September 2022 IPMU22-0037

E-models of inflation and primordial black holes

Daniel Frolovsky a, Sergei V. Ketov a,b,c,# and Sultan Saburov a

a Interdisciplinary Research Laboratory, Tomsk State University

36 Lenin Avenue, Tomsk 634050, Russia

b Department of Physics, Tokyo Metropolitan University

1-1 Minami-ohsawa, Hachioji-shi, Tokyo 192-0397, Japan

c Kavli Institute for the Physics and Mathematics of the Universe (WPI)

The University of Tokyo Institutes for Advanced Study,

Kashiwa 277-8583, Japan

#<EMAIL_ADDRESS>
Abstract

We propose and study the new (generalized) E-type $\alpha$-attractor models of
inflation, in order to include formation of primordial black holes (PBHs). The
inflaton potential has a near-inflection point where slow-roll conditions are
violated, thus leading to large scalar perturbations collapsing to PBHs later.
An ultra-slow roll (short) phase exists between two (longer) phases of slow-
roll inflation. We numerically investigate the phases of inflation, derive the
power spectrum of scalar perturbations and calculate the PBHs masses. For
certain values of the parameters, the asteroid-size PBHs can be formed with
the masses of $10^{17}\div 10^{19}$ g, beyond the Hawking evaporation limit
and in agreement with current CMB observations. Those PBHs are a candidate for
(part of) dark matter in the present universe, while the gravitational waves
induced by the PBHs formation may be detectable by the future space-based
gravitational interferometers.

##  1 Introduction

Measurements of the Cosmic Microwave Background (CMB) radiation by the Planck
mission provide tight observational constraints on cosmological inflation in
the early Universe [1, 2, 3]. Nevertheless, the simple Starobinsky model of
inflation [4], proposed the long time ago, is still consistent with the
current precision measurements of the CMB spectral tilt $n_{s}$ of scalar
perturbations [1, 2, 3],

$n_{s}=0.9649\pm 0.0042\quad(68\%~{}{\rm C.L.})$ (1)

The Starobinsky model also gives a prediction for the value of the CMB tensor-
to-scalar ratio $r$ up to an uncertainty in the duration of inflation measured
by the number of e-folds $N_{e}$ as

$r_{\rm S}\approx{{12}\over{N^{2}_{e}}}~{}~{},\quad{\rm where}\quad
N_{e}=\int^{t_{\rm end}}_{t_{\rm initial}}H(t)dt~{},$ (2)

and $H(t)$ is the Hubble function. The current observational bound [1, 2, 3]

$r<0.036\quad(95\%~{}{\rm C.L.})$ (3)

is already fulfilled for $N_{e}>20$, whereas the duration of inflation is
expected at $N_{e}=55\pm 10$. This estimate for $N_{e}$ comes from the
predicted value of $n_{s}$ in the Starobinsky model via the Mukhanov-Chibisov
formula [5]

$n_{s}\approx 1-{{2}\over{N_{e}}}~{}.$ (4)

Equations (2) and (4) for the tilts $r$ and $n_{s}$ show only the leading
terms with respect to the inverse e-folds number $N_{e}$. Given higher
precision of the $n_{s}$-measurements, the subleading terms may also be
important. For example, in the case of the Starobinsky model, one finds [6]

$n_{s}=1-{{2}\over{N_{e}}}+{{3\ln
N_{e}}\over{2N_{e}^{2}}}-{{4}\over{N^{2}_{e}}}+{\cal
O}\Bigg{(}{{\ln^{2}N_{e}}\over{N_{e}^{3}}}\Bigg{)}~{}.$ (5)

The scalar potential of the canonical inflaton field $\phi$ in the Starobinsky
model reads 111See e.g., Refs. [7, 8, 9] for details about the Starobinsky
model, various extensions and applications. We do not reproduce here the
standard equations describing background dynamics, perturbations and their
power spectrum in single-field inflation, because they are well known and
easily can be found in the literature.

$V_{S}(\phi)=\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}M^{2}_{\rm
Pl}M^{2}(1-y_{S})^{2}~{},$ (6)

where we have introduced the dimensionless field

$y_{S}=\exp\left(-\sqrt{{{2}\over{3}}}{{\phi}\over{M_{\rm Pl}}}\right)$ (7)

and the inflaton mass $M\sim 10^{-5}M_{\rm Pl}$, whose value is determined by
the known CMB amplitude. The scale of inflation can be estimated by the Hubble
function $H$ during slow-roll, which is related to the (unknown) tensor-to-
scalar ratio $r$. As regards the Starobinsky inflation, the scale of inflation
$H_{S}\sim M$ corresponds to super-high energy physics far beyond the electro-
weak scale and not far from the GUT scale.

The flatness of the inflaton potential during slow roll is guaranteed by the
smallness of $y_{S}$ during inflation. Therefore, the inflationary observables
for CMB will be essentially the same (in the leading approximation with
respect to $N^{-1}_{e}$ or $N^{-2}_{e}$) after a generalization of the scalar
potential (6) to

$V_{\zeta}(\phi)=\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}M^{2}_{\rm
Pl}M^{2}\left[1-y_{S}+y_{S}^{2}\zeta(y_{S})\right]^{2}~{},$ (8)

where $\zeta(y_{S})$ is a function regular at $y_{S}=0$. Some generalizations
of the Starobinsky model, like Eq. (8), were studied in Ref. [8]. In this
paper, we take the inflaton potential to be a real function squared because it
can always be minimally embedded into supergravity as a single-field
inflationary model [10].

Another simple way of generalizing the Starobinsky model of inflation is given
by the cosmological $\alpha-attractors$ [11, 12] that come in two families
called E-models and T-models. The E-models have the same scalar potential
$V(y)$ as in Eq. (6) but in terms of the new variable

$y=\exp\left(-\sqrt{{{2}\over{3\alpha}}}{{\phi}\over{M_{\rm Pl}}}\right)$ (9)

that depends upon the parameter $\alpha>0$. The Starobinsky model corresponds
to $\alpha=1$. The E-models lead to the same Eq. (4) for the tilt $n_{s}$ but
significantly change the tilt $r$ as

$r_{\rm\alpha}\approx{{12\alpha}\over{N^{2}_{e}}}~{},$ (10)

thus making this theoretical prediction more flexible against future
measurements.

An opportunity of changing the inflaton potential by arbitrary function
$\zeta(y)$ can be exploited in order to generate primordial black holes (PBHs)
[13, 14] at smaller values of $\phi$ or, equivalently, at lower energy scales.
Those energy scales (below the scale of inflation) are not tightly constrained
by observations yet. Technically, the PBHs production can be engineered by
demanding a near-inflection point in the potential within the double inflation
scenario with an ultra-slow-roll phase between two slow-roll regimes of
inflation, leading to an enhancement of the power spectrum of scalar
perturbations [15, 16, 17, 18]. 222See Ref. [19] for a current review of PBHs
formation in single-field inflationary models. The PBHs born in the very early
Universe are considered as a candidate for cold dark matter in the present
Universe [20, 21, 22, 23].

A generalization of the Starobinsky model for PBHs formation was proposed and
studied in Ref. [24] by using a model very different from the
$\alpha$-attractors. As regards the generalized T-models of
$\alpha$-attractors, the PBHs production was studied in Refs. [25, 26] for
single-field inflation with the scalar potentials

$V_{T}(\phi)=f^{2}\left(\tanh{{\phi/M_{\rm
Pl}}\over{\sqrt{6\alpha}}}\right)~{},$ (11)

where $f$ is a regular function. In this paper, we propose and investigate the
generalized E-models of inflation with a near-inflection point along similar
lines.

Our paper is organized as follows. In Sec. 2 we introduce our model and
investigate its scalar potential. Section 3 is devoted to the slow-roll
approximation during the first stage of inflation relevant to CMB. In Sec. 4
we give our results for the power spectrum of scalar perturbations and its
enhancement leading to PBHs formation. Our conclusion is Sec. 5.

##  2 The model

Let us consider the following potential of the canonical inflaton $\phi$:

$V(\phi)=\hbox{\large{${{\textstyle{{3}\over\vphantom{2}\smash{\raise
0.72331pt\hbox{$\scriptstyle{{4}}$}}}}}$}}(M_{\rm
Pl}M)^{2}\left[1-y+y^{2}(\beta-\gamma y)\right]^{2}~{},$ (12)

with the dimensionless parameters $(\alpha,\beta,\gamma)$, where the function
$y(\phi)$ is given by

$y=\exp\left[-\sqrt{{{2}\over{3\alpha}}}{{(\phi+\phi_{0})}\over{M_{\rm
Pl}}}\right]~{}.$ (13)

Compared to Eqs. (8) and (9), we have Taylor-expanded the function $\zeta(y)$
up to a linear term, $\zeta(y)=\beta-\gamma y$, and have shifted the field
$\phi$ by a constant $\phi_{0}$ in order to have a Minkowski minimum at
$\phi=0$ with $V(0)=0$. Hence, the $\phi_{0}$ is fixed by other parameters. We
do not give here an explicit formula for $\phi_{0}$ because it is not very
illuminating.

Demanding the existence of a near-inflection point in the potential with a
coordinate $\phi_{i}$ allows us to replace the parameters $(\beta,\gamma)$ by
the new dimensionless parameters $(\phi_{i},\xi)$ as follows:

$\beta={{1}\over{1-\xi^{2}}}\exp\left[\sqrt{{{2}\over{3\alpha}}}{{(\phi_{i}+\phi_{0})}\over{M_{\rm
Pl}}}\right]~{},\quad\gamma={{1}\over{3(1-\xi^{2})}}\exp\left[2\sqrt{{{2}\over{3\alpha}}}{{(\phi_{i}+\phi_{0})}\over{M_{\rm
Pl}}}\right]~{}.$ (14)

The parameters $(\phi_{i},\xi)$ have the clear meaning: when $\xi=0$, the
potential has the inflection point at $\phi=\phi_{i}$ only; when $0<\xi\ll 1$,
the potential has a local minimum $y_{\rm ext}^{-}$ on the right hand side of
the inflection point $\phi_{i}$ and a local maximum $y_{\rm ext}^{+}$ on the
left hand side of the inflection point $\phi_{i}$, while both extrema are
equally separated from the inflection point,

$y^{\pm}_{\rm ext}=y_{i}\left(1\pm\xi\right)~{}.$ (15)

Equations (14) and (15) are easily derivable from considering extrema of the
cubic polynomial inside the square brackets in (12), which leads to a
quadratic equation (cf. Ref. [26]). The inverse relations are given by

$\sqrt{{{2}\over{3\alpha}}}{{(\phi_{i}+\phi_{0})}\over{M_{\rm
Pl}}}=\ln{{3\gamma}\over{\beta}}~{},\quad\xi^{2}=1-{{3\gamma}\over{\beta^{2}}}~{}.$
(16)

In terms of the new parameters our scalar potential takes the form

$\begin{split}V(\phi)=&{\textstyle{3\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}(MM_{\rm
Pl})^{2}\Bigg{\\{}1-\exp\Bigg{[}-\sqrt{{{2}\over{3\alpha}}}{{(\phi+\phi_{0})}\over{M_{\rm
Pl}}}\Bigg{]}+{{1}\over{1-\xi^{2}}}\exp\Bigg{[}\sqrt{{{2}\over{3\alpha}}}{{(\phi_{i}-2\phi_{0}-2\phi)}\over{M_{\rm
Pl}}}\Bigg{]}-{}\\\
&-{{1}\over{3(1-\xi^{2})}}\exp\Bigg{[}\sqrt{{{2}\over{3\alpha}}}{{(2\phi_{i}-3\phi_{0}-3\phi)}\over{M_{\rm
Pl}}}\Bigg{]}\Bigg{\\}}^{2}~{}~{}.\end{split}$ (17)

An example of the scalar potential leading to viable inflation and PBHs
formation is given in Fig. 1. The potentials in the original E-models of
$\alpha$-attractors, arising in the case of $\beta=\gamma=\phi_{0}=0$, do not
have a near-inflection point and thus do not lead to PBHs formation. Our
potential (17) has the small bump, associated with the local maximum, and the
small dip, associated with the local minimum, with both being close to the
inflection point, similarly to the models of Ref. [27].

Figure 1: A profile of the scalar potential (on the left) and the inflaton
dynamics (on the right) for the parameters $\alpha=0.739$,
$\phi_{i}+\phi_{0}=0.664M_{\rm Pl}$ and $\xi=0.012$ with the vanishing initial
velocity. The location of the inflection point is specified by the value of
$(\phi_{i}+\phi_{0})/M_{\rm Pl}$.

##  3 Slow-roll inflation

Since the flatness of the scalar potential during inflation, the standard
slow-roll approximation well describes both the inflaton dynamics and the
power spectrum of perturbations away from the inflection point and the end of
inflation. We employ the slow-roll approximation in order to calculate the
observables relevant to CMB and estimate the power spectrum of scalar
perturbations. It is known that the slow-roll approximation generically fails
in the ultra-slow-roll (non-attractor) regime near an inflection point [25,
26]. Therefore, after having fixed our parameters in the slow-roll
approximation, we numerically recalculate the power spectrum near the
inflection point by using the Mukhanov-Sasaki (MS) equation [28, 29] leading
to a correct answer.

The (running) number of e-folds in the slow-roll approximation is given by

$N_{e}=\int_{t}^{t_{end}}H(t)dt\approx{{1}\over{M_{\rm
Pl}^{2}}}\int_{\phi_{\rm
end}}^{\phi}{{V(\phi)}\over{V^{\prime}(\phi)}}d\phi~{},$ (18)

where the prime denotes differentiation with respect to the given argument.
The integral can be taken analytically in the case of our potential (17). We
find

$N_{e}(\phi)+N_{0}\approx{{3\alpha}\over{4}}\exp\left(\sqrt{{{2}\over{3\alpha}}}{{(\phi+\phi_{0})}\over{M_{\rm
Pl}}}\right)-{{3\sqrt{\alpha}}\over{4}}\Bigg{[}1-2\exp\left(\sqrt{{{2}\over{3\alpha}}}{{\phi_{i}}\over{M_{\rm
Pl}}}\right)\Bigg{]}\sqrt{{{2}\over{3}}}{{(\phi+\phi_{0})}\over{M_{\rm
Pl}}}~{},$ (19)

where $N_{0}$ is an integration constant close to one. We ignore this constant
for simplicity in what follows because it merely shifts counting of $N_{e}$.
The standard slow-roll parameters are given by

$\epsilon={{M_{\rm
Pl}^{2}}\over{2}}\Bigg{(}{{V^{\prime}(\phi)}\over{V(\phi)}}\Bigg{)}^{2}={{3\alpha}\over{4N^{2}_{e}}}+{\cal
O}\Bigg{(}{{\ln^{2}N_{e}}\over{N_{e}^{3}}}\Bigg{)}$ (20)

and

$\eta=M_{\rm
Pl}^{2}{{V^{\prime\prime}(\phi)}\over{V(\phi)}}=-{{1}\over{N_{e}}}+3\alpha(1-2e^{\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3\alpha}$}}}}}{\textstyle{\phi_{i}\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{M_{\rm Pl}}$}}}}}){{\ln
N_{e}}\over{4N_{e}^{2}}}+{{3\alpha(1-2e^{\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3\alpha}$}}}}}{\textstyle{\phi_{i}\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{M_{\rm
Pl}}$}}}}})\ln{\textstyle{4\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3\alpha}$}}}}}\over{4N_{e}^{2}}}+{\cal
O}\Bigg{(}{{\ln^{2}N_{e}}\over{N_{e}^{3}}}\Bigg{)}~{}.$ (21)

It yields

$n_{s}=1+2\eta-6\epsilon=1-{{2}\over{N_{e}}}+a{{3\ln
N_{e}}\over{2N_{e}^{2}}}+{{b}\over{N^{2}_{e}}}+{\cal
O}\Bigg{(}{{\ln^{2}N_{e}}\over{N_{e}^{3}}}\Bigg{)}~{},$ (22)

whose coefficients are given by

$a=\alpha\left(1-2e^{\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3\alpha}$}}}}}{\textstyle{\phi_{i}\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{M_{\rm Pl}}$}}}}}\right)\quad{\rm and}\quad
b={{3\alpha}\over{2}}\left[\left(1-2e^{\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3\alpha}$}}}}}{\textstyle{\phi_{i}\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{M_{\rm
Pl}}$}}}}}\right)\ln{{4}\over{3\alpha}}-3\right]~{}.$ (23)

Equation (20) reproduces Eq. (10) because $r=16\epsilon$. When choosing
$\alpha=1$ and $\phi_{i}\to-\infty$, Eq. (5) is also recovered up to a small
correction $(=0.05)$ in the value of the coefficient $b$ due to our
approximation.

##  4 Power spectrum and PBH masses

We numerically solve the inflaton equation of motion by using initial
conditions with the vanishing initial velocities and then substitute the
background solutions into the equations for perturbations. All our
inflationary solutions are attractors (during slow roll) by construction. The
initial inflaton field value is fixed by a desired number of e-folds, see
e.g., Refs. [7, 19] for details.

A typical numerical solution to the Hubble function during double inflation is
given on the left-hand-side of Fig. 2. Demanding a peak in the power spectrum
of scalar perturbations, required for PBHs production, we find the parameter
$\alpha$ has to be restricted to the interval between 0.5 and 0.9, whereas the
parameter $\phi_{i}$ also has to be fixed, as is shown on the right-hand-side
of Fig. 2. There is a short phase of ultra-slow-roll between the two stages of
slow-roll inflation (corresponding to two plateaus), which leads to large
perturbations in the power spectrum and PBHs production.

Figure 2: The Hubble function $H$ (on the left) and the relation between the
parameters $\phi_{i}$ and $\alpha$ for the power spectrum enhancement and PBHs
production (on the right).

The standard formula for the power spectrum of scalar perturbations in the
slow-roll approximation [15]

$P_{R}={{H^{2}}\over{8M^{2}_{\rm Pl}\pi^{2}\epsilon}}$ (24)

is useful for analytic studies of the power spectrum and its dependence upon
the parameters. However, it cannot be used in the ultra-slow-roll phase where
the slow-roll conditions are violated. Instead, one should use the MS equation
[28, 29]. We used both in our calculations in order to see a difference
between the two methods.

The scalar tilt $n_{s}$ is related to the power spectrum by a relation
$n_{s}={{d\ln P_{R}}\over{d\ln k}}$, where $k=aH=da/dt$ and $a(t)$ is the
cosmic factor in the Friedman-Lemaitre-Robertson-Walker metric. Our results
for the power spectrum are given in Fig. 3 for a particular choice of the
parameters. Our results are qualitatively similar for other values of the
parameters, see the right-hand-side of Fig. 2.

Figure 3: The power spectrum $P_{R}(k)$ of scalar perturbations from a
numerical solution to the MS equation (in red) versus an analytic derivation
from Eq. (24) in the slow-roll approximation (in black), with the same
parameters as in Fig. 1.

As is clear from Fig. 3, the exact results based on the MS equation versus the
slow-roll approximation increase the hight of the peak by one or two orders of
magnitude, whereas the amplification of the peak versus the CMB spectrum (on
the very left-hand-side of the power spectrum) is given by the seven orders of
magnitude.

The PBHs masses can be estimated from the peaks as follows [30]:

$M_{\rm PBH}\simeq{{M_{\rm Pl}^{2}}\over{H(t_{\rm peak})}}\exp\left[2(N_{\rm
total}-N_{\rm peak})+\int_{t_{\rm peak}}^{t_{\rm
total}}\epsilon(t)H(t)dt\right]~{}~{}.$ (25)

The right-hand-side of this equation is mainly sensitive to the value of
$\Delta N=N_{\rm total}-N_{\rm peak}$, whereas the integral gives a sub-
leading correction.

Our findings are summarized in the Table below where we give the values of the
CMB tilts $n_{s}$ and $r$ associated with the values of the parameters
$\alpha$, $\phi_{i}$ and $\xi$, together with the corresponding values of
$\Delta N$ and PBHs masses $M_{\rm PBH}$ in our model.

@$\mathbf{n_{s}}$@ |  @$\mathbf{r}$@ |  @$\alpha$ @ |  @$\xi$ @ |  @$\phi_{\textbf{\text{i}}}$@ |  @$\phi_{\textbf{\text{i}}}+\phi_{\textbf{\text{0}}}$@ |  @$\mathbf{\Delta N}$@ |  @ $\mathbf{M_{\textbf{\text{PBH}}}}$ @
---|---|---|---|---|---|---|---
0.95452 | 0.00307 | 0.5 | 0.0102 | \- 0.334 | 0.606 | 15.08 |  $1.06\cdot 10^{19}$ g
0.95491 | 0.00360 | 0.6 | 0.0106 | \- 0.455 | 0.633 | 15.35 |  $1.04\cdot 10^{19}$ g
0.95658 | 0.00409 | 0.739 | 0.0122 | \- 0.611 | 0.664 | 13.28 |  $1.89\cdot 10^{17}$ g
0.95672 | 0.00439 | 0.8 | 0.0115 | \- 0.671 | 0.677 | 13.96 |  $7.75\cdot 10^{17}$ g
0.95650 | 0.00496 | 0.9 | 0.0111 | \- 0.765 | 0.696 | 13.74 |  $8.84\cdot 10^{17}$ g

The $n_{s}$ values below 0.9545 are certainly excluded by CMB observations, so
we do not include our results for the lower values of $n_{s}$, see Eq. (1).
The values of $n_{s}$ above 0.9565 are in good agreement with CMB observations
at the 95% C.L. The values of the tensor-to scalar ratio $r$ in the Table are
well inside the current observational bound (3). We also found that lowering
the value of the parameter $\alpha$ leads to narrowing the peaks in the scalar
perturbations spectrum. The PBHs masses are very sensitive to the value of
$\Delta N$.

PBHs may be part of the present dark matter when the PBH masses are beyond the
Hawking evaporation limit of $10^{15}$ g, which is required for survival of
those PBHs in the present universe. However, consistency with the measured CMB
value of $n_{s}$ restricts $\Delta N$ from above, as is clear from the Table.

##  5 Conclusion

Our approach is this paper is phenomenological and classical. However, it is
not excluded that our deformations of the E-models of inflation proposed in
this paper could appear as quantum corrections from a more fundamental theory
of quantum gravity.

We modified the scalar potential of the single-field E-models of
$\alpha$-attractors in order to allow PBHs formation in those models at lower
scales, while keeping success in the theoretical description of large-single-
field inflation in agreement with CMB measurements. We found that efficient
PBHs production consistent with CMB measurements restricts the $\alpha$
parameter to approximately $0.7\pm 0.2$ and leads to the asteroid-size PBHs
with masses of the order $10^{17}\div 10^{19}$ g. The masses of the PBHs
formed in the very early universe may grow further with time via accretion and
mergers.

A similar approach was realized in the T-models of $\alpha$-attractors [25,
26]. In terms of pole inflation [12] with a non-canonical inflaton field
having just a mass term, the kinetic terms in the E-models have a pole of
order two and exhibit the $SL(2,R)$ symmetry, whereas the kinetic terms in the
T-models also have a pole of order two but with the $SU(1,1)$ symmetry. Since
those symmetries are equivalent, the main predictions of the standard E- and
T-models for inflation are essentially the same. The generalized E-models of
inflation proposed in this paper simultaneously describe viable inflation and
PBHs formation.

The next generation of CMB measurements will probe deeper regions of parameter
space, leading to a discrimination among currently viable models of inflation,
which may falsify the Starobinsky model in particular. The $\alpha$-attractors
add more flexibility on the theoretical side, as regards the tensor-to-scalar
ratio. We demonstrated that certain deformations of the scalars potentials in
the E-models can also lead to efficient PBHs production capable to describe a
whole (or part of) dark matter in the present universe.

We tuned the parameters of our model in order to overcome the Hawking
radiation bound $10^{15}$ g for the PBHs masses, so that those PBHs may
contribute to the current dark matter. Remarkably, the PBHs with the masses
between $10^{17}$ g and $10^{19}$ g belong to the current observational mass
window where those PBHs may constitute the whole dark matter [22, 23]. With
lower PBHs masses we found no strong constraints on the parameters, but those
PBHs should all evaporate until now. Still, those PBHs may have dominated the
early universe, while their remnants could form dark matter at present.

The PBHs formation in the very early universe should lead to a stochastic
background of gravitational waves (GW) at present [31]. 333See e.g., Ref. [32]
for a current review. The frequency of those GW can be estimated as

$f_{\rm GW}\approx\left({{M_{\rm PBH}}\over{10^{16}~{}{\rm
g}}}\right)^{-1/2}~{}~{}{\rm Hz}~{}.$ (26)

It was argued in the literature [33, 34, 35] that those GW may be detectable
by the future space-based gravitational interferometers such as LISA [36],
TAIJI [37], TianQin [38] and DECIGO [39].

## Acknowledgements

This work was supported by Tomsk State University under the development
program Priority-2030. SVK was also supported by Tokyo Metropolitan
University, the Japanese Society for Promotion of Science under the grant No.
22K03624, and the World Premier International Research Center Initiative
(MEXT, Japan).

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