# The Superpotential Method and the Amplification of Inflationary
Perturbations

Alexander Y. Kamenshchik<EMAIL_ADDRESS>L.D. Landau
Institute for Theoretical Physics of the Russian Academy of Sciences
119334 Moscow, Russia  Dipartimento di Fisica e Astronomia and INFN, Via
Irnerio 46, 40126 Bologna, Italy  Ekaterina O. Pozdeeva pozdeeva@www-
hep.sinp.msu.ru Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow
State University, Leninskie Gory 1, 119991 Moscow, Russia  Augustin Tribolet
<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia and INFN,
Via Irnerio 46, 40126 Bologna, Italy  Institut d’Astrophysique et de
Géophysique, University of Liège, Allée du 6 août 19C, 4000 Liège, Belgium
Alessandro Tronconi<EMAIL_ADDRESS>Dipartimento di Fisica e
Astronomia and INFN, Via Irnerio 46, 40126 Bologna, Italy  Giovanni Venturi
<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia and INFN, Via
Irnerio 46, 40126 Bologna, Italy  Sergey Yu. Vernov<EMAIL_ADDRESS>Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Leninskie Gory 1, 119991 Moscow, Russia

###### Abstract

The superpotential method is a reconstruction technique which has proven
useful to build exact cosmological solutions. We here employ the
superpotential method in order to reconstruct the features necessary for the
inflaton potential to drive inflation and lead to the amplification of the
curvature perturbations. Such an amplification, at wavelengths shorter than
those observed in the CMB, is necessary in order to have a significant
formation of Primordial Black Holes after inflation ends. The technique is
applied to the cases of a minimally coupled inflaton, to the non-minimal
coupling case and to $f(R)$ theories of gravity. For such theories, a model
dependent analysis of the features leading to the scalar spectrum enhancement
is also presented.

##  1 Introduction

The hypothesis that black holes of primordial origin (PBHs) may constitute a
relevant fraction of the energy density in our Universe was introduced more
than fifty years ago in two seminal papers [1, 2]. Moreover, the direct
detections of gravitational waves (GW) by Ligo/Virgo [3] also attracts much
attention to PBHs [4, 5, 6, 7].

The observed GWs originate from the coalescence of astrophysical objects with
masses incompatible with our present knowledge of stellar collapse and are
therefore, perhaps, of primordial origin [4, 7]. Recently, the possible
existence of a large number of PBHs has therefore been reconsidered and
related to the Dark Matter content [5, 7, 8].

PBHs may be formed from the gravitational collapse of primordial density
inhomogeneities generated during inflation [9, 10, 11, 12, 13] (see also the
review [6]). Their abundance and their mass can then be related to the
features of the inflationary spectrum of curvature perturbations. A large
amplification of the amplitude of the inflationary spectrum at scales shorter
than those probed by CMB is necessary in order for perturbations to collapse
and form PBHs during the radiation domination era. A very finely tuned
inflationary potential is needed in order to achieve the desired
amplification. This tuning imposes quite severe constraints on the inflaton
evolution and, from a technical point of view, building models of inflation
leading to the desired scalar spectrum may be complicated. The conditions for
PBH formation can be created both in one-field and in two-field inflationary
models [14]. In this paper, we consider one-field models.

Scalar perturbations may then seed gravitational waves. This effect is tiny at
the scales probed by CMB, whose amplitude is very small ($\sim 10^{-9}$), but
may be relevant if some region of the inflationary spectrum is amplified by
several orders of magnitude. Thus, PBHs formation may be accompanied by a
stochastic GW background within a frequency range of $\sim{\rm nHz}$ and
related to the mass of the forming PBHs. A stochastic background of GWs in
that range has been recently observed by NANOGrav and other pulsar timing
array (PTA) collaborations. In the next few years increasingly precise
estimates of the spectrum of such GWs will possibly shed light on the
inflationary era at scales nowadays not accessible.

The scope of this article is to analytically reconstruct the inflationary
evolution leading to an amplification of the curvature perturbations. The
reconstruction procedure employs, whenever possible, the superpotential
technique [15, 16, 17, 18] and is applied to a set of modified gravity
theories [19, 20]. The aim of this paper is not to build a complete
inflationary potential but to find the general analytic features of
inflationary models leading to an amplification. The corresponding spectrum of
scalar perturbations is then estimated.

The article is organized as follows: in Section 2, we discuss the general
features needed in order to obtain amplification. In Section 3, we introduce
the superpotential method, which is then applied to single field models with
the inflaton minimally and non-minimally coupled to gravity. The case of
minimally coupled inflaton is considered in Section 4. In Section 5, we
investigate models with non-minimal coupling, in particular induced gravity
models. In Sections 6 and 7, we discuss $f(R)$ theories and finally, in
Section 8, we illustrate our conclusions.

##  2 Amplification of the spectrum

The constant-roll regime (CR) is a particular inflationary evolution wherein
slow-roll (SR) conditions are violated [21, 22, 23, 24]. In such a regime, one
still has $\epsilon_{1}<1$ corresponding to an accelerated expansion of the
Universe but in contrast with the SR regime the resulting inflationary spectra
can be an increasing function of the wavenumber $k$. This normally does not
occur for the SR inflation when the resulting spectral amplitude for any given
mode $k$ is proportional to $H^{2}$ evaluated at the time of the horizon
crossing $k/(aH)\sim 1$. Since the Hubble parameter $H$ decreases in time, the
amplitude is a (slowly) decreasing function of $k$.

In one-field minimally coupled models of inflation with $\phi(t)$ being the
homogeneous inflaton, the homogeneous Klein–Gordon equation for the constant
rolling inflaton has the form

$\ddot{\phi}-\beta H\dot{\phi}=0$ (1)

or equivalently

$\ddot{H}-2\beta H\dot{H}=0\,,$ (2)

where the dot denotes the differentiation with respect to the cosmic time $t$
and the Hubble parameter $H\equiv\dot{a}/a$, $a(t)$ is the FRW scale factor
defined by

${\rm d}s^{2}={\rm d}t^{2}-a^{2}(t){\rm d}\vec{x}^{2}.$ (3)

Slow-roll (SR) parameters hierarchy is used in the context of inflation as a
replacement of the homogeneous degrees of freedom $H$ and $\phi$ (inflaton).
The time variation of such parameters divided by $H$ is negligible if they are
much smaller than unity. Therefore, the adoption of the SR parameters instead
of $H$, $\phi$ and their derivatives is useful to obtain an approximate form
of the homogeneous dynamical equations. Moreover, the spectral indices of the
inflationary spectra can be expressed in terms of the SR parameter. Throughout
this article, we shall adopt two different hierarchies of the SR parameters.
The Hubble flow function hierarchy is recursively defined as

$\epsilon_{0}=\frac{H_{0}}{H},\;\epsilon_{i+1}=\frac{\dot{\epsilon_{i}}}{H\epsilon_{i}}$
(4)

and the scalar field flow function hierarchy is

$\delta_{0}=\frac{\phi}{\phi_{0}},\;\delta_{i+1}=\frac{\dot{\delta_{i}}}{H\delta_{i}}\,,$
(5)

where $H_{0}$ and $\phi_{0}$ are arbitrary constants.

The CR equation (2) may be rewritten in terms of the SR parameters as

$2\epsilon_{1}^{2}-\epsilon_{1}\epsilon_{2}+2\beta\epsilon_{1}=0\stackrel{{\scriptstyle\epsilon_{1}\neq
0}}{{\longrightarrow}}2\epsilon_{1}-\epsilon_{2}-2\beta=0,$ (6)

or equivalently

$\delta_{1}\left(\delta_{2}+\delta_{1}-\epsilon_{1}\right)-\beta\delta_{1}=0\stackrel{{\scriptstyle\delta_{1}\neq
0}}{{\longrightarrow}}\delta_{2}+\delta_{1}-\epsilon_{1}-\beta=0,$ (7)

and has a constant de Sitter solution ($\epsilon_{1}=0$) and a time-dependent
solution with $\epsilon_{1}\rightarrow 0$ and $\epsilon_{2}\rightarrow-2\beta$
in the $t\rightarrow\infty$ limit. Let us note that the second solution
describes the evolution approaching the de Sitter solution and the $-2\beta$
limit for $\epsilon_{2}$ is a consequence of the presence of the de Sitter
solution. Indeed (as we already discussed in [25]) if an inflaton-gravity
system approaches a different attractor with
$\epsilon_{1}\rightarrow\epsilon_{1,\infty}\neq 0$, then from the definition
of $\epsilon_{2}$ in (4), one has $\epsilon_{2}\rightarrow 0$. The existence
of a de Sitter solution is a necessary but not sufficient condition in order
to have such a non-trivial limit for $\epsilon_{2}$. Similarly, one needs
$\delta_{1}\rightarrow 0$ in order to have
$\delta_{2}\rightarrow\delta_{2,\infty}\neq 0$.

Differentiating (6), one finds

$2\epsilon_{1}-\epsilon_{3}=0,\;{\rm and}\quad
2\epsilon_{1}\epsilon_{2}-\epsilon_{3}\epsilon_{4}=0\Rightarrow\epsilon_{2}=\epsilon_{4}$
(8)

and then $\epsilon_{2i+1}=2\epsilon_{1}$, $\epsilon_{2i}=\epsilon_{2}$ with
$i$ a positive integer.

The existence of an Ultra Slow-Roll (USR) phase during inflation is frequently
considered in order to achieve the desired amplification of the inflationary
perturbations. Such a phase may occur when the minimally coupled inflaton
rolls close to an inflection point of its potential
($V_{,\phi}=V_{,\phi\phi}=0$) where its dynamics can be described by (1) with
$\beta=-3$.

##  3 Evolution Equations and the Superpotential Method

###  3.1 Evolution equations

Our overview will be general enough to include non-minimally coupled models,
however, through a frame transformation we can always recast our model in the
form of a minimally coupled inflaton-gravity system. Let us note that the
superpotential method is similar to the Hamilton–Jacobi approach [16, 26, 27]
since it uses the (homogeneous) scalar field values to parametrise the
evolution. Indeed, it can be applied to cosmological models in which the
evolution of the scalar field is monotonic. $f(R)$ gravity inflationary models
will also be discussed in this article.

$f(R)$ gravity inflationary models [28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40] as well as inflationary models with non-minimally coupled scalar
fields [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51] are actively studied.
Generally, the models considered can be described by the following action:

$S=\int
d^{4}x\sqrt{-g}\left[U(\sigma)R-\frac{c}{2}g^{\mu\nu}\sigma_{,\mu}\sigma_{,\nu}+V(\sigma)\right],$
(9)

where $U(\sigma)$ and $V(\sigma)$ are differentiable functions of the scalar
field $\sigma$, $g$ is the determinant of the metric tensor $g_{\mu\nu}$. We
shall only consider the case with $U(\sigma)>0$. The case with $c=0$
corresponds to $f(R)$ gravity, whereas the case with $c=1$ corresponds to a
standard scalar field. Indeed (see [35]), $f(R)$ gravity models with

$S_{R}=\int{\rm d}^{4}x\sqrt{-g}f(R).$ (10)

are equivalent to the following scalar-tensor gravity models

$S_{J}=\int d^{4}x\sqrt{-g}\left[f_{,\sigma}(R-\sigma)+f\right]~{},$ (11)

where $\sigma$ is a scalar field without kinetic term.

In Subsection 3.3 and Sections 4 and 5, we shall consider the case of a
standard scalar field and $c=1$. The $f(R)$ gravity models will be considered
in Sections 6 and 7.

For the evolution of a homogeneous scalar field on a spatially flat FRW
universe with the metric (3), the Einstein equations derived from the action
(9) are as follows

$6UH^{2}+6\dot{U}H=\frac{c}{2}{\dot{\sigma}}^{2}+V,$ (12)
$2U\left(2\dot{H}+3H^{2}\right)+4\dot{U}H+2\ddot{U}+\frac{c}{2}{\dot{\sigma}}^{2}-V=0,$
(13)

and the variation of (9) with respect to $\sigma$ gives the Klein–Gordon
equation,

$c\left(\ddot{\sigma}+3H\dot{\sigma}\right)+V_{,\sigma}=6\left(\dot{H}+2H^{2}\right)U_{,\sigma}\,,$
(14)

where the subscript ,σ indicates the derivative w.r.t. the scalar field
$\sigma$. Combining Eqs. (12) and (13), one obtains

$4U\dot{H}-2HU_{,\sigma}\dot{\sigma}+2\left(U_{,\sigma\sigma}{\dot{\sigma}}^{2}+U_{,\sigma}\ddot{\sigma}\right)+c{\dot{\sigma}}^{2}=0.$
(15)

###  3.2 Evolution of the inflationary spectra

In inflationary theories described by (9), the Mukhanov-Sasaki equation for
scalar inflationary perturbations has the form

$v^{\prime\prime}_{k}+\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0$
(16)

where the prime denotes the derivative w.r.t. conformal time and

$z\equiv\frac{\sqrt{c+3\frac{\dot{U}^{2}}{\dot{\sigma}^{2}U}}}{1+\frac{\dot{U}}{2HU}}\frac{a\dot{\sigma}}{H}.$
(17)

Correspondingly, the comoving curvature perturbation ${\mathcal{R}}_{k}$ is
related to $v_{k}$ by

${\mathcal{R}}_{k}=\frac{v_{k}}{z}$ (18)

and satisfies the following second order differential equation

$\zeta^{2}\frac{{\rm d}^{2}{\mathcal{R}}_{k}}{{\rm
d}\zeta^{2}}+\left[\frac{\epsilon_{1}\epsilon_{2}-2\left(1-\epsilon_{1}\right)\frac{{\rm
d}\ln z}{{\rm d}N}}{\left(1-\epsilon_{1}\right)^{2}}\right]\zeta\frac{{\rm
d}{\mathcal{R}}_{k}}{{\rm
d}\zeta}+\frac{\zeta^{2}}{\left(1-\epsilon_{1}\right)^{2}}{\mathcal{R}}_{k}=0\
,$ (19)

with $\zeta\equiv k/(aH)$ and the e-folding number $N\equiv\ln(a/a_{0})$. In
the long wavelength limit $\zeta\rightarrow 0$, Eq. (19) admits a constant
solution and, for constant SR parameters, a solution of the form $\zeta^{B}$
with

$B=1-\frac{\epsilon_{1}\epsilon_{2}-2\left(1-\epsilon_{1}\right)\frac{{\rm
d}\ln z}{{\rm d}N}}{\left(1-\epsilon_{1}\right)^{2}}.$ (20)

This second solution evolves as ${\mathcal{R}}_{k}\simeq{\rm
e}^{-B(1-\epsilon_{1})N}$ and increases or decreases depending on the sign of

$\Phi\equiv
B\left(1-\epsilon_{1}\right)=\frac{\left(1-\epsilon_{1}\right)^{2}-\epsilon_{1}\epsilon_{2}+2\left(1-\epsilon_{1}\right)\frac{{\rm
d}\ln z}{{\rm d}N}}{\left(1-\epsilon_{1}\right)}.$ (21)

The existence of an increasing solution ($\Phi<0$) is a sufficient condition
for the amplification of the inflationary spectra. Moreover if $\Phi>0$ a
positive spectral index $n_{s}-1$ signals an amplification. In such a case and
for the general action (9), one has

$n_{s}-1=2-\left[\frac{\epsilon_{1}\epsilon_{2}}{(1-\epsilon_{1})^{2}}-1\right]-\sqrt{\left[\frac{\epsilon_{1}\epsilon_{2}}{(1-\epsilon_{1})^{2}}-1\right]^{2}+4\frac{z_{,NN}+(1-\epsilon_{1})z_{,N}}{z(1-\epsilon_{1})^{2}}}$
(22)

where the comma indicates differentiation. Let us note that $z$ depends on the
choice of the inflationary action (9) and $z_{,N}/z$, $z_{,NN}/z$ are model
dependent functions of the SR parameters $\epsilon_{i}$’s and $\delta_{i}$’s.
For the inflationary models described by the action (9) and in the limit
$\epsilon_{2i+1}\rightarrow 0$ and $\delta_{2i+1}\rightarrow 0$ one finds

$\Phi=3+2\delta_{2}\quad{\rm and}\quad n_{s}-1=3-\sqrt{\Phi^{2}}=-2\delta_{2}$
(23)

and the relation between $\delta_{2}$ and the other SR parameters depends on
the specific model considered.

###  3.3 Superpotential method for models with non-minimal coupling

In this subsection, we describe the superpotential method in its variant
proposed for models with a non-minimally coupled scalar field [19] ($c=1$). In
the context of inflation, the superpotential method has been proposed in Ref.
[16] (see also Refs. [26, 52]).

If we introduce the function $G(\sigma)$ defined as

$\dot{\sigma}=G(\sigma)$ (24)

and $\sigma=\sigma(t)$ is monotonic, then by the chain rule one has

$\frac{{\rm d}}{{\rm d}t}=G(\sigma)\frac{{\rm d}}{{\rm d}\sigma}$ (25)

and Eq. (14) takes the form

$4UH_{,\sigma}+2(G_{,\sigma}-H)U_{,\sigma}+(2U_{,\sigma\sigma}+1)G=0.$ (26)

Equation (26) contains two unknown functions: $G(\sigma)$ and $H(\sigma)$.
Given one of them, the second one can be obtained as the solution of a linear
first order differential equation. If $G$ is given, then $H(\sigma)$ can be
obtained as

$H(\sigma)={}-\left[\int\limits^{\sigma}\frac{2G_{,\tilde{\sigma}}U_{,\tilde{\sigma}}+(2U_{,\tilde{\sigma}\tilde{\sigma}}+1)G}{4U^{3/2}}\,{\rm
d}\tilde{\sigma}+c_{0}\right]\\!\sqrt{U(\sigma)},$ (27)

where $c_{0}$ is an integration constant. Similarly, on using some given
$H(\sigma)$, we can find $G(\sigma)$ as:

$G(\sigma)=\left[\int\limits^{\sigma}\frac{U_{,\tilde{\sigma}}H-2UH_{,\tilde{\sigma}}}{U_{,\tilde{\sigma}}}e^{\Upsilon}{\rm
d}\tilde{\sigma}+\tilde{c}_{0}\right]e^{-\Upsilon(\sigma)},$ (28)

where

$\Upsilon(\sigma)\equiv\frac{1}{2}\int\limits^{\sigma}\frac{2U_{,\tilde{\sigma}\tilde{\sigma}}+1}{U_{,\tilde{\sigma}}}\,d\tilde{\sigma}$

and $\tilde{c}_{0}$ is an integration constant. Let us note that, in the
minimally coupled case ($U={\rm const}$), Eq. (28) no longer holds. Indeed,
Eq. (26) becomes an algebraic equation, which fixes $G$ in terms of
$H_{,\sigma}$. In such a case, for any function $H(\sigma)$ the reconstruction
procedure can be analytically fulfilled, and the potential can be obtained as
a function of $\sigma$. Once the three functions $H$, $G$ and $U$ of $\sigma$
are known, the corresponding potential $V(\sigma)$ can be easily obtained by
inverting the Friedmann equation (12):

$V(\sigma)=6UH^{2}+6U_{,\sigma}GH-\frac{1}{2}G^{2}=6H^{2}U\left(1+3\frac{U_{,\sigma}^{2}}{U}\right)-\frac{\left(G-6U_{,\sigma}H\right)^{2}}{2}.$
(29)

In principle, the time evolution of $\sigma(t)$ and $H(t)$ can also be found
by integrating Eq. (24), but this goes beyond the scope of this paper.

##  4 Minimally coupled inflaton

###  4.1 Inflationary model

Let us first consider the minimally coupled inflaton case111We conventionally
use $\phi$ to indicate the scalar field minimally coupled to gravity and
$\sigma$ to indicate the non-minimally coupled inflaton. with $U=U_{0}={\rm
M_{\rm P}}^{2}/2$. As we already observed, in such a case, Eq. (26) reduces to
the acceleration equation

$\dot{H}={}-\frac{\dot{\phi}^{2}}{2{\rm M_{\rm P}}^{2}}\Rightarrow G={}-2{\rm
M_{\rm P}}^{2}H_{,\phi}.$ (30)

If $H(\phi)$ is fixed, then $G(\phi)$ can be simply obtained by
differentiating $H$ as illustrated by (30). Therefore, one can
straightforwardly build exact potentials starting from any given Hubble
parameter expression $H(\phi)$. The expression (29) simplifies and one thus
obtains

$V(\phi)=3{\rm M_{\rm P}}^{2}H^{2}-2{\rm M_{\rm P}}^{4}H_{,\phi}^{2}.$ (31)

For example for

$H=h_{0}\cosh\left[A\left(\frac{\phi}{\phi_{0}}-1\right)\right]$ (32)

and $A^{2}=3\phi_{0}^{2}/(2{\rm M_{\rm P}}^{2})$ one recovers the constant
potential and USR evolution. The SR parameters can be calculated in terms of
$H(\phi)$ and its derivatives [26]. One has

$\epsilon_{1}={}-\frac{\dot{H}}{H^{2}}={}-G\frac{H_{,\phi}}{H^{2}}=2{\rm
M_{\rm
P}}^{2}\frac{H_{,\phi}^{2}}{H^{2}},\;\delta_{1}=\frac{\dot{\phi}}{H\phi}={}-2{\rm
M_{\rm P}}^{2}\frac{H_{,\phi}}{H\phi}$ (33)

and

$\epsilon_{2}=\frac{\dot{\epsilon_{1}}}{H\epsilon_{1}}={}-4{\rm M_{\rm
P}}^{2}\frac{H_{,\phi\phi}}{H}+2\epsilon_{1},\;\delta_{2}=\frac{\dot{\delta_{1}}}{H\delta_{1}}=\frac{\epsilon_{2}}{2}-\delta_{1}\,,$
(34)

where expressions (34) are very similar to the constant-roll conditions (6)
and (7). We are interested in solutions with the scalar field and the Hubble
parameter asymptotically approaching some fixed point:
$\phi\rightarrow\phi_{0}$, $H\rightarrow H_{0}$. These solutions are realised
if

$\lim_{\phi\rightarrow\phi_{0}}\dot{\phi}=0,\quad\Rightarrow\quad\lim_{\phi\rightarrow\phi_{0}}H_{,\phi}\rightarrow
0.$ (35)

If $H(\phi)$ is regular in $\phi_{0}\neq 0$, then its Taylor expansion around
$\phi_{0}$ has the following form:

$H(\phi)=\sum_{n=0}^{\infty}h_{n}\left(\frac{\phi}{\phi_{0}}-1\right)^{n}\equiv\sum_{n=0}^{\infty}h_{n}\left(\frac{\delta\phi}{\phi_{0}}\right)^{n}\,,$
(36)

and the condition (35) is satisfied if $h_{1}=0$. Correspondingly
$\delta_{1},\epsilon_{1}\rightarrow 0$ and, to the leading order, one has

$\dot{\phi}\simeq-4{\rm M_{\rm
P}}^{2}\frac{h_{2}}{\phi_{0}}\left(\frac{\phi}{\phi_{0}}-1\right).$ (37)

We observe from (37) that $\phi_{0}$ is an attractor if $h_{2}/\phi_{0}>0$.
Let us note that $h_{0}>0$ is a necessary condition in order to have inflation
close to the attractor. If such conditions are satisfied, then one can
calculate the SR parameters (34) in the $\phi\rightarrow\phi_{0}$ limit and
obtain

$\lim_{\phi\rightarrow\phi_{0}}\epsilon_{2}={}-8\frac{{\rm M_{\rm
P}}^{2}}{\phi_{0}^{2}}\frac{h_{2}}{h_{0}},\quad\lim_{\phi\rightarrow\phi_{0}}\delta_{2}={}-4\frac{{\rm
M_{\rm P}}^{2}}{\phi_{0}^{2}}\frac{h_{2}}{h_{0}}\equiv{}-\gamma.$ (38)

Let us note that $\epsilon_{2}$ and $\delta_{2}$ are different from zero if
$h_{2}\neq 0$. The coefficient $h_{2}$ must be positive in order for
$\epsilon_{1}$ and $\delta_{1}$ to be decreasing functions of time close to
$\phi_{0}$.
The expressions (38) seem to indicate that $\epsilon_{2}$ and $\delta_{2}$ may
take arbitrary values, depending on the choice of $h_{2}$ and $h_{0}$.
However, on calculating the potential around the attractor $\phi_{0}$ and
imposing the condition for the stability of the solutions, one finds that for
consistency $\gamma>-6$ and correspondingly $-6<\epsilon_{2}<0$.

On calculating the successive SR parameters and evaluating their value at
$\phi_{0}$, one finds a succession of zero and constant values in analogy with
the CR case, as illustrated in our preceding article [25].
Indeed, on integrating (37), one finds

$\delta_{0}=\frac{\phi}{\phi_{0}}\simeq 1+{\rm e}^{-4{\rm M_{\rm
P}}^{2}\phi_{0}^{-2}h_{2}t}\simeq 1+{\rm e}^{-\gamma N}\quad{\rm and}\quad
H\sim h_{0}+h_{2}{\rm e}^{-2\gamma N}.$ (39)

The succession of zero and constant values can be verified by explicitly
calculating the $\delta_{i}$’s and the $\epsilon_{i}$’s starting from (39).

Once the SR parameters close to the attractor are calculated, one can
calculate the parameter $\Phi$ and the spectral index $n_{s}-1$. One has

$\Phi=3+\epsilon_{2}=3-2\gamma$ (40)

and, when $\gamma>3/2$, $\Phi$ is negative, signalling the presence of an
increasing solution of the Mukhanov-Sasaki equation (and thus of an
amplification similarly to what occurs for the USR regime). In contrast, if
$\gamma<3/2$, then the increasing solution is not present and one can
calculate the spectral index

$n_{s}-1={}-\epsilon_{2}=2\gamma$ (41)

and observe that a blue-tilted spectrum (increasing) is obtained for
$0<\gamma<3/2$.

###  4.2 Stability

Let us now study the stability of the de Sitter solution ($\phi=\phi_{0}$)
w.r.t. homogeneous and isotropic perturbations. On expanding the homogeneous
Klein-Gordon equation around the scalar field fixed point $\phi_{0}$, to
linear order one finds

$3{\rm M_{\rm
P}}^{2}V_{,\phi}(\phi_{0})+\left(V(\phi_{0})\delta\phi_{,NN}+3V(\phi_{0})\delta\phi_{,N}+3{\rm
M_{\rm P}}^{2}V_{,\phi\phi}(\phi_{0})\delta\phi\right)=0$ (42)

where the first term is zero, $\delta\phi=\phi-\phi_{0}$, $V(\phi_{0})=3{\rm
M_{\rm P}}^{2}h_{0}^{2}$,

$V_{,\phi\phi}(\phi_{0})=12\frac{{\rm M_{\rm
P}}^{2}}{\phi_{0}^{2}}h_{0}h_{2}\left(1-\frac{4}{3}\frac{h_{2}}{h_{0}}\frac{{\rm
M_{\rm P}}^{2}}{\phi_{0}^{2}}\right)=\frac{V(\phi_{0})}{{\rm M_{\rm
P}}^{2}}\left(\gamma-\frac{\gamma^{2}}{3}\right).$ (43)

and $h_{0}$ and $h_{2}$ are the Taylor coefficients of the expansion of $H$
around $\phi_{0}$. Let us note that $V(\phi_{0})$ and
$V_{,\phi\phi}(\phi_{0})$ are invariant w.r.t. the substitution
$h_{2}\rightarrow-h_{2}+3\phi_{0}^{2}h_{0}/(4{\rm M_{\rm P}}^{2})$, where
$\phi_{0}^{2}$ is the scalar field attractor and the coefficient $3/(4{\rm
M_{\rm P}}^{2})$ is $-1/2$ the ratio between the coefficient in front $H^{2}$
and that in front of $H_{,\phi}^{2}$ in (31). This last result is quite
general and will be also applied to the non-minimal coupling case. This
invariance is simply related to the fact that for any given potential, on
linearizing the KG equation close to $\phi_{0}$, one finds two independent
solutions. Indeed, the linearized equation for $\delta\phi$ is easily solved
by

$\delta\phi=c_{1}{\rm e}^{-N\left(3-\gamma\right)}+c_{2}{\rm e}^{-\gamma N}.$
(44)

The solution (44) is stable if

$0<\gamma<3,$ (45)

and this result coincides with the stability condition for constant-roll
solutions. If the condition (45) is satisfied, $\delta\phi$ decreases in time
for any choice of the initial conditions $c_{1}$ and $c_{2}$. The second
exponential decreases slower than the first if $0<\gamma<3/2$ and dominates
close to the attractor. It may also dominate, for a non-negligible amount of
time, if $3/2<\gamma<3$ and $c_{1}/c_{2}\ll 1$. In such a case, one
essentially recovers the solution (39) and the expressions (40) and (41). The
first exponential dominates over the second close to the attractor if
$3/2<\gamma<3$ or, for a non-negligible period, if $0<\gamma<3/2$ and
$c_{2}/c_{1}\ll 1$. In this case, one finds

$\delta_{1}\sim 0,\quad\delta_{2}\sim-3+\gamma\quad{\rm
and}\quad\epsilon_{2}\sim 2\delta_{2}$ (46)

and correspondingly

$\Phi={}-3+2\gamma\quad{\rm and}\quad n_{s}-1=6-2\gamma.$ (47)

##  5 Non-minimally coupled inflaton

When the case with a non-minimal coupling is considered, the superpotential
method can be applied but some of the above simplifying assumptions do not
hold. Indeed, one cannot easily relate $G$ and $H_{,\sigma}$ and express
easily the potential and the SR parameters in terms of a single function and
its derivative. This occurs since (26) contains the first derivatives of both
$G$ and $H$ and for any given $H(\sigma)$, then $G(\sigma)$ must be obtained
by integration (not differentiation). In such a case, it is more convenient to
reformulate the superpotential method in terms of a different set of functions
as suggested in [53] (see also Ref. [54]). One may define an effective
potential

$V_{\rm eff}(\sigma)=\frac{U_{0}^{2}V(\sigma)}{U^{2}(\sigma)},$ (48)

the new, dimensionless, function

$A=\frac{U_{0}^{2}}{U^{2}}\left(1+\frac{3U_{,\sigma}^{2}}{U}\right)$ (49)

and

$Y=\sqrt{\frac{U_{0}}{U}}\left(H+\dot{\sigma}\frac{U_{,\sigma}}{2U}\right)\,.$
(50)

In terms of these functions, Eqs. (12) and (15) take the following form:

$6U_{0}Y^{2}=\frac{A}{2}{\dot{\sigma}}^{2}+V_{\rm eff},$ (51)

which has the same structure as the Friedmann equation (12) for a minimally
coupled inflaton. The corresponding acceleration equation is

$\dot{Y}={}-\frac{A\sqrt{U}}{4U_{0}^{3/2}}\,{\dot{\sigma}}^{2}\Leftrightarrow
Y_{,\sigma}=-\frac{A\sqrt{U}}{4U_{0}^{3/2}}\,G$ (52)

with $G=\dot{\sigma}$. Furthermore, the Klein–Gordon equation (14) takes the
following form:

$\ddot{\sigma}={}-3\sqrt{\frac{U}{U_{0}}}Y\dot{\sigma}-\frac{A_{,\sigma}}{2A}{\dot{\sigma}}^{2}-\frac{{V_{\rm
eff}}_{,\sigma}}{A}.$ (53)

Therefore, the condition ${V_{\rm eff}}_{,\sigma}=0$ clearly corresponds to de
Sitter solutions. It has been shown in Ref. [53] that such solutions are
stable for ${V_{\rm eff}}_{,\sigma\sigma}>0$ and unstable for ${V_{\rm
eff}}_{,\sigma\sigma}<0$, provided the condition $U(\sigma)>0$ is satisfied.
Let us note that any non-minimally coupled model can be transformed to the
corresponding EF through a conformal transformation of the metric and a field
redefinition. In the EF, the effective potential $V_{eff}$ is equal to the
potential $V$ and $Y(\sigma)$ is the Hubble parameter.

Once the functional dependence of $U(\sigma)$ and $Y(\sigma)$ is properly
chosen, one easily gets the corresponding effective potential

$V_{\rm
eff}(\sigma)=2U_{0}\left(3Y^{2}-\frac{4U_{0}^{2}}{A\,U}Y_{,\sigma}^{2}\right)$
(54)

and the associated inflaton potential

$V(\sigma)=2\frac{U^{2}}{U_{0}}\left(3Y^{2}-\frac{4U_{0}^{2}}{A\,U}Y_{,\sigma}^{2}\right)=6\frac{U^{2}}{U_{0}}Y^{2}-\frac{8U^{4}}{U_{0}\left(3U_{,\sigma}^{2}+U\right)}Y_{,\sigma}^{2}.$
(55)

Finally, one finds from (50) and (52) that

$H=\sqrt{\frac{U}{U_{0}}}\left(Y+\frac{2U\,U_{,\sigma}Y_{,\sigma}}{U+3U_{,\sigma}^{2}}\right),\quad
G={}-\frac{4U^{5/2}Y_{,\sigma}}{\sqrt{U_{0}}\left(U+3U_{,\sigma}^{2}\right)}$
(56)

Thus, on following the same procedure as used for the minimally coupled case,
one can find, for any given choice of $U(\sigma)$, the conditions for $Y$
leading to specific inflationary evolutions in the Jordan Frame.

###  5.1 Induced Gravity Models

To illustrate the proposed method let us work out in detail the Induced
Gravity (IG) case with

$U=\frac{\xi\sigma^{2}}{2}.$ (57)

The IG term naturally arises in inflationary model inspired by particle
theories, when quantum effects are taken into account [41, 42, 44, 46, 47, 48,
50, 51]. From Eq. (56), one has

$G={}-\alpha\sigma^{3}Y_{,\sigma},\qquad H=\beta\sigma
Y+\alpha\sigma^{2}Y_{,\sigma}\,,$ (58)

with $\beta\equiv\sqrt{\xi}/{\rm M_{\rm P}}$ and $\alpha\equiv
2\xi\beta/(1+6\xi)$. One can easily write the corresponding potential222 Using
$\sigma=\sigma_{0}\exp\left(\frac{\xi\phi}{\sqrt{2U_{0}(\xi+6\xi^{2})}}\right)$
and Eq. (48), one can calculate the potential in the EF in terms of $\phi$.

$V=\frac{\xi^{2}\sigma^{4}\left[3(1+6\xi)Y^{2}-2\xi\sigma^{2}Y_{,\sigma}^{2}\right]}{(1+6\xi){\rm
M_{\rm P}}^{2}},$ (59)

and calculate the SR parameters

$\delta_{1}\equiv\frac{\dot{\sigma}}{H\sigma}={}-\frac{\alpha\sigma
Y_{,\sigma}}{\beta Y+\alpha\sigma
Y_{,\sigma}}={}-\frac{2\xi\sigma\,Y_{,\sigma}}{2\xi\sigma\,Y_{,\sigma}+\left(6\xi+1\right)Y},$
(60)
$\delta_{2}\equiv\frac{\dot{\delta_{1}}}{H\delta_{1}}={}-\frac{\beta\,Y\,\delta_{1}+\sigma\,Y_{,\sigma}\left(2\alpha(1+\delta_{1})+\beta\delta_{1}\right)+\alpha(1+\delta_{1})\sigma^{2}\,Y_{\sigma\sigma}}{\beta\,Y+\alpha\sigma\,Y_{,\sigma}}$
(61)

in terms of $Y$ and its derivatives. If the inflaton evolves to some fixed
point $\sigma_{0}$ (without loss of generality we consider $\sigma_{0}>0$) and
we Taylor expand $Y$ around it

$Y=\sum_{n=0}^{\infty}y_{n}\left(\frac{\sigma}{\sigma_{0}}-1\right)^{n}\,,$
(62)

one may then easily observe that $\delta_{1}(\sigma_{0})=0$, provided the
first two coefficients of (62) satisfy the conditions $y_{0}\neq 0$ and
$y_{1}=0$. Then one has $\delta_{2}=-1-2\alpha/\beta\left(y_{2}/y_{0}\right)$
at $\sigma_{0}$. The conditions $\delta_{1}(\sigma_{0})=0$,
$\delta_{2}(\sigma_{0})\neq 0$ are CR-like conditions which may be associated
to an amplifying inflationary phase.

A very simple potential possibly leading to an amplification can be obtained
by choosing $Y=y_{0}+y_{2}\left(\sigma/\sigma_{0}-1\right)^{2}$. In such a
case,

$\sigma
Y_{,\sigma}=2y_{2}\frac{\sigma}{\sigma_{0}}\left(\frac{\sigma}{\sigma_{0}}-1\right)$
(63)

and the potential is

$V=\frac{\xi^{2}\sigma^{4}y_{2}^{2}\left\\{3(1+6\xi)\left[\frac{y_{0}}{y_{2}}+\left(\frac{\sigma}{\sigma_{0}}-1\right)^{2}\right]^{2}-8\xi\frac{\sigma^{2}}{\sigma_{0}^{2}}\left(\frac{\sigma}{\sigma_{0}}-1\right)^{2}\right\\}}{{\rm
M_{\rm P}}^{2}(1+6\xi)}.$ (64)

For IG, the expression for $\Phi$ close to the de Sitter attractor
($\delta_{1}\rightarrow 0$) is

$\Phi=3+2\delta_{2}=3-\frac{8\xi}{1+6\xi}\frac{y_{2}}{y_{0}}.$ (65)

If $y_{2}/y_{0}>3(1+6\xi)/(8\xi)$ the presence of an increasing solution of
the Mukhanov-Sasaki equation determines an amplification of the spectrum. In
contrast, when $y_{2}/y_{0}<3(1+6\xi)/(8\xi)$ the spectral index is

$n_{s}-1=-2\delta_{2}=\frac{8\xi}{1+6\xi}\frac{y_{2}}{y_{0}}$ (66)

and is then positive for $0<y_{2}/y_{0}<3(1+6\xi)/(8\xi)$. Let us note that
the discussion in subsection (3.3) about solutions and stability can be
straightforwardly applied to this case. In particular, concerning stability,
it can be deduced from that of the corresponding solutions in the EF.
Moreover, given the expression for the potential (59) and on substituting

$y_{2}\rightarrow-y_{2}+\frac{3\left(1+6\xi\right)}{4\xi}y_{0}$ (67)

one finds the second solutions close to $\sigma_{0}$. When such a solution
dominates, one has

$\Phi=-3+\frac{8\xi}{1+6\xi}\frac{y_{2}}{y_{0}}\quad{\rm and}\quad
n_{s}-1=6-\frac{8\xi}{1+6\xi}\frac{y_{2}}{y_{0}}.$ (68)

Let us also obtain the (general) behavior close to $\sigma_{0}\,$, where
$Y\sim y_{0}+y_{2}\left(\sigma/\sigma_{0}-1\right)^{2}$. One has

$\dot{\sigma}\simeq-2y_{2}\alpha\sigma_{0}^{2}\left(\frac{\sigma}{\sigma_{0}}-1\right),H\simeq\beta\sigma_{0}y_{0}\left[1+\left(1+2\frac{\alpha
y_{2}}{\beta y_{0}}\right)\left(\frac{\sigma}{\sigma_{0}}-1\right)\right]\,,$
(69)

where we observe that $H$ now approaches the constant value linearly on
varying the field $\sigma$ and $\delta_{0}$ has a “time” dependence similar to
(39).

##  6 $f(R)$ Theories

The approach illustrated in previous sections can also be applied to the
$f(R)$ theories, described by the action (10).

Considering flat FRW metric in absence of additional sources of matter, the
dynamical equations reduce to the Friedmann-like equation

$3FH^{2}=\frac{1}{2}\left(FR-f\right)-3H\dot{F}\,,$ (70)

where the dot denotes the differentiation w.r.t. the cosmic time and
$F(R)={\rm d}f/{\rm d}R$. For such theories, the Jordan to Einstein frame
transformation consists of two steps. Starting from the action (10) one
rewrites it as the action (9) with $c=0$. Then by the conformal transformation

$\tilde{g}_{\mu\nu}=\frac{2F(\sigma)}{{\rm M_{\rm P}}^{2}}g_{\mu\nu}$ (71)

and the following redefinition of the scalar field

$\phi=\sqrt{\frac{3}{2}}{\rm M_{\rm P}}\ln\frac{2F(\sigma)}{{\rm M_{\rm
P}}^{2}}\,,$ (72)

one obtains the EF action

$S_{E}=\int{\rm d}^{4}x\sqrt{-\tilde{g}}\left[\frac{{\rm M_{\rm
P}}^{2}}{2}\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\tilde{V}\right]$
(73)

where the tilde indicates physical quantities in the EF and $\phi$ is the
minimally coupled “inflaton”. Let us note that the theories described by
actions (10) and (11) are equivalent. If one calculates some observable in one
of the two “frames” the result does not change. In contrast, the EF action
(73) is not equivalent to (10). In particular from (72) it follows that $F(R)$
should be positive. In principle, one may proceed as illustrated for the non-
minimal coupling case discussed in the previous section. In this context,
however, our aim is to search for the possible form of $f(R)$ leading to an
amplification.

We note that once the potential or the Hubble parameter in the EF are
calculated, the reconstruction of $f(R)$ is not straightforward [35]. Indeed,
the function $f(R)$ can be expressed only parametrically in terms of the field
$\phi$ as follows [23]:

$R=\left(\frac{\sqrt{6}\tilde{V}_{,\phi}}{{\rm M_{\rm
P}}}+\frac{4\tilde{V}}{{\rm M_{\rm P}}^{2}}\right){\rm
e}^{\sqrt{\frac{2}{3}}\frac{\phi}{{\rm M_{\rm P}}}}$ (74)

and

$f=\frac{{\rm M_{\rm P}}^{2}}{2}\left(\frac{\sqrt{6}\tilde{V}_{,\phi}}{{\rm
M_{\rm P}}}+\frac{2\tilde{V}}{{\rm M_{\rm P}}^{2}}\right){\rm
e}^{2\sqrt{\frac{2}{3}}\frac{\phi}{{\rm M_{\rm P}}}}\,.$ (75)

Since the expression (74) cannot always be inverted, the analytic form of the
$f(R)$ function can only be reconstructed in particular cases. The problem of
the reconstruction of $f(R)$ models with exact solutions has already been
studied in Ref. [20], where $f(R)$ was obtained for an EF potential of the
form

$\tilde{V}=\frac{{\rm M_{\rm P}}^{2}}{2}\left(C_{2}{\rm
e}^{-2\sqrt{\frac{2}{3\,}}\frac{\phi}{{\rm M_{\rm P}}}}+C_{1}{\rm
e}^{{}-\sqrt{\frac{2}{3\,}}\frac{\phi}{{\rm M_{\rm P}}}}+C_{3}{\rm
e}^{\omega\sqrt{\frac{2}{3\,}}\frac{\phi}{{\rm M_{\rm P}}}}\right),$ (76)

where $C_{i}$ and $\omega$ are constants. This potential corresponds to

$f(R)=\frac{{\rm M_{\rm
P}}^{2}}{2}\left[C_{3}(\omega+1)\left(\frac{R-C_{1}}{C_{3}(\omega+2)}\right)^{\frac{\omega+2}{\omega+1}}-C_{2}\right].$
(77)

On following a procedure similar to that illustrated for the IG case, we can
calculate the Hubble flow function SR parameters in the JF in terms of the EF
Hubble parameter $\tilde{H}(\phi)$ and its derivatives. If $t$ is the cosmic
time in the JF and $\tilde{t}$ is the cosmic time in the EF, then from Eqs.
(71) and (72) it follows that

$\frac{d\tilde{t}}{dt}={\rm e}^{\frac{\phi}{\sqrt{6}{\rm M_{\rm P}}}}$ (78)

and one has

$H={\rm e}^{\frac{\phi}{\sqrt{6}{\rm M_{\rm
P}}}}\left(\tilde{H}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi}\right).$ (79)

On further differentiating, one finds expressions for the JF slow-roll
parameters:

$\epsilon_{1}=2{\rm M_{\rm
P}}^{2}\tilde{H}_{,\phi}\frac{\frac{\tilde{H}}{\sqrt{6}{\rm M_{\rm
P}}}+\frac{4}{3}\tilde{H}_{,\phi}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi\phi}}{\left(\tilde{H}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi}\right)^{2}}$ (80)

and

$\begin{split}\epsilon_{2}&={}-\frac{2{\rm M_{\rm
P}}^{2}\tilde{H}_{,\phi}}{\tilde{H}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi}}\times\\\
&\left(\frac{\tilde{H}_{,\phi\phi}}{\tilde{H}_{,\phi}}-2\frac{\tilde{H}_{,\phi}+\sqrt{\frac{2}{3}}{\rm
M_{\rm P}}\tilde{H}_{,\phi\phi}}{\tilde{H}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi}}+\frac{\frac{\tilde{H}_{,\phi}}{\sqrt{6}{\rm M_{\rm
P}}}+\frac{4}{3}\tilde{H}_{,\phi\phi}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi\phi\phi}}{\frac{\tilde{H}}{\sqrt{6}{\rm M_{\rm
P}}}+\frac{4}{3}\tilde{H}_{,\phi}+\sqrt{\frac{2}{3}}{\rm M_{\rm
P}}\tilde{H}_{,\phi\phi}}\right).\end{split}$ (81)

On using Eq. (31), one obtains the potential (76) from

$\tilde{H}(\phi)=h_{a}{\rm e}^{a\sqrt{\frac{3}{2}}\frac{\phi-\phi_{0}}{{\rm
M_{\rm P}}}}+h_{b}{\rm e}^{b\sqrt{\frac{3}{2}}\frac{\phi-\phi_{0}}{{\rm M_{\rm
P}}}},$ (82)

if the constants $a$ and $b$ take certain values (see the list of such values
in Ref. [20] and Table 1). Therefore, the reconstruction can be analytically
performed only in certain cases. For example, for $a={}-2/3$ and $b=0$, we get

$f(R)=\frac{{\rm M_{\rm
P}}^{2}}{2}\left(\frac{1}{24h_{b}^{2}}\,R^{2}-\frac{h_{a}}{h_{b}}R+\frac{8}{3}h_{a}^{2}\right)\,.$
(83)

Let $\phi_{0}$ be an attractor in the EF, then $\epsilon_{1}=0$ if
$\tilde{H}_{,\phi}(\phi_{0})=0$ and in order for $\epsilon_{2}\rightarrow{\rm
const}$ at $\phi_{0}$ one needs $\tilde{H}_{,\phi\phi}(\phi_{0})\neq 0$ and
$\tilde{H}(\phi_{0})>0$. On imposing such conditions on the expression (82),
one obtains

$\tilde{H}_{,\phi}(\phi_{0})=\sqrt{\frac{3}{2}}\left(\frac{a\,h_{a}}{{\rm
M_{\rm P}}}+\frac{b\,h_{b}}{{\rm M_{\rm P}}}\right)=0\Rightarrow
h_{b}={}-\frac{a}{b}h_{a}$ (84)

and

$\tilde{H}(\phi_{0})=\frac{\left(b-a\right)h_{a}}{b},\;\tilde{H}_{,\phi\phi}(\phi_{0})=\frac{3a\left(a-b\right)\,h_{a}}{2{\rm
M_{\rm P}}^{2}}.$ (85)

Since we are interested in solutions where $\phi_{0}$ is an attractor, then
close to $\phi_{0}$

$\tilde{G}(\phi)\equiv\frac{{\rm d}\phi}{{\rm
d}\tilde{t}}\sim-\tilde{\alpha}(\phi-\phi_{0})$ (86)

with $\tilde{\alpha}>0$. From (30) one then has

$\tilde{G}(\phi)\sim-2{\rm M_{\rm
P}}^{2}\tilde{H}_{,\phi\phi}(\phi_{0})\left(\phi-\phi_{0}\right)=3ab\tilde{H}(\phi_{0})(\phi-\phi_{0})$
(87)

and $\tilde{\alpha}={}-3ab\tilde{H}(\phi_{0})$ is positive if $a$ and $b$ have
opposite signs (and we only consider solutions with a de Sitter attractor,
$\tilde{H}(\phi_{0})>0$). On summing up, the above requirement restricts the
solutions to those having $h_{a}>0$, $(b-a)/b>0$ and $a\,b<0$. Such solutions
are listed in Table 1 with $W_{a}\equiv
h_{a}\exp\left(-a\sqrt{3/2}\,\phi_{0}/{\rm M_{\rm P}}\right)$. Values of
$\epsilon_{2}$ in Table 1 are calculated in the limit $\epsilon_{1}\rightarrow
0$.

Table 1: Reconstructed $f(R)$ models $a$ | $b$ | $\epsilon_{2}^{(1)}$ | $\epsilon_{2}^{(2)}$ | $2f(R)/{\rm M_{\rm P}}^{2}$
---|---|---|---|---
$-\frac{2}{3}$ | $1$ | $-2$ | $-1$ | $20W_{a}^{2}\left(\frac{3R}{100}W_{a}^{2}\right)^{5/3}-\frac{10}{3}W_{a}^{2}$
$-1$ | $\frac{1}{3}$ | $-2$ | $-1$ | $96W_{a}^{2}\left(\frac{R}{144W_{a}^{2}}-\frac{1}{3}\right)^{3/2}$
$-\frac{1}{3}$ | $1$ | $-2$ | $-1$ | $\frac{32}{3}W_{a}^{2}\left(\frac{R}{16W_{a}^{2}}-\frac{1}{3}\right)^{3/2}$
$-\frac{7}{3}$ | $1$ | $-7$ | $-$ | $160W_{a}^{2}\left(\frac{3R}{400W_{a}^{2}}\right)^{5/6}-\frac{280}{3}W_{a}^{2}$
$-\frac{5}{3}$ | $1$ | $-5$ | $-$ | $\frac{128}{3}W_{a}^{2}\left(\frac{R}{32W_{a}^{2}}-\frac{5}{3}\right)^{3/4}$

##  7 de Sitter solutions in $f(R)$ theories

###  7.1 Stability

Given the limitations for applying the reconstruction method to $f(R)$
theories, we henceforth adopt a less general approach compared with what we
illustrated so far. Such an approach consists in studying some of the specific
models discussed in the literature and verifying for each of them the
existence of a de Sitter attractor and the existence of CR-like solutions
leading to an enhancement of the primordial spectra. Indeed, a plethora of
$f(R)$ theories has been proposed in order to explain inflation or the present
cosmic acceleration and the existence of a de Sitter solution is crucial in
order to possibly have the amplification. In particular, let the attractor be
$R=R_{0}>0$. The existence of such an attractor is consistent with Eq. (70),
which can be recast in the form

$1=\left(2-\epsilon_{1}\right)\frac{n_{f}-1}{n_{f}}-\frac{\dot{R}}{HR}n_{F}\,,\quad{\rm
where}\quad n_{g}\equiv\frac{R}{g}\frac{{\rm d}g}{{\rm d}R},$ (88)

hence,

$n_{f}(R_{0})=2.$ (89)

From Eq. (88) and the expression

$\dot{R}={}-6\epsilon_{1}H^{3}\left[\epsilon_{2}+2\,(2-\epsilon_{1})\right],$

it follows that

$\epsilon_{2}=\left(2-\epsilon_{1}\right)\left(\frac{n_{f}\left(\epsilon_{1}-1\right)+2-\epsilon_{1}}{\epsilon_{1}\,n_{F}\,n_{f}}-2\right).$
(90)

The stability of the solution w.r.t. homogeneous perturbations can be studied
[55] starting from the KG-like equation

$\ddot{F}+3H\dot{F}+\frac{2f-RF}{3}=0$ (91)

which can be easily obtained by differentiating (88). In Eq. (91), analogously
to the minimally coupled case, $F$ plays the role of the field and the last
term is the potential gradient $V_{,F}$. The stability of a de Sitter solution
(which indeed can be found on solving for $V_{,F}=0$) is equivalent to the
condition $V_{,FF}>0$. This last conditions becomes

$V_{,FF}=\frac{{\rm d}R}{{\rm d}F}\frac{{\rm d}V_{,F}}{{\rm
d}R}=R\frac{1-n_{F}}{3n_{F}}>0\Rightarrow 0<\left.n_{F}\right|_{R=R_{0}}<1\,,$
(92)

provided $R_{0}>0$. We note that $\epsilon_{1}=2-R/(6H^{2})$, hence, $R>0$
during inflation.

The stability of de Sitter solutions can be also studied by using the
equivalent scalar-tensor action (11). It has been shown that a de Sitter
solution corresponds to the extrema of the following potential

$W\equiv\frac{RF-f}{F^{2}},$ (93)

If we only consider models that can be transformed to the EF model with a
standard scalar field, hence assuming $F(R)>0$, de Sitter solutions are the
solutions of

$W_{,R}=\frac{(2f-RF)F_{,R}}{F^{3}}=0$ (94)

and the stable de Sitter solutions correspond to minima of $W$ (whereas
unstable ones correspond to maxima [56]). It can be easily shown that this
stability condition is equivalent to Eq. (92).

In order to calculate the second SR parameter $\epsilon_{2}$ we consider
expression (90) in the limit $\epsilon_{1}\rightarrow 0$ and $n_{f}\rightarrow
2$ and we get the equation

$\epsilon_{2}^{2}+3\epsilon_{2}-4\left(1-n_{F}^{-1}\right)=0\,,$ (95)

that has the following two solutions for $\epsilon_{2}$:

$\epsilon_{2}^{(1,2)}={}-\frac{1}{2}\left(3\pm\sqrt{25-\frac{16}{n_{F}}\,}\,\right)\,.$
(96)

We observe that, provided the stability condition (92) is satisfied, the limit
for $\epsilon_{2}$ exists only when $n_{F}\geq 16/25$. In particular, one has
that $-3<\epsilon_{2}^{(1)}<-3/2$ and $-3/2<\epsilon_{2}^{(2)}<0$.
If $0<n_{F}<16/25$, the solutions are stable, $\epsilon_{1}\rightarrow 0$ and
$\epsilon_{2}$ has an oscillatory behavior with a constant amplitude. Even if
this behavior may have interesting consequences on the shape of the
inflationary spectra, we shall not discuss it further in the present article.

If the stability condition is not satisfied, one still has a solution with
decreasing $\epsilon_{1}$ corresponding to $\epsilon_{2}^{(1)}$. In
particular, for $n_{F}>1$ one finds $-3<\epsilon_{2}^{(1)}<-4$ and for
$n_{F}<0$ then $-\infty<\epsilon_{2}^{(1)}<-4$.

Let us finally note that, for large times and close to the attractor one has

$\Phi^{(1,2)}=3+2\epsilon_{2}^{(1,2)}=\mp\sqrt{25-\frac{16}{n_{F}}}$ (97)

and

$n_{s}^{(1,2)}-1=-2\epsilon_{2}^{(1,2)}=3\pm\sqrt{25-\frac{16}{n_{F}}}.$ (98)

Therefore, the Mukhanov-Sasaki equation has an increasing solution ($\Phi<0$)
if $\epsilon_{2}<-3/2$. If $-3/2<\epsilon_{2}<0$ increasing solutions not
exist and the spectral index is blue-tilted.

###  7.2 Scale invariant model

The easiest model for amplification is $f(R)=\alpha R^{2}$ with $n_{f}=2$ and
$n_{F}=1$ exactly. This implies that $R_{0}$ can take all positive values. For
such a case, one finds

$\epsilon_{2}=-3$ (99)

and $\Phi=-3<0$. The presence of an increasing solution for the curvature
perturbations determines the amplification of the spectrum. The stability
condition (92) is not satisfied analogously to what occurs for the USR
inflation and indeed the second solution to $\epsilon_{2}$ corresponds to
$\epsilon_{2}=0$. Let us note that the corresponding model in the EF has a
constant potential.

###  7.3 Second order polynomial in $R$

Let us now consider the model
$f(R)=\alpha_{0}M^{4}+\alpha_{1}M^{2}R+\alpha_{2}R^{2}$. For some values of
the constants $\alpha_{i}$, we get model (83) with an exact solution. The de
Sitter solution exists if $\alpha_{0}\alpha_{1}<0$ and is

$R_{0}=-2M^{2}\frac{\alpha_{0}}{\alpha_{1}}.$ (100)

Correspondingly, one finds

$n_{F}=\frac{1}{1-\frac{\alpha_{1}^{2}}{4\alpha_{0}\alpha_{2}}}$ (101)

and the stability condition is satisfied for
$0<\alpha_{1}^{2}/(4\alpha_{0}\alpha_{2})<1$.

###  7.4 n-order trinomial

Let us now consider the following, more general form of $f$:

$f(R)=M^{2}R+\alpha\,R^{2}+\beta\,\frac{R^{n}}{M^{2n}}.$ (102)

The de Sitter solution exists if $\beta(n-2)>0$ and is

$R_{0}=M^{2}\left[\frac{1}{\beta(n-2)}\right]^{1/(n-1)}.$ (103)

Correspondingly, one obtains

$n_{F}=\frac{1+\frac{n}{2}\nu}{1+\nu}\,,\quad{\rm
where}\quad\nu=\frac{M^{2}(n-1)}{R_{0}\alpha(n-2)}\,.$ (104)

Stability of the de Sitter solutions in models described by (102) have been
considered in [57]. Using condition (92), we distinguish 3 different cases. If
$0<n<2$, the stability condition requires $\nu>0$. If $n>2$, then the de
Sitter solution is stable at $-2/n<\nu<0$. Finally, for $n<0$, one needs
$0<\nu<-2/n$ for stability.

When $\alpha=0$, the general expression (102) becomes a binomial with
different powers of $R$. For such a case, one finds

$n_{F}=\frac{n}{2}$ (105)

and stable solutions therefore exist if $0<n<2$ that corresponds to $\beta<0$.
When $n>2$ and $n<0$ the solution (103) exists but is not stable. We also
remind that the condition $F(R_{0})>0$ should be satisfied. In order to have a
definite limit for $\epsilon_{2}$ one also needs $n\geq 32/25$ and the stable
solutions lead to

$\Phi=\mp\sqrt{25-\frac{32}{n}}\quad{\rm and}\quad
n_{s}-1=3\pm\sqrt{25-\frac{32}{n}}.$ (106)

Let us note that for $n>2$ and $n<0$ one solution which approaches the de
Sitter limit still exists and has

$\epsilon_{2}=-\frac{1}{2}\left(3+\sqrt{25-\frac{32}{n}}\right).$ (107)

##  8 Conclusions

Nowadays, primordial black holes are very attractive candidates to explain
dark matter [5, 7, 8] and the origin of the large masses of the BH binaries
observed by the LIGO/Virgo collaboration a few years ago. Furthermore, PBHs
may explain the spectrum of the stochastic GW background which has recently
been observed by NanoGRAV and other experiments and give further insights on
the physics of inflation.
Indeed, large primordial curvature perturbations can be generated during
inflation and collapse after inflation ends forming PBHs. Several mechanisms
to amplify the inflationary perturbations are currently studied. An
inflationary period with a phase of USR or CR at energy scales smaller than
those probed by CMB has been considered in diverse inflationary models. During
such a phase the homogeneous inflaton-gravity system rapidly evolves towards a
de Sitter attractor wherein the Mukhanov-Sasaki equation governing the
evolution of the curvature perturbation has a growing solution or may generate
a blue-tilted spectrum. In this paper, we discussed the general features of
some of the inflationary models which employ this mechanism to generate the
perturbations enhancement. In particular, whenever possible, we applied the
superpotential method to reconstruct the inflaton potential generating an USR
or a CR evolution. These inflaton potential features are found for a minimally
coupled inflaton field and in some modified gravity models involving a non-
minimal coupling and $f(R)$ theories. This article is the generalization of a
previous work [25] wherein a different technique was employed for the
reconstruction. We found that the superpotential method is a far more powerful
method in this context, predicting the shape of the inflaton potential close
to the attractor for a large class of models. The stability of the evolution
in such models is also studied. Moreover, reconstruction could also be applied
to $f(R)$ theories, but has a restricted range of applicability in such cases.
Therefore, in such a context, besides illustrating how reconstruction could
work, a set of $f(R)$ inflationary models were also considered and discussed
as a possible source of an amplification of the primordial spectra.

## References

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