the
metastable vacuum (as can be seen in our example, this vacuum could be the
only existing minimum initially. The true vacuum could develop as inflation
proceeds). As the first field rolls, either the nucleation rate from the meta-
stable vacuum to the true one in the perpendicular direction grows and reaches
the critical value [69] needed for the bubbles of true vacuum to percolate and
cease inflation, or the second direction becomes tachyonic and destabilize the
rolling field toward the true minimum. The latter case is famously known as
”hybrid” inflation [76, 55]. However, even before inflation ends this way, the
bubbles of true vacuum are nucleated as soon as the two minima become
degenerate, and they would collapse and lead to the formation of PBHs, as in
[41, 42]. We will use the setup of double-field inflation to construct an
explicit inflationary model that satisfies the latest Planck 2018 constraints
at the CMB scales [25] and generate PBHs with proper mass fraction after the
end of inflation through the slow-roll violation. Following [54], we assume
the double-field potential takes the form,

$V(\phi,\psi)=V_{0}+V_{1}(\phi)+V_{2}(\phi,\psi),$ (2.1)

where $V_{0}$ is the constant vacuum energy, and $\phi$ is the rolling field,
whose potential $V_{1}(\phi)$, along with the vacuum energy, drives inflation.
$\psi$ is the field, which facilitates settling down to the true vacuum either
through tachyonic instability or first-order PT.

We focus on a variation of extended hybrid Inflationary potential [55, 54], in
which the potential along the rolling direction is replaced with an inflection
point inflation potential [57, 58, 59]

$\displaystyle V(\phi,\psi)$ $\displaystyle=$ $\displaystyle
V_{0}+\frac{1}{2}m^{2}{\phi}^{2}-\frac{4A\lambda_{{}_{3}}}{3}{\phi}^{3}+\lambda_{{}_{3}}^{2}{\phi}^{4}+$
(2.2) $\displaystyle\frac{1}{4}\lambda\psi^{4}-\frac{1}{3}\gamma
M\psi^{3}+\frac{1}{2}\lambda^{\prime}\phi^{2}\psi^{2}+\frac{1}{2}\alpha
M^{2}\psi^{2}$

which is supplemented by the cubic term for the field $\psi$ to provide the
possibility of tachyonic instability or first order PTs at/after the end of
inflation. For large values of $\phi$, the potential has one minimum in both
$\phi$ and $\psi$ direction. As inflation proceeds and $\phi$ rolls toward its
vacuum, second minimum along the $\psi$ direction develops, if
$\gamma^{2}>4\alpha\lambda$ at

$\phi_{{}_{\rm
I}}^{2}=M^{2}\frac{\gamma^{2}-4\alpha\lambda}{4\lambda^{\prime}\lambda}\,.$
(2.3)

When the value of $\psi$ drops below $\phi_{{}_{\rm I}}$, the potential
$V_{2}$ acquires two minima, located at $\psi_{{}_{\rm{false}}}=0$ and
$\psi_{{}_{\rm{true}}}$, which is given by

$\displaystyle\psi_{{}_{\rm{true}}}=\frac{2M\gamma+\sqrt{4M^{2}\gamma^{2}-36M^{2}\alpha\lambda-36\lambda\lambda^{\prime}\phi^{2}}}{2\lambda}\,.$

The two minima are separated by a barrier during inflation that allows for the
nucleation of bubbles of true vacuum while inflaton rolls and can even allow
for the termination of inflation with a first order PT. The expression in the
second line plays the role of $V_{2}(\phi,\psi)$, which couples the inflaton
to the field that creates the false vacuum in the $\psi$ direction and
facilitates the phase PT. In principle, $V_{1}(\phi)$ determines the
predictions of the model at large scales in the limiting case that the
cosmological constant goes to zero and can be chosen such that the model is
compatible with the CMB observables at cosmological scales [25]. We later add
a constant to this direction to lift the inflating direction.

The model we are interested in exploits the inflection point inflationary
potential [57, 58, 77] for the rolling direction,

$\displaystyle V_{1}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}{\phi}^{2}-\frac{A\lambda_{3}}{3}{\phi}^{3}+\lambda_{3}^{2}\phi^{4}\,.$
(2.4)

This form of inflationary potential was initially proposed in the context of
the potentials from supersymmetry breaking of the flat directions in the
minimal supersymmetric standard model (MSSM), where the parameter $m$ is in
the $1-10$ TeV energy scale. The $A$-term (like the mass term) is a soft
supersymmetry breaking parameter while the last one in (2.4) comes from a
minimal globally supersymmetric model. It was also realized in the context of
Matrix inflation [59] that such an inflationary scenario is possible, although
mainly the symmetry breaking potentials was focused on [78]. The advantage of
the near inflection point potential is that while the scalar spectral index
and the amplitude of the density perturbations at the CMB scales can match the
observational data, the scale of inflation –and hence the tensor-to-scalar
ratio $r$– can be lowered as much as desired. The other benefit of this kind
of inflationary potential is that the displacements of the field remain sub-
Planckian. Here we assumed that such potential could be realized within the
landscape of string theory [46]. Regardless of the origin of such a potential,
we assume that the scale of inflation is such that the Hubble parameter is in
the range $10^{-7}~{}{\rm GeV}\lesssim H\lesssim 10^{-3}~{}{\rm GeV}$, which
is desirable to produce PBHs as dark matter.

Theoretically, inflation can occur near a point $\phi_{0}$ if one sets the
parameters such that $V^{\prime}_{1}$ and $V^{\prime}_{2}$ remain very small
around this point. Indeed, choosing

$\displaystyle\lambda_{3}$ $\displaystyle=$ $\displaystyle\frac{m}{2\phi_{0}}$
(2.5) $\displaystyle A$ $\displaystyle=$ $\displaystyle 4m\,,$

both the first and second derivatives of the potential exactly become zero at
inflection point $\phi_{0}$. For $\phi$ near the inflection point,
$V^{\prime}_{1}\approx 0$ and the background space-time is effectively de
Sitter. However as time goes on, the classical friction gradually leads to the
slow-rolling evolution and eventually the inflationary phase ends with
$\epsilon=\frac{1}{2}(\frac{V^{\prime}_{1}}{V_{1}})^{2}\sim\mathcal{O}(1)$.
Because of this extreme flat potential, the velocity of the rolling field is
extremely small. Therefore, we expect that in the low energy scales, the
potential (2.4) with the parameters (2.5) gives rise to adequate amplitude for
density perturbations explaining the Planck 2018 normalization [25]. However,
as discussed in [58], for such inflection potential the number of e-folds,
$N_{e}$, of inflation after which the observable universe leaves the horizon
is related to the spectral index $n_{s}$ as

$\displaystyle n_{{}_{S}}\approx 1-4/N_{e}\,.$ (2.6)

If the scale of inflation is about $10^{8}$ GeV and reheating is instantaneous
and efficient, the required number of e-folds needed to solve the SBB
cosmology problems is $N_{e}\approx 40$, for which for which
$n_{{}_{S}}=0.90$. Of course, this would have been an obstacle for such
inflection point inflation model. One way to obtain larger scalar spectral
index is to assume that one deviates from inflection point condition such that
parameter $A$ is perturbed as

$\displaystyle A=4m\left(1-\frac{1}{4}\nu^{2}\right)\,.$ (2.7)

Above, $\nu\ll 1$ is a dimensionless parameter that quantifies deviation from
the inflection condition we had before. One can show that such potential still
possesses an inflection point $\phi_{i}$ at which
$V^{\prime\prime}_{1}(\phi_{i})=0$ and up to quadratic order of $\nu$ is
written as,

$\displaystyle\phi_{i}=\phi_{0}\Big{(}1-\frac{1}{4}\nu^{2}+\mathcal{O}(\nu^{4})\Big{)}\,.$
(2.8)

We should emphasize that due to the deviation of $A$, the condition
$V^{\prime}_{1}=0$ is not satisfied any longer at the new inflection point.
When inflation is driven near $\phi_{i}$, one can approximate the potential
(2.4) in the vicinity of the inflection point with its cubic Taylor expansion
as222Comparing this expansion with that of [58], the reader immediately
recognizes that the coefficients of expansion are fairly different. This is
because the coefficient of $\nu^{2}$ in the expansion of $\phi_{i}$ has been
erroneously evaluated in [58].

$\displaystyle V_{1}(\phi)$ $\displaystyle\approx$
$\displaystyle\frac{1}{12}m^{2}\phi_{0}^{2}\Big{(}1+\nu^{2}\Big{)}+\frac{\nu^{2}}{4}m^{2}\phi_{0}\Big{(}\phi-\phi_{i}\Big{)}+\frac{2m^{2}-m^{2}\nu^{2}}{6\phi_{0}}\Big{(}\phi-\phi_{i}\Big{)}^{3}\,,$
(2.9)

in which we have replaced $\lambda_{3}$ from (2.5) and $A$ from (2.7).

Apart from $V_{0}$, The parameters of inflection point potential $V_{1}$
should be determined such that we retain the compatibility with the CMB
observations at cosmological scales. Thus, for remaining the observational
prediction of $V_{1}$ intact, it is crucial to assume that $|V_{0}|\ll|V_{1}|$
during inflation. Moreover, for having a zero vacuum constant after the PT at
the minimum of total potential, which happens in say $\phi=0$, we must impose
the following relation between parameters of the potential $V_{2}$ and $V_{0}$
term in (2.4)

$V_{2}(\psi_{{}_{\rm{true}}},\phi=0)=-V_{0}\,.$ (2.10)

Below we have taken $V_{0}\simeq\frac{V_{1}}{100}$. This will leave the CMB
predictions derived for $V_{1}$ intact to $\mathcal{O}(10^{-4})$.

In the classical field theory, when the potential of a scalar field possess
two local minima, $\phi_{\pm}$, for which $V(\phi_{+})>V(\phi_{-})$, the
vacuum at $\phi_{-}$ would be the unique classical vacuum of lowest energy.
This minimum also defines the quantum theory’s unique vacuum state (true
vacuum) in perturbation theory. The classical vacuum, $\phi_{+}$, known as
false vacuum, is rendered unstable by barrier penetration due to the quantum
fluctuations. These quantum fluctuations cause bubbles of true vacuum to
materialize in the sea of the false one, causing it to decay if the bubbles
coalesce. When the bubble nucleation rate per unite four-volume $\Gamma$
becomes significant, the system undergoes a first-order PT due to bubbles
collision and percolation.

Figure 2: Evolution of $V_{2}(\phi,\psi)$ with rolling inflaton. The first
blue line shows the $\phi={\rm const.}$ profile, when the potential attains
its new minimum at $\phi=\phi_{{}_{\rm I}}$. The second blue curve illustrates
the profile when the field $\phi$ rolls further and the newly formed second
minimum falls more. The red curve shows the profile, for $\phi_{critical}$
when the two minimum become degenerate and $\delta=2$ . Finally the green
curve shows the profile at $\phi<\phi_{{}_{\rm critical}}$, when the second
minimum drops bellow the first one and quantum tunneling from the first one to
true one acquires a nonzero probability.

According to [62], one can derive the following relation for the bubble
nucleation rate at the leading order using of WKB approximation:

$\Gamma={\cal A}\exp(-S_{{}_{\rm E}})\,,$ (2.11)

where $S_{{}_{\rm E}}$ is the on-shell four-dimensional Euclidean action
satisfying the required boundary conditions of solution at the two minima. The
prefactor ${\mathcal{A}}$ with mass dimension four can be calculated using the
path integral techniques, and so depending on the action, it may be completely
difficult to elaborate explicitly [62]. Nonetheless [79] suggests that the
pre-exponential factor may be approximated as $M^{4}_{\psi}$ when
$M^{2}_{{}_{\psi}}>2H^{2}$. Furthermore, it can be shown that when
$M^{2}_{{}_{\psi}}<2H^{2}$, then the magnitude of this prefactor is of the
order of $H^{4}$ [80] 333In [54], it was realized that the correct estimate
for this prefactor can have substantial effects on the allowed parameter space
for the extended hybrid models that exit inflation through a first order PT..
One notices that once the nucleation rate is found, we can determine the
probability of PT in the early Universe,

$p=\frac{\Gamma}{H^{4}}\,,$ (2.12)

where $H$ is the Hubble parameter at the time of PT. Fortunately, for the
quartic potential $V_{2}$ in the second line of (2.2), the Euclidean action
for a first-order PT has been parameterized extrapolating the numerical
analysis with analytic results [81],

$S_{{}_{\rm
E}}=\frac{4\pi^{2}}{3\lambda}(2-\delta)^{-3}(\alpha_{1}\delta+\alpha_{2}\delta^{2}+\alpha_{3}\delta^{3})\,,$
(2.13)

where $\alpha_{1}=13.832,~{}\alpha_{2}=-10.819,~{}\alpha_{3}=2.0765$, and
$\delta$ is a $\phi$-depended function which has defined as

$\delta=\frac{9\lambda\alpha}{\gamma^{2}}+\frac{9\lambda\lambda^{\prime}\phi^{2}}{\gamma^{2}M^{2}}\,.$
(2.14)

This relation is only valid in the allowed range, $0<\delta<2$, and therefore
we should be careful that the legitimate parameter space for PT satisfies this
condition. Once $\phi$ meets $\phi_{{}_{\rm I}}$, potential $V_{2}$ starts to
develop another minimum along the $\psi$ direction, which is not energetically
favorable for quantum tunneling. This can be observed by noticing that for
such values of $\phi$, $\delta$ is bigger than $2$. However, when inflaton
rolls down more along the $\phi$ direction, the other minimum goes down
further and finally becomes degenerate with the one at $\psi=0$, at
$\phi_{{}_{\rm crit.}}$. At this point, we have $\delta=2$, and the Euclidean
action blows up, leading to $p=0$. When the rolling field drops below its
critical value further, $V_{2}$ at the new minimum turns into a true vacuum
which is when $\delta$ departs from $2$ and moves towards zero such that the
probability of PT, $p$, becomes more appreciable (see Fig. 2). It can be shown
that [69] the true vacuum percolates effectively provided that $p\gtrsim
p_{c}=0.24$. This is the critical value which we assume for the first order PT
to occur and complete.

Three scenarios are conceivable about how inflation ends and how the field
ends up in the true vacuum: In one, the end of inflation takes place when the
first slow-roll parameter, $\epsilon$, for the inflaton potential $V_{1}$
becomes one. This is the case when the following relation between the
parameters holds:

$\epsilon\equiv\frac{1}{2}\left(\frac{V_{1}^{\prime}}{V_{1}}\right)^{2}=1\,.$
(2.15)

If we indicate the physical solution of the above polynomial equation as
$\phi_{{}_{\rm e}}$, when the inflaton field meets this value, inflation will
terminate. In this scenario, the field continues rolling along the ridge, and
it could be that (a) the field settles to the true vacuum due to a tachyonic
instability when $M_{{}_{\psi}}^{2}<0$. The other way to trigger settling to
the true vacuum is (b) through the first order PT after the termination of
inflation via slow-roll violation at some $0<\phi_{{}_{\rm pt}}<\phi_{{}_{\rm
e}}$ when $p>p_{c}$. One can also envisage a situation where (c) triggering
the false vacuum decay occurs before the slow-roll violation occurs, namely
inflation abruptly stops via a first order PT when the value $\phi_{{}_{\rm
pt}}$ is bigger than the slow-roll violating value $\phi_{{}_{\rm e}}$ in the
absence of the $\psi$ direction 444One can also consider a scenario (d), in
which inflation and decay of the false vacuum direction happen through the
waterfall field. However, this scenario is observationally indistinguishable
from the scenario (a), at least from the aspects we assess in this paper.. In
all cases, the number of e-folds can be evaluated as

$\displaystyle N(\phi_{{}_{\rm CMB}},\phi_{\rm e})=-\int_{\phi_{{}_{\rm
CMB}}}^{\phi_{{}_{\rm e}}}\frac{V_{1}}{V_{1}^{\prime}}\,d\phi\,.$ (2.16)

Although this seems a complicated integral, it can be calculated exactly for
(2.9). Here $\phi_{\rm CMB}$ is the initial value for inflaton field which can
be determined by large scale observations. Another important parameter is the
scalar spectral index, $n_{s}$,

$\displaystyle n_{s}-1$ $\displaystyle=$ $\displaystyle-6\epsilon+2\eta\,,$
(2.17)

where

$\eta\equiv\frac{V^{\prime\prime}_{1}}{V_{1}}\,.$ (2.18)

Figure 3: Illustration of inflection point potential $V_{1}(\phi)$ with
parameters given by (2.24).

Moreover, for the inflection point inflation, $V_{1}$, the start and the end
of inflationary era happen in a neighborhood of inflection point $\phi_{i}$.
Making use of (2.9), one can expand relations (2.15)-(2.18) at the leading
order of $\nu$ and drive the following relation for the spectral tilt [77]

$n_{s}=1-4\beta\,\cot\beta N_{e}\,,$ (2.19)

where $\beta$ is a dimensionless parameter and has been defined as

$\beta\equiv\frac{6\nu}{\phi_{0}^{2}}\,.$ (2.20)

In a similar fashion, the power spectrum of fluctuations in terms of $\beta$
and $N_{e}$ is also given by 555We have used the standard definition of power
spectrum $\mathcal{P}_{\phi}=\frac{V_{1}}{24\pi^{2}\epsilon}|_{k=aH}$.,

$\mathcal{P}_{\phi}^{1/2}\approx
0.625\,\frac{m}{\phi_{0}}\Big{(}\frac{\sin^{2}\beta
N_{e}}{\beta^{2}}\Big{)}\,.$ (2.21)

Concerning the Planck observation [25] for spectral index i.e. with the mean
value $n_{s}\approx 0.965$ and the equation (2.19), the value of $\beta$ can
be specified for a given $N_{e}$. On the other hand, for the inflection point
inflation the Hubble parameter can be approximated as

$H^{2}\approx\frac{1}{36}m^{2}\phi_{0}^{2}\,.$ (2.22)

With this estimation and from the normalization for the amplitude of scalar
perturbations, (2.21), the parameters in $V_{1}$ i.e. $\phi_{0},\,m,\,\nu$.
can be completely fixed.

As will be discussed in the next section, the mass of PBHs resulting from the
collapsing bubbles or bubbles collisions peaks mainly around a value inversely
proportional to the Hubble parameter at the formation time. On the other hand,
from the recent observational constraints, it is realized that those PBHs with
the masses lying within the range of $10^{-17}-10^{-13}$ solar mass can
compensate for $10\%$-$100\%$ of the dark matter[44, 45]. This would be
equivalent to placing the Hubble parameter in the range of

$1.3\times 10^{-25}M_{\rm P}\lesssim H\lesssim 1.3\times 10^{-21}M_{\rm P}\,.$
(2.23)

Although, such $H$’s are very small, it would be always possible to tune the
parameters in inflection point potential such that all observational criteria
are satisfied. For instance, here we study the possibility of PBH creation
with $\sim 10^{-14}-10^{-13}$ solar mass. For such a typical mass, the Hubble
parameter turns out to be about $H\approx 10^{-24}M_{\rm P}$ which, and
assuming an instant reheating, it requires $N_{e}\approx 40$ to solve the
problems of SBB. Accordingly, from Planck observations for the amplitude of
fluctuations and the spectral index and also making use of (2.19)-(2.22), we
tune $V_{1}$ such that

$\displaystyle m=1.8048357604\times 10^{-18}\,$ ,
$\displaystyle\qquad\phi_{0}=3.3077384399035\times 10^{-6},\qquad$ (2.24)
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle 5.970452\times 10^{-14}$

For these values of parameters, the points $\phi_{{}_{\rm CMB}}$ and
$\phi_{{}_{\rm e}}$666A ssuming that inflation ends when $\epsilon=1$. are
given by

$\phi_{{}_{\rm e}}=3.30566217\times 10^{-6},\,\qquad\phi_{{}_{\rm
CMB}}=3.3077384399034\times 10^{-6}\,.$ (2.25)

We first focus on the scenario (a) above, where the field becomes tachyonic
after inflation ends via slow-roll violation along $\phi$. As we will see
later, the advantage of this scenario is that the small bubbles (subcritical)
find enough time to collapse. Moreover, in this case, the large bubbles
(supercritical) also find the opportunity to inflate and connect to the parent
FRW universe via a wormhole, where the wormhole pinches off with black holes
at the mouths. Therefore they can contribute to the PBH mass fraction. We have
tuned the parameters in $V_{2}$ such that during the course of inflation, the
probability $p(t)\sim\mathcal{O}(10^{-18})$ and remains almost constant. The
direction $\psi$ becomes tachyonic immediately after the slow-roll violation.
Our parameters are tuned as

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle 2.6934\times
10^{-10}\,,\,\,\gamma=1.7310\times 10^{-14}\,,$ (2.26)
$\displaystyle\lambda^{\prime}$ $\displaystyle=$
$\displaystyle-\alpha=5.0\times 10^{-28}\,,\,\,M=2.975\times 10^{-6}\,.$

For such values of the parameters space, $M_{\psi}^{2}$ becomes negative when
$\phi$ drops below $\phi_{\rm tac}\approx 2.975\times 10^{-6}$.

For scenario (b), where the first-order PT kicks in after the end of the
inflationary era with the slow-roll violation, we tune other parameters in
$V_{2}$. We assume that the true-vacuum percolation happens at $\phi_{{}_{\rm
pt}}=10^{-11}$ for $V_{0}\approx 3.0\times 10^{-50}$. For that, we choose the
following set for the parameters of $V_{2}$,

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle 1.3011\times
10^{-9}\,,\,\,\gamma=4.7311\times 10^{-14}\,,$ (2.27)
$\displaystyle\lambda^{\prime}$ $\displaystyle=$
$\displaystyle\alpha=5.0\times 10^{-28}\,,\,\,M=3.546946\times 10^{-6}\,.$

Inflation ends at $\phi_{{}_{\rm e}}$ via slow-roll violation and the
background becomes decelerating since then, until the field rolls to
$\phi_{{}_{\rm pt}}$. At this time, the field configuration makes a first
order PT to the true vacuum. As we will see, this model can produce a GW
signal from the first order PT that is observable by LISA. Also, many of the
subcritical bubbles produced during inflation, namely the ones with the radius
smaller or about the true vacuum Hubble radius, find enough time to collapse
to produce the required PBHs that account for part or all dark matter later.
As depicted in Fig. 4, for these parameters set, the Euclidean action in term
of $\phi$ will be decreasing during inflation. Since the inflaton field is
decreasing as a function of time, the Euclidean action will be a decreasing
function of time too. As soon as the inflaton reaches close to its critical
value, $\phi_{{}_{\rm pt}}$ where the probability of bubble nucleation reaches
$p_{c}$, more new bubbles of true vacuum are produced; however, these bubbles
do not find time to collapse and coalesce. Instead, they collide to terminate
inflation by converting the bubble walls’ energy to radiation. In the last
section, we will show that the bubble collision can not produce many PBHs to
alter the mass distribution of PBHs produced from the collapse of the bubbles.
One should also note that the extrapolation of the Euclidean action beyond the
time the first order PT becomes complete does not make sense.

Figure 4: The Euclidean action in term of $\phi$ .

The maximum reheating temperature happens in the case of instant reheating,
when the kinetic energy in the bubbles and the inflaton energy are transformed
almost completely to radiation. This reheating temperature is given by

$T_{*}=\Big{(}\frac{90H_{f}^{2}M_{\rm p}^{2}}{\pi^{2}g_{*}}\Big{)}^{1/4}\,,$
(2.28)

where $g_{*}\approx 106$, is the total number of relativistic degrees of
freedom, and $H_{f}$ denotes the value of the Hubble parameter calculated at
the end of inflation. For the parameter set in (2.24)) , we obtain
$T_{*}\approx 5.99\times 10^{-13}$.

As mentioned before, scenario (c), where inflation terminates via the first
order PT, can happen in this framework too. In the model at hand, like
scenario (a), this would not betide unless one assumes $\alpha<0$. Such a
scenario can happen, for example, if we tune the parameters as follows

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle 8.2345\times
10^{-9}\,,\qquad\gamma=1.9950\times 10^{-13}\,,$ (2.29)
$\displaystyle\lambda^{\prime}$ $\displaystyle=$
$\displaystyle-\alpha=5.0\times 10^{-20}\,,\qquad M=3.305\times 10^{-6}\,.$

In the next sections, we first obtain the SGWB resulting from the first order
PT in cases (b) and (c), and then we compute the fraction of PBHs produced
during inflation in cases (a) and (b). We also estimate the PBHs produced from
bubble collisions in cases (b) and (c) in the last section.

##  3 Stochastic gravitational wave background from first order phase
transition

Since some inflationary model with the parameters obtained in the last
section, ends with a first order PT, we first obtain the SGWB signature that
these models generate. We employ the gravitational waves spectrum results
generated from a first order PT [82, 66]. Although the end of inflation
happens sometime before first order PT (and meantime, some reheating may ensue
from the coupling of inflaton to other species), still, the real reheating
does not begin until the first order PT completes via bubble collision. This
will allow us to use SGWB generation’s formalism from a bubble collision at
zero temperature because we expect inflation to have supercooled the Universe.
During a first order PT, gravitational waves may be produced by bubble
collisions or turbulent plasma motion if the temperature is nonzero. The
numerical analysis for two-bubble collisions was first developed by [82] and
extended for more bubbles in [66]. As discussed there, around the peak
frequency, $f_{m}$, the amplitude of gravitational waves takes the shape of an
asymmetric wedge form, which raises in the region with $f<f_{m}$ as $f^{2.3}$
and declines as $f^{-1}$ in the region $f>f_{m}$.

After considering post-inflationary redshifts, the today peak frequency
$f_{m}$ can be written in term of the reheating temperature as [66],

$f_{\rm m}=3\times
10^{-10}{\left(\frac{g_{\ast}}{100}\right)}^{1/6}\left(\frac{T_{\ast}}{1\rm
GeV}\right)^{1/4}\left(\frac{\beta}{H_{{}_{\rm pt}}}\right)\,,$ (3.1)

where $H_{{}_{\rm pt}}$ stands for the Hubble parameter at the time of the
first order PT. Moreover, using the Euclidean action, we can also obtain a
relation for the amplitude at the above peak frequency. This amplitude is
given by

$\Omega_{{}_{\rm
GW}}h^{2}(f_{m})=10^{-6}\left(\frac{g_{\ast}}{100}\right)^{1/3}{\left(\frac{H_{{}_{\rm
pt}}}{\beta}\right)}^{2},$ (3.2)

where $\beta$ is

$\beta\equiv\frac{dS_{{}_{\rm E}}}{dt}=\frac{dS_{{}_{\rm
E}}}{d\phi}\frac{d\phi}{dt}.$ (3.3)

The quantity $1/\beta$ provides an estimation of the time span of how long the
first-order PT lasts777To find the velocity of rolling direction in (3.3) at
the PT time after the inflation, one has to integrate the field equation
$\ddot{\phi}+3H\dot{\phi}+\frac{dV_{1}}{d\phi}=0$ from the end of inflation to
the point of PT, which we did. If the first-order PT occurs right before that
inflation ends with slow-roll violation, the velocity is given by
$\dot{\phi}^{2}=\frac{2\epsilon V_{1}}{3-\epsilon}|_{\phi=\phi_{\rm pt}}$..
The above analysis for GWs would be reliable if the time duration for which
the PT takes to complete is much smaller than the expansion rate i.e.
$\beta/H_{f}\gg 1$. For the parameters set in (2.24), the ratio is about
$\beta/H_{f}\sim 16$. This confirms the above discussed numerical tuning and
leads to the following values for amplitude and peak frequency

$\displaystyle\Omega_{{}_{\rm GW}}h^{2}(f_{m})$ $\displaystyle\approx$
$\displaystyle 3.55\times 10^{-9}\,,$ (3.4) $\displaystyle f_{m}$
$\displaystyle=$ $\displaystyle 0.0074\,{\rm Hz}\,.$

Interestingly this falls in the LISA sensitivity band.

Figure 5: The SGWB signal resulting from the case (b) where inflation ends via
slow-roll and first order PT completes some time after inflation. Both the low
and high frequency branches of the produced SGWB spectrum could in principle
be observed by LISA. The colored curves depict the power law sensitivity for
the six LISA configurations: green A1M5, green dashed A1M2, blue A2M5, blue
dashed A2M2, red A5M5, red dashed A5M2 [83].

Interestingly, the peak frequency amplitude and the low and high frequency
branches of the SGWB spectrum falls within the sensitivity band of the Laser
Interferometer Space Antenna (LISA) gravitational wave probe which is aimed to
detect the gravitational wave signature in the range $10^{-4}-10^{-1}$ Hz,
please see fig. 5. As we will argue in the next section, some bubbles
nucleated during inflation, whose size is very close to the Hubble radius of
the true vacuum, find the opportunity to collapse and constitute part of the
dark matter energy density in this example.

The scenario (c) happens if inflation terminates abruptly with the first-order
PT before slow-roll violation or tachyonic instability takes place. As
mentioned before, in the model at hand, such a scenario would not occur unless
one assumes $\alpha<0$. We have to be careful of this assumption because when
the value of inflaton decreases below $\phi_{\rm PT}$, the mass $M_{\psi}^{2}$
also gradually decreases and finally becomes negative. To avoid settling to
the true vacuum through rolling, instead of a first order PT, we must ensure
that the time for rolling to the vacuum is much larger than the time it takes
for the first order PT to complete. This is guaranteed if tune the parameters
in $V_{2}$ such that

$\beta\gg|\alpha|^{1/2}M\,.$ (3.5)

This condition guarantees that the time scale for rolling when $M_{\psi}^{2}$
becomes negative is much smaller than the time of PT. Such a scenario can
happen, for example, if we tune the parameters as follows

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle 8.2345\times
10^{-9}\,,\qquad\gamma=1.9950\times 10^{-13}\,,$ (3.6)
$\displaystyle\lambda^{\prime}$ $\displaystyle=$
$\displaystyle-\alpha=5.0\times 10^{-20}\,,\qquad M=3.305\times 10^{-6}\,.$

For this set of data, the peak frequency of SGWB and its amplitude is,

$\displaystyle f_{m}$ $\displaystyle=$ $\displaystyle 9.73\times 10^{10}\,{\rm
Hz}\,,$ (3.7) $\displaystyle\Omega_{\rm GW}h^{2}(f_{m})$ $\displaystyle=$
$\displaystyle 2.10\times 10^{-35}\,,$

which is completely out of the frequency range and sensitivity of any planned
probes. The important thing is that in such cases where inflation terminates
with a first-order PT, the production of PBHs only happens through the bubble
collision process. As shown in the next section, this kind of inflation ending
never leads to an appreciable number of PBHs.

##  4 Mass fraction of PBHs

In this section, we try to compute if the model with the parameters discussed
in the previous section can yield a significant PBH mass fraction compensating
for all or part of the missing dark matter. In the context of the bubble
nucleation scenario in the early Universe, PBHs can be formed in two ways: The
collapsing bubbles and the bubble collision. Below, we will study scenarios
related to the models we discussed in the last section.

###  4.1 PBHs from collapsing bubbles

While the field rolls along the $\phi$ direction, inflation proceed. As soon
as the second minimum along the $\psi$ direction forms and becomes degenerate
with the one at $\psi=0$, bubbles of true vacuum nucleate and then expand
during inflation. Since the nucleation rate enhances very slowly with time in
our model, more of such bubbles gradually form during the course of inflation.
Here, we simply take the inflationary energy density to be given by $\rho_{f}$
of the false vacuum, which changes very slowly during inflation. In the
orthogonal direction, the true vacuum also varies with time, and we denote its
energy density by $\rho_{t}$. The nucleated bubbles, with the energy density
of $\rho_{t}$ inside, initially have a radius smaller than the inflationary
Hubble length, $H_{t}^{-1}$, with $H_{t}=\sqrt{\rho_{t}/3}$ at the formation
time, but they expand very fast and can become large, some even larger than
$H_{t}^{-1}$. This is more probable, particularly for the bubbles that
nucleate much earlier than the end of inflation. The tension of bubbles are
related to the Euclidean action at the time of nucleation of the bubble
through the relation,

$\sigma\simeq\frac{S_{{}_{\rm E}}H_{\rm t}^{3}}{2\pi^{2}}\,.$ (4.1)

One can define a Hubble parameter related to this tension as

$H_{\sigma}=\frac{\sigma}{4M_{\rm P}^{2}}\,,$ (4.2)

which is the Hubble parameter resulting from the repulsive self-gravitational
energy of the bubble wall. In our models, $S_{E}\sim{\rm few}\times 100$, so
$\sigma\sim H_{t}^{3}$ and $H_{\sigma}\sim\left(H_{t}/H_{M_{\rm
P}^{2}}\right)^{2}H_{t}\sim 10^{-48}H_{t}\ll H_{t}$.

We first focus on scenario (a) that we described above, in which inflation
ends with the slow-roll violation and settling to the true vacuum occurs with
tachyonic rolling. If inflation ends at time $t_{e}$ due to violation of the
slow-roll, which for example, happened in the first scenario we described
above, depending on whether by the end of inflation, the radius of the bubble,
$R_{e}$, is bigger or smaller than $H_{t}^{-1}$, two scenarios are
conceivable. For the details of the following estimations for the parameters,
we refer the reader to [84]. If $R_{e}\lesssim H_{t}^{-1}$, which are dubbed
as subcritical, initially the bubble starts accelerating relative to the
surrounding matter and acquire a large Lorentz factor estimated to be about

$\gamma_{i}\simeq\frac{H_{f}}{H_{\sigma}}\sim 10^{48}\,,$ (4.3)

where the last approximation used the parameters of the scenario (a).
Interacting with the matter surrounding the bubble wall, it loses its kinetic
energy and comes to rest with respect to the outside FRW Universe in the time
scale, which is much smaller than $H_{f}^{-1}$. For our case, this time scale
is of order [84]

$\Delta t\sim\frac{H_{\sigma}}{H_{t}^{2}}\sim 10^{-48}H_{t}^{-1}\,.$ (4.4)

The radius of the bubble grows only by the factor $\frac{\Delta
R_{e}}{R_{e}}\approx\frac{\sigma
H_{f}}{\rho_{t}}\approx\frac{H_{t}^{2}}{M_{\rm P}^{2}}\sim 10^{-48}$, before
it starts to decouple from the Hubble flow and recede to collapse to form a
black hole. Due to the low inflationary scale in our model, the change in the
black hole radius will be negligible for subcritical bubbles. For the
subcritical bubbles, one can simply estimate the mass of the resulting PBHs as

$M_{\rm bh}\simeq\Big{(}\dfrac{4\pi}{3}\rho_{t}+4\pi\sigma
H_{f}\Big{)}R_{e}^{3}=\kappa R_{e}^{3}~{}\,.$ (4.5)

Here, $\sigma$ and $R_{e}$ are the tension of the bubble wall the bubble
radius at $t\approx t_{e}$, respectively. The parameter $\kappa$ in (4.5)
depends on the Hubble parameter of the true and false vacuum and the tension
of the bubble walls. It can be written down as [42]

$\kappa=\frac{1}{2}H_{t}^{2}+2H_{\sigma}\left(\sqrt{H_{f}^{2}-H_{t}^{2}}-H_{\sigma}\right)\,,$
(4.6)

The subcritical bubbles lead to the black holes with mass smaller than the
transition mass, $M_{\ast}$ defined as [42, 84]

$M_{\ast}\sim\frac{8\pi H_{t}^{3}}{\kappa^{2}}\,,$ (4.7)

Since in our model, $H_{t}\sim\mathcal{O}(10^{-24})M_{\rm P}$, we expect that
$\kappa\approx\frac{1}{2}H_{t}^{2}$ and $M_{\ast}\sim 32\pi M_{\rm
P}^{2}H^{-1}_{t}$.

On the other hand, if the bubble size exceeds $H_{t}^{-1}$, the bubble
continues to inflate. The inflating baby universe is connected to the outside
FRW universe by a wormhole, that eventually pinches off after a time scale
$\sim\frac{M_{\rm bh}}{M_{\rm P}^{2}}\gtrsim H_{t}^{-1}$ and black holes form
on two mouths of the wormhole. It can be deduced from causality that the
region influenced by the Schwarzschild radius of the resulting PBHs cannot be
larger than the Hubble radius of the parent Universe, with
$t_{h}=H_{f}R_{e}^{2}$ 888This is because during the radiation dominated era,
the parent Universe with scale factor $a(t)=(t/t_{e})^{1/2}$ admits Hubble
radius given by $t_{h}=a(t_{h})R_{e}$, where $R_{e}$ is the radius of the
bubble at the end of inflation.. This implies that the mass of such PBHs
should be of the order of

$M_{\rm bh}\simeq\frac{4\pi}{3}\rho(t_{h})H_{f}^{-3}(t_{h})=H_{f}R_{e}^{2}\,,$
(4.8)

Nonetheless in both supercritical and subcritical cases, it is supposed that a
bubble that nucleates at $t=t_{n}$ has a radius negligible compared to the
Hubble radius, $H_{t}^{-1}$. This easily helps us to approximate the radius of
the bubbles of true vacuum as 999Below we will use the fact that in our case
to a good approximation, $H_{f}\approx H_{t}$

$R(t)\approx H_{f}^{-1}\left(e^{H_{f}(t-t_{n})}-1\right)\,.$ (4.9)

The number of nucleated bubbles in the comoving four-volume element is

$\displaystyle dN=p(t_{n})H_{f}^{4}e^{3H_{f}t_{n}}d^{3}\mathbf{x}dt_{n}\,.$
(4.10)

Using relations (4.9) and (4.10), we can determine the number density of
bubbles with radius $(R_{e},R_{e}+dR_{e})$ during the radiation era as

$dn(t)=p(t_{n})\frac{dR_{e}}{(R_{e}+H_{f}^{-1})^{4}}\left[\frac{a(t_{e})}{a(t)}\right]^{3}\,.$
(4.11)

The last cubic factor in the right hand side of (4.11) comes from the fact
that for the black holes formed during the radiation dominated epoch, the
corresponding bubble number density should be diluted by the cosmic expansion
factor $(a(t_{e})/a(t))^{3}$ where $a(t)\propto t^{1/2}$. Now, to calculate
the black hole mass distribution, we define the standard total fraction of
cold dark mater (CDM) which are in the form of PBHs as,

$f_{\rm{PBH}}\equiv\frac{\rho_{\rm{PBH}}(t)}{\rho_{\rm{CDM}}(t)}=\int\frac{dM}{M}f(M)\,,$
(4.12)

where $\rho_{\rm{PBH}}(t)$ and $\rho_{\rm{CDM}}(t)$ are the energy density of
PBHs and CDM. The CDM energy density during the radiation era changes with
time as $\rho_{{}_{\rm{CDM}}}(t)\sim\dfrac{1}{Bt^{3/2}{\cal M}_{{}_{\rm
eq}}^{1/2}}$, where $B\sim 10$ and ${\cal M}_{{}_{\rm eq}}\sim
10^{17}M_{\odot}$ is of the order of the CDM mass in the Hubble radius at the
equality time. Above, $f(M)$ is called the standard mass function, which is
defined as

$f(M)=\frac{M^{2}}{\rho_{{}_{\rm{CDM}}}(t)}\frac{dn(t)}{dM},$ (4.13)

It can be shown that the while mass function of PBHs of subcritical mass
remains constant, it decreases with $M$ for the supercritical cases. In a more
exact form, [42]

$f(M)\sim B~{}p(t_{n}){\cal M}_{{\rm
eq}}^{1/2}\begin{cases}M_{\ast}^{-1/2}&M<M_{\ast}\\\
M^{-1/2}&M>M_{\ast}\,,\end{cases},$ (4.14)

Figure 6: The mass distribution of PBHs formed from the bubbles nucleated
during the double field inflation in scenario (a). The mass distribution for
subcritical bubbles is almost constant for $M_{{}_{\rm cr}}\lesssim M\lesssim
M_{\ast}$, but it drops like $M^{-1/2}$ for $M\gtrsim M_{\ast}$.

where (4.14). one may realize a lower bound mass for the subcritical PBHs due
to the shape fluctuations of the bubbles. In fact, quantum fluctuations may
cause the nucleated bubbles of true vacuum may not to be exactly spherical at
the time of nucleation [85]. With the collapse of a subcritical bubble, the
shape fluctuations increase and might finally be large enough before the
bubble becomes a PBH. In such case, the bubble has a different fate: instead
of ending up as a black hole, it will be shattered into relativistic
particles. This sets a lower bound for the mass of the subcritical bubbles as
following [42]

$M_{min}\sim\begin{cases}max\\{M_{H},M_{F}\\},&M_{F}<M_{cr}\\\
M_{cr},&M_{F}>M_{cr}\end{cases},$ (4.15)

where

$\displaystyle M_{F}$ $\displaystyle\equiv$
$\displaystyle\rho_{t}\left(\frac{\rho_{f}}{\rho_{t}\sigma}\right)^{3/2},$
(4.16) $\displaystyle M_{H}$ $\displaystyle\equiv$ $\displaystyle\kappa
H_{f}^{-3},$ $\displaystyle M_{\rm{cr}}$ $\displaystyle\sim$ $\displaystyle
4\pi\min\\{H_{t}^{-1},H_{\sigma}^{-1}\\}.$

In our case since $\rho_{t}\approx\rho_{f}=3H_{f}^{2}$, it is easy to check
that $M_{\rm min}\sim M_{\rm cr}\sim 4\pi M_{\rm{p}}^{2}H_{f}^{-1}$. Therefore
the subcritical PBHs lies within the narrow mass band $4\pi M_{\rm
p}^{2}H_{f}^{-1}\lesssim M_{\rm sub}\lesssim 32\pi M_{\rm p}^{2}H_{f}^{-1}$.
On the other hand, from (4.12) and (4.14) one can find a simple relation for
total mass fraction as

$f_{{\rm PBH}}\sim Bp(t_{n})\left(\frac{{\cal M}_{{\rm
eq}}}{M_{*}}\right)^{1/2}\left[\ln\left(\frac{M_{*}}{M_{{\rm
min}}}\right)+1\right].$ (4.17)

It is again easy to see that since during the course of inflation $p(t)\sim
10^{-18}-10^{-17}$ for the parameter set (2.27), the total mass fraction of
CDM becomes $f_{PBH}\sim 1$. We have depicted the mass distribution of PBHs
resulting from scenario (a).

In scenario (b), where first order PT occurs $\sim 10^{-6}H_{f}^{-1}$ after
the end of inflation and typical time scale of phase transition is
$\sim\frac{H_{f}^{-1}}{16}$, supercritical bubbles that need about $\sim
H_{f}^{-1}$ to form black holes, will collide with each other with other
bubbles and get destroyed. However, much of the subcritical ones will find the
opportunity to collapse in the meantime in particular that the ones
contributing to the mass function are those whose radius is very close to
$H_{f}^{-1}$.

In scenario (c), where inflation ends and settles in the true vacuum with a
first order phase transition in a time scale $\beta^{-1}\sim
10^{-9}H_{f}^{-1}$, we do not expect that the nucleated bubbles, either
critical or subcritical, will have time to form black holes and all get
destroyed and their wall energy is converted to radiation. Both in scenarios
(b) and (c), there is a chance that bubble collision leads to concentration of
the energy and the formation of the bubbles, but as we will show in the next
subsection, the contribution of PBHs from this process is small.

###  4.2 PBHs from bubble collisions

In case of bubble collision giving rise to the PBHs formation, we need to know
the probability that $n$ bubbles collide at a point in space-time. According
to [86], The probability that no bubble has nucleated in the past of a point p
in space-time is approximately given by

$P_{0}=e^{-\frac{4\pi}{3}p(t_{n})H(t_{\rm p}-t_{n})}\,.$ (4.18)

Imagine a ball $B$ of radius $r<H^{-1}$ in the neighborhood of p. If O be the
region to the past of B such that it does not coincide with the past of p,
then its volume turns out to be of the order of

$V({\rm O})\approx\frac{4\pi}{3}rH^{-3}\,.$ (4.19)

Now, if there are $n$ bubbles which have been nucleated in the region O, the
energy of their walls confined within the ball B is

$E_{\rm wall}\approx n\pi r^{2}HM^{2}_{\rm P}\,.$ (4.20)

With the consideration that the energy that gives rise to the gravitational
collapse if it is bigger than $4\pi rM^{2}_{\rm P}$, we finds

$n>\frac{4}{rH}$ (4.21)

Figure 7: Past light cone of the ball B with radius $r$ with the center at the
collision point p at time $t_{p}$, excluding the region which lies in the past
of light cone p. $t_{n}$ is the nucleation time of the bubble. The picture is
taken from [86].

We should note that the above criterion for gravitational collapse holds only
for regions whose radius $r$ is small compared to the Hubble radius i.e.
$H^{-1}$ and hence we can neglect the effects of the curvature and expansion.
It is reasonable to assume $r\lesssim\frac{1}{2}H^{-1}$ and thus

$n\geq 8.$ (4.22)

Therefore, at least $8$ bubbles should collide in region B to produce a PBH
whose mass is $\dfrac{2\pi}{H}M^{2}_{\rm P}$. The probability of the collision
of eight bubble walls within the ball with $r=\frac{1}{2}H^{-1}$ is simply the
product of the probability $P_{0}$ that bubbles don’t nucleate in the past of
p with the probability of nucleation of eight bubbles in the region O. This
probability is of order

$P_{\rm col}\,\sim\,e^{-\frac{4\pi}{3}p(t_{n})H(t_{\rm
p}-t_{n})}\left(\dfrac{2\pi}{3}p(t_{n})\right)^{8}\,.$ (4.23)

To find the mass fraction $f(M)$ for the PBHs from collision, first, we simply
write down the number density of bubbles with radius $(R_{e},R_{e}+dR_{e})$ at
the end of inflation, $t_{e}$, as

$dn(t_{e})=\frac{dN}{dV}=\frac{p(t_{n})a(t_{n})^{3}d^{3}Xdt_{n}}{a(t_{e})^{3}d^{3}X}.$
(4.24)

Using this number density one can obtain the mass fraction normalized to the
DM density:

$f(M)=\dfrac{M_{PBH}}{\rho_{\rm
DM}(t_{e})}\int_{t_{i}}^{t_{p}}\frac{dN}{dV}P_{\rm col}dt_{n}\,,$ (4.25)

where $t_{p}$ is the time at which the first order PT completes and $t_{i}$ is
the beginning of PT. To find the order of magnitude of mass fraction $f(M)$
for PBHs formed from bubble collisions, it is enough to use relation (4.23)
and (4.24) and calculate the following integral for the given model

$f(M)\,\sim\,\Big{(}\dfrac{2\pi}{3}\Big{)}^{9}\dfrac{3M^{2}_{\rm P}}{H(t_{\rm
p})\rho_{\rm DM}}\int_{t_{i}}^{t_{\rm
p}}p^{9}(t^{\prime})H^{4}(t^{\prime})e^{-\frac{4\pi}{3}p(t^{\prime})H(t_{\rm
p}-t^{\prime})}dt^{\prime}\,,$ (4.26)

if we use the set of parameters in (2.27), the model predicts PBHs of bubble
collision provide us only $f(10^{-14}M_{\odot})\sim 10^{-50}$. Hence, we
conclude that due to the first order PT time being much shorter than the
Hubble time, $\beta/H\gg 1$, and the low energy scale of $H$, the potential
$V_{2}$ in (2.4) does not let for those PBHs coming from the bubble collision
to compose for a considerable ratio of dark matter.

##  5 Concluding remarks

Among a host of PBH formation scenarios, black holes may also form during
inflation through the nucleations of bubbles of true vacuum, with a
cosmological constant smaller than the one at the false vacuum that drives
inflation. This scenario is quite probable in the string theory landscape,
where there would be a lot of rolling and tunneling directions simultaneously
around. Under the assumption that inflation has happened in such a landscape
and enough PBHs as dark matter has been produced during inflation from
nucleating bubbles, we constructed a realistic double field model similar to
extended hybrid inflation in which the rolling direction is replaced with a
near inflection-point inflation potential. The model is capable of producing
observables compatibles with the latest CMB data and produce enough PBHs in
the mass range $10^{-17}~{}M_{\odot}\lesssim M_{\rm PBH}\lesssim
10^{-13}~{}M_{\odot}$, where PBHs can constitute up to almost all dark matter
energy density. This constrains the scale of inflation to be in the range
$10^{-7}~{}{\rm GeV}\lesssim H\lesssim 10^{-3}~{}{\rm GeV}$, which makes the
possibility of detection of B modes at the CMB scales fruitless.

During inflation, due to the existence of another direction with smaller
vacuum energy, bubbles of true vacuum with small radii, often much smaller
than the Hubble radius in the true vacuum $H_{t}^{-1}$, form and then stretch
to much larger lengths due to the inflationary dynamics. Some even become
larger than $H_{T}^{-1}$ and start inflating again due to the vacuum energy
inside the bubble. The resulting inflating baby universe is connected to the
parent radiation dominated FRW universe after inflation through a wormhole,
which pinches off on the time scales of $t_{\rm pinch}\simeq\frac{M_{\rm
bh}}{M_{\rm P}^{2}}$. These bubbles are called supercritical. For our specific
inflationary model it turned out that $t_{\rm pinch}\simeq H_{f}^{-1}$. On the
other hand, bubbles that remain smaller than $H_{t}^{-1}$ lose their energy on
a very short time scale and start collapsing to black holes very fast.

Therefore, how inflation in such a model ends or how the transition from the
false to the true vacuum occurs, have consequences for what sections of the
mass fraction of the black holes survive the post-inflationary phase. In one
of the scenarios examined in this paper, inflation ended with a first order
PT. Also, the time scale for the first order PT to complete is much smaller
than the Hubble time during such a PT. Since all such the sub- and
supercritical bubbles produced during the course of inflation will collide and
get destroyed to convert the energy difference between two vacua to radiation,
it is expected that almost all these bubbles will not find enough time to
collapse, in the case of subcritical bubbles, or form during the wormhole
pinch-off, in the case of supercritical ones. Also, since the time scale of PT
will be quite short, the gravitational spectrum from the collision of the
bubbles is shifted to very high frequencies and small intensities. This will
make the detection of such an SGWB very difficult.

On the other hand, if inflation ends via a tachyonic instability, as in hybrid
inflation [76], both the subcritical and supercritical branches of the mass
distribution of PBHs survive. There will be no SGWB from a first order PT in
these models, although it could be that the coupling of the inflaton after it
rolls toward its true minimum give rise to SGWB from the usual preheating
mechanism [87, 88, 89, 90]. However, the characteristic of such SGWB spectrum
is often different from the ones generated from the bubble collisions. In
particular, the rise and fall off in, respectively, the high and low-frequency
branches of the spectrum is different. Also, the SGWB spectrum resulting from
preheating often shows a double bump feature, in contrast to the single bump
feature of the SGWB generated from a first order phase transition.

There is an intermediate scenario in which both these two signatures from
inflation within the landscape remain. That is when the first order PT happens
sometime after the end of inflation via slow-roll violation, and the first
order phase transition time scale is not much smaller than the inflationary
Hubble time. In this case, much of the subcritical bubbles find enough time to
collapse and contribute to the PBH mass fraction. We tuned the parameters in
this case such that we would get the whole dark matter energy density from
PBHs. From the collision of the bubbles, we also obtained an SGWB signal that
falls both in terms of frequency and intensity in the LISA sensitivity band.
The energy scale of inflation in our case is typical of an SGWB signal that
falls in the LISA sensitivity band, if the time scale of PT, parameterized in
terms of $\beta$ takes reasonable values, i.e. ${\rm
few}\lesssim\frac{\beta}{H}\lesssim 50$. Faster PT will shift the maximum
frequency to higher values and suppress the SGWB amplitude.

It seems under the assumption that inflation has happened in the string theory
landscape and PBHs constitute a sizable portion of dark matter energy density,
we can gain invaluable information about the structure of the landscape around
us. Although here we focused mainly on the PBHs mass window,
$10^{-17}~{}M_{\odot}\lesssim M_{\rm PBH}\lesssim 10^{-13}~{}M_{\odot}$, one
can design a similar double field potential at other energy scales. For
example this could have been done for the $10^{-22}~{}{\rm GeV}\lesssim
H\lesssim 10^{-20}~{}{\rm GeV}$ inflationary model. The reason for the lower
bound of $H\simeq 10^{-22}~{}{\rm GeV}$ is that we have demanded the reheating
temperature resulting from the PT be bigger than $10^{-2}$ GeV needed for BBN.
The range of PBHs produced from such an inflationary scenario is between
$4M_{\odot}\lesssim M_{\rm bh}\lesssim 3650M_{\odot}$. Interestingly this
covers the gap produced by (pulsational) pair-instability supernova processes
too [5]. Hence, inflation at such a low scale within the landscape can account
for PBHs observed by LIGO, including the ones in the mass gap too.
Interestingly the signal from such an accelerated phase of expansion within
the double field model we described can account for the 12.5-year signal
observed by the NANOgrav probe [91]. We plan to return to this in more detail
in future work.

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