#  Constant-roll $f(R)$ inflation compared with Cosmic Microwave Background
anisotropies and swampland criteria

Mehdi Shokri<EMAIL_ADDRESS>Department of Physics, Campus of
Bijar, University of Kurdistan, Bijar, Iran  Mohammad Reza Setare
<EMAIL_ADDRESS>Department of Physics, Campus of Bijar, University of
Kurdistan, Bijar, Iran  Salvatore Capozziello<EMAIL_ADDRESS>Dipartimento di Fisica E. Pancini", Universitá di Napoli Federico II", Via
Cinthia, I-80126, Napoli, Italy  Istituto Nazionale di Fisica Nucleare (INFN),
sez. di Napoli, Via Cinthia 9, I-80126 Napoli, Italy  Scuola Superiore
Meridionale, Largo S. Marcellino, I-80138, Napoli, Italy  Laboratory for
Theoretical Cosmology, Tomsk State University of Control Systems and
Radioelectronics (TUSUR), 634050 Tomsk, Russia  Jafar Sadeghi<EMAIL_ADDRESS>Department of Physics, University of Mazandaran, P. O. Box 47416-95447,
Babolsar, Iran  School of Physics, Damghan University, P. O. Box 3671641167,
Damghan, Iran

###### Abstract

Inflationary models derived from $f(R)$ gravity, where the scalaron rolls down
with a constant rate from the top to the minimum of the effective potential,
are considered. Specifically, we take into account three $f(R)$ models i.e.
Starobinsky $R^{2}$, $R^{2p}$ and the logarithmic corrected models. We compare
the inflationary parameters derived from the models with the observational
data of CMB anisotropies i.e. the Planck and Keck/array datasets in order to
find observational constraints on the parameters space. We find that although
our $f(R)$ constant-roll models for $\gamma=0$ show observationally acceptable
values of $r$, they do not predict favoured values of the spectral index. In
particular, we have $n_{s}>1$ for the Starobinsky $R^{2}$ and $R^{2p}$ models
and $0.996<n_{s}<0.999$ for logarithmic model. Finally, we study the models
from the point of view of Weak Gravity Conjecture adopting the swampland
criteria.

PACS: 04.50.$-$h; 98.80.Cq; 98.80.$-$K.
Keywords: Extended theories of gravity; Constant-roll inflation; Cosmic
Microwave Background.

##  I Introduction

Cosmological inflation is the most straightforward paradigm to describe the
early universe phenomena. In particular, the inflationary scalar and tensor
perturbations are unanimously considered responsible for structure formation
and primordial gravitational waves respectively Guth ; Sato:1980yn ;
Kazanas:1980tx ; Linde:1981my ; Albrecht:1982wi ; Lyth:1998xn . The simplest
inflationary model is based on a single scalar field, the so-called inflaton,
rolling down slowly from the top to the minimum point of potential under the
slow-roll approximation. Then, the inflationary epoch terminates when inflaton
decays to standard particles at the final step of inflation due to the
reheating process Kofman2 ; Shtanov . The single field models have been
studied broadly in inflationary literature and also have been compared with
the observational datasets. Consequently, some of them are restricted or even
ruled out martin  and some models are still viable with the recent high
precision observations staro ; barrow ; bezrukov ; kallosh1 . Moreover, this
group of models does not show any non-Gaussianity in their primordial spectrum
because of uncorrelated modes of the spectrum Chen . In such a case, if the
future observations predict non-Gaussianity in the perturbations spectrum,
then the single field models will be situated in an unstable status. Recently,
a new approach to inflation has been proposed in which inflaton is rolling
down with a constant rate martin2 ; Motohashi1 ; Motohashi2 . Hence,
$\ddot{\varphi}$ as a non-negligible term can be expressed as

$\ddot{\varphi}=\beta H\dot{\varphi}$ (1)

where $\beta=-(3+\alpha)$ and $\alpha$ is a non-zero parameter. For
$\alpha=-3$, the model is reduced to the standard slow-roll. Going beyond of
the slow-roll approximation, we can consider a ultra slow-roll regime where
the term of $\ddot{\varphi}$ is non-negligible in the Klein-Gordon equation as
$\ddot{\varphi}=3H\dot{\varphi}$. The ultra slow-role models show a limited
amount for the non-decaying mode of curvature perturbations Inoue  and also
predict a large $\eta$ but unable to solve the $\eta$ problem introduced in
supergravity for the hybrid inflationary models Kinney . Moreover, the ultra
solutions are located in the non-attractor phase of inflation but reveal a
scale-invariance of the scalar perturbation spectrum. Besides, the main
problem of ultra models is that the non-Gaussinaity consistency relation of
single field models is violated in the presence of ultra conditions through
super-Hubble evolution of the scalar perturbation Namjoo . The fast-roll
models are another class of inflationary models introduced to go beyond the
slow-roll approximation in which a fast-rolling stage is considered at the
start of inflation and can be connected to the standard slow-roll only after a
few e-folds Contaldi ; Lello .

Constant-roll approach has opened a new window to analyze cosmic inflation due
to a constant rate of rolling for inflaton. One can find a wide range of
inflationary models investigated in the context of constant-roll idea Odintsov
; Cicciarella ; Awad ; Anguelova ; Ito ; Yi ; Morse ; karam1 ; Ghersi ; Lin ;
Micu ; Oliveros ; Motohashi3 ; Kamali ; diego ; setare ; new2 , in particular,
models coming from $f(R)$ gravity, where new scalar fields are not required to
drive inflation and one only takes into account geometric extensions of the
Hilbert-Einstein action. In Nojiri , the authors considered two different
approaches. First, they studied the constant-roll evolution of a scalar-tensor
theory in the presence of $f(R)$ gravity and, as second approach, they applied
the constant-roll condition directly into $f(R)$ gravity. Generalizations of
the costant-roll condition in scalar-tensor gravity Oikonomou:2021yks , in
Gauss-Bonnet gravity Oikonomou:2020oil  and in connection to reheating in
$f(R)$ gravity Oikonomou:2017bjx  have been also developed. On the other hand,
logarithmic-corrected $R^{2}$ gravity in presence of Kalb-Ramond fields has
been considered in Elizalde:2018now . This model, i.e logarithmic $f(R)$ \+
Kalb-Ramond, seems to be consistent with the observable values of inflationary
parameters. Specifically, the authors consider both $\gamma=0$ and non-zero
$\gamma$. Finally, the constant slow-roll condition in Palatini formalism has
been taken into account in Antoniadis:2020dfq . In Motohashi9 , the authors
introduced the constant-roll condition in the Jordan frame, and by calculating
the potential of scalaron in the Einstein frame, they obtained the form of
$f(R)$ in the Jordan frame.

The main aim of the present manuscript is to find the observational
constraints from Cosmic Microwave Background (CMB) anisotropies on the
parameters space of some $f(R)$ models introduced in the context of the
constant-roll inflation. In practice, we use as CMB data, the Planck and the
BICEP2/Keck array data releases cmb ; bicep . Also, we compare the predictions
of the models with observations in order to find the effects of the new
approach. As a secondary purpose, we investigate the $f(R)$ constant-roll
inflationary models in view of swampland conditions since $f(R)$ models
contains some stringy-like corrections to GR and also $f(R)$ gravity can be
realized as a natural class of theories emerging from the string landscape
because of the existence of Noether symmetries in this class of extended
theories of gravity Gionti ; Benetti .

First, we study the Starobinsky $R^{2}$ model as the most well-known
inflationary model in $f(R)$ gravity containing a quadratic term coming from
higher-order curvature added to Ricci scalar in the Hilbert-Einstein action as
${\displaystyle f(R)=R+\frac{R^{2}}{6M^{2}}}$ staro . The Starobinsky model is
also well-known as one of the most successful inflationary models being in
good agreement with current observations. Hence, it is now considered as a
"target" model for several future CMB experiments as, for example, the Simons
Observatory Aguirre , CMB-S4 Abazajian , and the LiteBIRD satellite experiment
Suzuki . Also, $R^{2}$ model has been proposed as one of the possible
alternatives to the cosmological constant of the concordance $\Lambda$CDM
model a ; b ; c ; c ; d ; d ; e ; f ; g ; h ; i ; k . Despite the mentioned
achievements, the Starobinsky model predicts a very tiny tensor-to scalar
ratio $r\simeq 0.003$ for 60 e-folds which is out of the current and probably
future observations. The goal of these future experiments is therefore to
improve the experimental sensitivity to measure such a signal with enough
statistical significance with $\delta s<0.001$. Since the predicted value of
$r$ is the first approximation, we might consider some theoretical corrections
to the Starobinsky model in order to improve the obtained value. Therefore, we
focus on a generalized form of the Starobinsky inflation, the so-called
$R^{2p}$ model (with $p\approx 1$). These inflationary models were first
proposed by Schmidt ; Maeda  in the context of higher-derivative theories and
subsequently were applied to inflation providing a simple and elegant
generalization of the $R^{2}$ inflation Muller ; Felice ; Ringeval ; Trotta ;
new0 . Moreover, we consider the logarithmic corrected Starobinsky model
$f(R)=R+\lambda R^{2}+\upsilon R^{2}\ln R$ where logarithmic corrections come
from quantum gravity. Logarithmic $f(R)$ gravity can be considered a prototype
model with quantum corrections, able to describe primordial and current
accelerated expansions of the universe. Inflationary examples of logarithmic
$f(R)$ models can be found in Refs. Starobinsky3 ; Sotiriou ; Felice ; a1 ; a2
; a3 ; a4 ; a5 ; a6 ; a7 ; a8 ; a9 ; a11 ; a12 ; a15 ; a16 ; a17 ; a18 . For
other cosmological situations see a21 ; a22 ; a23 ; a25 ; a26 ; a27 ; a29 ;
a30 ; a32 ; a34 ; a35 .

The above discussion motivates us to arrange the paper as follows. In §II, we
introduce $f(R)$ gravity and its main properties in both Jordan and Einstein
frames with the connecting relations of parameters in the two frames. §III is
devoted to the study of the constant-roll inflation in $f(R)$ gravity. In §IV,
we focus on the $f(R)$ models and obtain the corresponding potentials and also
the inflationary parameters. In §V, we analyze the obtained results by
comparing with the observational datasets coming from the Planck and the
BICEP2/Keck array satellites. Also, we investigate the models from the
viewpoint of the Weak Gravity Conjecture using the swampland criteria. In §VI,
we conclude the analysis of the models and draw our future outlooks.

##  II $f(R)$ Gravity in Jordan and Eistein frame

Let us start with the general form of $f(R)$ gravity action

$S=\int{d^{4}x\sqrt{-g}\frac{f(R)}{2}}+\int{d^{4}x\mathcal{L}_{M}(g_{\mu\nu},\Psi_{M})}$
(2)

where $g$ is the determinant of the metric $g_{\mu\nu}$,
$R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar as gravitational sector and
$\mathcal{L}_{M}$ is the Lagrangian of matter fields $\Psi_{M}$ filling the
universe as perfect fluid Report ; Mantica ; CANTATA . Also, here we suppose
$\kappa^{2}\equiv 8\pi G=1$. By varying the action (2) respect to the metric,
the Einstein field equation is given by

$\mathcal{G}_{\mu\nu}\equiv
FR_{\mu\nu}-\frac{g_{\mu\nu}f}{2}-\nabla_{\mu}\nabla_{\nu}F+g_{\mu\nu}\Box
F=T_{\mu\nu}^{m}$ (3)

where $F$, d’Alembert $\Box$ and the energy-momentum tensor of matter fields
$T_{\mu\nu}^{m}$ are expressed by

$F\equiv\frac{df(R)}{dR},\hskip
28.45274pt\Box=\frac{1}{\sqrt{-g}}\partial_{\nu}[\sqrt{-g}g^{\mu\nu}\partial_{\mu}],\hskip
28.45274ptT_{\mu\nu}^{m}=-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{M}}{\delta
g_{\mu\nu}}.$ (4)

The conservation law of the energy-momentum tensor
$\nabla^{\mu}T_{\mu\nu}^{m}=0$ is valid when
$\nabla^{\mu}\mathcal{G}_{\mu\nu}=0$. Also, the field equation (3) can be
rewritten in terms of the Einstein tensor $G_{\mu\nu}$ as

$G_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=T_{\mu\nu}^{m}+T_{\mu\nu}^{c}$ (5)

where $T_{\mu\nu}^{c}$ including the terms coming from $f(R)$ modifications
takes the following form

$T_{\mu\nu}^{c}\equiv(1-F)R_{\mu\nu}+\frac{1}{2}(f-R)g_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}F-g_{\mu\nu}\Box
F$ (6)

and the conservation law of the energy-momentum tensor
$\nabla^{\mu}T_{\mu\nu}^{m}=0$ is valid as a consequence of the Bianchi
identities $\nabla^{\mu}G_{\mu\nu}=0$ if $\nabla^{\mu}T_{\mu\nu}^{c}=0$.
Obviously, in the case of $f(R)=R$, both approaches reduce to Einstein
gravity. In the following we consider a spatially flat universe described by a
Friedmann-Robertson-Walker (FRW) metric as

$ds^{2}=-dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2})$ (7)

where $t$ and $a$ depict to cosmic time and scale factor, respectively. By
using the above metric and the definition of the energy-momentum tensor of
perfect fluid as gravitational and matter sectors of the Eq. (3), we obtain
the dynamical equations as

$3FH^{2}=\frac{(FR-f)}{2}-3H\dot{F}+\rho_{m},\hskip
28.45274pt2F\dot{H}=-\ddot{F}+H\dot{F}-(\rho_{m}+P_{m})$ (8)

where $H\equiv\frac{\dot{a}}{a}$ is Hubble parameter and dot represents time
derivation. Moreover, $\rho_{m}$ and $P_{m}$ are energy density and pressure
of the perfect fluid. In the rest of the paper, we confine ourselves to
inflationary analysis in $f(R)$ gravity by removing the role of matter fields
in the main action (2) since we don’t require to consider any type of matter
in order to push inflation. Hence, the inflationary action in $f(R)$ gravity
takes the simple form

$S=\int{d^{4}x\sqrt{-g}\frac{f(R)}{2}}$ (9)

with the dynamical equations

$3FH^{2}=\frac{(FR-f)}{2}-3H\dot{F},\hskip
28.45274pt2F\dot{H}=-\ddot{F}+H\dot{F}.$ (10)

By using the conformal transformation as a useful mathematical tool, we can
move from Jordan frame (main frame) to Einstein frame in order to escape from
difficulties of $f(R)$ gravity. The metrics in the two frames are connected
with

$\hat{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}\quad\quad
with\quad\quad\Omega^{2}=F=e^{\sqrt{\frac{2}{3}}\varphi}.$ (11)

Hence, the form of action in the Einstein frame takes the standard form as

$S_{E}=\int{d^{4}x\bigg{(}\hat{R}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V(\varphi)}\bigg{)}$
(12)

where hat denotes to the parameters in the Einstein frame. Clearly, in the
Einstein frame, we deal with a scalar field $\varphi$ so that called scaleron
with the potential

$V(\varphi)=\frac{RF-f}{2F^{2}}.$ (13)

Conversely, the Ricci scalar and the function of $f(R)$ in the Jordan frame
can be connected to the potential in the Einstein frame by

$R=e^{\sqrt{\frac{2}{3}}\varphi}\bigg{(}\sqrt{6}\frac{\partial
V}{\partial\varphi}+4V\bigg{)},\quad\quad\quad
f(R)=e^{2\sqrt{\frac{2}{3}}\varphi}\bigg{(}\sqrt{6}\frac{\partial
V}{\partial\varphi}+2V\bigg{)}.$ (14)

Moreover, the dynamical equations in the Einstein frame are defined by

$\hat{H}^{2}=\frac{1}{3}\bigg{(}\frac{1}{2}(\frac{d\varphi}{d\hat{t}})^{2}+V(\varphi)\bigg{)},\hskip
28.45274pt\frac{d\hat{H}}{d\hat{t}}=-\frac{1}{2}(\frac{d\varphi}{d\hat{t}})^{2},\hskip
28.45274pt\frac{d^{2}\varphi}{d\hat{t}^{2}}+3\hat{H}(\frac{d\varphi}{d\hat{t}})+\frac{\partial
V}{\partial\varphi}=0.$ (15)

Also, the parameters in the two frames are connected by

$H=e^{\frac{\varphi}{\sqrt{6}}}\bigg{(}\hat{H}-\frac{1}{\sqrt{6}}\frac{d\varphi}{d\hat{t}}\bigg{)},\hskip
28.45274pt\frac{d\varphi}{dt}=e^{\frac{\varphi}{\sqrt{6}}}\frac{d\varphi}{d\hat{t}},\hskip
28.45274pt\frac{d^{2}\varphi}{dt^{2}}=e^{\sqrt{\frac{2}{3}}\varphi}\bigg{(}\frac{d^{2}\varphi}{d\hat{t}^{2}}+\frac{1}{\sqrt{6}}(\frac{d\varphi}{d\hat{t}})^{2}\bigg{)}.$
(16)

Note that the above expressions are obtained due to
$dt=e^{-\frac{\varphi}{\sqrt{6}}}d\hat{t}$ and
$a=e^{-\frac{\varphi}{\sqrt{6}}}\hat{a}$.

##  III Constant-roll $f(R)$ inflation

Deviations from slow-roll approximation can be found when we restrict our
attention to ultra slow-roll inflationary models in which $\alpha=0$ or even
fast-roll models in which a fast-rolling period is considered at the beginning
of inflation and then will be transited to the slow-roll stage just after a
few e-folds. As a new and more complete approach, we can introduce constant-
roll inflation in which inflaton rolls down with a constant rate from top of
the potential to the minimum point. This viewpoint has been proposed to
provide non-Gaussianity properties in the single field models since these
models do not predict any non-Gaussianity in the context of slow-roll
approximation. Recently, diverse inflationary models have been investigated in
the presence of the constant-roll condition. In this paper, we focus on some
inflationary models motivated by $f(R)$ gravity. Hence, we consider a natural
generalization of the constant-roll condition (1) in $f(R)$ gravity expressed
in the Jordan frame as

$\ddot{F}=\beta H\dot{F}$ (17)

which is reduced to the standard slow-roll approximation for $\beta=0$. Since
the inflationary potential of $f(R)$ is defined in the Einstein frame, we
follow the mechanism in the Einstein frame using the connecting relations
between two frames (16). By the definition of $F$ (11), the constant-roll
condition (17) in the Einstein frame is rewritten by

$\frac{d^{2}\varphi}{d\hat{t}^{2}}+\frac{3+\beta}{\sqrt{6}}\bigg{(}\frac{d\varphi}{d\hat{t}}\bigg{)}^{2}-\beta\hat{H}\frac{d\varphi}{d\hat{t}}=0.$
(18)

Then, by using the Einstein equation (15), the above equation takes the
following form

$\frac{d\hat{H}}{d\varphi}\bigg{(}\frac{d^{2}\hat{H}}{d\varphi^{2}}+\frac{3+\beta}{\sqrt{6}}\frac{d\hat{H}}{d\varphi}+\frac{\beta}{2}\hat{H}\bigg{)}=0$
(19)

which includes two separated cases. The first case
$\frac{d\hat{H}}{d\varphi}=0$ is associated to the case of $\hat{H}=const.$
which leads to $V=const.$ and then $f(R)=R-const.$ The second case is a second
order differential equation with a general solution

$\hat{H}(\varphi)=C\bigg{(}\gamma
F^{-\frac{3}{2}}+F^{-\frac{\beta}{2}}\bigg{)}$ (20)

and by using the Friedmann equation (15), the potential and the evolution of
the scaleron are obtained as

$V(\varphi)=\frac{3-\beta}{3}C^{2}\bigg{(}6\gamma
F^{-\frac{(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)},\quad\quad\quad\frac{d\varphi}{d\hat{t}}=-2\frac{d\hat{H}}{d\varphi}=\frac{2}{\sqrt{6}}\bigg{(}3\gamma
F^{\frac{-3}{2}}+\beta F^{-\frac{\beta}{2}}\bigg{)}$ (21)

where $F=e^{\sqrt{\frac{2}{3}}\varphi}$. The parameters $C$ and $\gamma$ are
the integration constants mass dimension 1 and dimensionless, respectively.
The amplitude of $C$ can be fixed by the CMB normalization and we follow the
paper in the unit where $C=1$. Also, the value of $\gamma$ can be normalized
by redefining $C$ and $\varphi$ so that it can be considered for three value
$+1,0,-1$. In ref. Motohashi9 , the authors claimed that only the parameters
region with $\beta=-0.02$ and $\gamma=-1$ are viable, cosmologically. In other
words, these parameters are situated in a region where inflaton shows an
attractor-like behavior. By combination of the potential (21) and the
connection relations (14), the Ricci scalar takes the following form

$R=(\beta-3)\bigg{(}2\gamma(\beta-1)e^{-\frac{(1+\beta)\varphi}{\sqrt{6}}}+\frac{2}{3}(\beta-2)(\beta+3)e^{\frac{2(1-\beta)\varphi}{\sqrt{6}}}\bigg{)}.$
(22)

Now, we can calculate the slow-roll parameters as

$\epsilon\equiv\frac{1}{2}\bigg{(}\frac{V^{\prime}}{V}\bigg{)}^{2}=\frac{(3+\beta)^{2}F^{\prime
2}\bigg{(}3\gamma F^{\frac{-(5+\beta)}{2}}+\beta
F^{-(\beta+1)}\bigg{)}^{2}}{2\bigg{(}6\gamma
F^{\frac{-(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)}^{2}},$ (23)
$\eta\equiv\frac{V^{\prime\prime}}{V}=\frac{-(3+\beta)\bigg{(}F^{\prime\prime}(3\gamma
F^{\frac{-(5+\beta)}{2}}+\beta F^{-(\beta+1)})-F^{\prime
2}(\frac{3\gamma(5+\beta)}{2}F^{-\frac{(7+\beta)}{2}}+\beta(1+\beta)F^{-(2+\beta)})\bigg{)}}{\bigg{(}6\gamma
F^{\frac{-(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)}},$ (24)
$\displaystyle\\!\zeta^{2}\equiv\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}}=\frac{(3+\beta)^{2}F^{\prime}\bigg{(}3\gamma
F^{\frac{-(5+\beta)}{2}}+\beta F^{-(\beta+1)}\bigg{)}}{\bigg{(}6\gamma
F^{\frac{-(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)}^{2}}\times$
$\displaystyle\\!\times\bigg{(}F^{\prime\prime\prime}(3\gamma
F^{\frac{-(5+\beta)}{2}}+\beta
F^{-(\beta+1)})-3F^{\prime\prime}F^{\prime}(\frac{3\gamma(5+\beta)}{2}F^{-\frac{(7+\beta)}{2}}+$
$\displaystyle\\!+\beta(1+\beta)F^{-(2+\beta)})+F^{\prime
3}(\frac{3\gamma(5+\beta)(7+\beta)}{4}F^{-\frac{(9+\beta)}{2}}+\beta(1+\beta)(2+\beta)F^{-(3+\beta)})\bigg{)}$
(25)

where prime implies to the derivation with respect to the scalar field
$\varphi$ and inflation ends when the condition $\epsilon=1$ or $\eta=1$ is
fulfilled. Also, we can apply the swampland conditions $\sqrt{2\epsilon}\geq
c$ and $|\eta|\leq-c^{\prime}$, first derived in Garg:2018reu  and then in K1
, for the $f(R)$ models by

$\frac{(3+\beta)F^{\prime}\bigg{(}3\gamma F^{\frac{-(5+\beta)}{2}}+\beta
F^{-(\beta+1)}\bigg{)}}{\bigg{(}6\gamma
F^{\frac{-(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)}}\geq c,$ (26)
$\Bigg{|}\frac{-(3+\beta)\bigg{(}F^{\prime\prime}(3\gamma
F^{\frac{-(5+\beta)}{2}}+\beta F^{-(\beta+1)})-F^{\prime
2}(\frac{3\gamma(5+\beta)}{2}F^{-\frac{(7+\beta)}{2}}+\beta(1+\beta)F^{-(2+\beta)})\bigg{)}}{\bigg{(}6\gamma
F^{\frac{-(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)}}\Bigg{|}\leq c^{\prime}$
(27)

where $c$ and $c^{\prime}$ are unit orders. Based on our purpose, we need to
calculate the number of e-folds in the Einstein frame using

$N\equiv\int^{\varphi_{i}}_{\varphi_{f}}{\frac{1}{\sqrt{2\epsilon}}}d\varphi=\int^{\varphi_{i}}_{\varphi_{f}}\frac{\bigg{(}6\gamma
F^{\frac{-(3+\beta)}{2}}+(3+\beta)F^{-\beta}\bigg{)}}{(3+\beta)F^{\prime}\bigg{(}3\gamma
F^{\frac{-(5+\beta)}{2}}+\beta F^{-(\beta+1)}\bigg{)}}d\varphi$ (28)

where the subscribes "$i$" and "$f$" denote the value of the scaleron field at
the beginning and the end of inflation, respectively. The spectral parameters
i.e. the first order of spectral index, the first order of running spectral
index and the tensor-to-scalar ratio can be obtained by

$n_{s}=1-6\epsilon+2\eta,\quad\quad\quad\alpha_{s}=\frac{dn_{s}}{d\ln
k}=16\epsilon\eta-24\epsilon^{2}-2\zeta^{2},\quad\quad\quad r=16\epsilon.$
(29)

##  IV $f(R)$ models

In this section, we shall examine the constant-roll inflation for some $f(R)$
models i.e. Starobinsky $R^{2}$ , $R^{2p}$ and logarithmic corrected models.

Figure 1: The potential, evolution of scaleron and Hubble parameter (32)
plotted versus $\varphi$ for $\gamma=+1,0,-1$ with $\beta=-0.02$ and
$\lambda=1$.

###  IV.1 Starobinsky $R^{2}$ model

First, we start with the Starobinsky $R^{2}$ model as the most successful
inflationary model which contains a quadratic term coming from higher-order
curvature given by

$f(R)=R+\lambda R^{2}$ (30)

where $\lambda$ is an arbitrary constant. By using the form of the Ricci
scalar (22), $F=\frac{df}{dR}$ for the Starobinsky model can be driven by

$F_{1}=1+2\lambda(\beta-3)\bigg{(}2\gamma(\beta-1)e^{-\frac{(1+\beta)\varphi}{\sqrt{6}}}+\frac{2}{3}(\beta-2)(\beta+3)e^{\frac{2(1-\beta)\varphi}{\sqrt{6}}}\bigg{)}.$
(31)

Now the Hubble parameter (20), the potential of scalar field and the evolution
of the scalar field (21) are given by

$\hat{H}_{1}(\varphi)=\bigg{(}\gamma
F_{1}^{-\frac{3}{2}}+F_{1}^{-\frac{\beta}{2}}\bigg{)},\quad
V_{1}(\varphi)=\frac{3-\beta}{3}\bigg{(}6\gamma
F_{1}^{-\frac{(3+\beta)}{2}}+(3+\beta)F_{1}^{-\beta}\bigg{)},\quad\frac{d\varphi_{1}}{d\hat{t}}=\frac{2}{\sqrt{6}}\bigg{(}3\gamma
F_{1}^{\frac{-3}{2}}+\beta F_{1}^{-\frac{\beta}{2}}\bigg{)}.$ (32)

Now let’s review the appropriate plots of the Starobinsky $R^{2}$ model. The
panel (a) of Fig. 1 presents the behaviour of scaleron potential (32) for
three values of $\gamma=+1,0,-1$ with $\beta=-0.02$ and $\lambda=1$. For
$\gamma=+1$ the potential rolls down to a minimum point and then tilts
upwards. For $\gamma=0$ the potential moves in a steady manner from negative
points of scaleron and then gradually shows an increasing behavior. For
$\gamma=-1$, the potential starts moving from the origin and tilts upward with
a decreasing slope. Besides this information, the panel (a) also reveals that
all three values of $\gamma$ act the same with a constant rate at the last
steps of inflation. The panels (b) and (c) of Fig. 1 show the phase diagram of
scaleron and the behaviour of the Hubble parameter (32) for the values
$\gamma=+1,0,-1$ with $\beta=-0.02$ and $\lambda=1$, respectively. Notice that
in the rest of paper, we study the inflationary paradigm in the case of
$\gamma=0$. Now let’s find the slow-roll parameters (23 \- 25) of the
Starobinsky $R^{2}$ model for the case of $\gamma=0$ by

$\epsilon=\frac{\beta^{2}}{2}\bigg{(}\frac{F^{\prime}_{1}}{F_{1}}\bigg{)}^{2},\quad\eta=-\beta\bigg{(}\frac{F_{1}^{\prime\prime}}{F_{1}}-(\beta+1)(\frac{F_{1}^{\prime}}{F_{1}})^{2}\bigg{)},\quad\zeta^{2}=\beta^{2}\frac{F_{1}^{\prime}}{F_{1}^{2}}\bigg{(}F_{1}^{\prime\prime\prime}-3(\beta+1)\frac{F_{1}^{\prime\prime}F_{1}^{\prime}}{F_{1}}+(\beta+1)(\beta+2)\frac{F_{1}^{\prime
3}}{F_{1}^{2}}\bigg{)}.$ (33)

Also, the number of e-folds (28) of the model can be found as

$N=\int^{\varphi_{i}}_{\varphi_{f}}\frac{1}{\beta}\frac{F_{1}}{F^{\prime}_{1}}d\varphi=\frac{9}{8\lambda\beta(\beta-1)^{2}(\beta-2)(3-\beta)(\beta+3)}\bigg{(}e^{\frac{2(\beta-1)\varphi_{i}}{\sqrt{6}}}-e^{\frac{2(\beta-1)\varphi_{f}}{\sqrt{6}}}\bigg{)}+\frac{\sqrt{6}}{2\beta(1-\beta)}(\varphi_{i}-\varphi_{f})$
(34)

and then by neglecting the value of scalar field at the end of inflation
($\varphi_{f}\ll\varphi_{i}$), the above expression is reduced to

$N\simeq\frac{9}{8\lambda\beta(\beta-1)^{2}(\beta-2)(3-\beta)(\beta+3)}e^{\frac{2(\beta-1)\varphi_{i}}{\sqrt{6}}}.$
(35)

Now using the slow-roll parameters (33) and the above expression for the
number of e-folds, the inflationary parameters (29) of the Starobinsky $R^{2}$
model can be calculated as shown in the appendix (49 \- 51).

Figure 2: The potential, evolution of scaleron and Hubble parameter (38)
plotted versus $\varphi$ for $p=1.05,1,0.95$ with $\gamma=-1$, $\beta=-0.02$
and $\lambda=1$.

###  IV.2 $R^{2p}$ model

As second case, we study $R^{2p}$ model as a generalization of Starobinsky
$R^{2}$ model which was first introduced in the context of higher derivative
theories as

$f(R)=R+\lambda R^{2p}$ (36)

where $\lambda$ and $p$ are free parameters. In this case, $F$ is obtained by

$F_{2}=1+2p\lambda\bigg{(}2\gamma(\beta-3)(\beta-1)e^{-\frac{(1+\beta)\varphi}{\sqrt{6}}}+\frac{2}{3}(\beta-3)(\beta-2)(\beta+3)e^{\frac{2(1-\beta)\varphi}{\sqrt{6}}}\bigg{)}^{2p-1}$
(37)

and now the Hubble parameter (20), the potential of scaleron and also its
evolution (21) can be expressed by

$\hat{H}_{2}(\varphi)=\bigg{(}\gamma
F_{2}^{-\frac{3}{2}}+F_{2}^{-\frac{\beta}{2}}\bigg{)},\quad
V_{2}(\varphi)=\frac{3-\beta}{3}\bigg{(}6\gamma
F_{2}^{-\frac{(3+\beta)}{2}}+(3+\beta)F_{2}^{-\beta}\bigg{)},\quad\frac{d\varphi_{2}}{d\hat{t}}=\frac{2}{\sqrt{6}}\bigg{(}3\gamma
F_{2}^{\frac{-3}{2}}+\beta F_{2}^{-\frac{\beta}{2}}\bigg{)}.$ (38)

The Fig. 2 shows the potential, the evolution of scaleron and the Hubble
parameter (38) versus $\varphi$ for different values of $p$ in $R^{2p}$ model
with $\gamma=-1$, $\beta=-0.02$ and $\lambda=1$. From the panel (a), we
recover the Starobinsky model in the case of $p=1$. For the values $p=1.05$
and $p=0.95$, the potential behaves the same with $p=1$. It seems that a tiny
deviation from the Starobinsky model $p=1$ does not provide a remarkable
change when inflation is considered in the context of constant-roll approach.
This fact also can be understood from the panels (b) and (c) which show the
phase diagram of the scaleron and the Hubble parameter of the model,
respectively. The slow-roll parameters (23 \- 25) of the $R^{2p}$ model for
the case of $\gamma=0$ are given by

$\epsilon=\frac{\beta^{2}}{2}\bigg{(}\frac{F^{\prime}_{2}}{F_{2}}\bigg{)}^{2},\quad\eta=-\beta\bigg{(}\frac{F_{2}^{\prime\prime}}{F_{2}}-(\beta+1)(\frac{F_{2}^{\prime}}{F_{2}})^{2}\bigg{)},\quad\zeta^{2}=\beta^{2}\frac{F_{2}^{\prime}}{F_{2}^{2}}\bigg{(}F_{2}^{\prime\prime\prime}-3(\beta+1)\frac{F_{2}^{\prime\prime}F_{2}^{\prime}}{F_{2}}+(\beta+1)(\beta+2)\frac{F_{2}^{\prime
3}}{F_{2}^{2}}\bigg{)}$ (39)

and the number of e-folds (28) of the $R^{2p}$ model can be calculated as

$N=\int^{\varphi_{i}}_{\varphi_{f}}\frac{1}{\beta}\frac{F_{2}}{F^{\prime}_{2}}d\varphi=\frac{-3\bigg{(}\frac{2}{3}(\beta-2)(\beta-3)(\beta+3)\bigg{)}^{1-2p}}{4\lambda\beta
p(2p-1)^{2}(1-\beta)^{2}}\bigg{[}\bigg{(}e^{\frac{2(1-\beta)\varphi_{i}}{\sqrt{6}}}\bigg{)}^{1-2p}-\bigg{(}e^{\frac{2(1-\beta)\varphi_{f}}{\sqrt{6}}}\bigg{)}^{1-2p}\bigg{]}+\frac{3(\varphi_{i}-\varphi_{f})}{\sqrt{6}\beta(1-\beta)(2p-1)}$
(40)

and by removing the role of scalar field when inflation ends
($\varphi_{f}\ll\varphi_{i}$), $N$ takes the following form

$N\simeq\frac{-3\bigg{(}\frac{2}{3}(\beta-2)(\beta-3)(\beta+3)e^{\frac{2(1-\beta)\varphi_{i}}{\sqrt{6}}}\bigg{)}^{1-2p}}{4\lambda\beta
p(2p-1)^{2}(1-\beta)^{2}}.$ (41)

By combining the slow-roll parameters (39) and the number of e-folds (41), the
inflationary parameters (29) of the $R^{2p}$ model are calculated as shown in
(52 \- 54).

Figure 3: The potential, evolution of scaleron and Hubble parameter (44)
plotted versus $\varphi$ for different signs of $\upsilon=0,|0.02|\lambda$
with $\lambda=1.2\times 10^{-9}$, $\gamma=-1$ and $\beta=-0.02$.

###  IV.3 Logarithmic corrected model

Let us consider now a logarithmic corrected $f(R)$ model

$f(R)=R+\lambda R^{2}+\upsilon R^{2}\ln R$ (42)

where the phenomenological parameters $\lambda$, $\upsilon$ will be fixed
momentarily. In fact, the extra terms come from the leading quantum gravity
corrections. See Birrell ; Shapiro  for details. For the model (42), $F$ takes
the following form

$\displaystyle\\!F_{3}=1+\bigg{(}2\gamma(\beta-3)(\beta-1)e^{-\frac{(1+\beta)\varphi}{\sqrt{6}}}+\frac{2}{3}(\beta-3)(\beta-2)(\beta+3)e^{\frac{2(1-\beta)\varphi}{\sqrt{6}}}\bigg{)}\times$
$\displaystyle\\!\times\Bigg{\\{}2\lambda+\upsilon+2\upsilon\ln{\bigg{(}2\gamma(\beta-3)(\beta-1)e^{-\frac{(1+\beta)\varphi}{\sqrt{6}}}+\frac{2}{3}(\beta-3)(\beta-2)(\beta+3)e^{\frac{2(1-\beta)\varphi}{\sqrt{6}}}\bigg{)}}\Bigg{\\}}\hskip
14.22636pt$ (43)

and now the Hubble parameter (20), then the potential and also the evolution
of scaleron (21) are obtained by

$\hat{H}_{3}(\varphi)=\bigg{(}\gamma
F_{3}^{-\frac{3}{2}}+F_{3}^{-\frac{\beta}{2}}\bigg{)},\quad
V_{3}(\varphi)=\frac{3-\beta}{3}\bigg{(}6\gamma
F_{3}^{-\frac{(3+\beta)}{2}}+(3+\beta)F_{3}^{-\beta}\bigg{)},\quad\frac{d\varphi_{3}}{d\hat{t}}=\frac{2}{\sqrt{6}}\bigg{(}3\gamma
F_{3}^{\frac{-3}{2}}+\beta F_{3}^{-\frac{\beta}{2}}\bigg{)}.$ (44)

The panel (a) of Fig. 3 exhibits the behaviour of potential and also the phase
diagram of the scaleron (44) for different signs of $\upsilon$ in the
logarithmic $f(R)$ model. The potential behaves almost the same for different
signs of $\upsilon$. However, $\upsilon<0$ shows a different behaviour at the
end of inflation compared to two other cases. Similar to the previous models,
the potential shows a constant rate of rolling which is connected to the
standard slow-roll. The slow-roll parameters (23 \- 25) of the logarithmic
corrected model for the case of $\gamma=0$ are obtained as

$\epsilon=\frac{\beta^{2}}{2}\bigg{(}\frac{F^{\prime}_{3}}{F_{3}}\bigg{)}^{2},\quad\eta=-\beta\bigg{(}\frac{F_{3}^{\prime\prime}}{F_{3}}-(\beta+1)(\frac{F_{3}^{\prime}}{F_{3}})^{2}\bigg{)},\quad\zeta^{2}=\beta^{2}\frac{F_{3}^{\prime}}{F_{3}^{2}}\bigg{(}F_{3}^{\prime\prime\prime}-3(\beta+1)\frac{F_{3}^{\prime\prime}F_{3}^{\prime}}{F_{3}}+(\beta+1)(\beta+2)\frac{F_{3}^{\prime
3}}{F_{3}^{2}}\bigg{)}$ (45)

and the number of e-folds (28) of the logarithmic model discussed in this
section is introduced by

$N=\int^{\varphi_{i}}_{\varphi_{f}}\frac{1}{\beta}\frac{F_{3}}{F^{\prime}_{3}}d\varphi=\frac{6e^{\frac{2\lambda+3\upsilon}{2\upsilon}}}{8\beta(1-\beta)^{2}\upsilon}\mbox{Ei}\bigg{(}-\frac{3\upsilon+2\lambda+2\upsilon\ln\bigg{(}\frac{2}{3}(\beta-1)(\beta-3)(\beta+3)e^{\frac{2(1-\beta)\varphi}{\sqrt{6}}}\bigg{)}}{2\upsilon}\bigg{)}\bigg{|}^{\varphi_{i}}_{\varphi_{f}}$
(46)

where Ei is the exponential integral that can be expressed by the Puiseux
series

$\mbox{Ei}(x)=\rho+\ln(x)+x+\frac{x^{2}}{4}+\frac{x^{3}}{18}+\frac{x^{4}}{96}+...\quad\quad
with\quad\quad e^{2\rho}=3.17221895...$ (47)

along the positive real axis. Finally, by keeping the logarithmic term and
also neglecting the value of scaleron at the end of inflation
($\varphi_{f}\ll\varphi_{i}$), $N$ is written as

$\displaystyle
N\simeq\frac{6e^{\frac{2\lambda+3\upsilon}{2\upsilon}}}{8\beta(1-\beta)^{2}\upsilon}\ln\Bigg{\\{}-\frac{3\upsilon+2\lambda+2\upsilon\ln\Big{(}\frac{2}{3}(\beta-1)(\beta-3)(\beta+3)e^{\frac{2(1-\beta)\varphi_{i}}{\sqrt{6}}}\Big{)}}{2\upsilon}\Bigg{\\}}.$
(48)

Figure 4: The marginalized joint 68% and 95% CL regions for $n_{s}$ and $r$ at
$k=0.002$ Mpc-1 from Planck 2018 combining with BK15+BAO data cmb  and the
$n_{s}-r$ constraints on different $f(R)$ inflationary models described in the
context of the constant-roll idea. The dashed and solid lines represent $N=50$
and $N=60$, respectively. The results are obtained for some allowed values of
$\beta$ when $\gamma=0$ for Starobinsky $R^{2}$ model (a), $\gamma=0$ and
$\lambda=1$ for $R^{2p}$ model (b), $\gamma=0$ and $\upsilon=0.02\lambda$ for
logarithmic corrected model (c).

By considering the slow-roll parameters (45) and the number of e-folds (48),
the inflationary parameters (29) of the logarithmic corrected model are
obtained as shown in (56 \- 58).

##  V Comparison with observations

In this section, we compare the obtained results of $f(R)$ models considered
in the paper with the inflationary observations coming from temperature and
polarization anisotropies of CMB.

In Figure 4, we present the $n_{s}-r$ constraints coming from the marginalized
joint 68% and 95% CL regions of the Planck 2018 in combination with BK15 and
BAO data on the $f(R)$ inflationary models i.e. the Starbionsky $R^{2}$ model,
the $R^{2p}$ model and the logarithmic corrected model described in the
context of the constant-roll approach. The panels are drawn for some allowed
values of $\beta$ in the cases $N=50$ (dashed line) and $N=60$ (solid line).
Panel (a) is belonged to the Starobisnky $R^{2}$ model (30) when $\gamma=0$ in
the Eq. (20). As we can see, the obtained value of tensor-to-scalar ratio $r$
for the allowed values of $\beta$ is in good agreement with the Planck 2018
cmb  constraint $r<0.064$, in particular, for the cases of $\beta=0.05$ and
$\beta=0.07$ which show $r=0.047341$ and $r=0.021806$, respectively. The
negative cases of $\beta$ predict a tiny tensor-to-scalar ratio
$r=\mathcal{O}(10^{-3})$ which is beyond the reach of the current
observations. Despite the mentioned success, the Starobinsky $R^{2}$ model
does not show a scale-invariant spectrum since the value of the spectral index
$n_{s}$ for the allowed values of $\beta$ is bigger than the unit. Panel (b)
is dedicated to show the Planck constraints on the $R^{2p}$ model (36) when
$\gamma=0$ in the Eq. (20) and $\lambda=1$ for two interesting cases of
$p=1.05$ and $p=0.95$. The panel shows that by considering a tiny variation
from the original Starobinsky model ($p=1$), the obtained values of $r$ are
still in good agreement with the Planck constraint in exception the case of
$\beta=0.05$ in $p=0.95$ which presents a disfavoured value $r\simeq 0.077$.
Also, the negative values of $\beta$ reveal a tiny tensor-to-scalar ratio
$r=\mathcal{O}(10^{-3})$ for both cases of $p$. Similar to the Starobinsky
model, the $R^{2p}$ model shows observationally disagreeable values of the
spectral index $n_{s}>1$ for both cases of $p$. In overall, by taking a look
at the obtained values of $r$ and $n_{s}$, we find that the case of $p=0.95$
is more desirable than $p=1.05$ since the values of $r$ are closer to the
upper limit ($r<0.064$) of the current observations. In contrast, by focusing
on the $n_{s}$, one can find that the deviation from $n_{s}=1$ in the case of
$p=1.05$ is remarkably less than the case of $p=0.95$. Finally, panel (c) is
corresponded to the interesting case of the logarithmic corrected model (42)
when $\gamma=0$ in the Eq. (20) and $\upsilon=0.02\lambda$. From the panel, we
can see that the obtained values of $r$ for the allowed values of $\beta$ are
compatible with the observational constraint coming from Planck 2018. This is
valid for both positive and negative values of $\beta=0.05$, $0.07$ and
$-0.07$ which show $r=0.012266$, $0.023041$ and $0.0305$, respectively.
Concerning the spectral index, the panel tells us that the logarithmic
corrected model shows a scale-invariant spectrum ($n_{s}<1$) but still
situated in a disfavorate region $0.996\leq n_{s}\leq 0.999$ which is far from
the Planck constraint $n_{s}=0.9649\pm 0.0042$. In comparison with the
previous models, here the negative values of $\beta$ predict more desirable
results of $r$ and $n_{s}$ than the positive values of $\beta$.

On the other hand, we can study the issue from the viewpoint of the swampland
criteria (26) and (27). In Table 1, we present the values of the swampland
parameters $c$ and $c^{\prime}$ for all three $f(R)$ constant-roll models. The
results are obtained for some allowed values of $\beta$ with $N$ between 50
and 60 when $\gamma=0$ for the Starobinsky $R^{2}$ model, $\gamma=0$ and
$\lambda=1$ for the $R^{2p}$ model and $\gamma=0$ and $\upsilon=0.02\lambda$
for the logarithmic corrected model. Also, we investigate the behaviour of the
swampland parameters $c$ and $c^{\prime}$ versus the tensor-to-scalar ratio
$r$ for the considered models in Figure 5. Now, let’s review the swampland
conditions for our $f(R)$ constant-roll inflationary models.

Model | $\beta$ | $N$ | $c\leq$ | $c^{\prime}\leq$
---|---|---|---|---
| $0.05$ | $50-60$ | $0.077065$ | $-0.18394$
Starobinsky $R^{2}$ | $0.07$ | $50-60$ | $0.052303$ | $-0.081309$
| $-0.07$ | $50-60$ | $0.016687$ | $-0.010866$
| $-0.05$ | $50-60$ | $0.015134$ | $-0.0086153$
| $0.05$ | $50-60$ | $0.052024$ | $-0.101226$
$R^{2p}(p=1.05)$ | $0.07$ | $50-60$ | $0.04055$ | $-0.059005$
| $-0.07$ | $50-60$ | $0.015865$ | $-0.011918$
| $-0.05$ | $50-60$ | $0.014602$ | $-0.0097295$
| $0.05$ | $50-60$ | $0.15984$ | $-0.64812$
$R^{2p}(p=0.95)$ | $0.07$ | $50-60$ | $0.075377$ | $-0.13836$
| $-0.07$ | $50-60$ | $0.017365$ | $-0.0096579$
| $-0.05$ | $50-60$ | $0.015475$ | $-0.0073956$
| $0.05$ | $50-60$ | $0.039158$ | $-0.0015361$
Logarithmic corrected | $0.07$ | $50-60$ | $0.053666$ | $-0.0028838$
| $-0.07$ | $50-60$ | $0.061746$ | $-0.0038075$
Table 1: The values of the swampland parameters $c$ and $c^{\prime}$ coming
from the swampland conditions (26) and (27) for different $f(R)$ inflationary
models described in the context of the constant-roll approach. The results are
obtained for some allowed values of $\beta$ with $N=50$ when $\gamma=0$ for
the Starobinsky $R^{2}$ model, $\gamma=0$ and $\lambda=1$ for the $R^{2p}$
model, $\gamma=0$ and $\upsilon=0.02\lambda$ for the logarithmic corrected
model.

Figure 5: The behaviour of the swampland parameters $c$ and $c^{\prime}$ (26)
and (27) versus the tensor-to-scalar ratio $r$ for different $f(R)$
inflationary models described in the context of the constant-roll approach.
The results are obtained for some allowed values of $\beta$ when $\gamma=0$
for the Starobinsky $R^{2}$ model (a), $\gamma=0$ and $\lambda=1$ for the
$R^{2p}$ model (b), $\gamma=0$ and $\upsilon=0.02\lambda$ for the logarithmic
corrected model (c).

In the Starobinsky $R^{2}$ model, we find that the swampland condition of the
parameter $c$ shown in the table leads to some desirable values of the tensor-
to-scalar ratio $r$, in particular, for the positive cases of $\beta$ and this
coincidence is distorted for tiny values of $c$ since the corresponding values
of $r$ are not situated in the range of our current observations (see the
first panel of Figure 5). For another swampland parameter $c^{\prime}$, the
table tells us in the case of $\beta=0.05$, $c^{\prime}$ takes the value
$-0.18394$ which is connected to an observationally acceptable value $r\simeq
0.05$. By considering the swampland condition $c^{\prime}$, the value of $r$
approaches the observational upper limit and then reaches disfavoured regions
of $r$ by crossing from the limit. For other values of $\beta$, we find that
the values of $c^{\prime}$ are so tiny and lead to the small values of $r$. By
setting the swampland condition $c^{\prime}$, the situation becomes better
since the corresponding value of $r$ approaches the upper limit (see panel (a)
of Figure 5). In the $R^{2p}$ model with $p=1.05$, the situation of $c$ is
almost similar to the Starobinsky $R^{2}$ model while the parameter
$c^{\prime}$ reveals some interesting features. The values of
$\beta=0.05,0.07,-0.05$ show $r\simeq 0.02,0.01,0.02$ which are close to the
lower limit of the observations and by considering the swampland condition,
$r$ approaches the upper limit. For $\beta=-0.07$, we find $r\simeq 0.7$ which
is situated beyond the upper limit (see panel (b) of Figure 5). In the case of
$p=0.95$, the parameter $c$ behaves analogous to the Starobinsky $R^{2}$ model
in exception the case of $\beta=0.05$ which leads to $r$ bigger than the upper
limit ($r<0.064$). Also, for $\beta=0.07,-0.07,-0.05$, the parameter
$c^{\prime}$ shows the values of $r\simeq 0.03,0.05,0.02$ so that by setting
the swampland condition, they approach the upper limit. Note that for
$\beta=0.05$, the parameter $c^{\prime}$ predicts a very large $r$ which is
not compatible with the observations. In the logarithmic corrected model, the
obtained values of $c$ present $r$ below the upper limit and again by
considering the swampland condition, they approach the lower limit. For the
parameter $c^{\prime}$, $\beta=-0.07$ shows $r\simeq 0.04$ while
$\beta=0.05,0.07$ predict tiny $r$ (see panel (c) of Figure 5).

##  VI Discussion and conclusions

In the present work, we focused on some $f(R)$ inflationary models i.e. the
Starobinsky $R^{2}$, the $R^{2p}$ and the logarithmic corrected models
introduced in the context of the constant-roll where inflaton rolls down with
a constant rate as $\ddot{\varphi}=\beta H\dot{\varphi}$. We have investigated
the inflationary dynamics for the three $f(R)$ models in presence of constant-
roll condition. Specifically, we calculated the spectral parameters of the
models i.e. the spectral index, its running and the tensor-to-scalar ratio.
Then, we compared the obtained results with Planck 2018 data combined with
BK15+BAO data. Results can be summarized as follows :

  * •

We studied the Starobinsky $R^{2}$ model for $\gamma=0$ and found that, for
positive values of $\beta$, $r$ is in good agreement with observations while,
for negative values of $\beta$, it shows a tiny $r=\mathcal{O}(10^{-3})$.
Furthermore, the Starobinsky model does not present a scale-invariant spectrum
since the obtained values of the spectral index $n_{s}$ are bigger than the
unit.

  * •

In the $R^{2p}$ model, we investigated the model for the two cases $p=1.05$
and $p=0.95$ when $\gamma=0$ and $\lambda=1$. We found that the obtained
values of $r$ are compatible with the observational constraint on $r$, except
the case $\beta=0.05$ in $p=0.95$ which predicts $r\simeq 0.077$ bigger than
the upper limit ($r<0.064$). Also, a negative $\beta$ presents a tiny
$r=\mathcal{O}(10^{-3})$ for both cases of $p$. Moreover, the model shows the
spectral index $n_{s}>1$ which is not in good agreement with the Planck
constraint. In summary, we found the case of $p=0.95$ more desirable than
$p=1.05$ since the values of $r$ are closer to the upper limit ($r<0.064$). In
contrast, the deviation from $n_{s}=1$ in the case of $p=1.05$ is less than
the case of $p=0.95$.

  * •

As a modified form of the Starobinsky model, we have considered the
logarithmic corrected model when $\gamma=0$ and $\upsilon=0.02\lambda$. We
have found that the obtained values of $r$ for positive and negative values of
$\beta$ are in good agreement with the observations. Also, the model predicts
the spectral index $n_{s}<1$ but it is still situated in a disfavoured region
$0.996<n_{s}<0.999$.

According to the above results, one can conclude that although all three
constant-roll $f(R)$ inflationary models i.e. the Starobinsky $R^{2}$ model,
the $R^{2p}$ model and the logarithmic corrected model are observationally
compatible with the values of $r$, they do not show acceptable values for the
spectral index $n_{s}$. Consequently, the considered constant-roll $f(R)$
inflationary models are disfavored models when $\gamma=0$ in the Eq. (20).
This result can be referred to Ref. Motohashi9  where the authors studied the
behavior of constant-roll $f(R)$ inflation in a general approach. They found
that the model can lead to an inflationary solution only for
$-0.1\leq\beta\leq 0$ when $\gamma=-1$ in the Eq. (20).

Furthermore, we have studied the models from the viewpoint of the Weak Gravity
Conjecture using the swampland criteria. By setting the condition of $c$, we
found that the scalar-to-tensor ratio $r$ starts from an observationally
acceptable value and approaches the lower limit. Finally, by crossing from the
lower limit, it tilts to the disfavored areas of $r$. Also, by setting the
condition of $c^{\prime}$, the value of $r$ begins from a desirable value and
finally, by crossing from the upper limit, it shifts to the disagreeable
regions. Hence, we can conclude that our $f(R)$ inflationary models introduced
in the context of the constant-roll idea are not fully compatible with the
swampland criteria.

## References

  * (1) A. H. Guth, “The inflationary universe: A possible solution to the horizon and flatness problems,” Phys. Rev. D, vol. 23, p. 347, (1981).
  * (2) K. Sato, “First order phase transition of a vacuum and expansion of the universe,” Mon. Not. Roy. Astron. Soc., vol. 195, p. 467, (1981).
  * (3) D. Kazanas, “Dynamics of the universe and spontaneous symmetry breaking,” Astrophys. J., vol. 241, p. L59, (1980).
  * (4) A. D. Linde, “A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy problems,” Phys. Lett. B, vol. 108, p. 389, (1982).
  * (5) A. Albrecht and P. J. Steinhardt, “Cosmology for grand unified theories with radiatively induced symmetry breaking,” Phys. Rev. Lett., vol. 48, p. 1220, (1982).
  * (6) D. H. Lyth and A. Riotto, “Particle physics models of inflation and the perturbation,” Phys. Rept., vol. 314, p. 1, (1999).
  * (7) L. Kofman, A. D. Linde, and A. A. Starobinsky, “Reheating after inflation,” Phys. Rev. Lett., vol. 73, p. 3195, (1994).
  * (8) Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, “Universe reheating after inflation,” Phys. Rev. D, vol. 51, p. 5438, (1995).
  * (9) J. Martin, “What have the planck data taught us about inflation?,” Class. Quant. Grav., vol. 33, p. 034001, (2016).
  * (10) A. A. Starobinsky, “A new type of isotropic cosmological models without singularity,” Phys. Lett. B, vol. 91, p. 99, (1980).
  * (11) J. D. Barrow and S. Cotsakis, “Inflation and the conformal structure of higher order gravity theories,” Phys. Lett. B, vol. 214, p. 515, (1988).
  * (12) F. L. Bezrukov and M. Shaposhnikov, “The standard model higgs boson as the inflaton,” Phys. Lett. B, vol. 659, p. 703, (2008).
  * (13) R. Kallosh and A. Linde, “Universality class in conformal inflation,” JCAP, vol. 07, p. 002, (2013).
  * (14) X. Chen, “Primordial non-gaussianities from inflation models,” Adv. Astron., vol. 2010, p. 638979, (2010).
  * (15) J. Martin, H. Motohashi, and T. Suyama, “Primordial non-gaussianities from inflation models,” Phys. Rev. D, vol. 87, p. 023514, (2013).
  * (16) H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Inflation with a constant rate of roll,” JCAP, vol. 09, p. 018, (2015).
  * (17) H. Motohashi and A. A. Starobinsky, “Constant-roll inflation: confrontation with recent observational data,” Europhys. Lett., vol. 117, p. 39001, (2017).
  * (18) S. Inoue and J. Yokoyama, “Curvature perturbation at the local extremum of the inflaton’s potential,” Phys. Lett. B, vol. 524, p. 15, (2002).
  * (19) W. H. Kinney, “Horizon crossing and inflation with large $\eta$,” Phys. Rev. D, vol. 72, p. 023515, (2005).
  * (20) M. H. Namjoo, H. Firouzjahi, and M. Sasaki, “Violation of non-gaussianity consistency relation in a single field inflationary model,” Europhys. Lett., vol. 101, p. 39001, (2013).
  * (21) C. R. Contaldi, L. K. M. Peloso, and A. D. Linde, “Suppressing the lower multipoles in the cmb anisotropies,” JCAP, vol. 0307, p. 002, (2003).
  * (22) L. Lello and D. Boyanovsky, “Tensor to scalar ratio and large scale power suppression from pre-slow roll initial conditions,” JCAP, vol. 1405, p. 029, (2014).
  * (23) S. Odintsov and V. Oikonomou, “Inflationary dynamics with a smooth slow-roll to constant-roll era transition,” Phys. Rev. D, vol. 96, p. 024029, (2017).
  * (24) F. Cicciarella, J. Mabillard, and M. Pieroni, “New perspectives on constant-roll inflation,” JCAP, vol. 01, p. 024, (2018).
  * (25) A. Awad, W. E. Hanafy, G. Nashed, S. Odintsov, and V. Oikonomou, “Constant-roll inflation in f(T) teleparallel gravity,” JCAP, vol. 07, p. 026, (2017).
  * (26) L. Anguelova, P. Suranyi, and L. Wijewardhana, “Systematics of constant roll inflation,” JCAP, vol. 02, p. 004, (2018).
  * (27) A. Ito and J. Soda, “Anisotropic constant-roll inflation,” Eur. Phys. J. C, vol. 78, p. 55, (2018).
  * (28) Z. Yi and Y. Gong, “On the constant-roll inflation,” JCAP, vol. 1803, p. 052, (2018).
  * (29) M. J. P. Morse and W. H. Kinney, “Large $\eta$ constant-roll inflation is never an attractor,” Phys. Rev. D, vol. 97, p. 123519, (2018).
  * (30) A. Karam, L. Marzola, T. Pappas, A. Racioppi, and K. Tamvakis, “Constant-roll (quasi-)linear inflation,” JCAP, vol. 05, p. 011, (2018).
  * (31) J. T. G. Ghersi, A. Zucca, y, and A. V. Frolov, “Observational constraints on constant roll inflation,” JCAP, vol. 05, p. 030, (2019).
  * (32) W.-C. Lin and M. J. P. Morsey, “Dynamical analysis of attractor behavior in constant roll inflation,” JCAP, vol. 09, p. 063, (2019).
  * (33) A. Micu, “Two-field constant roll inflation,” JCAP, vol. 11, p. 003, (2019).
  * (34) A. Oliveros and H. E. Noriega, “Constant-roll inflation driven by a scalar field with nonminimal derivative coupling,” Int. J. Mod. Phys. D, vol. 28, p. 1950159, (2019).
  * (35) H. Motohashi and A. A. Starobinsky, “Constant-roll inflation in scalar-tensor gravity,” JCAP, vol. 11, p. 025, (2019).
  * (36) V. Kamali, M. Artymowski, and M. R. Setare, “Constant roll warm inflation in high dissipative regime,” JCAP, vol. 07, p. 002, (2020).
  * (37) M. Guerrero, D. Rubiera-Garcia, and D. S.-C. Gomez, “Constant roll inflation in multifield models,” Phys. Rev. D, vol. 102, p. 123528, (2020).
  * (38) M. Shokri, J. Sadeghi, M. R. Setare, and S. Capozziello, “Nonminimal coupling inflation with constant slow roll,” To be publish in Int. J. Mod. Phys. D, (2021).
  * (39) M. Shokri, J. Sadeghi, and M. R. Setare, “The generalized $sl(2,\textit{R})$ and $su(1,1)$ in non-minimal constant-roll inflation,” Ann. Phys., vol. 429, p. 168487, (2021).
  * (40) S. Nojiri, D. Odintsov, and V. K. Oikonomou, “Constant-roll inflation in f(R) gravity,” Class. Quant. Grav., vol. 34, p. 245012, (2017).
  * (41) V. K. Oikonomou, “Generalizing the Constant-roll Condition in Scalar Inflation,” 6 2021.
  * (42) V. K. Oikonomou and F. P. Fronimos, “A Nearly Massless Graviton in Einstein-Gauss-Bonnet Inflation with Linear Coupling Implies Constant-roll for the Scalar Field,” EPL, vol. 131, no. 3, p. 30001, 2020.
  * (43) V. K. Oikonomou, “Reheating in Constant-roll $F(R)$ Gravity,” Mod. Phys. Lett. A, vol. 32, no. 33, p. 1750172, 2017.
  * (44) E. Elizalde, S. D. Odintsov, V. K. Oikonomou, and T. Paul, “Logarithmic-corrected $R^{2}$ Gravity Inflation in the Presence of Kalb-Ramond Fields,” JCAP, vol. 02, p. 017, 2019.
  * (45) I. Antoniadis, A. Lykkas, and K. Tamvakis, “Constant-roll in the Palatini-$R^{2}$ models,” JCAP, vol. 04, no. 04, p. 033, 2020.
  * (46) H. Motohashi and A. A. Starobinsky, “f(R) constant-roll inflation,” Eur. Phys. J. C, vol. 77, p. 538, (2017).
  * (47) Y. Akrami et al., “Planck 2018 results. constraints on inflation,” [astro-ph: 1807.06211].
  * (48) H. C. Chiang et al., “Measurement of cmb polarization power spectra from two years of bicep data,” Astrophys. J. , vol. 711, p. 1123, (2010).
  * (49) S. Capozziello, S. J. Gionti, Gabriele, and D. Vernieri, “String duality transformations in f(R) gravity from Noether symmetry approach,” JCAP, vol. 1601, p. 015, (2016).
  * (50) M. Benetti, S. Capozziello, and L. L. Graef, “Swampland conjecture in f(R) gravity by the noether symmetry approach,” Phys. Rev. D, vol. 100, p. 084013, (2019).
  * (51) J. Aguirre et al., “The simons observatory: Science goals and forecasts,” JCAP, vol. 1902, p. 056, (2019).
  * (52) K. N. Abazajian et al., “Cmb-s4 science book, first edition,” [arXiv:1610.02743].
  * (53) A. Suzuki et al., “The litebird satellite mission: Sub-kelvin instrument,” Journal of Low Temperature Physics, vol. 193, p. 1048, (2018).
  * (54) H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Phantom boundary crossing and anomalous growth index of fluctuations in viable f(R) models of cosmic acceleration,” Prog. Theor. Phys., vol. 123, p. 887, (2010).
  * (55) H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Future oscillations around phantom divide in f(R) gravity,” JCAP, vol. 1106, p. 006, (2011).
  * (56) R. Gannouji, B. Moraes, and D. Polarski, “The growth of matter perturbations in f(R) models,” JCAP, vol. 0902, p. 034, (2009).
  * (57) H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Analytic solution for matter density perturbations in a class of viable cosmological f(R) models,” Int. J. Mod. Phys. D, vol. 18, p. 1731, (2009).
  * (58) S. Tsujikawa, B. M. R. Gannouji, and D. Polarski, “The dispersion of growth of matter perturbations in f(R) gravity,” Phys. Rev. D, vol. 80, p. 084044, (2009).
  * (59) H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Matter power spectrum in f(R) gravity with massive neutrinos,” Phys. Rev. D, vol. 124, p. 541, (2010).
  * (60) H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Cosmology based on f(R) gravity admits 1 ev sterile neutrinos,” Phys. Rev. Lett., vol. 110, p. 121302, (2013).
  * (61) S. Tsujikawa, “Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints,” Phys. Rev. D, vol. 77, p. 023507, (2008).
  * (62) S. Appleby and R. Battye, “Aspects of cosmological expansion in f(R) gravity models,” JCAP, vol. 0805, p. 019, (2008).
  * (63) T. Kobayashi and K. Maeda, “Relativistic stars in f(R) gravity, and absence thereof,” Phys. Rev. D, vol. 78, p. 064019, (2008).
  * (64) H. Schmidt, “Variational derivatives of arbitrarily high order and multi-inflation cosmological models,” Class. Quant. Grav., vol. 6, p. 557, (1989).
  * (65) K. Maeda, “Towards the einstein-hilbert action via conformal transformation,” Phys. Rev. D, vol. 39, p. 3159, (1989).
  * (66) V. Muller, H. Schmidt, and A. A. Starobinsky, “Power-law inflation as an attractor solution for inhomogeneous cosmological models,” Class. Quant. Grav., vol. 7, p. 1163, (1990).
  * (67) A. D. Felice and S. Tsujikawa, “f(R) theories,” Living Rev. Rel., vol. 13, p. 3, (2010).
  * (68) J. Martin, C. Ringeval, and V. Vennin, “Encyclopedia inflationaris,” Phys. Dark Univ., vol. 5, p. 75, (2014).
  * (69) J. Martin, C. Ringeval, R. Trotta, and V. Vennin, “The best inflationary models after planck,” JCAP, vol. 03, p. 039, (2014).
  * (70) F. Renzi, M. Shokri, and A. Melchiorri, “What is the amplitude of the gravitational waves background expected in the starobinski model?,” Phys. Dark Univ., vol. 27, p. 100450, (2020).
  * (71) A. A. Starobinsky, “A new type of isotropic cosmological models without singularity,” Phys. Lett. B, vol. 91, p. 99, (1980).
  * (72) T. P. Sotiriou and V. Faraoni, “f(R) theories of gravity,” Rev. Mod. Phys., vol. 82, p. 451, (2010).
  * (73) M. Ivanov and A. V. Toporensky, “Stable super-inflating cosmological solutions in f(R)-gravity,” Int. J. Mod. Phys. D, vol. 21, p. 1250051, (2012).
  * (74) I. Ben-Dayan, S. Jing, M. Torabian, A. Westphal, and L. Zarate, “R2logR quantum corrections and the inflationary observables,” JCAP, vol. 1409, p. 005, (2014).
  * (75) S. V. Ketov and N. Watanabe, “The f(R) gravity function of the linde quintessence,” Phys. Lett. B, vol. 741, p. 242, (2014).
  * (76) M. Rinaldi, G. Cognola, L. Vanzo, and S. Zerbini, “Inflation in scale-invariant theories of gravity,” Phys. Rev. D, vol. 91, p. 123527, (2015).
  * (77) B. J. Broy, F. G. Pedro, and A. Westphal, “Disentangling the f(R)\- duality,” JCAP, vol. 1503, p. 029, (2015).
  * (78) S. Capozziello, F. Occhionero, and L. Amendola, “The phase space view of inflation. 2: Fourth order models,” Int. J. Mod. Phys. D, vol. 1, p. 615, (2015).
  * (79) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini, “A class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion,” Phys. Rev. D, vol. 77, p. 046009, (2008).
  * (80) S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from f(R) theory to lorentz non-invariant models,” Phys. Rept., vol. 505, p. 059, (2011).
  * (81) E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini, “Non-singular exponential gravity: a simple theory for early- and late-time accelerated expansion,” Phys. Rev. D, vol. 83, p. 086006, (2011).
  * (82) H. Motohashi and A. Nishizawa, “Reheating after f(R) inflation,” Phys. Rev. D, vol. 86, p. 083514, (2012).
  * (83) K. Bamba, A. Lopez-Revelles, R. Myrzakulov, S. D. Odintsov, and L. Sebastiani, “The universe evolution in exponential f(R)-gravity,” TSPU Bulletin, vol. 2012, p. 22, (2012).
  * (84) L. Sebastiani, G. Cognola, R. Myrzakulov, S. D. Odintsov, and S. Zerbini, “Nearly starobinsky inflation from modified gravity,” Phys. Rev. D, vol. 89, p. 023518, (2014).
  * (85) J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, “Starobinsky-like inflation in dilaton-brane cosmology,” Phys. Lett. B, vol. 732, p. 380, (2014).
  * (86) M. Artymowski and Z. Lalak, “Inflation and dark energy from f(R) gravity,” JCAP, vol. 1409, p. 036, (2014).
  * (87) S. Capozziello, M. D. Laurentis, and O. Luongo, “Connecting early and late universe by f(R) gravity,” Int. J. Mod. Phys. D, vol. 24, p. 1541002, (2014).
  * (88) H. Alavirad and J. M. Weller, “Modified gravity with logarithmic curvature corrections and the structure of relativistic stars,” Phys. Rev. D, vol. 88, p. 124034, (2013).
  * (89) A. V. Frolov and J.-Q. Guo, “Small cosmological constant from running gravitational coupling,” [astro-ph:1101.4995].
  * (90) H.-H. Meng and P. Wang, “Palatini formation of modified gravity with lnR terms,” Phys. Lett. B, vol. 584, p. 1, (2004).
  * (91) L. Iorio, “Astronomical constraints on some long-range models of modified gravity,” Adv. High Energy Phys, vol. 2007, p. 90731, (2007).
  * (92) S. A. Appleby and R. A. Battye, “Aspects of cosmological expansion in f(R) gravity models,” JCAP, vol. 019, p. 0805, (2008).
  * (93) Z. Girones, A. Marchetti, O. Mena, C. Pena-Garay, and N. Rius, “Cosmological data analysis of f(R) gravity models,” JCAP, vol. 1011, p. 004, (2010).
  * (94) M. Ivanov and A. V. Toporensky, “Stable super-inflating cosmological solutions in f(R)-gravitys,” Int. J. Mod. Phys. D, vol. 21, p. 1250051, (2012).
  * (95) J.-Q. Guo and A. V. Frolov, “Cosmological dynamics in f(R) gravity,” Phys. Rev. D, vol. 88, p. 124036, (2013).
  * (96) A. V. Astashenok, S. Capozziello, and S. D. Odintsov, “Further stable neutron star models from f(R) gravity,” JCAP, vol. 1312, p. 040, (2013).
  * (97) S. V. Ketov and N. Watanabe, “The f(R) gravity function of the linde quintessence,” Phys. Lett. B, vol. 741, p. 242, (2015).
  * (98) M. Rinaldi, G. Cognola, L. Vanzo, and S. Zerbini, “Inflation in scale-invariant theories of gravity,” Phys. Rev. D, vol. 91, p. 123527, (2015).
  * (99) S. Capozziello and M. De Laurentis, “Extended Theories of Gravity,” Phys. Rept., vol. 509, p. 167, (2011).
  * (100) S. Capozziello, C. A. Mantica, and L. G. Molinari, “Cosmological perfect-fluids in f(R) gravity,” Int. J. Geom. Meth. Mod. Phys., vol. 16, no. 01, p. 1950008, (2018).
  * (101) E. N. Saridakis et al., “Modified Gravity and Cosmology: An Update by the CANTATA Network,” 5 2021.
  * (102) S. K. Garg and C. Krishnan, “Bounds on Slow Roll and the de Sitter Swampland,” JHEP, vol. 11, p. 075, 2019.
  * (103) K. S. Garg, C. Krishnan, and M. Zaid, “Bounds on slow roll at the boundary of the landscape,” JHEP, vol. 03, p. 029, (2019).
  * (104) N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.  Cambridge Monographs on Mathematical Physics, Cambridge, UK: Cambridge Univ. Press, 2 1984.
  * (105) I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective action in quantum gravity.  Bristol, UK: IOP, 1992.

##  Appendix A The spectral parameters of $f(R)$ models

###  A.1 Starobinsky $R^{2}$ model

The spectral index of the Starobinsky $R^{2}$ model:

$\displaystyle\\!n_{s}=\frac{1}{\Big{(}2N\beta^{3}-4N\beta^{2}+2N\beta-3\Big{)}^{2}}\Bigg{\\{}(4N^{2}+8N)\beta^{6}-(16N^{2}+32N)\beta^{5}+(24N^{2}+48N-6)\beta^{4}+$
$\displaystyle\\!+(-16N^{2}-44N+12)\beta^{3}+(4N^{2}+32N-6)\beta^{2}-12\beta
N+9\Bigg{\\}}.$ (49)

The tensor-to-scalar ratio of the Starobinsky $R^{2}$ model:

$r=\frac{48(\beta-1)^{2}\beta^{2}}{\Big{(}2N\beta^{3}-4N\beta^{2}+2N\beta-3\Big{)}^{2}}.$
(50)

The running spectral index of the Starobinsky $R^{2}$ model:

$\alpha_{s}=-\frac{32N\Big{(}N\beta^{2}-\beta
N-\frac{3}{2}\Big{)}\beta^{3}(\beta-1)^{7}}{\Big{(}2N\beta^{3}-4N\beta^{2}+2N\beta-3\Big{)}^{4}}.$
(51)

###  A.2 $R^{2p}$ model

The spectral index of the $R^{2p}$ model:

$\displaystyle\\!n_{s}=\frac{1}{\Bigg{(}3p\lambda
4^{p}9^{-p}\mathcal{W}^{-1}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{2p}+\beta^{3}-2\beta^{2}-9\beta+18\Bigg{)}^{2}}\Bigg{\\{}-3p^{2}4^{2p}9^{-2p}\lambda^{2}\mathcal{W}^{-2}\times$
$\displaystyle\\!\times\bigg{(}-3+(p-\frac{1}{2})^{2}\Big{(}16\beta^{3}-32\beta^{2}+16\beta\Big{)}\bigg{)}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{4p}-16(\beta-3)(\beta-2)4^{p}9^{-p}\lambda
p(\beta+3)\mathcal{W}^{-1}\times$
$\displaystyle\\!\times\bigg{(}-\frac{3}{8}+(p-\frac{1}{2})^{2}\Big{(}\beta^{3}-2\beta^{2}+\beta\Big{)}\bigg{)}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{2p}-24(\beta-2)\Bigg{(}(p-\frac{1}{2})^{2}\beta\mathcal{W}^{-2}\lambda^{2}p^{2}81^{-p}16^{p}\times$
$\displaystyle\\!\times(\beta-1)^{2}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{4p}-\frac{(\beta-2)(\beta-3)^{2}(\beta+3)^{2}}{24}\Bigg{)}\Bigg{\\}}.$
(52)

The tensor-to-scalar ratio of the $R^{2p}$ model:

$r=\frac{192(p-\frac{1}{2})^{2}\beta^{2}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{4p}\mathcal{W}^{-2}\lambda^{2}p^{2}81^{-p}(\beta-1)^{2}16^{p}}{\Bigg{(}3p\lambda
4^{p}9^{-p}\mathcal{W}^{-1}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{2p}+\beta^{3}-2\beta^{2}-9\beta+18\Bigg{)}^{2}}.$
(53)

The running spectral index of the $R^{2p}$ model:

$\displaystyle\\!\alpha_{s}=\frac{1}{\Bigg{(}-3p\lambda
4^{p}\mathcal{W}^{-1}\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{2p}+3^{(2+2p)}\beta-(\beta^{3}-2\beta^{2}+18)9^{p}\Bigg{)}^{4}}\Bigg{\\{}-128(p-\frac{1}{2})^{4}\beta^{2}p^{2}\times$
$\displaystyle\\!\times(\beta-2)(\beta-3)(\beta+3)(\beta-1)^{4}\lambda^{2}\Bigg{(}\mathcal{W}^{-2}1296^{p}(\beta-2)(\beta-3)(\beta+3)\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{4p}+3p64^{p}9^{p}\lambda\times$
$\displaystyle\\!\times\mathcal{W}^{-3}(\beta-1)\Big{(}(\beta-2)(\beta^{2}-9)\Big{)}^{6p}\Bigg{)}\Bigg{\\}}$
(54)

where

$\mathcal{W}=-\frac{2\beta p\lambda
9^{-p}(-1+b)^{2}(2p-1)^{2}N4^{p}(\beta^{3}-2\beta^{2}-9\beta+18)^{2p}}{(\beta-2)(\beta^{2}-9)}.$
(55)

###  A.3 Logarithmic corrected mode

The spectral index of the Logarithmic corrected model:

$\displaystyle\\!n_{s}=\frac{1}{12\Bigg{(}e^{\mathcal{Z}}\upsilon(\beta-2)\ln\Big{(}{\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}}\Big{)}+(\lambda+\frac{\upsilon}{2})(\beta-2)e^{\mathcal{Z}}+\frac{(\beta-1)}{2}\Bigg{)}^{2}}\Bigg{\\{}-8e^{2\mathcal{Z}}\upsilon^{2}\ln\Big{(}\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}\Big{)}^{2}\times$
$\displaystyle\\!\times(\beta-2)^{2}\big{(}\beta^{4}-2\beta^{3}+\beta^{2}-\frac{3}{2}\big{)}-16\Bigg{(}\bigg{(}(\lambda+\frac{3\upsilon}{2})\beta^{4}-(2\lambda+3\upsilon)\beta^{3}+(\lambda+\frac{3\upsilon}{2})\beta^{2}+-\frac{3}{4}(2\lambda+\upsilon)\bigg{)}(\beta-2)e^{2\mathcal{Z}}+$
$\displaystyle\\!+\frac{\Big{(}\beta^{3}-2\beta^{2}+\beta-\frac{3}{2}\Big{)}e^{\mathcal{Z}}(\beta-1)}{2}\Bigg{)}(\beta-2)\upsilon\ln\Big{(}\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}\Big{)}-8\bigg{(}(\lambda+\frac{3\upsilon}{2})^{2}\beta^{4}-(2\lambda^{2}+6\lambda\upsilon+\frac{13}{2}\upsilon^{2})\beta^{3}+(\lambda^{2}+$
$\displaystyle\\!+3\lambda\upsilon+\frac{25}{4}\upsilon^{2})\beta^{2}-2\beta\upsilon^{2}-\frac{3(\lambda+\frac{\upsilon}{2})^{2}}{2}\bigg{)}(\beta-2)^{2}e^{2\mathcal{Z}}-8\bigg{(}(\beta-2)\Big{(}(\lambda+\frac{5\upsilon}{2})\beta^{3}-(2\lambda+5\upsilon)\beta^{2}+(\lambda+\frac{5\upsilon}{2})\beta-$
$\displaystyle\\!-\frac{3}{4}(2\lambda+\upsilon)\Big{)}e^{\mathcal{Z}}-\frac{3}{8}(\beta-1)\bigg{)}(\beta-1)\Bigg{\\}}.$
(56)

The tensor-to-scalar ratio of the Logarithmic corrected model:

$r=\frac{16\beta^{2}(\beta-2)^{2}\bigg{(}\upsilon\ln\Big{(}\frac{(\beta-2)e^{Z}}{\beta-1}\Big{)}+\lambda+\frac{3\upsilon}{2}\bigg{)}^{2}e^{2\mathcal{Z}}(\beta-1)^{2}}{3\bigg{(}e^{\mathcal{Z}}\upsilon(\beta-2)\ln\Big{(}\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}\Big{)}+(\lambda+\frac{\upsilon}{2})(\beta-2)e^{\mathcal{Z}}+\frac{\beta-1}{2}\bigg{)}^{2}}.$
(57)

The running spectral index of the Logarithmic corrected model:

$\displaystyle\\!\alpha_{s}=-\frac{1}{9\Bigg{(}e^{\mathcal{Z}}\upsilon(\beta-2)\ln\Big{(}{\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}}\Big{)}+(\lambda+\frac{\upsilon}{2})(\beta-2)e^{\mathcal{Z}}+\frac{(\beta-1)}{2}\Bigg{)}^{4}}\Bigg{\\{}4\beta^{2}\Bigg{(}e^{\mathcal{Z}}\upsilon^{2}(\beta-2)(\beta-1)^{2}\ln\Big{(}\frac{(\beta-2)e^{\mathcal{Z}}}{(\beta-1)}\Big{)}^{2}+$
$\displaystyle\\!+2\bigg{(}-\beta\upsilon^{2}(\beta-2)^{2}e^{2\mathcal{Z}}+\Big{(}\frac{1}{4}+(\lambda+2\upsilon)(\beta-2)e^{Z}\Big{)}(\beta-1)^{2}\bigg{)}\upsilon\ln\Big{(}\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}\Big{)}-2\Big{(}(\frac{3\beta}{2}-2)n+\beta\lambda\Big{)}\upsilon^{2}(\beta-2)^{2}e^{2\mathcal{Z}}+$
$\displaystyle\\!+(\beta-1)\bigg{(}\Big{(}\big{(}\frac{15\beta}{4}-\frac{31}{4}\big{)}\upsilon^{2}+4\lambda\upsilon(\beta-1)+\lambda^{2}(\beta-1)\Big{)}(\beta-2)e^{\mathcal{Z}}+\frac{\big{(}\lambda+\frac{7\upsilon}{2}\big{)}(\beta-1)}{2}\bigg{)}\Bigg{)}(\beta-2)^{2}\bigg{(}\upsilon\ln{\Big{(}\frac{(\beta-2)e^{\mathcal{Z}}}{\beta-1}\Big{)}}+$
$\displaystyle\\!+\lambda+\frac{3\upsilon}{2}\bigg{)}e^{2\mathcal{Z}}(\beta-1)^{4}\Bigg{\\}}$
(58)

where

$\mathcal{Z}=\frac{2\upsilon\exp{\Big{(}{\frac{8(1-\beta)^{2}\beta\upsilon
N}{6e^{\frac{2\lambda+3\upsilon}{2\upsilon}}}}\Big{)}+3\upsilon+2\lambda}}{2\upsilon}.$
(59)


$\displaystyle-2\cos(\theta)+4\frac{c_{1}d_{1}a_{2}b_{2}}{c_{1}^{2}a_{2}^{2}+d_{1}^{2}b_{2}^{2}}\sin(\theta),$
(D20) $\displaystyle c_{14}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{22}]$
$\displaystyle=$ $\displaystyle
2\cos(\theta)+4\frac{c_{1}d_{1}c_{2}d_{2}}{c_{1}^{2}c_{2}^{2}+d_{1}^{2}d_{2}^{2}}\sin(\theta),$
(D21)

where
$\hat{\rho}_{ij}=\textrm{Tr}_{BC}[M_{BC}|\Phi_{i}\rangle_{AB}|\Psi_{j}\rangle_{CD}\langle\Psi_{j}|\langle\Phi_{i}|]/\textrm{Tr}[M_{BC}|\Phi_{i}\rangle_{AB}|\Psi_{j}\rangle_{CD}\langle\Psi_{j}|\langle\Phi_{i}|]$,
$\textrm{Tr}_{BC}$ denotes the partial trace operations performed on the
systems $BC$. From the equations (D18)-(D21), it follows from the equation
(D9) that

$\displaystyle\sum_{i,j=1,2}p_{i}q_{j}c_{ij}=$ $\displaystyle
2\alpha\cos(\theta)+2\beta\sin(\theta)$ $\displaystyle=$ $\displaystyle
2\sqrt{\alpha^{2}+\beta^{2}}$ $\displaystyle>$ $\displaystyle 2$ (D22)

for some $a_{i},b_{i},c_{i},d_{i},p_{i},q_{i}$s, where
$\cos(\theta):=\alpha/\sqrt{\alpha^{2}+\beta^{2}}$,
$\alpha=(p_{1}-p_{2})(q_{1}-q_{2})$, and
$\beta=p_{1}q_{1}\frac{2a_{1}b_{1}a_{2}b_{2}}{a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}}+p_{1}q_{2}\frac{2a_{1}b_{1}c_{2}d_{2}}{a_{1}^{2}c_{2}^{2}+b_{1}^{2}d_{2}^{2}}+p_{2}q_{1}\frac{2c_{1}d_{1}a_{2}b_{2}}{c_{1}^{2}a_{2}^{2}+d_{1}^{2}b_{2}^{2}}+p_{2}q_{2}\frac{c_{1}d_{1}c_{2}d_{2}}{c_{1}^{2}c_{2}^{2}+d_{1}^{2}d_{2}^{2}}\leq
1$ from the inequalities $2a_{1}b_{1}a_{2}b_{2}\leq
a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}$, $2a_{1}b_{1}c_{2}d_{2}\leq
a_{1}^{2}c_{2}^{2}+b_{1}^{2}d_{2}^{2}$, $2c_{1}d_{1}a_{2}b_{2}\leq
c_{1}^{2}a_{2}^{2}+d_{1}^{2}b_{2}^{2}$ and $2c_{1}d_{1}c_{2}d_{2}\leq
c_{1}^{2}c_{2}^{2}+d_{1}^{2}d_{2}^{2}$. Note that the inequality (D22) holds
for special quantum states such as $a_{1}=b_{1},c_{1}=d_{1}$ and
$a_{2}d_{2}=b_{2}c_{2}$.

Example 2. Assume that a $\Lambda$-shaped quantum network consists of two
bipartite Werner states as:

$\displaystyle\rho_{AB}=\frac{1-p}{4}\mathbb{I}_{4}+p|\Phi\rangle_{AB}\langle\Phi|,$
(D23)
$\displaystyle\rho_{CD}=\frac{1-q}{4}\mathbb{I}_{4}+q|\Psi\rangle_{CD}\langle\Psi|,$
(D24)

where $|\Phi\rangle_{AB}=a_{1}|00\rangle+b_{1}|11\rangle$,
$|\Psi\rangle_{CD}=a_{2}|00\rangle+b_{2}|11\rangle$, $\mathbb{I}_{4}$ as the
identity matrix of rank $4$ can be viewed as a mixed fully separable state,
and $0\leq p,q\leq 1$. Define
$\rho_{11}=\frac{1}{16}\mathbb{I}_{4}\otimes\mathbb{I}_{4}$,
$\rho_{12}=\frac{1}{4}\mathbb{I}_{4}\otimes|\Psi\rangle_{CD}\langle\Psi|$,
$\rho_{21}=\frac{1}{4}|\Phi\rangle_{AB}\langle\Phi|\otimes\mathbb{I}_{4}$, and
$\rho_{22}=|\Phi\rangle_{AB}\langle\Phi|\otimes|\Psi\rangle_{CD}\langle\Psi|$.
Let
$\hat{\rho}_{ij}=\textrm{Tr}_{BC}[M_{BC}\rho_{ij}]/\textrm{Tr}[M_{BC}\rho_{ij}]$
for $i,j=1,2$, where $M_{BC}$ denotes the local POVM of Bob.

Similar to Example 1 with the same quantum observables for three observers, we
can easily obtain

$\displaystyle c_{11}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{11}]$
$\displaystyle=$ $\displaystyle 0,$ (D25) $\displaystyle c_{12}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{12}]$
$\displaystyle=$ $\displaystyle 0,$ (D26) $\displaystyle c_{21}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{21}]$
$\displaystyle=$ $\displaystyle 0,$ (D27) $\displaystyle c_{14}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{22}]$
$\displaystyle=$ $\displaystyle
2\cos(\theta)+4\frac{a_{1}b_{1}a_{2}b_{2}}{a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}}\sin(\theta),$
(D28)

From the equations (D25)-(D28), it follows from the equation (D9) or (D10)
that

$\displaystyle\sum_{i,j=1,2}p_{i}q_{j}c_{ij}=$ $\displaystyle
2pq(\cos(\theta)+\beta\sin(\theta))$ $\displaystyle=$ $\displaystyle
2pq\sqrt{1+\beta^{2}}$ $\displaystyle>$ $\displaystyle 2$ (D29)

for some $a_{i},b_{i},c_{i},d_{i},p,q$s, where
$\cos(\theta):=1/\sqrt{1+\beta^{2}}$, and
$\beta=2a_{1}b_{1}a_{2}b_{2}/(a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2})\leq 1$.
For example, the inequality (D29) holds for the following condition

$\displaystyle pq>\frac{1}{\sqrt{1+\beta^{2}}}.$ (D30)

Specially, when $\beta=1$, i.e.,
$2a_{1}b_{1}a_{2}b_{2}=a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}$ including
$a_{i}=b_{i}=1/\sqrt{2}$ (Werner state of EPR state), it follows from the
inequality (D30) that $p>\frac{1}{\sqrt{2}}$ and $q=1$, or
$q>\frac{1}{\sqrt{2}}$ and $p=1$.

## References

  * (1) F. Buscemi, All entangled quantum states are nonlocal, Phys. Rev. Lett. 108, 200401 (2012).
  * (2) J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Phys. 1, 195 (1964).
  * (3) S. Popescu and D. Rohrlich, Generic quantum nonlocality, Phys. Lett. A 166, 293 (1992).
  * (4) B. S. Cirel’son, Quantum generalizations of Bell’s inequality, Lett. Math. Phys. 4, 93 (1980).
  * (5) M. Horodecki, P. Horodecki, and R. Horodecki, Inseparable two spin-(1/2) density matrices can be distilled to a singlet form, Phys. Rev. Lett. 78, 574-577 (1997).
  * (6) N. Gisin, Q. Mei, A. Tavakoli, M. O. Renou, and N. Brunner, All entangled pure quantum states violate the bilocality inequality, Phys. Rev. A 96, 020304(R) (2017).
  * (7) M.-X. Luo, Computationally efficient nonlinear Bell inequalities for quantum networks, Phys. Rev. Lett. 120, 140402 (2018).
  * (8) J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  * (9) N. Gisin, Bell’s inequality holds for all non-product states, Phys. Lett. A 154, 201 (1991).
  * (10) S. Yu, Q. Chen, C. Zhang, C. H. Lai, and C. H. Oh, All entangled pure states violate a single Bell’s inequality, Phys. Rev. Lett. 109, 120402 (2012).