# The Trans-Planckian Censorship Conjecture in different frameworks of viable
inflation

Bruno Sanna1111E-mail address<EMAIL_ADDRESS>Lorenzo
Sebastiani2222E-mail address<EMAIL_ADDRESS>

1Dipartimento di Fisica, Universitá di Pisa, Largo B. Pontecorvo 3, 56127
Pisa, Italy.
1Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo
3, 56127 Pisa, Italy.

###### Abstract

We review the recently proposed _Trans-Planckian Censorship Conjecture_ that
stems from the Trans-Planckian problem of cosmological perturbations. We
analyze the implications and constraints that TCC introduces in different
frameworks of viable inflation. We revisit the case of slow-roll scalar field
inflation and we investigate the cases of slow-roll $f(R)$ and
$f(R,\phi)$-gravity. Finally, we consider the conjecture in the context of
constant-roll scalar field inflation.

##  1 Introduction

The inflationary paradigm, according to which the universe underwent a brief
period of early-time rapid expansion, was initially introduced several years
ago by Starobinsky [1, 2], Guth [3] and Sato [4] and later by Linde [5],
Albrech and Steinhardt [6]; in the last decades several theories have been
suggested in order to describe inflation (see Ref. [7] for a nice review).
Despite the fact that the arena of inflationary models is quite large, the
huge amount of observational data [8, 9] can be used to discriminate between
them the viable ones. In particular, inflation provides a causal mechanism to
generate the primordial inhomogeneities across the matter distribution in our
universe, which evolve and persist in the universe today and which are object
of cosmological observations.

Fluctuations in both matter and gravitational waves are believed to have a
quantum mechanical origin in terms of vacuum perturbations which originate
inside the Hubble radius (or horizon) at the beginning of inflation. During
inflation, when they cross the Hubble horizon, they become classical and later
re-enter the horizon [1, 10]. The study of these perturbations can be carried
out by making use of field-theory computations without invoking any trans-
Planckian physics. However, inflationary cosmology generally suffers from the
so called ‘trans-Planckian problem’, which appears if the macroscopic
fluctuations which cross the Hubble horizon trace back to trans-Planckian
wavelengths at very early times. Since these fluctuations would contribute to
the power spectrum, their computation involves low energy physics into regions
where this physics is not applicable, which is clearly not desirable [11, 12,
13, 14, 15], unless one admits the possibility that trans-Planckian effects
manifest themselves in the form of ultra-high energy particles at any point in
time [16, 17]. Recently, Bedroya and Vafa have proposed an alternative
viewpoint that avoids the Trans-Planckian problem [18]. Their work is
motivated by string theory, and is connected to the broader Swampland
scenario, which encodes the low-energy effective field theories of gravity
that are not compatible with (super)string theory [19, 20]. In this respect
different Swampland conditions have been formulated during the years, such as
the de Sitter Conjecture [21] and the Distance Conjecture [22], limiting the
number of theories that admit an ultraviolet completion (or that belong to the
"_string landscape_ "). Some examples of constraints emerging from the
Swampland criteria can be found in Refs. [23, 24].

The Trans-Planckian Censorship Conjecture (TCC) proposed by Bedroya and Vafa
in the seminal paper [18] states that “a field theory consistent with a
quantum theory of gravity does not lead to a cosmological expansion where any
perturbation with length scale greater than the Hubble radius traces back to
trans-Planckian scales at an earlier time”. In other words, the TCC forbids
Planck-scale perturbations to ever cross the Hubble horizon and enter the
power spectrum. This statement can be formulated in the following mathematical
form (in reduced Planck units),

$\frac{a(t)}{a(t_{0})}l_{Pl}<\frac{1}{H(t)}\longrightarrow\frac{a(t)}{a(t_{0})}<\frac{M_{Pl}}{H(t)}\,,$
(1)

where $a(t)$ and $H\equiv H(t)=\dot{a}(t)/a(t)$ are the scale factor of the
universe and the Hubble parameter, respectively, at a generic time $t$, the
dot being the derivative with respect to time, and $a(t_{0})$ is the scale
factor of the universe at the early-time $t_{0}$, when quantum fluctuations
take place. The Planck length $l_{Pl}$ is related to the reduced Planck Mass
$M_{Pl}$ as $l_{Pl}=1/M_{Pl}$. As a consequence, when the TCC holds true, the
length scales which exit the Hubble horizon preserve a wavelength bigger than
the Planck length back into the past and the trans-Planckian quantum
fluctuations remain quantum.

As an immediate consequence of the TCC we have that $H(t)<M_{Pl}$ in an
expanding universe. Moreover, if the expansion is decelerated only, the TCC is
never violated due to the fact that

$\dot{a}(t)<\dot{a}(t_{0})<a(t_{0})M_{Pl}\longrightarrow
a(t)H(t)<a(t_{0})M_{Pl}\,.$ (2)

Therefore, a possible violation of the TCC takes place if there is an
accelerating expansion somewhere along the way.

As it is well known, the expansion of our universe today is accelerating and
the so called ‘dark energy’ epoch is well described by the Cosmological
Constant which implies a constant Hubble parameter (de Sitter space-time).
Thus, if we assume the validity of the TCC we must accept that the de Sitter
expansion cannot continue for an infinite amount of time and that there should
be an upper bound for the lifetime of the universe (in Ref. [18] it is
estimated as $\sim 2.4$ trillion years). However, if the implications of the
TCC on late-time universe are purely speculative, it is clear that they are
extremely strong for inflation, which describes the early-time cosmic
acceleration. In Ref. [25] it has been found that by assuming the TCC and in
order to obtain a successful inflationary scenario for structure formation of
galaxies, the energy scale of inflation has to be lower than $109$ GeV.
Moreover, for slow-roll inflationary scalar field models, a negligible
amplitude of primordial gravitational waves is predicted with a severe fine-
tuning of initial conditions.

In this paper we would like to generalize these studies. We will investigate
the impact of the TCC on different models and different frameworks of viable
inflation. By ‘viable’ we mean in agreement with cosmological observations,
which constrain the values of the power spectrum of primordial fluctuations,
the spectral index of scalar perturbations and the tensor-to-scalar power
spectra ratio. Our aim is to analyze if and under which conditions inflation
free of trans-Planckian problem can be realized by starting from the TCC. In
Sec. 2 we will revisit the consequences of the TCC in the classical picture of
slow-roll scalar field inflation. In Secs. 3 and 4 we will consider the cases
of $f(R)$-gravity and $f(R,\phi)$-gravity, respectively. In Sec. 5 we will
study the constant-roll scalar field inflationary scenario. Conclusions and
final remarks are given in Sec. 6.

In our convention, the speed of light and the reduced Planck constant are
$c=\hbar=1$.

##  2 Scalar field slow-roll inflation

Let us start by considering a scalar field theory whose action is given by,

$I=\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left(\frac{M_{Pl}^{2}}{2}R-\frac{g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi}{2}-V(\phi)\right)\,,$
(3)

where $g$ is the determinant of the metric tensor and $V(\phi)$ is the
potential of the scalar field $\phi$. The metric of a flat Friedmann-
Robertson-Walker (FRW) space-time is given by,

$ds^{2}=-dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2})\,.$ (4)

Thus, the first Friedmann equation results to be,

$3H^{2}M_{Pl}^{2}=\frac{\dot{\phi}^{2}}{2}+V(\phi)\,,$ (5)

with the associated field conservation law,

$\ddot{\phi}+3H\dot{\phi}=-\frac{dV(\phi)}{d\phi}\,.$ (6)

The dynamic of slow-roll inflation is described by the slow-roll parameters,

$\epsilon_{1}=-\frac{\dot{H}}{H^{2}}\,,\quad\epsilon_{2}=\frac{\dot{\epsilon}_{1}}{H\epsilon_{1}}\,,$
(7)

which should be small during inflation. Thus, the scale-invariant power
spectrum of primordial fluctuations when their wavelength amplitudes are equal
to the horizon size reads,

$\mathcal{P}=\frac{1}{8\pi^{2}\epsilon_{1}}\left(\frac{H^{2}}{M_{Pl}^{2}}\right)_{k=aH}\,,$
(8)

where $k$ is the wavenumber of perturbation.

Let us introduce the $e$-folds parameter,

$N=\ln\left[\frac{a(t_{e})}{a(t)}\right]\,,$ (9)

where $t_{e}$ is the time when inflation ends. In order to solve the problem
of the initial conditions of our Friedmann universe the perturbations must
cross the horizon at $N\sim 55-65$ before the inflation ends. Thus, $N=55-65$
is the minimum expansion rate required for viable inflation. According with
the inhomogeneities observed in our universe, $\mathcal{P}\sim 10^{-9}$, while
the Planck data [8] constrain the spectral index of scalar perturbations
$n_{s}$ and the tensor-to-scalar power spectra ratio $r$ as $n_{s}=0.9649\pm
0.0042$ at $68\%$ CL and $r<0.06$ at $95\%$ CL. These quantities are given by
(in first order approximation),

$n_{s}=1-2\epsilon_{1}-\epsilon_{2}\,,\quad r=16\epsilon_{1}\,,$ (10)

and must be evaluated at $N=55-65$.

Now we will see how the TCC (1) introduces an upper bound for the Hubble
parameter. We make use of the effective Equation of State (EoS) parameter,

$\omega_{\text{eff}}=\frac{\dot{\phi}^{2}-2V(\phi)}{\dot{\phi}^{2}+2V(\phi)}\,.$
(11)

The slow-roll approximation $\epsilon_{1}\,,|\epsilon_{2}|\ll 1$ in Eqs.
(5)–(6) leads to,

$3H^{2}M_{Pl}^{2}\simeq V(\phi)\,,\quad
3H\dot{\phi}\simeq-\frac{V(\phi)}{d\phi}\,,$ (12)

such that

$\epsilon_{1}=\frac{3(1+\omega_{\text{eff}}(N))}{2}\,,\quad\epsilon_{2}=-\frac{d}{dN}\ln(1+\omega_{\text{eff}}(N))\,,$
(13)

where we used the fact that $d/dt=-Hd/dN$ and made explicit the dependence of
$\omega_{\text{eff}}$ on $N$. As a consequence we obtain

$(1-n_{s})=3(1+\omega_{\text{eff}}(N))-\frac{d}{dN}\ln(1+\omega_{\text{eff}}(N))\,,\quad
r=24(1+\omega_{\text{eff}}(N))\,,$ (14)

with $N=55-65$.

From (8) together with (1) we get,

$\epsilon_{1}\simeq\frac{10^{9}}{8\pi^{2}}\left(\frac{H^{2}}{M_{Pl}^{2}}\right)|_{k=aH}<\frac{10^{9}}{8\pi^{2}}\text{e}^{-2\mathcal{N}}\,,$
(15)

where we assumed $\mathcal{P}\sim 10^{-9}$ and where $\epsilon_{1}$ must be
evaluated at $N=55-65$. Here, $\mathcal{N}$ is the total e-folds from the
beginning of inflation and in order to satisfy the TCC we minimized the Hubble
horizon $1/H(t)$ in Eq. (1). Note that in any case the Hubble parameter should
be almost a constant all through the inflation. Moreover, in order to check
whether we meet the TCC condition, by taking into account that in slow-roll
inflation the $\epsilon_{1}$ slow-roll parameter decreases with the $e$-folds
number, we will take $\mathcal{N}$ as the e-folds when perturbations cross the
horizon and we will pose $\mathcal{N}=60$. Thus, we arrive to the following
inequality,

$1>(12\pi^{2})10^{-9}\text{e}^{2\mathcal{N}}(1+\omega_{\text{eff}}(\mathcal{N}))\sim
10^{45}(1+\omega_{\text{eff}}(\mathcal{N}))\,,$ (16)

which is our starting point to analyze viable scalar field inflation in terms
of the effective EoS parameter. We will use a reconstructive approach
following Refs. [26, 27].

Inflation corresponds to a (quasi) de Sitter space-time, when the effective
EoS parameter can be taken close but not equal to the value of minus one.
Since we need an exit from inflation we also must require
$\omega_{\text{eff}}>-1$ (quintessence inflation), due to the fact that, if
$\omega_{\text{eff}}$ passes through the value of minus one, the corresponding
(exact) de Sitter space-time becomes a final attractor of the system and
inflation never ends. Furthermore, we need $\omega_{\text{eff}}$ to approach
$-1/3$ in order eventually end acceleration. A reasonable ansatz for the EoS
parameter in terms of the $e$-folds number is given by (see Ref. [26]),

$\omega_{\text{eff}}(N)=\frac{\beta}{N^{\alpha}}-1\,,\quad
0<\alpha\,,\beta\,,$ (17)

where $\alpha$ and $\beta$ are positive numbers. For large values of $N$ we
have $\omega_{\text{eff}}\simeq-1$, while acceleration ends when $N\rightarrow
0$. As a consequence, the spectral index and the tensor-to-scalar spectra
ratio (14) are derived as,

$(1-n_{s})=\frac{3\beta}{\mathcal{N}^{\alpha}}+\frac{\alpha}{\mathcal{N}}\,,\quad
r=\frac{24\beta}{\mathcal{N}^{\alpha}}\,.$ (18)

Since we are considering $\mathcal{N}=60$, the spectral index $n_{s}$
satisfies the Planck constraint only if $\alpha=1$ or $\alpha=2$, but in the
first case $\beta$ should be $\beta\simeq 1/3$ and the tensor-to-scalar ratio
is ruled out by observations. Scalar field equation with effective EoS
parameter in the form of (17) with $\alpha=1$ corresponds to power-law
potentials [27] and the choice $\beta=1/3$ leads to a quadratic potential,
whose viability fell down due to the incompatibility with the observed tensor-
to-scalar spectra ratio.

Thus, we will focus on the case $\alpha=2$, namely

$\omega_{\text{eff}}(N)=\frac{\beta}{N^{2}}-1\,,\quad\beta>0\,,$ (19)

in order to have

$(1-n_{s})\simeq\frac{2}{\mathcal{N}}\,,\quad
r=\frac{24\beta}{\mathcal{N}^{2}}\,,$ (20)

which are in general in agreement with the Planck data.

The TCC condition (16) reads,

$\beta^{-1}>\frac{10^{45}}{\mathcal{N}^{2}}\simeq 3\times 10^{41}\,,$ (21)

and we find the following upper bound on the parameter $\beta$,

$\beta<3\times 10^{-42}\,.$ (22)

This result brings to

$r<10^{-44}\,,$ (23)

confirming the severe fine-tuning of initial conditions found in Ref. [25].
However, we can explicitly reconstruct a viable model which is compatible with
the TCC predicting a strong suppression of the amplitude of primordial
gravitational waves. As a matter of fact, the EoS parameter (19) corresponds
to an exponential potential, as we can easily verify. By using the prime index
to denote the derivative with respect to the $e$-folds number, in slow-roll
approximation (12) we derive,

$(1+\omega_{\text{eff}}(N))\simeq 2H^{\prime}/(3H)\,,$ (24)

such that by using (19) we obtain a differential equation for $H$ whose
solution reads

$H^{2}\simeq\frac{\rho_{0}}{3M_{Pl}^{2}}\text{e}^{-\frac{3\beta}{N}}\,.$ (25)

Here, $\rho_{0}$ is an integration constant whose physical meaning is the
effective energy density of the universe at the beginning of inflation, when
$\mathcal{N}$ is quite large. Now, by equaling $3H^{2}M_{Pl}^{2}$ to $V(\phi)$
we obtain, in slow-roll approximation,

$\omega_{\text{eff}}\simeq-1+\frac{1}{9\beta}\ln^{2}\left[\frac{V(\phi)}{\rho_{0}}\right]\,,$
(26)

and together with Eq. (11) we get

$\dot{\phi}\simeq-\frac{\sqrt{V(\phi)}}{3\sqrt{\beta}}\ln\left[\frac{V(\phi)}{\rho_{0}}\right]\simeq\frac{\sqrt{V(\phi)}}{3\sqrt{\beta}}\left[\frac{\rho_{0}}{V(\phi)}-1\right]\,,$
(27)

where we assumed $\dot{\phi}>0$ during inflation, when $V(\phi)$ is smaller
and close to the initial (effective) energy density $\rho_{0}$. Now, by making
use of the second equation in (12) we are able to reconstruct the full form of
the scalar field potential as

$V(\phi)=\rho_{0}\left(1-c_{1}\text{e}^{\frac{1}{\sqrt{3\beta}}\frac{\phi}{M_{Pl}}}\right)\,,$
(28)

where $c_{1}$ is a positive dimensional integration constant and we are taking
$\phi<0$.

In terms of the cosmological time the explicit solutions $H\equiv H(t)$ and
$\phi\equiv\phi(t)$ of (12) are given by,

$H(t)^{2}=\frac{\rho_{0}}{3M_{Pl}^{2}}\left(1-\frac{3\sqrt{3}\beta
M_{Pl}}{\sqrt{\rho_{0}}(t_{e}-t)}\right)\,,\quad\phi(t)=-\sqrt{3\beta}M_{Pl}\ln\left[\frac{c_{1}\sqrt{\rho_{0}}}{3\sqrt{3}\beta
M_{Pl}}(t_{e}-t)\right]\,,$ (29)

where $t_{e}$ is approximately the time when inflation ends and $\beta
M_{Pl}/(c_{1}\sqrt{\rho_{0}})\ll t_{e}$ such that $|\phi|/M_{Pl}\gg 1$ at the
beginning of inflation. The $e$-folds $N\equiv N(t)$ is given by,

$N(t)\simeq\frac{1}{M_{Pl}^{2}}\int^{\phi(t)}_{\phi(t_{e})}\frac{V(\phi)}{V^{\prime}(\phi)}d\phi\simeq\frac{3\beta}{c_{1}}\text{e}^{-\sqrt{\frac{1}{3\beta}}\frac{\phi(t)}{M_{Pl}}}=\sqrt{\frac{\rho_{0}}{3}}\frac{(t_{e}-t)}{M_{Pl}}\,,$
(30)

and the total amount of inflation results to be

$\mathcal{N}\equiv N(0)\simeq\sqrt{\frac{\rho_{0}}{3}}\frac{t_{e}}{M_{Pl}}\,.$
(31)

Finally, the slow-roll parameters (7) in terms of the cosmological time read

$\epsilon_{1}=\frac{9\beta
M_{Pl}^{2}}{2\rho_{0}(t_{e}-t)^{2}}\,,\quad\epsilon_{2}=\frac{2}{(t_{e}-t)}\sqrt{\frac{3M_{Pl}^{2}}{\rho_{0}}}\,.$
(32)

As we observed above, when $\mathcal{N}=60$, the TCC condition is satisfied
for $\beta<3\times 10^{-42}$. It means that the Hubble parameter is a constant
during almost all the early-time acceleration and only at the very end of
inflation goes to zero.

As a last remark we note that the model with $c_{1}=2$ and $\beta=1/2$
corresponds to a scalar field inflation of the Starobinsky model in the
Einstein frame [27], which clearly violates the TCC condition. In the next
sections, we will investigate the TCC in inflationary modified gravity
theories frameworks.

##  3 The case of $f(R)$-gravity

A different approach to inflation is given by the modified theories of
gravity, where the gravitational Lagrangian is described as a general function
of some curvature invariants. Generally speaking, one expects that at the
early time some corrections to Hilbert-Einstein action arise, maybe related to
quantum effects at high curvature [28, 29, 30]. In this Section we would like
to analyze the simplest class of such models, namely $f(R)$-gravity, where the
Lagrangian depends on the Ricci scalar only [31, 32, 33, 34, 35, 36].

Let us consider the gravitational action,

$I=\frac{M_{Pl}^{2}}{2}\int_{\mathcal{M}}d^{4}\sqrt{-g}\,f(R)\,,$ (33)

where $f(R)$ is a function of the Ricci scalar $R$. The first Friedmann-like
equation is given by,

$3FH^{2}=\frac{(FR-f)}{2}-3H\dot{F}\,,$ (34)

with $F=df/dR\,,f\equiv f(R)$.

In the framework of $f(R)$-gravity slow-roll inflation is described by the
slow-roll parameters [37, 38, 39],

$\epsilon_{1}=-\frac{\dot{H}}{H^{2}}\simeq-\epsilon_{3}(1-\epsilon_{4})\,,\quad\epsilon_{4}=-3\epsilon_{1}+\frac{\dot{\epsilon}_{1}}{H\epsilon_{1}}\,,$
(35)

whose magnitude is assumed to be small during inflation333 In the next section
we will consider the more general framework of $f(R\,,\phi)$-gravity which
includes $f(R)$-gravity as a special case. Thus, $\epsilon_{1}\,,\epsilon_{3}$
and $\epsilon_{4}$ are labelled according with the corresponding slow-roll
parameters in $f(R\,,\phi)$-gravity.. We note that at the first order
approximation the $\epsilon_{3}$ slow-roll parameter coincides with the
opposite value of the $\epsilon_{1}$ slow-roll parameter and in the following
expressions for the power spectrum and the spectral index we will pose
$\epsilon_{3}\simeq-\epsilon_{1}\simeq\dot{H}/H^{2}$. However, the tensor-to-
scalar spectra ratio must be evaluated at the second leading order of
$\epsilon_{1}+\epsilon_{3}\simeq(\dot{H}/H^{2})\epsilon_{4}$, which implicitly
defines $\epsilon_{3}$.

The power spectrum of cosmological perturbations results to be [31]

$\mathcal{P}=\frac{1}{24\pi^{2}F\epsilon_{1}^{2}}\left(\frac{H}{M_{Pl}}\right)^{2}_{k=aH}\simeq\frac{1}{24\pi^{2}F\epsilon_{3}^{2}}\left(\frac{H}{M_{Pl}}\right)^{2}_{k=aH}\,,$
(36)

while the spectral index and the tensor-to-scalar power spectra scalar ratio
read (in the first and second order approximations),

$n_{s}=1-4\epsilon_{1}+2\epsilon_{3}-2\epsilon_{4}\simeq
1-6\epsilon_{1}-2\epsilon_{4}\,,\quad r=16(\epsilon_{1}+\epsilon_{3})\simeq
48\epsilon_{1}^{2}\,.$ (37)

As well as in the previous case, it is convenient to introduce an effective
equation of state parameter as in Eq. (24). In terms of the $e$-folds we get,

$1-n_{s}=-2\frac{d}{dN}\ln(1+\omega_{\text{eff}}(N))\,,\quad
r=108(1+\omega_{\text{eff}}(N))^{2}\,,$ (38)

with $N=55-65$. Thus, the TCC condition holds true if

$1>54\pi^{2}10^{-9}\text{e}^{2\mathcal{N}}F(\mathcal{N})(1+\omega_{\text{eff}}(\mathcal{N}))^{2}\sim
6\times 10^{45}F(\mathcal{N})(1+\omega_{\text{eff}}(\mathcal{N}))^{2}\,.$ (39)

Since we are interested in the sufficient condition to meet the TCC condition,
we posed again the total $e$-folds from the beginning of inflation equal to
the $e$-folds when perturbations cross the horizon, namely $\mathcal{N}=60$,
and we considered the implicit form of $F$ as a function of $\mathcal{N}$.

As in §2, we can assume the ansatz (17) for the effective EoS parameter
$\omega_{\text{eff}}(N)$. As a consequence we obtain

$(1-n_{s})=\frac{2\alpha}{\mathcal{N}}\,,\quad
r=\frac{108\beta^{2}}{\mathcal{N}^{2\alpha}}\,.$ (40)

In this case the Planck constraint on $n_{s}$ implies $\alpha=1$ [27, 34],

$\omega_{\text{eff}}(N)=\frac{\beta}{N}-1\,,\quad\beta>0\,,$ (41)

which leads to

$(1-n_{s})=\frac{2}{\mathcal{N}}\,,\quad
r=\frac{108\beta^{2}}{\mathcal{N}^{2}}\,.$ (42)

This choice corresponds to the Hubble parameter,

$H^{2}=\frac{1}{3M_{Pl}^{2}}\rho_{e}N^{3\beta}\,,$ (43)

which follows from Eq. (24). Here, $\rho_{e}$ is an integration constant
representing the effective energy density of the universe toward the end of
inflation (at $N\simeq 1$). Now we can infer the implicit form of $F(N)$ from
Eq. (34) which reads,

$-4HH^{\prime}(F-1)+2H^{2}F^{\prime}+2H^{2}F^{\prime\prime}+2HH^{\prime}F^{\prime}=\frac{2\rho_{e}\beta}{M_{Pl}^{2}}\,,$
(44)

with $F\equiv F(N)$. A simple analytic solution can be found for $\beta=1/3$,
namely

$F=c_{0}\left(\frac{1}{2}+N\right)\,.$ (45)

By taking into account that

$R=12H^{2}+6\dot{H}=12H^{2}-6HH^{\prime}\,,$ (46)

we easily derive

$F=\frac{c_{0}}{2}+\frac{3M_{Pl}^{2}c_{0}}{18\rho_{e}}R\,.$ (47)

The $f(R)$-model can be finally fully reconstructed as

$f=R+\frac{M_{Pl}^{2}}{6\rho_{e}}R^{2}\,,$ (48)

where we posed $c_{0}=2$ in order to recover the Hilbert-Einstein term of
General Relativity (GR). This model is nothing else but the Starobinski model
[2], which clearly violates the TCC condition (39). This fact is not
surprising, since the Starobinsky model in the Einstein-frame leads to the
scalar model with potential (28) and $\beta=1/2$, $c_{1}=2$. The conformal
transformation between the two frames is given by,

$\phi(N)=-\sqrt{\frac{3M_{Pl}^{2}}{2}}\ln F(N)\,.$ (49)

Now it is easy to verify that the power spectra of the two models coincide
after the identification $\rho_{e}=3\rho_{0}/2$ (note that the inflation
scales in the two frames do not coincide).

One may be interested to see if Eq. (44) admits some solutions for small
values of $\beta$, when the Hubble parameter reads

$H\simeq\sqrt{\frac{\rho_{e}}{3M_{Pl}^{2}}}\left(1+\frac{3\beta}{2}\log
N\right)\,.$ (50)

Thus, an implicit solution of Eq. (44) to the leading-order term of $\beta$
which allows to recover the Hilbert-Einstein contribution of GR is given by,

$F=1+3\beta N\,.$ (51)

In this case the TCC condition (39) is satisfied under the condition

$\beta<8\times 10^{-22}\,,$ (52)

which brings to

$r<2\times 10^{-44}\,.$ (53)

Due to the constraint on $\beta$ the Hubble parameter remains a constant
during inflation and starts to decrease at the very end of it. Moreover, as in
the case of scalar field slow-roll inflation, in this class of viable
$f(R)$-gravity models compatible with the TCC we have a strong suppression of
the amplitude of the primordial gravitational waves and a severe fine-tuning
of initial conditions.

Despite the fact that our analysis is not exhaustive of the wide variety of
$f(R)$-models for inflation, we can draw some conclusions. Since viable
$f(R)$-gravity reduces to Einstein gravity at small curvature, it is clear
that the TCC condition given in Eq. (39) introduces an upper bound on the
effective EoS parameter as $(1+\omega_{\text{eff}})\lesssim 10^{-23}$, such
that the tensor-to-scalar spectra ratio in (37) results to be at most $r\sim
10^{-44}$. This result is independent of the ansatz on $\omega_{\text{eff}}$
and is in agreement with (52)–(53). Thus, the suppression of the amplitude of
the primordial gravitational waves seems to be a general feature of viable
slow-roll $f(R)$-gravity compatible with TCC and since we have assumed as a
minimal requirement that the total amount of inflation coincides with the
$e$-folds at the perturbation horizon crossing, we get the fine-tuning problem
of initial conditions.

##  4 The case of $f(R,\phi)$-gravity: two specific examples of slow-roll
inflation

An important class of inflationary models is given by scalar-tensor theories,
where the gravitational interaction is mediated by both a scalar and a tensor
field [40, 41]. In what follows, in the attempt to investigate the TCC in this
framework, we will consider two specific examples of $f(R,\phi)$-slow-roll
inflation firstly presented in Ref. [42], where a scalar field is coupled with
the Ricci scalar.

The general action of $F(R,\phi)$-gravity is in the form,

$I=\int_{\mathcal{M}}dx^{4}\sqrt{-g}\left[\frac{M_{Pl}^{2}f(R,\phi)}{2}-\frac{g^{\mu\nu}\partial_{\mu}\phi\partial{\nu}\phi}{2}-V(\phi)\right]\,,$
(54)

but in what follows we will assume $V(\phi)=0$.

In terms of the $e$-folds number, the slow-roll parameters describing slow-
roll $f(R,\phi)$-inflation are given by [37],

$\epsilon_{1}=\frac{H^{\prime}}{H}\,,\quad\epsilon_{2}=-\frac{HH^{\prime}\phi^{\prime}+H^{2}\phi^{\prime\prime}}{H^{2}\phi^{\prime}}\,,\quad\epsilon_{3}=-\frac{F^{\prime}}{2F}\,,\quad\epsilon_{4}=-\frac{E^{\prime}}{2E}\,,$
(55)

where $F=\partial f/\partial R\,,f\equiv f(R,\phi)$, and

$E=F+\frac{3M_{Pl}^{2}F^{\prime 2}}{16\pi\phi^{\prime 2}}\,.$ (56)

The first Friedmann-like equation in slow-roll approximation leads to

$3FH^{2}\simeq\frac{1}{2}\left(RF-f\right)\,,$ (57)

while the conservation law related to the field reads,

$-3H^{2}\phi^{\prime}\simeq\frac{1}{2}\frac{\partial f}{\partial\phi}\,.$ (58)

The power spectrum of cosmological perturbations is given by,

$\mathcal{P}=\frac{1}{32\pi^{3}Q_{s}}H^{2}|_{k=aH}\,,$ (59)

with

$Q_{s}=\frac{\phi^{\prime 2}E}{F}\,.$ (60)

As a check, we observe that when $f(R,\phi)=R$ and therefore
$Q_{s}=\phi^{\prime 2}$, in slow-roll approximation we recover Eq. (8). On the
other hand, if $f(R,\phi)=f(R)$, $\phi^{\prime}=0$ and therefore
$Q_{s}=3M_{Pl}^{2}F^{\prime 2}/(16\pi F)$, we recover Eq. (36).

The TCC introduces the following constraint,

$Q_{s}<\frac{10^{9}}{32\pi^{3}}M_{Pl}^{2}\text{e}^{-2\mathcal{N}}\,,$ (61)

with $\mathcal{N}=60$, as per usual. Finally, we recall the expressions for
the spectral index and the tensor-to-scalar power spectra ratio,

$n_{s}=1-4\epsilon_{1}-2\epsilon_{2}+2\epsilon_{3}-2\epsilon_{4}\,,\quad
r=16(\epsilon_{1}+\epsilon_{3})\,,$ (62)

with $N=55-65$. As a check, we note that when $f(R,\phi)=R$ such that
$\epsilon_{3}=\epsilon_{4}=0$, by taking into account that, in slow-roll
approximation, $2\epsilon_{2}=-\epsilon_{1}-H^{\prime\prime}/H^{\prime}$, we
correctly find the results of slow-roll scalar field inflation (10) where
$\epsilon_{2}=H^{\prime}/H-H^{\prime\prime}/H$. Moreover, if we pose
$f(R,\phi)=f(R)$, $\phi^{\prime}=0$, we get [34] $\epsilon_{2}=0$,
$\epsilon_{1}\simeq-\epsilon_{3}(1-\epsilon_{4})$ and, by taking into account
that in slow-roll approximation $\epsilon_{1}\simeq-\epsilon_{3}$ and
$\epsilon_{4}\simeq-3\epsilon_{1}-\epsilon_{1}^{\prime}/\epsilon_{1}$, we
recover the results of pure $f(R)$-gravity (37).

In Ref. [42] inflation is realized thanks to a sort of switching on the
cosmological constant in two different models. The first model reads,

$f(R\,,\phi)=R-2\lambda\left(1-\text{e}^{b\left(\frac{1}{3M_{P}l^{2}}\right)^{3}\phi
R}\right)\,,\quad b>0\,,$ (63)

where $b$ is a positive parameter and $\lambda$ is a positive cosmological
constant (on curvature scale of inflation). Inflation starts with $\phi$
negative and very small ($|\phi|\ll 0$), such that we obtain a quasi-de Sitter
expansion with

$H^{2}\simeq\frac{\lambda}{3}\,.$ (64)

Thus, from Eq. (58) with $R\simeq 12H^{2}$ we obtain,

$\phi=-\frac{\log\left[16b^{2}\left(\frac{1}{3M_{P}l^{2}}\right)^{6}\lambda^{2}N\right]}{4b\left(\frac{1}{3M_{P}l^{2}}\right)^{3}\lambda}\,.$
(65)

The field grows during inflation which ends at $N\rightarrow 0$. A direct
evaluation of the slow roll parameters shows that

$\epsilon_{1}\,,\epsilon_{3}\,,\epsilon_{4}\sim\frac{M_{Pl}^{4}}{b^{2}\lambda^{2}N^{2}}\,,\quad\epsilon_{2}=\frac{1}{N}\,.$
(66)

Thus, a viable scenario with

$1-n_{s}\simeq\frac{2}{\mathcal{N}}\,,\quad
r\sim\frac{M_{Pl}^{4}}{b^{2}\lambda^{2}\mathcal{N}^{2}}\,,$ (67)

takes place. Moreover, since $Q_{s}\simeq\phi^{\prime 2}$, we observe that the
TCC condition (61) holds true if

$\frac{M_{Pl}^{4}}{b^{2}\lambda^{2}\mathcal{N}^{2}}<256\times
10^{9}\text{e}^{-2\mathcal{N}}\simeq 2\times 10^{-41}\,,$ (68)

and the amplitude of primordial gravitational waves is again strongly
suppressed. By taking into account that $R\sim\lambda$ condition (68) also
guarantees that the Hubble parameter is almost a constant until the very end
of inflation, due to the fact that $f$ turns out to behave as $f\simeq
R-2\lambda$, unless $N$ is very close to zero.

The second model under consideration reads,

$f(R\,,\phi)=R-2\lambda\left(1-\frac{1}{1+\left(-b\left(\frac{8\pi}{3M_{P}l^{2}}\right)^{3}\phi
R\right)^{n}}\right)\,,\quad n>0\,,b>0\,,$ (69)

with $n,b$ positive parameters and $\lambda$ a cosmological constant. Once
again, inflation is supported by a quasi-de Sitter solution with $\phi$
negative and very small such that

$H^{2}\simeq\frac{\lambda}{3}\,.$ (70)

The field behaves as,

$\phi=-\frac{(2+n)^{\frac{1}{2+n}}(nN)^{\frac{1}{2+n}}}{\left(4\lambda
b\left(\frac{1}{3M_{P}l^{2}}\right)^{3}\right)^{\frac{n}{2+n}}}\,,$ (71)

and the early-time acceleration ends when $N\rightarrow 0$. The slow-roll
parameters are derived as

$\epsilon_{1}\,,\epsilon_{3}\,,\epsilon_{4}\sim\frac{M_{Pl}^{4}}{b^{2}\lambda^{2}N^{2}}\,,\quad\epsilon_{2}=\frac{1}{N}\left(\frac{1+n}{2+n}\right)\,.$
(72)

As a consequence,

$1-n_{s}\simeq\frac{2}{N}\left(\frac{1+n}{2+n}\right)\simeq\quad
r\sim\frac{M_{Pl}^{4}}{b^{2}\lambda^{2}N^{2}}\,,$ (73)

and the spectral index $n_{s}$ is in agreement with the Planck observations
only for large values of $n$. In this case the TCC is satisfied under the
requirement

$\frac{M_{Pl}^{4}}{(n+2)^{2}b^{2}\lambda^{2}(\mathcal{N})^{2}}<8^{3}\times
10^{9}\text{e}^{-2\mathcal{N}}\simeq 4\times 10^{-41}\,,$ (74)

confirming the suppression of the amplitude of primordial gravitational waves
as the price to pay for the validity of the TCC condition.

Up to now we have considered models of slow-roll inflation. The slow-roll
approximation is valid if all the slow-roll parameters are small during
inflation. However, in order to obtain a constant Hubble parameter (or a flat
potential, in the classical scenario of scalar field inflation) it is enough
to require that $\epsilon_{1}\ll 1$, while the other horizon flow parameters
can also be not so small, but constant. In the next Section, we will study the
consequences of the TCC in the case of the so called ‘constant-roll’ inflation
scenario.

##  5 Constant-roll scalar field inflation

Constant-roll inflation has some important and interesting properties. For
example, it can generate large local non-gaussianities (which are negligible
in the case of slow-roll inflation) and the curvature perturbations may grow
on super-horizon scales [43, 44, 45, 46]. In Ref. [47] constant-roll scalar
field inflation has been studied and exact solutions for the inflaton
potential have been found (see also Refs. [48, 49, 50, 51] for constant-roll
inflation in modified gravity). We recall these results in the context of the
TCC.

The action of the theory is still given by (3). Scalar field constant-roll
inflation takes place when $\ddot{\phi}\sim H\dot{\phi}$, being not negligible
in Eq. (6). Following the prescription used in Ref. [47] we assume

$\ddot{\phi}=-(3+\alpha)H\dot{\phi}\,.$ (75)

For $\alpha=-3$ we obtain the slow-roll approximation. We will investigate two
models, for which all the possible values of $\alpha\neq-3$ are covered. The
first model reads,

$V(\phi)=3M^{2}M_{Pl}^{2}\left[1+\frac{\alpha}{6}\left(1-\cosh\sqrt{2(3+\alpha)}\frac{\phi}{M_{P}l}\right)\right]\,,\quad-3<\alpha\,,$
(76)

where $0<M$ is a generic mass constant. We assume $-\infty<\phi<0$ and
$0<\dot{\phi}$ during inflation. If $-3<\alpha<0$ the potential has a minimum
(i.e. an attractor point) for $\phi\rightarrow 0^{-}$, while if $0<\alpha$ the
field reaches a maximum of the potential when $\phi\rightarrow 0^{-}$.

The second model is described by the following field potential,

$V(\phi)=3M^{2}M_{Pl}^{2}\left[1+\frac{\alpha}{6}\left(1-\cos\sqrt{-2(3+\alpha)}\frac{\phi}{M_{P}l}\right)\right]\,,\quad\alpha<-3\,.$
(77)

Here, we are assuming $\infty>\phi>0\,,\dot{\phi}>0$ and the potential has a
maximum at $\phi=0^{+}$.

The exact solutions of the field equations (5)–(6) in the case of (76) are

$\displaystyle H(t)^{2}$ $\displaystyle=$ $\displaystyle
M^{2}\coth^{2}\left[(3+\alpha)Mt\right]\,,$ $\displaystyle\phi(t)$
$\displaystyle=$ $\displaystyle
M_{Pl}\sqrt{\frac{2}{3+\alpha}}\ln\left[\coth\left[\frac{(3+\alpha)}{2}Mt\right]\right]\,.$
(78)

When $t\rightarrow 0^{+}$ the Hubble parameter is not a constant (but we can
still have an acceleration). Nevertheless, since the Hubble parameter
approaches a constant when $t\rightarrow+\infty$ and the scale factor grows
exponentially, we eventually have inflation. Additionally, in this case a
transition phase at the end of inflation has to be assumed (see Refs. [52, 53,
54]).

The exact solutions of (5)–(6) in the case of (77) are

$\displaystyle H(t)^{2}$ $\displaystyle=$ $\displaystyle
M^{2}\tanh^{2}\left[(3+\alpha)Mt\right]\,,$ $\displaystyle\phi(t)$
$\displaystyle=$ $\displaystyle
2M_{Pl}\sqrt{-\frac{2}{3+\alpha}}\arctan\left[\text{e}^{-(3+\alpha)Mt}\right]\,,$
(79)

with $-\infty<t<0$, such that the Hubble parameter is almost a constant when
$t\rightarrow-\infty$ and we have inflation, while goes to zero when
$t\rightarrow 0^{-}$.

We will denote with $t_{0}\,,t_{e}$ the time when inflation starts and ends,
respectively. The $e$-folds number (9) reads,

$N=\int^{t_{e}}_{t}Hdt\,.$ (80)

For the model (76) we obtain,

$N=\frac{1}{(3+\alpha)}\log\left[\frac{(a(t_{e})/a(t_{0}))^{3+\alpha}}{\sinh[(3+\alpha)Mt]}\right]\rightarrow
t=\frac{\text{arcsinh}\left[\text{e}^{-(3+\alpha)N}(a(t_{e})/a(t_{0}))^{3+\alpha}\right]}{M(3+\alpha)}\,,$
(81)

where we have evaluated $a(t)=a(t_{0})(\sinh[(3+\alpha)Mt])^{1/(3+\alpha)}$
and we have posed $\sinh[(3+\alpha)Mt_{0}]=1$.

For the model (77) we derive,

$N=\frac{1}{(3+\alpha)}\log\left[\frac{1}{\cosh[(3+\alpha)Mt]}\right]\rightarrow
t=\frac{\text{arccosh}\left[\text{e}^{-(3+\alpha)N}\right]}{M(3+\alpha)}\,,$
(82)

where we have evaluated $a(t)=a(t_{e})(\cosh[(3+\alpha)Mt])^{1/(3+\alpha)}$
and we have posed $\cosh[(3+\alpha)Mt_{e}]=1$ (namely $t_{e}=0$).

We can now estimate the $\epsilon_{1}$ slow-roll parameter (7) in the two
models as

$\epsilon_{1}=\frac{3+\alpha}{\cosh^{2}\left[(3+\alpha)Mt\right]}=\frac{3+\alpha}{1+\text{e}^{2(3+\alpha)(\mathcal{N}-N)}}\,,\quad-3<\alpha\,,$
(83)
$\epsilon_{1}=-\frac{3+\alpha}{\sinh^{2}\left[(3+\alpha)Mt\right]}=\frac{3+\alpha}{1-\text{e}^{-2(3+\alpha)N}}\,,\quad\alpha<-3\,,$
(84)

where we have introduced the total $e$-folds number $\mathcal{N}$ through the
relation (9). Thus, for $-3<\alpha$, the bound of the $\epsilon_{1}$ slow-roll
parameter at the time $t=t_{0}$ (namely, $N=\mathcal{N}$) is given by
$\epsilon_{1}=(3+\alpha)/2$ and the parameter decreases with time through a
quintessence region. A remark is in order. When $\epsilon_{1}>1$ the
acceleration does not take place. However, it is clear that if
$(3+\alpha)\sim\mathcal{O}(1)$ the acceleration phase with $\epsilon_{1}\ll 1$
(namely, $H$ almost a constant) is immediately reached. For this reason we
still indicate with $\mathcal{N}$ the total amount of inflation. On the other
hand, for $\alpha<-3$, the $\epsilon_{1}$ parameter is negligible at the
beginning of inflation and increases with the time.

The power spectrum of scalar perturbations is given by [47],

$\mathcal{P}=\frac{1}{8\pi^{2}\epsilon_{1}}\left(\frac{H^{2}}{M_{Pl}^{2}}\right)_{k=aH}\frac{2^{2\nu-1}}{\pi}|\Gamma\left(\nu\right)|^{2}\,,\quad\nu=|\alpha+\frac{3}{2}|\,.$
(85)

As a check, note that when $\alpha=-3$ we recover the result of slow-roll
inflation. Furthermore, the spectral index $n_{s}$ is related to $\alpha$ as

$\alpha=\frac{1-n_{s}}{2}\,,\frac{n_{s}-7}{2}\,,$ (86)

such that in order to obtain $n_{s}=0.96$ with $\alpha>-3$ (model (76)) we
must fix $\alpha=0.02$, while with $\alpha<-3$ (model (77)) we must require
$\alpha=-3.02$, namely we are near the slow-roll approximation region.
Finally, the tensor-to-scalar power spectra ratio is still related to
$\epsilon_{1}$ as

$r=16\epsilon_{1}\,,$ (87)

where we remember that $\epsilon_{1}$ must be evaluated at the time when
perturbations cross the horizon, at $N=55-65$. In the case of $\alpha=0.02$ in
order to satisfy the Planck data with $r\lesssim 0.06$ we should require
$61.1\lesssim\mathcal{N}$ if $\epsilon_{1}$ is evaluated at $N=60$, namely the
total time of inflation must exceed by at least one the number of $e$-folds of
the perturbation horizon crossing. Moreover, for $\alpha=-3.02$ the tensor-to-
scalar ratio is in agreement with the Planck data, leading to $r\simeq 0.03$
if $\epsilon_{1}$ is evaluated at $N=60$.

Now, the TCC condition leads to

$\epsilon_{1}<\frac{10^{9}}{(8\pi)^{2}}\text{e}^{-2\mathcal{N}}\frac{2^{2\nu-1}}{\pi}|\Gamma\left(\nu\right)|^{2}\,,$
(88)

where $\epsilon_{1}$ is given by (83)–(84) with $N=60$.

Let us have a look for the viable model with $\alpha=-3.02$. The sufficient
condition for the validity of the TCC can be found as in the slow-roll
inflation scenario and is realized when $\mathcal{N}$ assumes the minimal
value, namely $\mathcal{N}=60$, when it is clearly violated. We remark that in
this case we are near the slow-roll approximation region. In Ref. [47] it is
argued that one can obtain $r\simeq 3\times 10^{-3}$ (like in Starobinsky
inflation) by setting $\phi\sim M_{Pl}$ at the beginning of inflation. Here
the result can be derived directly from (84) and (87) by assuming
$\mathcal{N}\simeq 60$. Thus, the model is affected by Trans-Planckian
problem.

The situation is different for constant-roll inflation with $\alpha=0.02$. In
this case the $\epsilon_{1}$-parameter decreases with the $e$-folds and the
sufficient condition for the validity of the TCC is realized when
$\mathcal{N}$ is much larger than its minimal value for viable inflation,
namely $\mathcal{N}\gg 61.1$. Specifically, we find that for

$86.4\lesssim\mathcal{N}\,,$ (89)

the TCC condition (88) is satisfied. The results show that in the constant-
roll inflationary scenario it is is possible to deal with viable inflation in
agreement with the TCC provided that inflation starts much before the time
when perturbations cross the Hubble horizon. Also in this case the tensor-to-
scalar ratio $r$,

$r<3\times 10^{-68}\,,$ (90)

is extremely small and the model predicts a strong suppression of the
amplitude of gravitational waves. However, at the beginning of inflation
$\epsilon_{1}=1.51$ and is large enough to avoid a fine-tuning problem of the
initial conditions, thanks to its peculiar behaviour.

##  6 Conclusions

In this paper we revisited the Trans-Planckian Censorship Conjecture in
different models of viable inflation. As already observed in Ref. [25] and,
more recently, in Ref. [55], the TCC tightly constrains slow-roll scalar field
inflation. Here, we first extended the result to different frameworks of slow-
roll inflation. For scalar field theory and $f(R)$-gravity, we used a general
approach which permits to reconstruct the models that lead to the power
spectrum of scalar perturbations, spectral index and tensor-to-scalar spectra
ratio in agreement with Planck data and where the TCC holds true. For
$f(R,\phi)$-inflation, we proposed the study of two viable models. In this
cases, we found that although under certain conditions we can obtain viable
inflation free of the trans-Planckian problem, we get a severe fine-tuning of
initial conditions. Moreover, these models predict a strong suppression of the
amplitude of primordial gravitational waves as a direct consequence of the
TCC. In the second part of the paper, moving away from slow-roll scenario, two
examples of constant-roll scalar field inflation have been analyzed. Here we
found that in principle it is possible to deal with viable inflation avoiding
both trans-Planckian problem and fine-tuning problem of initial conditions, by
asking that inflation starts much before the time when perturbations cross the
Hubble horizon. We should stress that the result is related to a peculiar
mechanism of inflation where the $\epsilon_{1}$ slow-roll parameter decreases
with cosmological time and the de Sitter expansion is an attractor of the
system, such that it is known that a transition phase at the end of inflation
must be introduced.

We may conclude that, since in order to avoid the presence of fluctuations
which trace back to quantities beyond the Planck scale in the classical power
spectrum the TCC imposes severe constraints on the majority of the
inflationary models, different mechanisms (as in slow-roll inflation) or
different approaches (like the cosmological bounce) are the natural
implications of the conjecture itself.

A last remark is in order. As briefly considered in the introduction, it
should be again emphasized that in the TCC, as well as in the consideration of
the trans-Planckian problem, the investigation of generation of metric and
fields is restricted only to the scenario where metric fluctuations become
large and quasi-classical, which may be thought of as a deficiency of the
conjecture. Specifically, the important case of particle creation when metric
and field fluctuations remain quantum but show themselves in the form of
ultra-high energy particles is neglected. In this instance, as shown in Ref.
[17], any deviation of the quantum state of trans-Planckian modes from the
adiabatic vacuum one would result in the appearance of super-high energy
particles in any expanding universe and at any time, including the present
time.

## Acknowledgement

The authors wish to thank D. Grasso and G. Marozzi for useful discussions.

## References

  * [1] A. A. Starobinsky, JETP Lett. 30, 682 (1979)
  * [2] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
  * [3] A. H. Guth, Phys. Rev. D 23, 347 (1981).
  * [4] K. Sato, Mon. Not. R. Astron. Soc. 195, 467 (1981); Phys. Lett. 99B, 66 (1981).
  * [5] A.D. Linde, Phys. Lett. B 108 (1982), n. 6, 389-393 doi:10.1016/0370-2693(82)91219-9
  * [6] A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982), 1220 doi:10.1103/PhysRevLett.48.1220
  * [7] R. Brandenberger, PoS(ICFI 2010)001 [arXiv:1103.2271 [astro-ph.CO]].
  * [8] N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * [9] P. A. R. Ade et al. [Planck], Astron. Astrophys. 594 (2016), A13 doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]].
  * [10] V. Mukhanov and G. Chibisov, JETP Lett. 33, 532 (1981)
  * [11] J. Martin and R. H. Brandenberger, Physical Review D, 63(12):123501, 2001.
  * [12] R. H. Brandenberger and J. Martin, Modern Physics Letters A, 16(15):999–1006, 2001.
  * [13] R. H. Brandenberger and J. Martin, Classical and Quantum Gravity, 30(11):113001, 2013.
  * [14] N. Kaloper, M. Kleban, A. Lawrence, S. Shenker and L. Susskind, Journal of High Energy Physics, 2002(11):037, 2003.
  * [15] R. Easther, B. R. Greene, W. H. Kinney and G. Shiu, Physical Review D, 66(2):023518, 2002.
  * [16] A.A. Starobinsky, JETP Lett. 73 (2001), 371-374 doi:10.1134/1.1381588 [arXiv:astro-ph/0104043]
  * [17] A.A. Starobinsky,I.I. Tkachev, JETP Lett. 76 (2002), 235-239 doi:10.1134/1.1520612 [arXiv:astro-ph/0207572]
  * [18] A. Bedroya and C. Vafa, JHEP 09 (2020), 123 doi:10.1007/JHEP09(2020)123 [arXiv:1909.11063 [hep-th]].
  * [19] T. Daniel Brennan, F. Carta, C. Vafa, [arXiv:1711.00864 [hep-th]]
  * [20] E. Palti, Fortsch. Phys. 67, no. 6, 1900037 (2019) doi:10.1002/prop.201900037 [arXiv:1903.06239 [hep-th]]
  * [21] G. Obied, H. Ooguri, L. Spodyneiko, C. Vafa, [arXiv:1806.08362 [hep-th]]
  * [22] H. Ooguri, E. Palti, G. Shiu, C. Vafa, doi:10.1016/j.physletb.2018.11.018 [arXiv:1810.05506 [hep-th]]
  * [23] Sumit K. Garg, C. Krishnan, JHEP 11 (2019), 075 doi:10.1007/JHEP11(2019)075 [arXiv:1807.05193 [hep-th]]
  * [24] L. Heisenberg, M. Bartelmann, R. Brandenberger, A. Refregier, Phys. Rev. D 98, 123502 (2018) doi: 10.1103/PhysRevD.98.123502 [arXiv:1808.02877 [astro-ph.CO]]
  * [25] A. Bedroya, R. Brandenberger, M. Loverde and C. Vafa, Phys. Rev. D 101 (2020) no.10, 103502 doi:10.1103/PhysRevD.101.103502 [arXiv:1909.11106 [hep-th]].
  * [26] V. Mukhanov, Eur. Phys. J. C 73 (2013), 2486 doi:10.1140/epjc/s10052-013-2486-7 [arXiv:1303.3925 [astro-ph.CO]].
  * [27] R. Myrzakulov, L. Sebastiani and S. Zerbini, Eur. Phys. J. C 75 (2015) no.5, 215 doi:10.1140/epjc/s10052-015-3443-4 [arXiv:1502.04432 [gr-qc]].
  * [28] K. Bamba, G. Cognola, S. D. Odintsov and S. Zerbini, Phys. Rev. D 90 (2014) no.2, 023525 doi:10.1103/PhysRevD.90.023525 [arXiv:1404.4311 [gr-qc]].
  * [29] R. Myrzakulov, S. Odintsov and L. Sebastiani, Nucl. Phys. B 907 (2016), 646-663 doi:10.1016/j.nuclphysb.2016.04.033 [arXiv:1604.06088 [hep-th]].
  * [30] E. Elizalde, S. D. Odintsov, L. Sebastiani and R. Myrzakulov, Nucl. Phys. B 921 (2017), 411-435 doi:10.1016/j.nuclphysb.2017.06.003 [arXiv:1706.01879 [gr-qc]].
  * [31] A. De Felice and S. Tsujikawa, Living Rev. Rel. 13 (2010), 3 doi:10.12942/lrr-2010-3 [arXiv:1002.4928 [gr-qc]].
  * [32] S. Capozziello and M. De Laurentis, Phys. Rept. 509 (2011), 167-321 doi:10.1016/j.physrep.2011.09.003 [arXiv:1108.6266 [gr-qc]].
  * [33] R. Myrzakulov, L. Sebastiani and S. Zerbini, “Some aspects of generalized modified gravity models,” Int. J. Mod. Phys. D 22 (2013), 1330017 doi:10.1142/S0218271813300176 [arXiv:1302.4646 [gr-qc]].
  * [34] L. Sebastiani and R. Myrzakulov, Int. J. Geom. Meth. Mod. Phys. 12 (2015) no.9, 1530003 doi:10.1142/S0219887815300032 [arXiv:1506.05330 [gr-qc]].
  * [35] S. Nojiri and S. D. Odintsov, Phys. Rept. 505 (2011), 59-144 doi:10.1016/j.physrep.2011.04.001 [arXiv:1011.0544 [gr-qc]].
  * [36] S. D. Odintsov, D. S. C. Gómez and G. S. Sharov, “Analyzing the $H_{0}$ tension in $F(R)$ gravity models,” [arXiv:2011.03957 [gr-qc]].
  * [37] J. Hwang and H. Noh, Phys. Rev. D 54, 1460 (1996).
  * [38] J. Hwang, Class. Quant. Grav. 14, 1981 (1997).
  * [39] J. Hwang, Class. Quant. Grav. 15, 1401 (1998).
  * [40] C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).
  * [41] Horndeski, G.W. Int. J. Theor. Phys. 10, 363 (1974).
  * [42] R. Myrzakulov, L. Sebastiani and S. Vagnozzi, Eur. Phys. J. C 75 (2015), 444 doi:10.1140/epjc/s10052-015-3672-6 [arXiv:1504.07984 [gr-qc]].
  * [43] S. Inoue and J. Yokoyama, Phys.Lett. B524, 15 (2002), hep-ph/0104083.
  * [44] W. H. Kinney, Phys.Rev. D72, 023515 (2005), gr- qc/0503017.
  * [45] M. H. Namjoo, H. Firouzjahi, and M. Sasaki (2012), 1210.3692.
  * [46] J. Martin, H. Motohashi, and T. Suyama, Phys. Rev. D 87, 023514 (2013), arXiv:1211.0083 [astro-ph.CO].
  * [47] H. Motohashi, A. A. Starobinsky, and J. Yokoyama, JCAP 1509, 018 (2015), arXiv:1411.5021 [astro-ph.CO].
  * [48] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Class. Quant. Grav. 34 (2017) no.24, 245012 doi:10.1088/1361-6382/aa92a4 [arXiv:1704.05945 [gr-qc]].
  * [49] S. D. Odintsov, V. K. Oikonomou and L. Sebastiani, Nucl. Phys. B 923 (2017), 608-632 doi:10.1016/j.nuclphysb.2017.08.018 [arXiv:1708.08346 [gr-qc]].
  * [50] H. Motohashi, A.A. Starobinsky, Euro. Phys. J. C 77 (2017), 538 doi:10.1140/epjc/s10052-017-5109-x [arXiv:1704.08188 [astro-ph.CO]]
  * [51] H. Motohashi, A.A. Starobinsky, J. Cosmol. Astropart. Phys. 11 (2019), 025 doi:10.1088/1475-7516/2019/11/025 [arXiv:1909.10883 [gr-qc]]
  * [52] J. D. Barrow, Phys.Rev. D49, 3055 (1994).
  * [53] A. D. Linde, Phys.Rev. D49, 748 (1994), arXiv:astro-ph/9307002 [astro-ph].
  * [54] H. Motohashi and W. Hu, Phys.Rev. D90, 104008 (2014), arXiv:1408.4813 [hep-th].
  * [55] R. Brandenberger, Trans-Planckian Censorship and Early-Universe Cosmology [arXiv:2102.09641 [hep-th]].