October 2022 IPMU22-0049

Pole inflation and primordial black holes
formation in Starobinsky-like supergravity

Shuntaro Aoki a, Ryotaro Ishikawa b, and Sergei V. Ketov b,c,d

a Department of Physics, Chung-Ang University, Seoul 06974, South Korea

b Department of Physics, Tokyo Metropolitan University

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

c Research School of High-Energy Physics, Tomsk Polytechnic University

2a Lenin Avenue, Tomsk 634028, Russian Federation

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

The University of Tokyo Institutes for Advanced Study, Kashiwa 277-8583, Japan

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Abstract

We extend the Cecotti-Kallosh model of Starobinsky inflation in supergravity
by adding a holomorphic function to the superpotential in order to generate a
large peak in the power spectrum of scalar (curvature) perturbations. In our
approach, the singular non-canonical kinetic terms are largely responsible for
inflation (as an attractor solution), whereas the superpotential is engineered
to generate a production of primordial black holes. We study the cases with
(i) a linear holomorphic function, (ii) a quadratic holomorphic function, and
(iii) an exponential holomorphic function, as regards the dependence of
inflation and primordial black holes production upon parameters of those
functions and initial conditions, as well as verify viability of inflation
with our superpotentials. We find that an efficient production of primordial
black holes consistent with CMB measurements is only possible in the second
(ii) case. We calculate the masses of the produced primordial black holes and
find that they are below the Hawking (black hole) evaporation limit, so that
they cannot be part of the current dark matter in our Universe.

##  1 Introduction

Cosmological inflation in the very early Universe is well motivated both
theoretically and experimentally. It solves the old problems of the standard
(Friedman) cosmology, proposes the origin of the large scale structure, and
gives correct predictions about fluctuations and anisotropy of the cosmic
microwave background (CMB) radiation.

However, the inflationary paradigm is not yet an established theory because
the scale of inflation, its driver (inflaton) and the origin of the inflaton
scalar potential are unknown. There are many inflationary models in the
literature so that there is a need for their discrimination. A necessary
requirement of consistency with CMB measurements still leaves a plenty of
viable inflationary models because the CMB window is very small, being limited
to scales (wavenumbers) $k\sim 10^{-4}\div 5\cdot 10^{-1}~{}{\rm Mpc}^{-1}$.
Amongst the additional theoretical requirements can be demands for an
attractor solution for inflation, the ultra-violet cutoff well beyond the
Hubble value during inflation, a minimal number of interactions and
parameters, a clear physical origin of inflaton, additional fundamental
symmetries, etc. It is, therefore, desirable to separate the well motivated
features of viable inflationary models from environmental (or model-dependent)
features specific to particular models of inflation, as well as identify the
leading candidate by applying the former conditions.

For this purpose, in this paper we employ the framework of pole inflation that
allows one to essentially describe inflation by a non-canonical kinetic term
having a pole, with a related unification of inflationary models into the
universality classes. This way of reasoning leads to the cosmological
$\alpha$-attractors [1, 2, 3]. Among the inflationary $\alpha$-attractors, the
famous Starobinsky model (1980) of inflation [4] still occupies the leading
position because (i) it employs only gravitational interactions, (ii)
Starobinsky’s inflaton (scalaron) is a physical excitation of the higher-
derivative gravity and can be interpreted as the Nambu-Goldstone boson
associated with spontaneous breaking of scale invariance in the no-scale and
no-ghost $R^{2}$-gravity, (iii) the ultra-violet cutoff in the Starobinsky
model is given by the (reduced) Planck mass $M_{\rm Pl}$ that is beyond the
Hubble function $H$ during inflation by the five orders of the magnitude, (iv)
the predictions of the Starobinsky model for CMB are in excellent agreement
with current (2021) measurements of CMB [5, 6], (v) the Starobinsky model has
only one parameter given by the inflaton mass $m$ that is fixed by the CMB
amplitude as $m={\cal O}(10^{-5})M_{\rm Pl}$, so that the Starobinsky model of
inflation has no free parameters at all, (vi) the Starobisky inflation offers
the universal mechanism for reheating after inflation, see e.g., Refs. [7, 8,
9] for a review of all these features.

The inflationary scale $H\sim{\cal O}(10^{14})$ GeV in the Starobinsky model
implies the necessity to include new physics well beyond the electro-weak
scale because the value of $H$ is not far from the Grand Unification scale
where gravitational interactions of elementary particles can no longer be
ignored. The appropriate framework in the theoretical high energy physics at
those scales is given by supergravity [10, 8]. In this paper, we employ the
Starobinsky inflation in the supergravity framework.

The use of pole inflation allows us to describe slow-roll inflation mainly by
the kinetic terms, with the scalar potential being largely undetermined. Then
we can use freedom in our choice of the scalar potential for engineering a
formation of primordial black holes (PBH) during or after inflation. In
supergravity theory, the kinetic terms are described by a Kähler potential,
and the scalar potential is governed by a superpotential also. We use the
supergravity description of the Starobinsky inflation in the minimal form of
the Cecotti-Kallosh model [11] but modify their superpotential by a
holomorphic function towards an inclusion of PBH production. It leads to the
two-field inflation whose consistency has to be checked again because the
modified superpotentials can destroy viable inflation. Multi-field inflation
and its applications in supergravity for PBH production were investigated in
the different models in Refs. [12, 13, 14, 15, 16].

Our paper is organized as follows. Section 2 is our setup where we introduce
our model, the parameterizations of the scalar fields used, and demonstrate
consistency with the earlier results. The main body of our paper is devoted to
the simplest Ansätze for the new functions $g$ and $\tilde{g}$ defined in Sec.
2. First, we try linear functions for them and investigate their parameter
space in Sec. 3, as regards the impact for double inflation and PBH formation.
Next, in Section 4, we investigate the case of a quadratic $g$-function that
turns out to be our main case. In Section 5 we derive the inflationary
trajectory, the slow roll parameters and the CMB observables in the case of
the quadratic $g$-function because other choices do not work. In Section 6 we
study the dependence of the observables upon initial conditions. In Section 7
we compute the power spectrum of scalar perturbations and the masses of the
generated PBH. Our conclusion is Section 8. We also study some other possible
cases, including the exponential functions for $g$ and $\tilde{g}$, and find
that they do not work for inflation and PBH production, see Appendices A and B
too.

We do not provide a historical overview or an introduction to cosmological
inflation and primordial black holes, skip some basic equations because they
can be easily found in many publications, see e.g., Refs. [8, 12, 14], and
confine our paper to original new results derived either analytically or
numerically, with a minimum of relevant references. We use the natural units
with $M_{\rm Pl}=1$ throughout the paper.

##  2 Setup

The basic idea of pole inflation [1, 2, 3] can be illustrated on the simplest
example of the single-field E-models of inflation, also known as the
cosmological $\alpha$-attractors in the literature, by starting from the
Lagrangian

${\cal L}[y]=-{{3\alpha}\over{4y^{2}}}\partial^{\mu}y\partial_{\mu}y-f^{2}(y)$
(1)

of the real field $y(x^{\mu})$ having the singular kinetic term (with a pole
of the 2nd order at $y=0$), the parameter $\alpha>0$ and the potential
$V(y)=f^{2}(y)$ in terms of a real non-constant function $f(y)$ that is
analytic at the origin $y=0$ with $f(0)\neq 0$. A non-linear field
redefinition

$y=\exp\left(-\sqrt{{{2}\over{3\alpha}}}\varphi\right)$ (2)

yields the canonical kinetic term of $\varphi$ with the scalar potential
$V(\varphi)=f^{2}(y(\varphi))$ that can be expanded in power series with
respect to $y$ that is supposed to be small during slow-roll inflation [14,
17]. The linear term (in $y$) of that expansion contributes to the leading
terms (with respect to the inverse powers of the e-folds number $N_{e}$) in
the cosmological observables known as the tilt $n_{s}$ of scalar perturbations
and the tensor-to-scalar ratio $r$, so that the rest of the expansion can be
chosen at will or for other purposes.

The pole inflation makes manifest the universality classes of the cosmological
$\alpha$-attractors, which are parametrized by $\alpha$ [18, 9]. Their
predictions for the CMB spectrum tilts,

$n_{s}=1-{{2}\over{N_{e}}}~{},\quad r={{12\alpha}\over{N_{e}^{2}}}~{}~{},$ (3)

comfortably fit observations [5, 6] for the parameter $\alpha$ values around
one and $N_{e}=55\div 10$. The Starobinsky inflation [4] appears in the case
of $\alpha=1$, while there exist the simple dual version of the model (1)
known as the modified $(R+R^{2})$ gravity, see e.g., Refs. [14, 17] for a
recent review. In this paper, we confine ourselves to $\alpha=1$ and dub our
modifications Starobinsky-like accordingly.

Our supergravity extension of the Starobinsky model of inflation in this paper
is described by the Kähler potential and the superpotential (cf. Refs. [19,
10, 11]) as follows:

$\displaystyle
K=-3\ln\left(T+\bar{T}-|C|^{2}+\zeta{{|C|^{4}}\over{T+\bar{T}}}\right),$ (4)
$\displaystyle W=MC(T-1)+g(T),$ (5)

in terms of the inflaton superfield $T$ and the goldstino superfield $C$. Here
$M=4m_{\rm inf.}$ is proportional to the mass $m_{\rm inf.}$ of Starobinsky’s
scalaron (inflaton), $m_{\rm inf.}\sim 10^{-5}M_{\rm Pl}$. The term with the
real coefficient $\zeta$ inside the logarithm is needed for stabilization of
the inflationary trajectory in the scalar field space with a $T$-independent
mass at $C=0$ [11]. When $\zeta=g(T)=0$, our model (5) reduces to the model in
Refs. [20, 21]. Unlike those references, we have added a new analytic function
$g(T)$ to the superpotential (5) that will be needed for PBH production.

In terms of the leading (scalar) field components, Eqs. (4) and (5) give the
Lagrangian

$\displaystyle\mathcal{L}=-{{3}\over{(T+\bar{T})^{2}}}\partial_{\mu}T\partial^{\mu}\bar{T}-V,$
(6) $\displaystyle V={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}{{M^{2}|T-1|^{2}}\over{(T+\bar{T})^{2}}}+{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}{{1}\over{(T+\bar{T})}}\left|{{dg}\over{dT}}\right|^{2}-{{{\textstyle{dg\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{dT}$}}}}\bar{g}+{{d\bar{g}}\over{d\bar{T}}}g}\over{(T+\bar{T})^{2}}}~{}~{},$
(7)

where we have set $C=0$, and the bars denote complex conjugation.

There is a pole in the kinetic term (6) at Re$~{}T=0$. However, unlike Eq.
(1), there are two real scalars, while their kinetic terms are field-
dependent, i.e. they form a non-linear sigma-model (NLSM) [22].

The half-(complex)-plane variable $T$ can be exchanged to the (Poincaré disk)
variable $Z$ via the Cayley (holomorphic) transformation accompanied by a
similar change of variables from $C$ to $S$ as follows:

$\displaystyle T={{1+Z}\over{1-Z}},\quad C={{\sqrt{2}S}\over{1-Z}}~{}~{},$ (8)

with the inverse transformation

$\displaystyle Z={{T-1}\over{1+T}},\quad S={{\sqrt{2}C}\over{1+T}}~{}~{}.$ (9)

Equations (4) and (5) can then be rewritten to

$\displaystyle
K=-3\ln\left(1-|Z|^{2}-|S|^{2}+\zeta{{|S|^{4}}\over{1-|Z|^{2}}}\right)+3\ln{{|1-Z|^{2}}\over{2}}~{},$
(10) $\displaystyle W=2\sqrt{2}{{MZS}\over{(1-Z)^{2}}}+g(Z)~{}.$ (11)

Next, after a Kähler transformation, $K\rightarrow K+\Lambda+\bar{\Lambda}$
and $W\rightarrow e^{-\Lambda}W$, by choosing $\Lambda=-3\log(1-Z)/\sqrt{2}$,
we obtain

$\displaystyle
K=-3\ln\left(1-|Z|^{2}-|S|^{2}+\zeta{{|S|^{4}}\over{1-|Z|^{2}}}\right)~{},$
(12) $\displaystyle W=MZ(1-Z)S+\tilde{g}(Z)~{},$ (13)

where we have introduced

$\tilde{g}(Z)={{(1-Z)^{3}g(Z)}\over{2\sqrt{2}}}~{}~{}.$ (14)

The scalar Lagrangian at $S=0$ reads

$\displaystyle\mathcal{L}=-{{3}\over{\left(1-|Z|^{2}\right)^{2}}}\partial_{\mu}Z\partial^{\mu}\bar{Z}-V,$
(15)

where the scalar potential is given by

$\displaystyle V=$ $\displaystyle\
{\textstyle{M^{2}\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}{{|Z|^{2}\left|1-Z\right|^{2}}\over{\left(1-|Z|^{2}\right)^{2}}}+{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}{{1}\over{(1-|Z|^{2})}}\left|{{d\tilde{g}}\over{dZ}}+{{3\bar{Z}}\over{(1-|Z|^{2})}}\tilde{g}\right|^{2}-{{3}\over{\left(1-|Z|^{2}\right)^{3}}}|\tilde{g}|^{2}$
$\displaystyle=$ $\displaystyle\
{\textstyle{M^{2}\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}{{|Z|^{2}\left|1-Z\right|^{2}}\over{\left(1-|Z|^{2}\right)^{2}}}+{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{24}$}}}}{{|1-Z|^{6}}\over{1-|Z|^{2}}}\left|{{dg}\over{dZ}}\right|^{2}-{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{8}$}}}}{{|1-Z|^{4}}\over{\left(1-|Z|^{2}\right)^{2}}}\left[(1-Z)^{2}{{dg}\over{dZ}}\bar{g}+(1-\bar{Z})^{2}{{d\bar{g}}\over{d\bar{Z}}}g\right].$
(16)

There is no fundamental reason to prefer any of the two parameterizations ($T$
or $Z$) above, so that we are going to use both of them. However, it matters
which function we choose in the superpotential, either $g$ or $\tilde{g}$ that
are related by Eq. (14).

It is convenient to parametrize

$Z=re^{i\theta}={\rm{tanh}}{{\varphi}\over{\sqrt{6}}}\,e^{i\theta}~{}~{}.$
(17)

Then the kinetic term of $Z$ can be rewritten to

$\displaystyle\mathcal{L}_{\rm{kin}}=-{\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}(\partial_{\mu}\varphi)^{2}-{\textstyle{3\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}{\rm{sinh}}^{2}{{2\varphi}\over{\sqrt{6}}}(\partial_{\mu}\theta)^{2},$
(18)

where the field $\varphi$ can be identified with the canonical inflaton
(scalaron), and $\theta$ is the additional physical scalar field (sinflaton).
It is not possible to canonically normalize both fields because the field
space (NLSM) curvature is non-vanishing. We get the NLSM metric

$\displaystyle G_{ab}=\left(\begin{array}[]{cc}1&0\\\
0&{\textstyle{3\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}{\rm{sinh}}^{2}{{2\varphi}\over{\sqrt{6}}}\end{array}\right),$
(21)

with $a,b=(1,2)=(\varphi,\theta)$. The non-vanishing Christoffel symbols are

$\displaystyle\Gamma_{\theta\theta}^{\varphi}=-{{3}\over{2\sqrt{6}}}{\rm{sinh}}{{4\varphi}\over{\sqrt{6}}}~{},\quad\Gamma_{\theta\varphi}^{\theta}=\Gamma_{\varphi\theta}^{\theta}={{2}\over{\sqrt{6}}}{\rm{coth}}{{2\varphi}\over{\sqrt{6}}}~{},$
(22)

which give rise to the NLSM Ricci scalar

$\displaystyle\mathcal{R}=-{{4}\over{3}}~{}.$ (23)

Therefore, the NLSM has the constant negative curvature in the field space,
i.e. a hyperbolic geometry.

In the case of $g=0$, the scalar potential (2) reduces to

$\displaystyle
V={{M^{2}}\over{12}}{\rm{sinh}}^{2}{{2\varphi}\over{\sqrt{6}}}\left(1+{\rm{tanh}}^{2}{{\varphi}\over{\sqrt{6}}}-2{\rm{tanh}}{{\varphi}\over{\sqrt{6}}}{\rm{cos}}\theta\right).$
(24)

Let us demonstrate that $\theta$ is stabilized with a heavy mass during slow-
roll inflation. The second derivative of the potential, $d^{2}V/d\theta^{2}$,
at $\theta=0$ is given by

$\displaystyle\left.{{d^{2}V}\over{d\theta^{2}}}\right|_{\theta=0}={{M^{2}}\over{6}}{\rm{sinh}}^{2}{{2\varphi}\over{\sqrt{6}}}{\rm{tanh}}{{\varphi}\over{\sqrt{6}}}~{}~{}.$
(25)

When $\varphi$ is approximately constant, the canonical normalization of the
$\theta$-field is

$\displaystyle\tilde{\theta}=\sqrt{{{3}\over{2}}}{\rm{sinh}}{{2\varphi}\over{\sqrt{6}}}\cdot\theta~{},$
(26)

so that we can read off the effective mass of $\tilde{\theta}$ from Eqs. (25)
and (26) as

$\displaystyle
m^{2}_{\tilde{\theta}}={{M^{2}}\over{9}}{\rm{tanh}}{{\varphi}\over{\sqrt{6}}}\simeq
4H^{2}~{},$ (27)

where we have used the Friedman equation. Since the $\tilde{\theta}$-field has
the mass beyond the Hubble scale $H$ during the first (Starobinsky) stage of
inflation, we can safely ignore isocurvature perturbations in the
$\tilde{\theta}$-direction (but not later: a tachyonic instability in the
$\theta$-direction will be needed for PBH production after Starobinsky
inflation, see next Sections).

Having stabilized the $\theta$-field during the first stage of inflation, we
get the effective single-field potential for inflation as follows:

$V_{\rm{eff}.}={{M^{2}}\over{3}}{{{\rm{tanh}}^{2}{{\varphi}\over{\sqrt{6}}}}\over{(1+{\rm{tanh}}{{\varphi}\over{\sqrt{6}}})^{2}}}={{M^{2}}\over{12}}\left(1-e^{-\sqrt{{{2}\over{3}}}\varphi}\right)^{2}={{4m_{\rm
inf.}^{2}}\over{3}}\left(1-e^{-\sqrt{{{2}\over{3}}}\varphi}\right)^{2}$ (28)

that coincides with the standard potential of the Starobinsky model. The shape
of the two-field potential and its slice at $\theta=0$ are given in Fig. 1.

|
---|---
Figure 1: The profile of the potential for $g=0$ (left) and its slice at
$\theta=0$ (right).

When the function $g\neq 0$, both scalars should be taken into account, and
single-field inflation becomes two-field inflation. The corresponding
equations of motion in the Friedman universe read

$\displaystyle
0=\ddot{\varphi}+3H\dot{\varphi}-\sqrt{{\textstyle{3\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{8}$}}}}}\sinh\left(\sqrt{{\textstyle{8\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi\right)\dot{\theta}^{2}+\partial_{\varphi}V~{}~{},$
(29) $\displaystyle
0=\ddot{\theta}+3H\dot{\theta}+2\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}{\mathrm{coth}}{\left(\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi\right)}\dot{\varphi}\dot{\theta}+{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}\mathrm{csch}^{2}\left(\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi\right)\partial_{\theta}V~{}~{},$
(30) $\displaystyle 0={\textstyle{1\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{2}$}}}}\dot{\varphi}^{2}+{\textstyle{3\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{4}$}}}}\sinh^{2}\left(\sqrt{{\textstyle{2\over\vphantom{2}\smash{\raise
0.60275pt\hbox{$\scriptstyle{3}$}}}}}\varphi\right)\dot{\theta}^{2}-3H^{2}+V~{}~{},$
(31)

with the potential

$\displaystyle V=$
$\displaystyle{{M^{2}}\over{3}}{{r^{2}(1-2r\cos\theta+r^{2})}\over{(1-r^{2})^{2}}}+{{1}\over{24}}{{(1-2r\cos\theta+r^{2})^{3}}\over{1-r^{2}}}\left|{{\mathrm{d}g}\over{\mathrm{d}Z}}\right|^{2}$
$\displaystyle-{{1}\over{8}}{{(1-2r\cos\theta+r^{2})^{2}}\over{(1-r^{2})^{2}}}\left[(1-re^{i\theta})^{2}{{\mathrm{d}g}\over{\mathrm{d}Z}}\bar{g}+(1-re^{-i\theta})^{2}{{\mathrm{d}\bar{g}}\over{\mathrm{d}\bar{Z}}}g\right]~{}~{}.$
(32)

##  3 Adding a linear superpotential

Having the $T$\- and $Z$-parameterizations on equal footing, we have to choose
either a $g$-function or a $\tilde{g}$-function, respectively. The simplest
choices are given by rational functions and exponentials. In Appendices A and
B we demonstrate that the exponentials are ruled out because they destabilize
inflation and do not lead to PBH production. Hence, we consider only
polynomial functions $g(Z)$ and $\tilde{g}(Z)$ in the main text. It follows
from Eq. (38) that a polynomial function $g(Z)$ leads to a polynomial function
$\tilde{g}(Z)$ but not vice versa in general.

###  3.1 A linear $g$-function

Let us use a linear function as the first trial,

$g(Z)=M(g_{0}+g_{1}Z)$ (33)

with the parameters $g_{0}$ and $g_{1}$. Actually, there is only one free
parameter because we always want to have a Minkowski vacuum in the potential.
Therefore, we fix the parameter $g_{0}$ by demanding

$\displaystyle\partial_{\varphi}V(g_{0})=\partial_{\theta}V(g_{0})=V(g_{0})=0~{}.$
(34)

The corresponding potential $V$ with the linear $g$-function (33) reads

$\displaystyle V=$ $\displaystyle\
{{M^{2}r^{2}}\over{3(1-r^{2})^{2}}}(1+r^{2}-2r\cos\theta)$
$\displaystyle-{{M^{2}}\over{24(1-r^{2})^{2}}}(1+r^{2}-2r\cos\theta)^{2}\left[A+B{\rm{cos}}\theta+C{\rm{cos}}2\theta\right],$
(35)

where we have introduced the notation

$\displaystyle A=6g_{0}g_{1}+g_{1}^{2}(-1-12r^{2}+r^{4}),\ \
B=-12g_{0}g_{1}r+4g_{1}^{2}r(2+r^{2}),\ \ C=6g_{0}g_{1}r^{2},$ (36)

and have used $r=\tanh{{\varphi}\over{\sqrt{6}}}$. The first line of Eq.
$(\ref{V_g_linear})$ corresponds to the Starobinsky potential, whereas the
second line shows the corrections caused by the $g$-function.

The second derivative of the potential with respect to $\theta$ is given by

$\displaystyle\left.{{d^{2}V}\over{d\theta^{2}}}\right|_{\theta=0}={{M^{2}r}\over{6(1-r^{2})^{2}}}\lfloor{\hskip
0.35pt}\\!\\!\\!\lceil
4r^{2}+3(1-r)^{4}\left(g_{1}^{2}-3g_{0}g_{1}+2r(g_{0}g_{1}-g_{1}^{2})\right)\rfloor{\hskip
0.35pt}\\!\\!\\!\rceil~{}.$ (37)

In order to have an instability in the $\theta$-direction after the
Starobinsky inflation, the second derivative in Eq. $(\ref{ddlgeq})$ should
become negative at some value of $r$. It is only possible when the signs of
the parameters $g_{0}$ and $g_{1}$ are opposite.

The potential $V$ in Eq. $(\ref{V_g_linear})$ is symmetric with respect to
interchange of signs of the parameters $g_{0}$ and $g_{1}$. We take $g_{1}$
positive and vary it from $0$ to $100$ with the increments of $10$ in order to
get the shape of the Hubble function derived from Eqs. (29), (30), (31) and
then determine whether double inflation occurs in each case. Having fixed the
$g_{1}$, the $g_{0}$ is automatically fixed from Eq. (34) by demanding the
potential to be have a Minkowski vacuum. The results are shown in Fig. 2 where
the initial conditions are set as $\theta(0)=0.001$ and $\varphi(0)=6$ with
the vanishing initial velocities.

Figure 2: The Hubble function for selected values of $g_{1}$ and $g_{0}$. The
initial conditions are $\theta(0)=0.001$ and $\varphi(0)=6$ with the vanishing
initial velocities.

According to Fig. 2, double inflation cannot be realized in the model with a
linear $g$-function. Hence, a more general $g$-function is needed.

###  3.2 A linear $\tilde{g}$-function

Since the functions $g$ and $\tilde{g}$ are non-linearly related, let us also
consider the case of a linear $\tilde{g}$-function,

$\displaystyle\tilde{g}(Z)=M(\tilde{g_{0}}+\tilde{g_{1}}Z),$ (38)

with the coefficients $\tilde{g_{0}}$ and $\tilde{g_{1}}$. With
$r={\rm{tanh}}{{\varphi}\over{\sqrt{6}}}$ and Eq. (38), the potential $V$ is
given by

$\displaystyle V=\ $
$\displaystyle{{M^{2}r^{2}}\over{3(1-r^{2})^{2}}}(1+r^{2}-2r\cos\theta)$
$\displaystyle-{{M^{2}}\over{3(1-r^{2})^{2}}}\left[9\tilde{g}_{0}^{2}+\tilde{g}_{1}^{2}(-1+4r^{2})+12\tilde{g}_{0}\tilde{g}_{1}r{\rm{cos}}\theta\right].$
(39)

We get a Minkowski vacuum by choosing $\tilde{g_{0}}$ properly. The second
derivative of the potential with respect to $\theta$ reads

$\displaystyle\left.{{d^{2}V}\over{d\theta^{2}}}\right|_{\theta=0}={{2M^{2}r}\over{3(1-r^{2})^{2}}}(r^{2}+6\tilde{g_{0}}\tilde{g_{1}})$
(40)

without using any approximation. Hence, the condition
$-1/6<\tilde{g_{0}}\tilde{g_{1}}<0$ is necessary for having a
$\theta$-instability.

To give our examples, we choose three values of the parameter $\tilde{g_{1}}$
as $-0.2$, $-0.1$ and $-0.03$. The sign of $g_{0}$ should be positive in all
these cases. The potential is the same after interchanging the signs between
$\tilde{g_{0}}$ and $\tilde{g_{1}}$, which can be seen from Eq. (39). The
shapes of the potential at $\theta=0$ and $\varphi=\mathrm{const.}$ near the
minimum of the potential are shown in Figs. 3 and 4, respectively. 111Since
the Starobinsky slow roll inflation takes place for the $\varphi$-values
between $5$ and $0.5$ [14, 17], we are not concerned by instabilities in the
trans-Planckian region where we expect new physics and our approach does not
apply.

Figure 3: The potential at $\theta=0$ for $\tilde{g_{1}}$=-0.2, -0.1, -0.03.

Figure 4: The potential at $\varphi=\mathrm{const.}$ for $\tilde{g_{1}}$=-0.2,
-0.1, -0.03.

As is clear from Fig. 3, when the value of the parameter $|\tilde{g_{1}}|$
gets larger, the length of the plateau of the potential gets shorter, so it
becomes harder to have enough e-folds. On the other hand, Fig. 4 shows that
the peak gets smaller when the parameter $|\tilde{g_{1}}|$ gets smaller.
Therefore, in order to have enough e-folds for the first stage of the
inflation, the value of the parameter $|\tilde{g_{1}}|$ should be smaller or,
in other words, closer to zero. However, when the parameter $\tilde{g_{1}}$
approaches zero, the $\theta$-instability will vanish, which results in the
absence of the second stage of the inflation.

The above arguments also can be checked directly by solving the equations of
motion in the Friedman universe and inspecting the shape of the Hubble
function. The shape of the Hubble function obtained from a solution to Eqs.
(29), (30) and (31) is shown in Fig. 5 where we have set the initial
conditions as follows: the $\varphi(0)$ is the value at the maximum of the
potential at $\theta=0$, 222Actually, the value of $\varphi(0)$ must be much
smaller than the field value at the maximum, in order to give an impetus to
the field (without it the field would stay at the maximum). The impetus
slightly changes the shape of the Hubble function but does not change our
conclusion. with $\theta(0)=0.001$ and the vanishing initial velocities.

Figure 5: The Hubble function in our examples with the initial conditions
$\theta(0)=0.001$ and $\varphi(0)$, near the maximal value of the potential at
$\theta=0$.

As is clear from Fig. 5, the required duration of inflation can be achieved by
lowering the value of $|\tilde{g_{1}}|$. However, having two plateaus cannot
be achieved in all cases. When the parameters approach zero, the Hubble
function apparently recovers the same shape as in the Starobinsky inflation.
Therefore, we conclude that in the linear $\tilde{g}$-case double inflation
cannot be achieved also, and thus a more general $\tilde{g}$-function is
needed.

##  4 The model with a quadratic $g$-function

Motivated by the negative findings in the previous Section, in this Section we
take a quadratic $g$ function,

$g(Z)=M(g_{0}+g_{1}Z+g_{2}Z^{2})~{},$ (41)

with three parameters $g_{0}$, $g_{1}$ and $g_{2}$, where the parameters
$g_{1}$ and $g_{2}$ are arbitrary and $g_{0}$ is fixed by demanding Minkowski
vacua, see Fig. 6.

We search for the proper values of the parameters in order to obtain double
inflation. By fixing one of the free parameters, the preferable shape of the
potential can be obtained by varying another free parameter. Increasing the
value of the parameters causes the two minima of the potential to become
closer to each other along the $\theta$ axis, which shortens duration of the
second stage of inflation.

The existing flexibility in the choice of the parameters is demonstrated by
Fig. 6 that shows the slices of the potential at fixed $\varphi$, which cross
the minima of the potential. In Fig. 6 we have set $g_{1}=-1.0$ and have
changed $g_{2}$ from $1.8$ to $2.2$ with the increment of $0.1$. The $g_{0}$
is derived by solving Eq. (34). Though a slice does not show the shape of the
potential at the inflection point, it does demonstrate that an instability can
be achieved by increasing the value of the parameter $g_{2}$, while the
maximum becomes lower and eventually becomes flat. Therefore, there should be
the upper and lower bounds on $g_{2}$ for each possible $g_{1}$. We can check
the actual range of the parameter $g_{2}$ by solving Eqs. (29), (30) and (31),
which allows us to get the shape of the Hubble function. We set the initial
values as $\varphi(0)=6$ and $\theta(0)=0.001$, with the vanishing initial
velocities. Our results are shown in Fig. 6. The straight line after $Mt=600$,
as in the $g_{2}=1.8$ and $g_{2}=1.85$ case, shows that an instability cannot
be achieved and the trajectory did not fall to a minimum. As can be seen from
Fig. 6, double inflation can be achieved for $1.9\leq g_{2}\leq 2.2$.

(a) |  (b)
---|---
Figure 6: (a) The sliced shape of the potential at fixed $\varphi$ passing
through the minima of the potential. (b) The Hubble function for $g_{1}=-1.0$
and the values of $g_{2}$ between $1.8$ and $2.2$. The initial values are set
to $\varphi(0)=6$ and $\theta(0)=0.001$ with the initial velocities all set to
zero. The legend on the right shows the values of the parameters $g_{0}$ and
$g_{2}$.

We also tested the case with fixed $g_{2}$ by changing $g_{1}$. Figure 7 shows
the case with $g_{2}=2.0$ and $g_{1}$ varied between $-1.25$ and $-0.85$, with
all other conditions being the same as above. It reveals the same trend,
namely, when increasing the value of $g_{1}$, a $\theta$-instability arises
and the height of the potential gets smaller. In all cases, the value of
$g_{0}$ decreases when $g_{1}$ and $g_{2}$ increase, while the potential gets
closer to the one in the original Cecotti-Kallosh model of inflation [11].

(a) |  (b)
---|---
Figure 7: (a) The sliced shape of the potential at fixed $\varphi$ passing
through the minima of the potential. (b) The Hubble function for $g_{2}=2.0$
and the values of $g_{1}$ from $-1.25$ to $-0.85$. The initial values are set
to $\varphi(0)=6$ and $\theta(0)=0.001$ with the vanishing initial velocities.
The legends on the right shows the values of the parameters $g_{0}$ and
$g_{1}$.

##  5 Inflationary trajectory, slow-roll parameters and
CMB observables

Having derived Figs. 6 and 7, let us study the model with the fixed parameters
$g_{1}=-1.0$, $g_{2}=2.0$ and $g_{0}=1.1073$, as a representative.

The scalar potential depending upon both fields $\varphi$ and $\theta$, the
inflationary trajectory, and the time evolution of both fields in that
potential are shown in Fig. 8. We also derived the Hubble function and the
slow-roll parameters $\epsilon$ and $\eta$ in the representative model, which
are shown in Fig. 9. The initial conditions are the same: $\varphi(0)=6$,
$\theta(0)=1.0\times 10^{-4}$ and the vanishing initial velocities.

(a) |  (b)
---|---
Figure 8: (a) The profile of the potential and the inflationary trajectory. (b) The solutions $\varphi(t)$ (blue) and $\theta(t)$ (red) to the equations of motion.  (a) |  (b)
---|---
Figure 9: (a) The Hubble function. (b) The slow-roll parameters $\epsilon$
(blue) and $\eta$ (orange).

As is clear from Fig. 8, inflation begins as the single-field slow-roll
Starobinsky inflation with $\varphi$ as the inflaton (the first stage of
double inflation). When the inflationary trajectory approaches the saddle
point where one of the scalars gets a negative mass squared (the signal of
tachyonic instability), the inflationary trajectory sharply changes its
direction. After that inflation is driven by $\theta$-field as the inflaton
(the second stage of double inflation).

As is clear from Fig. 9, there is double inflation indeed because of two
plateaus in the profile of the Hubble function, while the slow-roll parameter
$\epsilon$ becomes very small, which indicates the presence of the very short
ultra-slow-roll phase between the two slow-roll stages. The slow-roll
conditions are violated in the ultra-slow-roll regime.

We define the start of the second stage of inflation (or the end of the first
stage of inflation) by the time when the parameter $\eta$ first approaches one
(or when the parameter $\epsilon$ first reaches its maximum), and the end of
the second stage as the time when the parameter $\epsilon$ again reaches one
first. The duration of the second stage of the inflation $\Delta N_{2}$
appears to be approximately 22.24 e-folds.

The main (CMB) inflationary observables are given by the spectral index
$n_{s}$ and tensor-to-scalar ratio $r$ of scalar and tensor perturbations,
respectively. We derived them in order to confront our model against Planck
observational results.

The slow-roll parameters $\epsilon$ and $\eta_{\Sigma\Sigma}$ are defined by
($\phi^{A}\equiv\\{\varphi,\theta\\}$)

$\displaystyle\epsilon\equiv-{{\dot{H}}\over{H^{2}}}~{},\quad\eta_{\Sigma\Sigma}\equiv{{\mathcal{M}_{B}^{A}\Sigma_{A}\Sigma^{B}}\over{V}}~{}~{}~{},$
(42)
$\displaystyle\Sigma^{A}\equiv{{\dot{\phi}^{A}}\over{|\dot{\phi}|}}~{},\quad\mathcal{M}_{B}^{A}\equiv
G^{AC}\nabla_{B}\partial_{C}V-\mathrm{R}^{A}_{~{}CDB}\dot{\phi}^{C}\dot{\phi}^{D}~{},$
(43)

in terms of the metric $G_{AC}$ in the scalar field space and its Riemann-
Christoffel curvature $\mathrm{R}^{A}_{~{}CDB}$. The isocuvature parameter is
defined by

$\displaystyle\eta_{\Omega\Omega}\equiv{{\mathcal{M}_{B}^{A}\Omega_{A}\Omega^{B}}\over{V}}~{},\quad\mathrm{where}\quad\Omega^{A}\equiv{{{\omega}^{A}}\over{|{\omega}|}}~{},\quad\omega\equiv\dot{\Sigma^{A}}+\Gamma^{A}_{BC}\Sigma^{B}\dot{\phi}^{C}~{}.$
(44)

The transfer function are defined by

$\displaystyle
T_{SS}(t_{1},t_{2})\equiv\exp\left[\int_{t_{2}}^{t_{1}}dt^{\prime}\beta(t^{\prime})H(t^{\prime})\right]~{},$
(45) $\displaystyle T_{RS}(t_{1},t_{2})\equiv
2\int_{t_{2}}^{t_{1}}dt^{\prime}|\omega(t^{\prime})|T_{SS}(t_{1},t_{2})~{},$
(46)
$\displaystyle\mathrm{where}\quad\beta(t)\equiv-2\epsilon+\eta_{\Sigma\Sigma}-\eta_{\Omega\Omega}-{{4|\omega|^{2}}\over{3H^{2}}}~{}.$
(47)

These equations are enough to compute the observables by applying the
Mathematica packages for numerical calculations with the transport method in
multi-field models of inflation [23]. In particular, the observables $n_{s}$
and $r$ are given by

$\displaystyle n_{s}=1-6\epsilon+2\eta_{\Sigma\Sigma}\quad{\rm and}\quad
r={{16\epsilon}\over{1+T_{RS}^{2}}}~{}~{}.$ (48)

The values of the observables in our representative model at the CMB pivot
scale, which corresponds to $k_{\rm CMB}=0.05~{}{\rm Mpc}^{-1}$ (we set 70
e-folds as the duration of the whole double inflation), and the same initial
values mentioned above are shown in Table 1 together with the values of
$g_{0}$, the values of $\varphi$ and $\theta$ at the minimum, and the number
of e-folds for the second stage of inflation. Table 1 also contains the values
of the other parameters, $g_{1}$ and $g_{2}$.

The improved precision measurements of the CMB radiation [5, 6] give for the
spectral tilt $n_{s}$ of scalar perturbations the values

$n_{s}=0.9649\pm 0.0042\quad(68\%~{}{\rm C.L.})$ (49)

The current observational upper bound [5, 6] on the CMB tensor-to-scalar ratio
$r$ is given by

$r<0.036\quad(95\%~{}{\rm C.L.})$ (50)

As is clear from Table 1, the values of $r$ in our model are well below the
observational bound but the values of $n_{s}$ are outside the $1\sigma$ range
for the given values of $\Delta N_{2}$, except the case of $g_{1}=-0.25$. The
reason for that is our intension to maximize the value of $\Delta N_{2}$ which
is crucial for larger PBH masses, see Eq. (52) below. The $1\sigma$ agreement
can be achieved by changing the parameters at the expense of decreasing
$\Delta N_{2}$ to 10. 333For instance, it can be easily achieved by changing
the value of $g_{2}$. Nevertheless, it also follows from Table 1 that the
$3\sigma$ agreement with the observed value of $n_{s}$ is still possible for
$\Delta N_{2}\leq 22.24$.

$g_{1}$ | $-2.0$ | $-1.5$ | $-1.0$ | $-0.5$ | $-0.25$
---|---|---|---|---|---
$g_{2}$ | $2.9$ | $2.5$ | $2.0$ | $1.4$ | $1.05$
$g_{0}$ | $1.6292$ | $1.35754$ | $1.1073$ | $0.862908$ | $0.728364$
$\varphi_{min}$ | $1.33857$ | $1.23483$ | $1.13543$ | $0.999405$ | $0.752367$
$\theta_{min}$ | $0.832134$ | $0.869356$ | $0.938336$ | $1.05348$ | $1.03411$
$\Delta N_{2}$ | $31.50$ | $25.99$ | $22.24$ | $20.31$ | $12.69$
$n_{s}$ | $0.9416$ | $0.9496$ | $0.9537$ | $0.9551$ | $0.9616$
$r$ | $0.0092$ | $0.0069$ | $0.0059$ | $0.0055$ | $0.0041$
Table 1: The values of the parameters $(g_{1},g_{2},g_{0})$, the field values
($\varphi_{min},\theta_{min}$) at the minimum, the number of e-foldings for
the second stage of the inflation ($\Delta N_{2}$), and the CMB observables
$(n_{s},r)$. The initial values are $\varphi(0)=6$ and $\theta(0)=10^{-4}$,
with the vanishing initial velocities.

We found difficult to derive the power spectrum in our models due to field
oscillations along $\theta=0$ because our numerical calculations failed. This
”primordial feature” is likely caused due to resonances during inflation on
sub-horizon scales. Such resonances can occur when a massive scalar field
oscillates at the bottom of the potential [24, 25, 26], like in our model. To
avoid this problem, we computed the power spectrum in our model by considering
the first and second stages of double inflation separately, see the next
Sections. However, it required setting initial conditions at the beginning of
the second stage of inflation, which we did not determine and, hence, took
them randomly.

##  6 Dependence upon initial conditions

In this Section, we take a closer look on the dependence of the observables
upon initial conditions. In order to get the dependence upon the initial value
of $\theta$, we numerically computed the duration ($\Delta N_{2}$) of the
second stage of inflation (in e-folds) in the representative model for the
different values of $\theta(0)$ between $10^{-8}$ and $10^{-2}$. Our numerical
results are shown in Fig. 10. As is clear from Fig. 10, the values of $\Delta
N_{2}$ weakly depend upon small changes in the initial value of $\theta$. The
values of $\Delta N_{2}$ vary from about $20$ to $30$ that changes the
corresponding values of the observables by the order of $10^{-2}$. The similar
results for the CMB observables are given in Fig. 11, which are a bit more
sensitive to initial conditions.

Figure 10: The dependence of the duration $\Delta N_{2}$ of the second stage of inflation upon the value of initial condition $\theta(0)$ in our representative model. (a) |  (b)
---|---
Figure 11: The dependence of $n_{s}$ and $r$ upon the initial condition
$\theta(0)$ in our representative model.

The $\theta$-field is strongly suppressed at its minimum during the first
stage of the inflation but becomes sensitive to any additional small impact at
the saddle point where quantum corrections (for example, due to quantum
diffusion [27]) may become important while the impact of initial conditions
becomes negligible. Therefore, we study the first and second stages of
inflation separately, and assume a small non-vanishing value (kick) of the
$\theta$-velocity at the beginning of the second stage of inflation. We denote
$t_{1}$ to be the time at the end of the first stage inflation, and $t_{2}$ to
be the time of the end of the second stage of inflation. As regards the first
stage of inflation, we use our representative model studied in the preceding
Sections with the same initial value for $\varphi$ in order to get the
resulting values of $\varphi(t)$ and $\theta(t)$ and their velocities at the
end of the first stage of inflation.

Next, we check whether we can get enough e-folds $\Delta N_{2}$ by adding a
small $\theta$-velocity at the saddle point. In our representative model we
use the parameters $(g_{0},g_{1},g_{2})=(1.1496,-1.0,1.9)$ and the initial
values $(\varphi(0),\theta(0))=(6,0)$ with the vanishing initial velocities.
Then the values of $\varphi(t)$ and $\varphi^{\prime}(t)$ at the saddle point
are $|\varphi(t)|\sim 0.86$ and $\varphi^{\prime}(t)\lesssim\mathcal{O}(0.1)$,
respectively. Our result for the dependence of $\Delta N_{2}$ upon the kick
velocity $\theta^{\prime}(t_{1})$ at the saddle point is shown in Fig. 12. As
is clear from Fig. 12, $\Delta N_{2}\sim 20$ can be achieved with
$\theta^{\prime}(t_{1})\sim\mathcal{O}(10^{-10})$.

Figure 12: The dependence of duration of the second stage of inflation,
$\Delta N_{2}$, upon of the initial condition $\theta^{\prime}(t_{1})$ with
the parameters $(g_{0},g_{1},g_{2})=(1.1496,-1.0,1.9)$.

Having obtained those results, we derived the corresponding CMB observables
(with the total duration of inflation given by 70 e-folds) as follows:

$\displaystyle n_{s}=0.9634~{},\quad r=0.0038~{}.$ (51)

##  7 Power spectrum and PBH masses

In this Section we numerically compute the power spectrum of scalar
(curvature) perturbations by using the transport method with the Mathematica
package [23]. In order to efficiently produce the primordial black holes, the
enhancement (peak) in the power spectrum against its CMB value should be
$10^{6}$ times at least.

Given a peak, the masses of generated PBHs can be calculated by the equation
[28]

$\displaystyle
M_{\mathrm{PBH}}\simeq{{M_{\mathrm{Pl}}^{2}}\over{H(t_{*})}}\mathrm{exp}\left[2(N_{\mathrm{end}}-N_{*})+\int_{t_{*}}^{t_{\mathrm{exit}}}\epsilon(t)H(t)dt\right]~{},$
(52)

where $t_{*}$ and $N_{*}$ are the time and e-folds at the end of the first
stage of the inflation, $N_{\mathrm{end}}$ is e-folds at the end of (the
second stage) of inflation and $t_{\mathrm{exit}}$ is the time when the CMB
pivot scale exits the horizon. As is clear from Eq. (52), the PBH masses are
very sensitive to the value of $\Delta N_{2}=(N_{\mathrm{end}}-N_{*})$.

Our first example is the case with $g_{1}=1.0,g_{2}=1.9$, where we have
scanned the values of the $\theta$-velocity at the saddle point from $10^{-3}$
to $10^{-6}$. The power spectrum is shown in Fig. 13. As is clear from Fig.
13, the enhancement of the power spectrum strongly depends on the value of
$\theta^{\prime}(t_{1})$. In order to get the desired height of the peak from
the CMB value of $10^{-9}$ to $10^{-2}$, we have to choose the value of
$\theta^{\prime}(t_{1})$ within a small margin. We also give the corresponding
PBH masses (in grams) in Fig. 13.

Figure 13: The power spectrum of scalar perturbations in the case of
$g_{1}=1.0,g_{2}=1.9$ for the different values of $\theta^{\prime}(t_{1})$.
The initial conditions are $\varphi(0)=7,\theta(0)=0$ with the vanishing
initial velocities. The pivot scale $k_{*}$ corresponds to the end of the
first stage of inflation.

As another example, we choose $g_{1}={-1.0,-0.5,-0.25}$ and adjust the proper
values for $g_{2}$ and $g_{0}$. The $\theta^{\prime}(t_{1})$ is chosen to get
a peak up to $10^{-2}$ in the power spectrum. The resulting power spectrum is
given in Fig. 14.

Figure 14: The power spectrum of scalar perturbations for the different values
of $g_{1}$ with the peak height at $0.01$. The initial conditions are
$\varphi(0)=7,\theta(0)=0$ with the vanishing initial velocities. The pivot
scale $k_{*}$ corresponds to the end of the first stage of inflation.

The parameter values together with the values of the related observables
corresponding to Fig. 14 is collected in Table 2. As is clear from Table 2,
smaller values of $\left|g_{1}\right|$ lead to larger PBHs masses.

$g_{1}$ | $g_{0}$ | $g_{2}$ | $\theta^{\prime}$ | $\Delta N_{2}$ | $n_{s}$ | $r$ | $P_{enh}$ | $M_{\mathrm{PBH}}~{}(g)$
---|---|---|---|---|---|---|---|---
$-0.25$ | $0.7767$ | $0.98$ | $1.4\times 10^{-5}$ | $14.70$ | $0.9604$ | $0.0044$ | $1.9\times 10^{6}$ | $1.3\times 10^{14}$
$-0.5$ | $0.9106$ | $1.3$ | $1.7\times 10^{-5}$ | $12.32$ | $0.9623$ | $0.0040$ | $5.9\times 10^{6}$ | $7.4\times 10^{11}$
$-1.0$ | $1.1496$ | $1.9$ | $2.4\times 10^{-5}$ | $10.86$ | $0.9634$ | $0.0038$ | $1.1\times 10^{7}$ | $2.9\times 10^{10}$
Table 2: The parameters, the CMB observables, the enhancement of the power
spectrum, and the PBH masses in our models. The kick velocity
$\theta^{\prime}(t_{1})$ is chosen to have the peak summit at $10^{-2}$ in the
power spectrum. The initial conditions are $\varphi(0)=7$ and $\theta(0)=0$
with the vanishing initial velocities. The pivot scale is $k_{\rm
CMB}=0.05~{}{\rm Mpc}^{-1}$. The total duration of inflation is 70 e-folds.

To the end of this Section, we comment on the quadratic Ansatz for the
$\tilde{g}$-function in Eq. (2) as

$\tilde{g}(Z)=M(\tilde{g}_{0}+\tilde{g}_{1}Z+\tilde{g}_{2}Z^{2})$ (53)

in our search for proper values of its three parameters for double inflation.

We did not find the desired shape of the potential that would support double
inflation in this case. To examine the potential, we have set a space with the
three parameters $\tilde{g}_{0}$, $\tilde{g}_{1}$ and $\tilde{g}_{2}$ within
the range from $-100$ to $+100$, and have imposed the following conditions on
the potential: (i) the value of the potential must be positive, (ii) the
duration of inflation should be long enough, (iii) the potential along the
$\theta$-direction should be stable during the first stage of inflation, and
(iv) the height of the potential should be limited from above. By demanding
these conditions on the parameter space, no area was found to satisfy them.
Therefore, the model with the $\tilde{g}$-function (53) is ruled out. It is
worth noticing that the four conditions above are rather mild, e.g., in the
case of the quadratic $g$-function they are easily satisfied.

##  8 Conclusion

It follows from Table 2 that perfect consistency of our models with CMB
measurements restricts the possible PBH masses by $10^{14}$ g even after fine-
tuning of the parameters, which is below the Hawking (quantum black hole)
evaporation limit of $10^{15}$ g. It means that those PBH cannot be part of
the current dark matter.

This in contrast to the different extensions of Starobinsky model in
supergravity proposed and investigated in Refs. [13, 16, 29] where the
generated PBH masses can reach $10^{21}$ g. It is also different from the non-
supersymmetric Appleby-Battye-Starobinsky model [30] adapted in Ref. [31] for
PBH production, where the PBH masses can reach $10^{19}$ g.

Of course, all the above goes under the assumption that the PBH do not get
extra mass over time during the evolution of the Universe. It might be
possible that inflation generates a lot of PBH with masses of $10^{14}$ g,
which keep their masses and may get even larger masses via accretion and
mergers. However, then one would need a lot of those PBH in order to account
for a significant part of the current dark matter, which would contradict to
the non-detection of extra-galactic gamma-rays from their Hawking radiation.

We conclude that our supergravity models studied in this paper are consistent
with CMB measurements and do lead to the efficient PBH production with the
fine-tuned quadratic $g$-function in the superpotential, but those PBH cannot
be part of the current dark matter. We did not study more sophisticated
$g$-functions beyond quadratic polynomials and exponentials but we also did
not get any evidence that they would significantly change the outcome.

## Acknowledgements

SA was supported by Chung-Ang University and the Basic Science Research
Program through the National Research Foundation (NRF) funded by the Ministry
of Education, Science and Technology in South Korea under the grant No.
NRF-2022R1A2C2003567. RI and SVK were supported by Tokyo Metropolitan
University. SVK was also supported by the World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan, the Japanese Society for
Promotion of Science under the grant No. 22K03624, and the Tomsk Polytechnic
University Development Program Priority-2030-NIP/EB-004-0000-2022.

SVK is grateful to David I. Kaiser, Edward W. Kolb, Burt Ovrut and Misao
Sasaki for discussions and correspondence.

##  Appendix A Exponential $\tilde{g}$-function

In Appendices A and B we provide more support for our choice of polynomial
functions, $g(Z)$ and $\tilde{g}(Z)$, adopted in the main body of this paper,
by considering two alternatives (exponential- or KKLT-type [32]) and
demonstrating that those alternatives should be ruled out because inflation is
destabilized (too low e-folds).

Let us choose

$\displaystyle\tilde{g}(Z)=\tilde{C}+\tilde{A}e^{-\tilde{a}Z}~{},$ (54)

with the real parameters $\tilde{C},\tilde{A}$ and $\tilde{a}$. In Fig. 15, we
show the impact of $\tilde{C}$ and $\tilde{A}$ separately, when $\tilde{a}=1$
is fixed. As is clear from Fig. 15, large values of $\tilde{C}$ and
$\tilde{A}$ spoil the inflation trajectory at $\theta=0$. The approximate
upper bounds on those parameters are very low,

$\displaystyle|\tilde{A}|,~{}~{}|\tilde{C}|~{}~{}\lesssim 10^{-9}~{}.$ (55)

These constraints become even more restrictive for negative values of
$\tilde{a}$, while they also persist when both coefficients $\tilde{A}$ and
$\tilde{C}$ do not vanish. Within the range (55), the second derivative
$d^{2}V/d\theta^{2}$ always takes positive values, and there is no tachyonic
instability in the $\theta$-direction.

Figure 15: Left: the effective scalar potential at $\theta=0$ in the model
(54). The parameter $A$ is chosen to be $0$ (blue), $10^{-9}$ (yellow),
$10^{-8}$ (green) and $10^{-7}$ (red), with $C=0$ and $a=1$ being fixed.
Right: the parameter $C$ is chosen to be $0$ (blue), $10^{-9}$ (yellow),
$10^{-8}$ (green) and $10^{-7}$ (red), with $A=0$. The mass parameter $M$ is
fixed by $M=10^{-5}$.

##  Appendix B Exponential $g$-function

Here we consider an exponential-type $g$-function instead,

$\displaystyle g(Z)=C+Ae^{-aZ},$ (56)

where all parameters are real. First, in the case $C=0$, we derive Fig. 16. On
the top of Fig. 16, the inflation trajectory for small $\varphi$ around the
minimum is destroyed by the $A$-term in Eq. (56), depending upon the sign of
$a$. However, this can be cured when we include the parameter $C$ and tune it
accordingly. This is contrast to the case in Appendix A with the function
$\tilde{g}$ defined by Eq. (54), where both parameters $\tilde{A}$ and
$\tilde{C}$ affect the potential at large field values of $\varphi$, and the
destabilization of the potential cannot be cured by adjusting $\tilde{C}$.

In the middle of Fig. 16 we show the results of turning on the coefficient
$C$. As one can see in Fig. 16, a non-vanishing $C$ can uplift (or pull down)
the potential and realize a Minkowski vacuum. Finally, at the bottom of Fig.
16, we show the values of $d^{2}V/d\theta^{2}$ at $\theta=0$ for the same
parameter sets. We find that the $\theta$-direction is stable but these models
are not suitable for PBH production.

Figure 16: Top left: the effective scalar potential at $\theta=0$ with $a=1$
and $C=0$. The parameter $A$ is chosen as $0$ (blue), $10^{-5}$ (yellow),
$10^{-4}$ (green) and $10^{-3}$ (red). Top right: the same as in the left
figure but with $a=-1$. Middle left: the effective scalar potential at
$\theta=0$, with $a=1$ and $C$ being tuned to realize a Minkowski vacuum. The
parameters are chosen as $(A,C)=(10^{-5},-10^{-5})$,
$(A,C)=(10^{-4},-7.6\times 10^{-5})$ and $(A,C)=(10^{-3},-5.4\times 10^{-4})$.
Middle right: the same as in the left figure but with $a=-1$. The parameters
are chosen as $(A,C)=(10^{-5},-9\times 10^{-6})$, $(A,C)=(10^{-4},-1.5\times
10^{-4})$ and $(A,C)=(10^{-3},-2.1\times 10^{-3})$. Bottom left: the values of
the second derivative $d^{2}V/d\theta^{2}$ at $\theta=0$ with $a=1$ and the
same parameter set as in the middle-left figure. Bottom right: the same as in
the left figure but with $a=-1$. The mass parameter $M$ is fixed by
$M=10^{-5}$.

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