Constraints on cosmological parameters in light of the scalar-tensor theory of
gravity and swampland conjectures
S. Noori Gashti⋆,†111Email<EMAIL_ADDRESS>J.
Sadeghi⋆222Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this paper, we study various cosmological parameters and quantities in
scalar-tensor gravity from inflation and swampland conjecture. Therefore, by
selecting different models as power-law, exponential, and logarithmic in the
framework of scalar-tensor theory, we obtain potential, tensor-to-scalar
ratio, and the scalar spectral index. Next, we examine new constraints and
compare the corresponding results with the latest observable data. Here, we
take advantage of the obtained results and determine the compatibility or
incompatibility of the corresponding model with the swampland conjectures.

Keywords: Scalar-tensor gravity, Swampland conjecture, Power-law potential,
Exponential and logarithmic potential

###### Contents

  1. 1 Introduction
  2. 2 Overview of scalar-tensor gravitational model STG
  3. 3 Cosmological parameters in STG
  4. 4 Swampland conjectures and scalar-tensor gravity
  5. 5 The Models
    1. 5.1 Model I; Power- Law Potential
    2. 5.2 Model II; exponential Potential
    3. 5.3 Model III; logarithmic Potential
  6. 6 Conclusion

##  1 Introduction

As we know, researchers consider various inflationary models as a profound
concept to explain the structures of the present universe[1, 2]. The standard
model of cosmology has a series of principles, such as homogeneity in the FLRW
framework. It can explain many problems, such as expanding the universe and
cosmic microwave background radiation (CMBR). We note here that this model
always had problems, such as singularity, flatness, horizon, structure
formation, and dark energy. To solve these problems, researchers have used
various solutions in the past decades[3, 4, 5, 6, 7, 8]. For example, by
modifying the theory of general relativity, significant efforts have been made
to solve these problems. All of these concepts are consistent with recent
observable data. The inflation hypothesis shows an accelerated period of
expansion in the early universe, and during an inflationary period, a dS
universe is driven by cosmological constant[9, 10, 11, 12, 13, 14, 15]. So
far, researchers have introduced various types of inflation models. The most
famous is the $R^{2}$ Strabiansky model. Different types of modified theories
of gravity as $f(R),$ $f(R,T)$, and $f(Q)$ are used to explain the structures
of the universe. Inflation is also combined with various conjectures,
conditions and their cosmology concept has been studied, which can include the
slow-roll, constant-roll and etc[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].
On the other hand, non-minimal scalar-tensor theory has been comprehensively
studied for cosmological applications from the very early stages of inflation
to the last stages via a radiation-dominated era. It has shown that such a
particular theory has accepted a suitable inflation regime. We note here the
cosmological parameters such as the tensor-to-scalar ratio and the scalar
spectral index always have allowable values consistent with observable data.
Based on observable data and studies in this context, it seems that the non-
minimal coupled to a scalar-tensor gravitational theory is a very suitable
model for describing the history of universe evolution [27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. Given the above concepts, we will
study a specific type of inflationary scenario that has not been studied in
the literature. We want to check the inflation of the scalar-tensor gravity
model from swampland conjectures. On the other hand, the weak gravity
conjecture has two known concepts as swampland and landscape. These
conjectures are used to explain the quantum gravity [42, 43, 44, 45, 46, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77]. By selecting different inflation models, we want to study various
cosmological parameters and quantities in scalar-tensor gravity from inflation
and swampland conjecture. Therefore, we organized the continuation of this
article in the following form: In section 2, we review the model and introduce
some concepts of scalar-tensor gravity. In section 3, we present the
cosmological parameters in the concept of scalar-tensor gravity. In section 4,
we calculate cosmological parameters as scalar spectral index, tensor-to-
scalar ratio and potential. In that case, we select different inflation
models, as power-law, exponential and logarithmic forms, in the framework of
scalar-tensor gravity. Then we will call the swampland conjectures and combine
them with the scalar-tensor gravity in inflation. We will examine new
constraints and compare the results of these studies with the latest
observable data, such as Planck 2018, which has imposed new tighter
restrictions on cosmological parameters. Here also, we analyze the results and
determine the compatibility or incompatibility of this inflation model with
the swampland conjectures. And finally, in section 5, we present the result of
the corresponding model.

##  2 Overview of scalar-tensor gravitational model STG

In this section, we try to introduce and review the scalar-tensor
gravitational action with corresponding equations. So, the non-minimally
coupled scalar-tensor gravity can be expressed by the following action,

$S=\int
d^{4}x\sqrt{-g}\bigg{[}f(\phi)R-\frac{\omega(\phi)}{\phi}\phi_{,\mu}\phi^{,\mu}-V(\phi)-\mathcal{L}_{m}\bigg{]},$
(1)

where $\mathcal{L}_{m}$, $f(\phi)$ and $\omega(\phi)$ are the matter
Lagrangian density, coupling parameter, and Brans-Dicke parameter,
respectively. So the field equations are given by,

$\bigg{(}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\bigg{)}f(\phi)+g_{\mu\nu}\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}f(\phi)-f_{;\mu;\nu}-\frac{\omega(\phi)}{\phi}\phi_{,\mu}\phi_{,\nu}+\frac{1}{2}g_{\mu\nu}\bigg{(}\phi_{,\alpha}\phi^{,\alpha}+V(\phi)\bigg{)}=T_{\mu\nu},$
(2)

and

$Rf^{\prime}+2\frac{\omega(\phi)}{\phi}\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}\phi+\bigg{(}\frac{\omega^{\prime}(\phi)}{\phi}-\frac{\omega(\phi)}{\phi^{2}}\bigg{)}\phi_{,\mu}\phi^{,\mu}-V^{\prime}(\phi)=0,$
(3)

where $\hbox to0.0pt{$\sqcup$\hss}\sqcap$ is D’Alembertian as $\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}f(\phi)=f^{\prime\prime}\phi_{,\mu}\phi^{,\mu}-f^{\prime}\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}\phi$. In general, the coupling parameter, Brans-
Dicke parameter, etc., is generally selected to investigate the universe’s
evolution. However, the general conserved current and the above field
equations lead us to relate these parameters[37, 38, 39, 40, 41]. Hence,
according to equation (2), we will have,

$Rf-3\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}f-\frac{\omega(\phi)}{\phi}\phi_{,\mu}\phi^{,\mu}-2V=T^{\mu}_{\mu}=T,$
(4)

Now with respect to equations (3) and (4), one can obtain following equation,

$\bigg{(}3f^{\prime 2}+\frac{2\omega
f}{\phi}\bigg{)}^{\prime}\phi_{,\mu}\phi^{,\mu}+\bigg{(}3f^{\prime
2}+\frac{2\omega f}{\phi}\bigg{)}\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}\phi+2f^{\prime}V-fV=f^{\prime}T,$ (5)

The above equation can also be expressed by the following form,

$\bigg{[}\bigg{(}3f^{\prime 2}+\frac{2\omega
f}{\phi}\bigg{)}^{\frac{1}{2}}\phi^{;\mu}\bigg{]}_{,\mu}-\frac{f^{3}}{2\bigg{(}3f^{\prime
2}+\frac{2\omega
f}{\phi}\bigg{)}^{\frac{1}{2}}}\bigg{(}\frac{V}{f^{2}}\bigg{)}^{\prime}=\frac{f^{\prime}}{2\bigg{(}3f^{\prime
2}+\frac{2\omega f}{\phi}\bigg{)}^{\frac{1}{2}}}T,$ (6)

and finally we will have,

$\bigg{(}3f^{\prime 2}+\frac{2\omega
f}{\phi}\bigg{)}^{\frac{1}{2}}\bigg{[}\bigg{(}3f^{\prime 2}+\frac{2\omega
f}{\phi}\bigg{)}^{\frac{1}{2}}\phi^{;\mu}\bigg{]}_{,\mu}-f^{3}\bigg{(}\frac{V}{f^{2}}\bigg{)}^{\prime}=\frac{f^{\prime}}{2}T^{\mu}_{\mu},$
(7)

According to all the above concepts, the conserved current $J^{\mu}$ can be
expressed by[82, 83, 84],

$J^{\mu}_{;\mu}=\bigg{\\{}\bigg{(}3f^{\prime 2}+\frac{2\omega
f}{\phi}\bigg{)}^{\frac{1}{2}}\phi^{;\mu}\bigg{\\}}_{;\mu}=0,$ (8)

The importance of the above relations and especially derive this conserved
current is completely discussed in[82]. The trace-less matter field as
$T^{\mu}_{\mu}=T=0$ lead us to the following equation,

$V(\phi)\propto f(\phi)^{2},$ (9)

According to the following FRW metric,

$ds^{2}=-dt^{2}+a^{2}(t)\bigg{(}\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\bigg{)},$ (10)

the conserved current (8) is given by,

$\sqrt{\bigg{(}3f^{\prime 2}+\frac{2\omega
f}{\phi}\bigg{)}}a^{3}\dot{\phi}=\mathcal{C}_{1},$ (11)

where $a(t)$ and $\mathcal{C}_{1}$ are the scale factor and integration
constant respectively. The relations between coupling parameter and potential
is specified by equation (9), further using the relations and concepts
mentioned in[41, 82, 85, 86], we will have,

$3f^{\prime 2}+\frac{2\omega f}{\phi}=\omega_{0}^{2},$ (12)

where $\omega_{0}$ is a constant parameter. Here also, one can obtain,

$a^{3}\dot{\phi}=\frac{\mathcal{C}_{1}}{\omega_{0}}=\mathcal{C},$ (13)

Where $\mathcal{C}$ is a constant parameter. Given the above concept coupling
parameters, the potential and the Brans-Dicke parameter can be assumed
constant. Next, we assume $\frac{M_{pl}^{2}}{2}=c=1$ and also consider
different $f(\phi)$, which fixes the Brans-Dicke parameters and potential.
From these functions point of view, we focus all our attention on the
inflation in the early vacuum-dominated universe, but we also challenge
swampland conjectures. We add a term to the potential to relax the symmetry.
So that one of these parameters will act as a constant, which is related to
the cosmological constant. To be more consistent with observable data, we also
use some constants, which we will explain in more detail in the next section.
The number of parameters involved is increased to three
$(\omega_{0},V_{0},V_{1})$ to advance this work, which is necessary to fit
well with the latest observation data.

##  3 Cosmological parameters in STG

.

In this section, we provide more details on the scalar-tensor gravity theory
and introduce cosmological parameters. Therefore, to study inflation, we
consider a hypothesis as $p=0=\rho$. In this situation, the trace of the
matter field disappears, and symmetry is held. Hence, action (1) states as the
following form,

$S=\int
d^{4}x\sqrt{-g}\bigg{[}f(\phi)R-\frac{K(\phi)}{2}\phi_{,\mu}\phi^{,\mu}-V(\phi)\bigg{]},$
(14)

According to the conformal transformation $g_{E_{\mu\nu}}=f(\phi)g_{\mu\nu}$,
Einstein’s form of the corresponding action is [79].

$S=\int
d^{4}x\sqrt{-g_{E}}\bigg{[}R_{E}-\frac{1}{2}(\partial_{E}\sigma)^{2}-V_{E}(\sigma(\phi))\bigg{]},$
(15)

Where $E$ denotes Einstein’s frame. In this framework, the effective potential
$V_{E}$ as well as the field $\sigma$ can obtain in the following form,

$V_{E}=\frac{V(\phi)}{f^{2}(\phi)},$ (16)

and

$(\frac{d\sigma}{d\phi})^{2}=\frac{K(\phi)}{f(\phi)}+3\frac{f^{\prime
2}(\phi)}{f^{2}(\phi)}=\frac{2\omega(\phi)}{\phi f(\phi)}+3\frac{f^{\prime
2}(\phi)}{f^{2}(\phi)},$ (17)

In Einstein’s frame, according to the action (15), the field equations can be
obtained by,

$\ddot{\sigma}+3H\dot{\sigma}+V^{\prime}_{E}=0,$ (18)
$3H^{2}=\frac{1}{2}\dot{\sigma}^{2}+V_{E},$ (19)

where the above equations, $H$ is expressed by $H=\frac{\dot{a}_{E}}{a_{E}}$.
For cosmological studies, we use some important parameters as slow-roll
parameters and number of e-folds, which are given by,

$\epsilon=(\frac{V^{\prime}_{E}}{V_{E}})^{2}(\frac{d\sigma}{d\phi})^{-2},$
(20)
$\eta=2\bigg{[}(\frac{V^{\prime\prime}_{E}}{V_{E}})(\frac{d\sigma}{d\phi})^{-2}-(\frac{V^{\prime}_{E}}{V_{E}})(\frac{d\sigma}{d\phi})^{-3}(\frac{d^{2}\sigma}{d\phi^{2}})\bigg{]},$
(21)

and

$N=\int_{t_{i}}^{t_{f}}Hdt=\frac{1}{2\sqrt{2}}\int_{\phi_{i}}^{\phi_{e}}\frac{d\phi}{\sqrt{\epsilon}}\frac{d\sigma}{d\phi},$
(22)

To further explain the above equations, we can add some points, for example,
$t_{i}$ and $t_{f}$ show initiation and the end time, $\phi_{i}$ and
$\phi_{e}$ represent the scalar field at the beginning and end of inflation,
respectively. Also, we obtain a series of cosmological parameters such as the
tensor-to-scalar ratio $r$ and the scalar spectral index $n_{s}$. In that case
we compare the primordial curvature perturbation $P_{\zeta}(K)$ with the
primordial gravitational wave power spectrum $P_{t}(K)$, where
$r=\frac{P_{t}(K)}{P_{\zeta}(K)}$ and $n_{s}-1=2\eta-6\epsilon$. Also,
according to the latest observable data from TT,TE,EE+lowEB+lensing and
TT,TE,EE+lowE+lensing+BK14+BAO, different values are mentioned for these
parameters. They determined the tensor-to-scalar ratio is $r\leq 0.16$, $r\leq
0.07$ and the scalar spectral index is $n_{s}\leq 0.9569,n_{s}\leq 0.9815$
respectively [7].

##  4 Swampland conjectures and scalar-tensor gravity

We note here if the potential in the Einstein frame is constant, the flat part
of the effective potential will still be available[80, 81]. As mentioned in
the second section, symmetry causes the potential in the Einstein frame to be
constant, so we must relax the symmetry as stated in condition (9). So, by
adding a fixed parameter as $V_{0}$, the effective potential in the Einstein
frame is asymptotically constant. In this paper, we want to check inflation
from the scalar-tensor gravity (symmetry of the non-minimal coupled to the
scalar-tensor theory of gravity to fixing quantities and parameters such as
Brans-Dicke parameter and the potential) and swampland conjectures point of
view. Therefore, we calculate cosmological parameters as scalar spectral
index, tensor-to-scalar ratio and potential by selecting different inflation
models, i.e., power-law, exponential and logarithmic. Then we will call the
swampland conjectures and combine them with the scalar-tensor gravity in
inflation. We will examine new constraints and compare the results of these
studies with the latest observable data such as Planck 2018[7], which has
imposed new tighter constraints on cosmological parameters. Finally, we
analyze the results to determine the compatibility or incompatibility of this
inflation model with the swampland conjectures.
Swampland conjectures are used in various types of cosmological studies. In
recent decades, researchers have done a great deal of research on these
conjectures, also; in this paper, we use the refined swampland conjecture,
namely dS swampland[45, 46, 57, 58, 59, 60, 61, 62, 63, 64, 65], which is
given by,

$M_{pl}\frac{|V^{\prime}|}{V}>C_{1},\hskip
28.45274ptM_{pl}^{2}\frac{|V^{\prime\prime}|}{V}<-C_{2},$ (23)

Where $C_{1}$ and $C_{2}$ are constant parameters. We selected several models,
i.e., the power-law form, logarithmic and exponential form, and combined them
with various conditions. Here, first, we consider a power-law potential

##  5 The Models

To challenge the swampland conjecture from the tensor-scalar gravitational
theory perspective, we consider three general models to study the
inflation,i.e., we challenge the compatibility of the three issue viz the
swampland conjecture, the tensor-scalar gravitational theory and the some
important inflationary model by calculating cosmological parameters. Usually,
gravitational corrections are often overlooked in studies related to eternal
inflation, and exclusively the large field inflation models are assessed. To
consider quantum gravitational corrections, we must assume small field models,
which are also possible through swampland conjectures. Of course, there are a
few things to keep in mind about swampland conjectures, and the most important
is that this idea has not yet been accepted as a complete theory and has not
been widely accepted in the literature. However, it has been studied in many
cosmological concepts. These conjectures are somewhat consistent with
approaches such as warm inflation[78]. Of course, each of the swampland
conjectures is constantly being reviewed and completed, and even corrections
are made in each of these conjecture to be more compatible with the models and
the latest observable data[7]. However, these conjectures still face serious
challenges. But perhaps in the not-too-distant future, through observations,
the compatibility of this idea with theories can be increased, and it can even
find a place among physicists as an accepted idea. Of course, it is still
being studied as a nascent idea in studying many cosmic structures such as
inflation and the physics of black holes and dark energy. The next point is
that gravitational corrections are negligible in small field models and
potentially predominant in large field models[71]. For this reason, effective
field theories about large fields are not entirely reliable. Of course,
critical points can also be mentioned concerning eternal inflation. Matsui,
Takahashi, and Dimopoulos discussed the swampland conjectures with respect to
eternal inflation[87]. They showed the originally proposed de Sitter swampland
conjectures to be generically incompatible with eternal inflation. Also, there
are several studies about this view under certain conditions. For example,
more recently, William H. Kinney showed in his recent article[72] that the
swampland conjecture, which somehow applies weaker criteria to the potential
of the scalar field (Refined Swampland Conjecture) in inflation, is somewhat
compatible with eternal inflation. Therefore, if the refined conjecture is
valid, the presence of a landscape-based multiverse in string theory is not
inconsistent with a self-consistent UV completion, and it has remarkable
outcomes for model construction in string theory. (For further study, see [87,
88, 89, 90]). This paper will discuss the compatibility of cosmological
parameters in scalar-tensor gravitational theory with swampland conjectures.
Recently, however, Yuennan and Channuie, in an article entitled Further
refining Swampland Conjecture on inflation in general scalar-tensor theories
of gravity examined the consistency of extended swampland dS conjecture about
the general scalar-tensor theories, the results of which in turn can be
interesting[91].

###  5.1 Model I; Power- Law Potential

In this section, we consider a power-law potential, which is selected by form
of $f(\phi)=\phi^{n}$ and the potential is expressed by $V(\phi)=\phi^{2n}$.
Then we select $n=\frac{1}{2}$ and examine the corresponding model. Here, the
main purpose is to study the compatibility or incompatibility of combining the
studied model in the scalar-tensor gravity framework with the swampland
conjecture with respect to observable data. In the following, we can briefly
review for other powers of $n$ and discuss the overall effect of $n.$ However,
by using the above explanations and considering a constant $V_{0}$ for the
potential, we will have.

$V(\phi)=V_{1}\phi^{2n}+V_{0},$ (24)

According to the equations (12), (16), (17), (20), (21) and (22) and
$n=\frac{1}{2}$, we calculate the Brans-Dicke parameter, potential, the
expression for $\frac{d\sigma}{d\phi}$ and slow-roll parameters, which are
given by,

$\omega(\phi)=\frac{-3+4\phi\omega_{0}^{2}}{8\sqrt{\phi}},$ (25)
$V_{E}=V_{1}+V_{0}\phi^{-1},$ (26)

$\frac{d^{2}\sigma}{d\phi^{2}}=\frac{\omega_{0}^{2}}{\phi},$ (27)

$\epsilon=\frac{V_{0}^{2}\phi}{\omega_{0}^{2}(V_{0}\phi+V_{1}\phi^{2})^{2}},$
(28)

and

$\eta=\frac{3V_{0}}{\phi\omega_{0}^{2}(V_{0}+V_{1}\phi)},$ (29)

In the above equations, if we assume the value of the constant potential
$V_{0}$ set to be zero, the effective potential in the Einstein frame will be
equal to $V_{E}=V_{1}$ and it remains flat. Now we will use the swampland
conjectures in the corresponding model and examine its compatibility with
observable data. In that case, according to equation (26), we have to obtain
the first and second derivatives of the effective potential in terms of $\phi$
so that we will have,

$V^{\prime}(\phi)=-\frac{V_{0}}{\phi^{2}},\hskip
12.0ptV^{\prime\prime}(\phi)=\frac{2V_{0}}{\phi^{3}},$ (30)

Now with respect to equations (23), (26), and (30), one can calculate,

$\frac{-\frac{V_{0}}{\phi^{2}}}{V_{1}+V_{0}\phi^{-1}}>C_{1},\hskip
12.0pt\frac{\frac{2V_{0}}{\phi^{3}}}{V_{1}+V_{0}\phi^{-1}}<-C_{2},$ (31)

As we know, the tensor-to-scalar ratio $r$ and scalar spectral index $n_{s}$
are important quantities in cosmology. So, according to slow-roll parameters
in equations (28) and (29), we can have the following equations,

$n_{s}=1-\frac{6V_{0}^{2}\phi}{\omega_{0}^{2}(V_{0}\phi+V_{1}\phi^{2})^{2}}+\frac{6V_{0}}{\phi\omega_{0}^{2}(V_{0}+V_{1}\phi)},$
(32)

and

$r=\frac{16V_{0}^{2}\phi}{\omega_{0}^{2}(V_{0}\phi+V_{1}\phi^{2})^{2}},$ (33)

As mentioned in the previous section, observable data has created new
constraints on cosmological parameters such as $n_{s}$ and $r$. So, by
selecting the different values for free parameters as
$(V_{1},V_{0},\omega_{0})$, the important cosmological parameters obtain
different values compared with the latest observable data. According to the
component $f(\phi)$ also $\phi_{i}=1.9$, $V_{0}=0.9\times 10^{-13}T^{-2}$ and
$V_{1}=0.9\times 10^{-}{13}T^{-2}$ it can be checked that different values
obtained for the parameters under consideration. Hence we will have
$10<\omega_{0}<30$, $1.0124<\phi_{e}<1.0315$, $0.000215<r<0.00945$,
$0.9705<n_{s}<0.9922$ and $45<N<64$. Also this rout can be studied for
different values of the parameter $n$ according to the $f(\phi)$. As shown in
equations (32) and (33), the two cosmological parameters are expressed in the
scalar field $\phi$. Now, suppose we reverse these two equations so that the
scalar field $\phi$ is expressed in terms of two parameters $n_{s}$ and $r$
and then placed in equation (31). In that case, we can obtain the coefficients
of the swampland conjectures in terms of these two cosmological parameters. By
plotting some figures and according to the coefficients of the swampland
conjectures, we show the changes of these two cosmological parameters, also we
can compare them with the latest observable data. Therefore, to show these
variations, we will plot the figures $C_{1,2}-n_{s}$ and $C_{1,2}-r$,
respectively.

Figure 1: The plot of $C_{1,2}$ in terms of $n_{s}$ with respect to free
parameter values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times
10^{-13},\omega_{0}=13)$

Figure 2: The plot of $C_{1,2}$ in terms of $r$ with respect to free parameter
values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times 10^{-13},\omega_{0}=13)$
Figure 3: The plot of $r$ in term of $n_{s}$ with respect to the free
parameter values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times
10^{-13},\omega_{0}=13)$

As mentioned before, the swampland conjectures have been rewritten in terms of
two cosmological parameters $n_{s}$ and $r$. Here, by using different figures,
we explain the constraints on cosmological parameters, which are described by
following: As shown in figure (1), we plotted the constraints of the first and
second components of the swampland conjecture in terms of the scalar spectral
index concerning the free parameters as $(V_{0},V_{1},\omega_{0})$. The
associated range of the scalar spectral index which is found here must be
consistent with Planck’s observable data[7]. Also, the range of swampland
conjecture components is well defined and usually specified by the literature
in order 0.1 and 1. Also, as shown in figure (1), the coefficient of $C_{2}$
has smaller than the $C_{1}$. Similarly, in figure (2), new constraints are
examined, namely the constraint of swampland components to tensor-to-scalar
ratio. As shown in figure (2), the range associated with $r$ and the swampland
conjectures coefficient are also well defined, in which case the values are
consistent with the observable data[7]. To plot these figures, we also used
free parameters whose values we had determined. In figure(3), we have plotted
the restrictions of two cosmological parameters with respect to each other,
and the allowable range of these two parameters is well shown in figure (3).
As observed in the calculations and diagrams of this section, the model of
power-Law potential in combination with the scalar-tensor theory of gravity
and swampland conjectures is consistent with the observable data. Also, here
we see that the scalar-tensor theory of gravity can also be well compatible
with swampland conjectures. We determine the allowable range $C_{i}$ of this
model according to the explanations mentioned earlier and the swampland
conjecture in terms of cosmological parameters. Therefore, with respect to the
components of swampland conjecture, i.e., $C_{1}$ and $C_{2}$ as well as the
constant parameters mentioned in this paper and the observable values for $r$
and $n_{s}$, a relation can be determined as $f\geq C_{1}^{2}C_{2}^{2}$. As a
result, the allowable range of swampland conjectures for such a model is well
illustrated in figure (4). As mentioned concept specified the allowable
constraints of $C_{i}$ in figure(4). In the continuation of the article, we
will determine the allowable $C_{i}$ for other models according to the
cosmological parameters mentioned in the article. In that case, the best model
can determine by the above concepts.

Figure 4: The constraints for $C_{1}$, and $C{2}$ with respect to the free
parameter values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times
10^{-13},\omega_{0}=13)$ and tensor-to-scalar ratio $r$ and scalar spectral
index $n_{s}$

###  5.2 Model II; exponential Potential

Like the pervious section, here we consider a exponential potential. By
selecting $f(\phi)=\exp(\frac{n\phi}{2})$, the potential will be as
$V(\phi)=\exp(n\phi)$. Then we select $n=1$ and consider also constant of
$V_{0}$ for the potential, we will have,

$V(\phi)=V_{1}\exp(\phi)+V_{0},$ (34)

By using equations (12), (16), (17), (20), (21), and (22), we will calculate
the Brans-Dicke parameter and corresponding potential, which are given by,

$\omega(\phi)=\frac{(\omega_{0}^{2}-\frac{3}{4}\exp(\phi))\times\phi}{2\exp(\frac{\phi}{2})},$
(35) $V_{E}=V_{1}+V_{0}\exp(-\phi),$ (36)

As we know, the expression for $\frac{d\sigma}{d\phi}$ will be as,

$\frac{d^{2}\sigma}{d\phi^{2}}=\frac{\omega_{0}^{2}}{\exp(\phi)},$ (37)

and also, the slow-roll parameters in Einstein’s frame are calculated by the
following form,

$\epsilon=\frac{V_{0}^{2}\exp(\phi)}{\omega_{0}^{2}(V_{0}+V_{1}\exp(\phi))^{2}},$
(38)

and

$\eta=\frac{V_{0}\exp(\phi)}{\omega_{0}^{2}(V_{0}+V_{1}\exp(\phi))},$ (39)

Now we want to use the swampland conjectures and to examine its compatibility
with observable data. So, we need to obtain the first and second derivatives
of the effective potential in terms of $\phi$. According to equation (36), we
will have,

$V^{\prime}(\phi)=-V_{0}\exp(-\phi),\hskip
12.0ptV^{\prime\prime}(\phi)=V_{0}\exp(-\phi),$ (40)

Now we use the equations (23), (36), and (40), we calculate the following
expressions,

$\frac{-V_{0}\exp(-\phi)}{V_{1}+V_{0}\exp(-\phi)}>C_{1},\hskip
12.0pt\frac{V_{0}\exp(-\phi)}{V_{1}+V_{0}\exp(-\phi)}<-C_{2},$ (41)

Also, we note here, by considering the equations (38) and (39) the tensor-to-
scalar ratio $r$ and scalar spectral index $n_{s}$, can be obtained by the
following expressions,

$n_{s}=1-\frac{6V_{0}^{2}\exp(\phi)}{\omega_{0}^{2}(V_{0}+V_{1}\exp(\phi))^{2}}+\frac{2V_{0}\exp(\phi)}{\omega_{0}^{2}(V_{0}+V_{1}\exp(\phi))},$
(42)

and

$r=\frac{16V_{0}^{2}\exp(\phi)}{\omega_{0}^{2}(V_{0}+V_{1}\exp(\phi))^{2}},$
(43)

As mentioned before, the observable data has created new constraints on
cosmological parameters such as $n_{s}$ and $r$. So we consider the free
parameters as $V_{1}$, $V_{0}$, $\omega_{0}$. According to the component
$f(\phi)$ also $\phi_{i}=1.2$, $V_{0}=1.1\times 10^{-20}T^{-2}$ and
$V_{1}=1.1\times 10^{-20}T^{-2}$ it can be checked that different values
obtained for the parameters under consideration. Hence we will have
$10<\omega_{0}<30$, $0.02589<\phi_{e}<0.08569$, $0.015<r<0.065$,
$0.9615<n_{s}<0.9842$ and $30<N<78$. Also this rout can be studied for
differen $f(\phi)$. We see $n_{s}$ and $r$ are expressed in terms of the
scalar field $\phi$ in equations (42) and (43). Now, like the previous part,
if we reverse these two equations so that the scalar field $\phi$ is expressed
in terms of two parameters $n_{s}$ and $r$ and then placed in equation (41),
we can obtain the coefficients of the swampland conjectures in terms of these
two cosmological parameters. By plotting some figures and comparing the
results with observable data, it is possible to show the changes of these two
cosmological parameters, according to the coefficients of the swampland
conjectures. So, to show these variations, we will plot the figures
$C_{1,2}-n_{s}$ and $C_{1,2}-r$, respectively.

Figure 5: The plot of $C_{1,2}$ in terms of $n_{s}$ with respect to free
parameter values as $(V_{1}=-1.1\times 10^{-20},V_{0}=1.1\times
10^{-20},\omega_{0}=20)$

Figure 6: The plot of $C_{1,2}$ in terms of $r$ with respect to free parameter
values as $(V_{1}=-1.1\times 10^{-20},V_{0}=1.1\times 10^{-20},\omega_{0}=20)$
Figure 7: The plot of $r$ in term of $n_{s}$ with respect to the free
parameter values $(V_{1}=-1.1\times 10^{-20},V_{0}=1.1\times
10^{-20},\omega_{0}=20)$ Figure 8: The plot of $r$ in term of $\omega_{0}$
with respect to the free parameter values as $(V_{1}=-1.1\times
10^{-20},V_{0}=1.1\times 10^{-20}$

In this part, we perform the calculations for exponential potential. We have
plotted some figures about constraints related to swampland conjectures,
$n_{s}$ and $r$ like the previous section. The results are following: As shown
in figure (5), the constraints of the swampland conjecture, i.e., $C_{1,2}$
are plotted in terms of the scalar spectral index concerning the free
parameters as $(V_{0},V_{1},\omega_{0})$. The obtained range associated with
the scalar spectral index is consistent with Planck’s observable data[7].
Also, for the swampland conjecture coefficient, the range of these components
is well defined. Also, as shown in figure (5), the coefficient of $C_{2}$ is
smaller than $C_{1}$. Similarly, in figure (6), the constraint of swampland
components to the tensor-to-scalar ratio is specified. As shown in figure (6),
the range associated with $r$ and the swampland conjectures coefficient are
also well defined[7]. To plot these figures, we also used free parameters. In
figure(7), we show the restrictions of two cosmological parameters with
respect to each other, and the allowable range of these two parameters is also
well shown. As observed in the calculations and diagrams of this section, the
model of exponential potential in combination with the scalar-tensor theory of
gravity and swampland conjectures is consistent with the observable data.
Also, in figure (8), we show the limitations $r$ in terms of $\omega_{0}$. As
shown in figure (8), the variation of tensor-to-scalar ratio for different
values of the parameter $\omega_{0}$ is specified. Like the previous
subsection for the corresponding model, the allowable limit of $C_{i}$ can be
specified by figure(9).

Figure 9: The constraints for $C_{1}$, and $C{2}$ with respect to the free
parameter values as $(V_{1}=-1.1\times 10^{-20},V_{0}=1.1\times
10^{-20},\omega_{0}=20)$ and tensor-to-scalar ratio $r$ and scalar spectral
index $n_{s}$

###  5.3 Model III; logarithmic Potential

Here also we consider a logarithmic potential model. So by choosing the
$f(\phi)=(\log(\phi))^{\frac{n}{2}}$, the potential is
$V(\phi)=(\log(\phi))^{n}$. Therefore we will have.

$V(\phi)=V_{1}(\log(\phi))^{n}+V_{0},$ (44)

according to equations (12), (16), (17), (20), (21) and (22), one can obtain,

$\omega(\phi)=\frac{(\omega_{0}^{2}-\frac{3n^{2}(\log(\phi))^{-2+n}}{4\phi^{2}})\times\phi}{2(\log(\phi))^{\frac{n}{2}}},$
(45) $V_{E}=V_{1}+V_{0}(\log(\phi))^{-n},$ (46)

and the $\frac{d\sigma}{d\phi}$ is,

$\frac{d^{2}\sigma}{d\phi^{2}}=\frac{\omega_{0}^{2}}{(\log(\phi))^{-n}},$ (47)

Also here, the slow-roll parameters are given by,

$\epsilon=\frac{V_{0}^{2}n^{2}(\log(\phi))^{n-2}}{\omega_{0}^{2}\phi^{2}(V_{0}+V_{1}(\log(\phi))^{n})^{2}},$
(48)

and

$\eta=\frac{V_{0}n(\log(\phi))^{-2+n}(2+n+2(\log(\phi)))}{\omega_{0}^{2}\phi^{2}(V_{0}+V_{1}(\log(\phi))^{n})},$
(49)

We obtain the first and second derivatives of the effective potential
according to equation (46). So we will have,

$V^{\prime}(\phi)=-\frac{V_{0}n(\log(\phi))^{-1-n}}{\phi},\hskip
12.0ptV^{\prime\prime}(\phi)=\frac{V_{0}n(\log(\phi))^{-2-n}(1+n+\log(\phi))}{\phi^{2}},$
(50)

Now we use the equations (23), (46), (50) and calculate the following
expressions,

$\frac{-\frac{V_{0}n(\log(\phi))^{-1-n}}{\phi}}{V_{1}+V_{0}(\log(\phi))^{-n}}=>C_{1},\hskip
12.0pt\frac{\frac{V_{0}n(\log(\phi))^{-2-n}(1+n+\log(\phi))}{\phi^{2}}}{V_{1}+V_{0}(\log(\phi))^{-n}}<-C_{2},$
(51)

considering the equations (48) and (49), $r$ and $n_{s}$, one can obtain.

$n_{s}=1-\frac{6V_{0}^{2}n^{2}(\log(\phi))^{n-2}}{\omega_{0}^{2}\phi^{2}(V_{0}+V_{1}(\log(\phi))^{n})^{2}}+\frac{2V_{0}n(\log(\phi))^{-2+n}(2+n+2(\log(\phi)))}{\omega_{0}^{2}\phi^{2}(V_{0}+V_{1}(\log(\phi))^{n})},$
(52)

and

$r=\frac{16V_{0}^{2}n^{2}(\log(\phi))^{n-2}}{\omega_{0}^{2}\phi^{2}(V_{0}+V_{1}(\log(\phi))^{n})^{2}},$
(53)

Now by considering the mentioned constant values for the free parameters, the
scalar spectral index and tensor-to-scalar ratio for $n=3,4,5$ will be
$n_{s}=0.984,0.976,0.963$ and $r=0.066,0.053,0.038$, respectively. According
to the component $f(\phi)$ also $\phi_{i}=1.5$, $V_{0}=0.9\times
10^{-13}T^{-2}$ and $V_{1}=0.9\times 10^{-13}T^{-2}$ it can be checked that
different values obtained for the parameters under consideration. Hence we
will have $10<\omega_{0}<30$, $0.7124<\phi_{e}<1.4315$, $0.0.011<r<0.0.084$,
$0.9523<n_{s}<0.9892$ and $42<N<61$. Also this rout can be studied for
different values of the parameter $n$ according to the $f(\phi)$. Now, like
the pervious section, we reverse these equations (52) and (53) in terms of two
parameters $n_{s}$ and $r$ and then placed in equation (51). Then we will plot
the figures $C_{1,2}-n_{s}$ and $C_{1,2}-r$, and $r-n_{s}$ respectively. So,
according to the above description we have some figures,

Figure 10: The plot of $C_{1}$ in terms of $n_{s}$ and $\omega_{0}$ with
respect to free parameter values as $(V_{1}=-0.9\times
10^{-13},V_{0}=0.9\times 10^{-13},\omega_{0}=20)$

Figure 11: The plot of $C_{2}$ in terms of $n_{s}$ and $\omega_{0}$ with
respect to free parameter values as $(V_{1}=-0.9\times
10^{-13},V_{0}=0.9\times 10^{-13},\omega_{0}=20)$

Figure 12: The plot of $C_{1}$ in terms of $r$ and $\omega_{0}$ with respect
to free parameter values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times
10^{-13},\omega_{0}=20)$

Figure 13: The plot of $C_{2}$ in terms of $r$ and $\omega_{0}$ with respect
to free parameter values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times
10^{-13},\omega_{0}=20)$ Figure 14: The plot of $r$ in term of $n_{s}$ with
respect to the free parameter values as $(V_{1}=-0.9\times
10^{-13},V_{0}=0.9\times 10^{-13},\omega_{0}=20)$ Figure 15: The plot of $r$
in term of $\omega_{0}$ with respect to the free parameter values as
$(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times 10^{-13}$

As two previous sections, we consider the same limitations for the logarithmic
model. Of course, in this section, we also specify the role of the parameter
$n$. As mentioned above, we have no acceptable answers for $(n<3)$, but for
the $(n\geq 3)$, the result was comparable to the observable data[7]. So we
examined the extent of these constraints by considering the three different
values of $n$ viz the swampland conjectures in terms of two cosmological
parameters $n_{s}$ and $r$. In figure (10), the constraints of the $C_{1}$ are
plotted in terms of the scalar spectral index and the $\omega_{0}$. The range
of scalar spectral index concerning different values of $n$ as 3,4,5 is
consistent with Planck’s observable data[7]. Also, the changes of this
component in terms of the parameter $\omega_{0}$ for three different values of
$n$ are well specified. The range of the swampland conjecture components is
well defined, which is usually specified by the literature as order 0.1 and 1.
As shown in figures, coefficient of $C_{1}$ for larger values of the parameter
$\omega_{0}$, it assigns smaller values. Similarly, in figure (11),
constraints of $C_{2}$ are examined in terms of the scalar spectral index and
the $\omega_{0}$. As shown in figure (11), the range of $n_{s}$, $\omega_{0}$
and swampland conjectures coefficient are also well defined for various values
of $n$. In this figure, unlike the previous section, for larger values of
$\omega_{0}$, the $C_{2}$ has larger values. Also, as shown in figure (11a),
the allowable values of $C_{2}$ are smaller than $C_{1}$. To plot these
figures, we also used free parameters. We plotted the restrictions of $C_{12}$
in terms of the tensor-to scalar ratio and the $\omega_{0}$ by considering
different values of $n$ In figures (12) and (13). Also, in figure (12b), for
larger values of $n$, the values of $C_{1}$ become smaller. But in figure
(13b), for larger values of $n$, the values of $C_{1}$ become bigger. Also,
the values of the component $C_{2}$ in figure (13a) have smaller values than
the component $C_{1}$ in figure (12a). As shown in figure (14), we plot two
cosmological parameters with respect to each other. The allowable range of
these two parameters is well shown in this figure for different $n$. As shown
in figure (14), for larger values of $n$, the result will be closer to the
observable data[7]. Also, in figure (15), we plotted the change of this
cosmological parameter, i.e., $r$, in terms of different values of
$\omega_{0}$. The large values of $\omega_{0}$ are more acceptable than the
values for $r$ so that they are comparable to the observable data. In these
two figures, we also used the free parameters. We note here; these variations
are displayed for different values of $n$. According to the calculations and
figures plotted in this part, the logarithmic model in combination with the
scalar-tensor theory of gravity and swampland conjectures is consistent with
the observable data[7] This compatibility was more seen for larger values $n$
and $\omega_{0}$. Scalar-tensor theory of gravity can also be well compatible
with swampland conjectures. For this model, the allowable range of swampland
conjectures $C_{i}$ for different values of $n$ is determined according to the
constant parameters as well as the observable data $r$ and $n_{s}$. In figure
(16), we see the allowable ranges of swampland components or $C_{i}$ for the
corresponding model.

Figure 16: The constraints for $C_{1}$, and $C{2}$ with respect to the free
parameter values as $(V_{1}=-0.9\times 10^{-13},V_{0}=0.9\times
10^{-13},\omega_{0}=13)$, $n=3,4,5$ and tensor-to-scalar ratio $r$ and scalar
spectral index $n_{s}$

It is clear that from studying all three models, the logarithmic model with
comparing to the other models with increasing the values of $n$, the
cosmological parameters $r$ and $n_{s}$ more compatible with observable data.
After that, the exponential model and finally the power-law model is more
appropriate. The constraints of the $C_{1}$ and $C_{2}$ for these three models
are shown in figures (4), (9), and (16) viz they are satisfied by swampland
conditions. Here also, we note that $C_{i}s$ permitted for each model in a
certain range. The range of these components, i.e., $C_{1,2}$ is calculated
positive unit order and in some calculations is considered approximately $0.1$
in the literature. As you can see in figures (4), (9), and (16), the
exponential model is within the computational range of the literature.
However, we can not ignore the other two models; that is, swampland
conjectures can somehow accommodate a larger volume of the exponential model
and then the power-law and finally logarithmic.

##  6 Conclusion

Cosmic inflation with various theories studied from different perspectives and
conditions. In this paper, we checked inflation from the scalar-tensor gravity
perspective and swampland conjectures. Therefore, by selecting different
inflation models as power-law, exponential and logarithmic, we calculated
potential, scalar spectral index and tensor-to-scalar ratio in the framework
of scalar-tensor gravity. Then we employed the swampland conjectures on the
scalar-tensor gravity in inflation and examined new constraints such as
$C_{1,2}-n_{s}$, $C_{1,2}-r$ and $r-n_{s}$. We also determined the range of
these cosmological parameters and swampland conjectures coefficient and
compared these results to the latest observable data such as Planck 2018. It
has imposed new tighter constraints on cosmological parameters. Finally, we
analyzed the obtained results and achieved the compatibility or
incompatibility of this inflation model with the swampland conjectures. As
observed in the calculations and figures of this paper, we showed that the
logarithmic model with comparing to the other models with increasing the
values of $n$, the cosmological parameters $r$ and $n_{s}$ more compatible
with observable data. But from equations of (4), (9), and (16) and
corresponding figures, we observed that the exponential model is within the
computational range of the literature. However, we can not ignore the other
two models. But swampland conjectures are more compatible first to the
exponential model and then the power-law and finally logarithmic. So,
generally, one can say that the power law, exponential and logarithmic
potential in combination with the scalar-tensor theory of gravity and
swampland conjectures are consistent with the observable data[7]. Therefore,
the scalar-tensor theory of gravity can also be well compatible with swampland
conjectures. It may be interesting to check the above corresponding model for
the constant-roll with other conditions of the swampland program such as TCC.
Also, it can challenge these concepts for other cosmological structures and
theories, such as dark energy and dark matter. We leave the calculations
related to these structures and concepts to future work. Also, one critical
question that arises through these calculations is whether, assuming the
swampland constraints, is eternal inflation allowed for such scalar-tensor
models?

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