# Quintessential constant-roll inflation

Mehdi Shokri<EMAIL_ADDRESS>School of Physics, Damghan
University, P. O. Box 3671641167, Damghan, Iran  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  Saeed Noori Gashti<EMAIL_ADDRESS>Department
of Physics, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

We investigate a single field model in the context of the constant-roll
inflation in which inflaton moves down to the minimum point of the potential
with a constant rate of rolling. We use a quintessential inflationary model
obtained by a Lorentzian function which is dependent on the number of e-folds.
We present the inflationary analysis for the model and find the observational
constraints on the parameters space using the observations of CMB anisotropies
i.e. the Planck and Keck/array datasets. We find the observationally
acceptable values of the Width of the Lorentzian function as $0.3<\Gamma\leq
0.5$ at the $68\%$ CL and $\Gamma\leq 0.3$ at the $95\%$ CL when $\xi=120$,
$|\beta|=0.02$ and $N=60$. Also, we acquire the observationally favoured
values of the amplitude of the Lorentzian function as $400<\xi\leq 600$ at the
$68\%$ CL and $\xi\leq 400$ at the $95\%$ CL when $\Gamma=0.1$, $|\beta|=0.02$
and $N=60$. Moreover, we study the model from the Weak Gravity Conjecture
approach using the swampland criteria.

PACS: 98.80.Cq; 98.80.$-$K.
Keywords: Quintessential inflation; Constant-roll approach; Cosmic Microwave
Background.

##  I Introduction

The theory of inflation presents the most successful model to explain the
phenomena in the early universe by considering a rapid and large expansion in
the scale factor. In addition to removing the shortcomings of the hot big bang
model, inflation provides an acceptable mechanism to generate the primordial
density perturbations which are known as the main candidate for the structure
formation of the universe and also the temperature anisotropy of the cosmic
background photons. Moreover, the tensor perturbations which form the
primordial gravitational waves have been produced during the inflationary
epoch Guth ; Kazanas:1980tx ; Linde:1981my ; Albrecht:1982wi ; Lyth:1998xn .
In the standard picture, a single scalar field is the main responsible to push
the inflationary era. Inflaton slowly rolls down from the peak of the
potential to the down under the slow-roll approximation. Finally, the inflaton
decays to the particles and releases enough energy to reheat the supercooled
universe Kofman2 ; Shtanov . Using the inflationary observations coming from
the Planck and other related satellites, we find a few single field models in
good agreement with the observational datasets staro ; barrow ; kallosh1  and
the majority of them are ruled out or restricted martin . Also, these models
do not show non-Gaussianity in their spectrum because of the existence of the
uncorrelated modes. Chen . Therefore, if the future observations show a non-
Gaussianity feature in the spectrum, the single field models will be situated
in uncertainty. To escape such problems, it is proposed to go beyond the slow-
roll approximation in order to create some non-Gaussianity in the
perturbations spectrum of single field models. Hence, constant-roll inflation
is suggested in which the inflaton rolls down with a constant rate

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

where $\beta=-(3+\alpha)$ and $\alpha$ is a non-zero parameter martin2 ;
Motohashi1 ; Motohashi2 . Going beyond the slow-roll approach is also
addressed in the ultra slow-roll inflation in which we deal with a non-
negligible $\ddot{\varphi}$ in the Klein-Gordon equation
$\ddot{\varphi}=3H\dot{\varphi}$ Inoue ; Pattison ; Kinney ; Namjoo . It is
also found in the fast-roll realm where a fast-rolling phase is assumed at the
start of inflation and then is followed by the standard slow-roll after a few
e-folds Contaldi ; Lello ; Hazra . Recently, the constant-roll inflationary
approach has been intensively engaged so that one can find a wide range of the
inflationary models in this content, see, e.g., Odintsov ; Nojiri ; Sebastiani
; Motohashi9 ; Cicciarella ; Anguelova ; Ito ; karam1 ; Ghersi ; Lin ; Micu ;
Oliveros ; Motohashi3 ; oi1 ; neee ; oi2 ; Kamali ; diego ; sh1 ; sh2 ; oki ;
sh5 ; shokriii .

In this paper, we consider a quintessential inflationary model vil  introduced
by a Lorentzian function which is dependent on the number of e-folds in order
to obtain the slow-roll parameters. This approach studied in benn1 ; ben1 ;
ben2 ; benn2  for a single field model. It was shown that the corresponding
inflationary model in addition to explain the early time acceleration can
describe the late-time acceleration phase (dark energy), asymptotically. The
above discussion motivates us to arrange the paper as follows. In §II, we
present the constant-roll inflationary analysis for the model using the
Lorentzian function. We analyze the obtained results of the model by
comparison with the inflationary observations in §III. In §IV, we investigate
the model from viewpoint of the Weak Gravity Conjecture using the swampland
criteria. Conclusions are presented in §V.

##  II Constant-roll Inflation

We start with a Lorentzian function for the first slow-roll parameter ben1

$\epsilon_{1}\equiv\epsilon(N)=\frac{2\xi\Gamma}{\pi(4N^{2}+\Gamma^{2})},$ (2)

as a function of e-folds $N$ which is defined by $e^{N}=(a_{f}/a_{i})$. Also,
$\xi$ and $\Gamma$ are the amplitude and the width of the Lorentzian function.
Using the definition of the first slow-roll parameter
$\epsilon_{1}=-H^{\prime}/H$, we find an expression for the Hubble parameter

$H=C\exp{\Big{(}-\frac{\xi}{\pi}\arctan{(\frac{2N}{\Gamma})}\Big{)}},$ (3)

where $C$ is an integration constant. We can calculate other slow-roll
parameters using the generator formula $\epsilon_{n+1}=d\ln|\epsilon_{n}|/dN$
as follow

$\epsilon_{2}=-\frac{8N}{\Gamma^{2}+4N^{2}},\hskip
28.45274pt\epsilon_{3}=\frac{1}{N}-\frac{8N}{\Gamma^{2}+4N^{2}}.$ (4)

In order to study the model in the context of the constant-roll approach, we
set

$\epsilon_{2}=-\frac{\ddot{\varphi}}{H\dot{\varphi}}\equiv=-\beta,$ (5)

that leads to

$\beta=\frac{8N}{\Gamma^{2}+4N^{2}}.$ (6)

Now, let’s apply the formalism for a single field inflationary model. We
consider the action

$S=\int{d^{4}x\sqrt{-g}\bigg{(}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-V(\varphi)\bigg{)}}$
(7)

where $g$ is the determinant of the metric $g_{\mu\nu}$,
$R=g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar. Also, here we suppose
$\kappa^{2}\equiv 8\pi G=1$. Using the energy-momentum tensor of a perfect
fluid and Friedman-Robertson-Walker (FRW) metric
$ds^{2}=-dt^{2}+a(t)^{2}(dx^{2}+dy^{2}+dz^{2})$ for a homogeneous, isotropic
and spatially flat universe, the dynamical equations are given by

$3H^{2}=\frac{\dot{\varphi}^{2}}{2}+V,\hskip
28.45274pt2\dot{H}=-\dot{\varphi}^{2},$ (8)

where $H\equiv\dot{a}/a$ is the Hubble parameter, and the dot represents the
derivative with respect to $\varphi$. Moreover, by varying the action (7) with
respect to $\varphi$, we find the Klein-Gordon equation

$\ddot{\varphi}+3H\dot{\varphi}+\frac{dV}{d\varphi}=0.$ (9)

From the Friedmann equation (8) and using the obtained expression of the
Hubble parameter (3) and the relation (6), we find

$V(N)=Ce^{-\frac{2\xi}{\pi}\operatorname{arctanh}{\big{(}\frac{2N}{\Gamma}\big{)}}}\bigg{(}1-\frac{\Gamma\xi\beta}{12\pi
N}\bigg{)},$ (10)

Figure 1: The first slow-roll parameter $\epsilon$ (2) and the Hubble
parameter $H$ (3) plotted versus the number of e-folds $N$ (a and b). The
potential (12) plotted versus the scalar field $\varphi$ (c). The plots are
drawn for different values of the width $\Gamma$ when $|\beta|=0.02$ and the
amplitude $\xi$ is $\sim 120$. Also, $C=10^{-29}$ for the panel (b) and
$C=10^{-52}$ for the panel (c).

that using the number of e-folds

$N=\frac{\Gamma}{2}\sinh{\Big{(}\sqrt{\frac{\pi}{\xi\Gamma}}\varphi\Big{)}},$
(11)

the potential (10) takes the following form

$V(\varphi)=Ce^{-\frac{2\xi}{\pi}\operatorname{arctanh}{\Big{(}\sinh{\big{(}\sqrt{\frac{\pi}{\xi\Gamma}}\varphi\big{)}}\Big{)}}}\bigg{(}1-\frac{\xi\beta}{6\pi}\operatorname{csch}\big{(}\sqrt{\frac{\pi}{\xi\Gamma}}\varphi\big{)}\bigg{)},$
(12)

where $C$ is the integration constant. The panels (a) and (b) of Figure 1 show
the behaviour of the first slow-roll parameter $\epsilon$ (2) and the Hubble
parameter $H$ (3) versus the number of e-folds $N$. The panel (c) presents the
behaviour of the potential (12) versus the scalar field $\varphi$. The panels
are drawn for different values of the width $\Gamma$ when $|\beta|=0.02$ and
the amplitude $\xi\sim 120$. Also, $C=10^{-29}$ for the panel (b) and
$C=10^{-52}$ for the panel (c). In the presence of the constant-roll parameter
$\beta$, the potential still shows a quintessential inflationary manner since
it supplies the late-time acceleration condition, asymptotically. The slow-
roll parameters of the model

$\epsilon\equiv\frac{1}{2}\bigg{(}\frac{V^{\prime}}{V}\bigg{)}^{2},\hskip
28.45274pt\eta\equiv\frac{V^{\prime\prime}}{V},\hskip
28.45274pt\zeta^{2}\equiv\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}},$ (13)

are calculated as shown in (17 \- 19). Note that inflation ends when the
condition $\epsilon=1$ or $\eta=1$ is fulfilled. The number of e-folds of the
model takes the following form

$N\equiv\int^{\varphi_{i}}_{\varphi_{f}}{\frac{1}{\sqrt{2\epsilon}}}d\phi\simeq\frac{3\Gamma}{(\beta-12)}e^{\sqrt{\frac{\pi}{\xi\Gamma}}\varphi},$
(14)

and then the spectral index and tensor-to-scalar ratio

$n_{s}=1-6\epsilon+2\eta,\quad\quad\quad r=16\epsilon,$ (15)

are obtained as shown in (20) and (21).

##  III Constraints from CMB Anisotropies

In Figure 2, we present the $n_{s}-r$ constraints coming from the marginalized
joint 68% and 95% CL regions of the Planck 2018 in combination with BK14+BAO
data on the Lorentzian quintessential inflationary model (12) introduced in
the context of the constant-roll approach. The dashed and solid lines
represent $N=50$ and $N=60$, respectively.

The left panel of Figure 2 is plotted for different values of the width
$\Gamma$ when $|\beta|=0.02$ and the amplitude $\xi$ is $\sim 120$. For
$N=50$, the majority of the points show a reasonable value of tensor-to-scalar
ratio $r$ despite disfavored values of the spectral index $n_{s}$.

Figure 2: The marginalized joint 68% and 95% CL regions for $n_{s}$ and $r$ at
$k=0.002$ Mpc-1 from Planck in combination with BK14+BAO data cmb  and the
$n_{s}-r$ constraints on the model (12). The dashed and solid lines represent
$N=50$ and $N=60$, respectively. The left panel is plotted for different
values of the width $\Gamma$ when $|\beta|=0.02$ and the amplitude $\xi$ is
$\sim 120$. The right panel is plotted for different values of the amplitude
$\xi$ when $|\beta|=0.02$ and the width $\Gamma$ is $0.1$.

Figure 3: The behaviour of the swampland parameter $c$ versus the amplitude
$\xi$ for different widths $\Gamma$ (a). The behaviour of the swampland
parameter $c$ versus the width $\Gamma$ for different amplitudes $\xi$ (b).
The behaviour of the swampland parameter $c^{\prime}$ versus the amplitude
$\xi$ for different widths $\Gamma$ (c). The behaviour of the swampland
parameter $c^{\prime}$ versus the width $\Gamma$ for different amplitudes
$\xi$ (d). The plots are drawn for $|\beta|=0.02$ and $N=60$.

Considering the $68\%$ CL, we find that the obtained values of $r$ and $n_{s}$
for the width $\Gamma\leq 0.1$ are in good agreement with the observations.
Note that at the $95\%$, the panel does not show any compatibility with the
observational datasets. For $N=60$, the situation is quite better than $N=50$.
The panel reveals that the width $\Gamma>0.5$ is ruled out by the Planck data
while the widths $0.3<\Gamma\leq 0.5$ and $\Gamma\leq 0.3$ show
observationally acceptable values of $n_{s}$ and $r$ at the $68\%$ and $95\%$
CL, respectively. The right panel of Figure 2 is plotted for different values
of the amplitude $\xi$ when $|\beta|=0.02$ and the width $\Gamma$ is $0.1$. In
the case of $N=50$, the obtained values of $r$ and $n_{s}$ are out of the
observational region in exception the amplitude $\xi\leq 100$ that is
compatible with the observations at the $68\%$ CL. For $N=60$, the panel shows
that the results are fully consistent with the CMB observations for
$400<\xi\leq 600$ and $\xi\leq 400$ at the $68\%$ and $95\%$ CL, respectively.

##  IV Swampland Criteria

String theory as a developed theory in particle physics which can describe
many cosmological phenomena in the early universe without taking into account
a quantum gravity theory. String landscape and swampland regions included with
the low-energy effective field theories are compatible and non-compatible with
string theory, respectively. Hence, in order to embed the effective field
theories into stringy quantum gravity, we require to distinguish two regions
from each other. Weak Gravity Conjecture (WGC) is the most well-defined tool
to achieve the demanded separation Vafa ; Brennan ; Ooguri ; Palti ; Palti2 ;
Obied . Let’s study the model from the viewpoint of WGC using the conditions
of the swampland de sitter conjecture

$\sqrt{2\epsilon}\geq c,\hskip 28.45274pt|\eta|\leq-c^{\prime},$ (16)

where $c$ and $c^{\prime}$ are constant parameters. In Figure 3, we present
the behaviour of two swampland parameters $c$ and $c^{\prime}$ versus the
width $\Gamma$ and the amplitude $\xi$ of the Lorentzian function when
$|\beta|=0.02$ and $N=60$. From the panels (a) and (c), we find the swampland
conditions $c\leq 0.07$ and $c^{\prime}\leq-0.012$ when the observationally
favoured values $\Gamma=0.1$ and $\xi\leq 600$ are considered. Also, the
panels (b) and (d) almost present the same conditions when we focus on the
values $\xi=100$ and $\Gamma\leq 0.5$ which are compatible with the
observations.

##  V Conclusion

In the present manuscript, we have investigated a quintessential inflationary
model which is driven by a Lorentzian function in the presence of the
constant-roll condition. We have calculated the slow-roll parameters and then
the spectral inflationary parameters in order to present a complete study of
the model. Comparing the obtained results with the Planck in combination with
the BK14+BAO, we have found that the observationally allowed region of the
width of the Lorentzian function is $0.3<\Gamma\leq 0.5$ at the $68\%$ CL and
$\Gamma\leq 0.3$ at the $95\%$ CL when $|\beta|=0.02$ and the amplitude $\xi$
is $\sim 120$. This result is obtained for $N=60$ while the model is almost
ruled out by the observations for $N=50$. As a secondary analysis, we have
acquired the acceptable range of the amplitude $\xi$ of the Lorentzian
function when the width is fixed $\Gamma=0.1$, $|\beta|=0.02$ and $N=60$. We
have figured out that the amplitudes $400<\xi\leq 600$ and $\xi\leq 400$ are
in good agreement with the CMB observations at the $68\%$ and $95\%$ CL,
respectively. Finally, we have found the swampland conditions $c\leq 0.07$ and
$c^{\prime}\leq-0.012$ using the observational constraints on the width and
the amplitude of the Lorentzian. The issue would be investigated in our future
research, in particular, how works the reheating process in the model.
Moreover, studying the non-Gaussianity properties of the model could be very
interesting since the future observations of CMB will take into account the
non-Gaussianity factor in the power spectrum.

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##  Appendix A The slow-roll and spectral parameters of the model

The slow-roll parameters of the model:

$\epsilon=\frac{\pi^{2}(\beta-12)^{2}\cosh^{2}x+\Big{(}4\pi\beta\xi(\beta-12)\sinh
x+(24\beta-288)\pi^{2}+4\beta^{2}\xi^{2}\Big{)}\cosh^{2}x+48\pi\beta\xi\sinh
x-4\beta^{2}\xi^{2}+144\pi^{2}}{2\pi\Gamma\cosh^{2}x\sinh^{2}x\Big{(}36\pi^{2}\cosh^{2}x-12\pi\beta\xi\sinh
x+\beta^{2}\xi^{2}-36\pi^{2}\Big{)}},$ (17)
$\eta=\frac{\pi^{2}(12-\beta)\cosh^{4}x-\Big{(}6\pi\xi(\beta-4)\sinh
x+(\beta+24)\pi^{2}+4\beta\xi^{2}\Big{)}\cosh^{2}x+2\pi\xi(\beta-12)\sinh
x+4\beta\xi^{2}+12\pi^{2}}{\cosh^{2}x\Big{(}6\pi^{2}\Gamma\sinh^{3}x-\Gamma\pi\xi\beta\sinh^{2}x\Big{)}},$
(18)
$\displaystyle\\!\zeta^{2}=\frac{1}{{\pi^{2}\Gamma^{2}\cosh^{4}x\sinh^{4}x\Big{(}36\pi^{2}\cosh^{2}x-12\pi\beta\xi\sinh
x+\beta^{2}\xi^{2}-36\pi^{2}\Big{)}}}\Bigg{\\{}\pi^{2}(\beta-12)^{2}\cosh^{8}x+\pi^{2}\cosh^{6}x\Big{(}16(\beta-\frac{9}{2})\times$
$\displaystyle\\!\times(\beta-12)\pi\xi\sinh
x+5\pi^{2}(\beta-144)+4\xi^{2}(13\beta^{2}-120\beta+144)\Big{)}+\cosh^{4}x\Big{(}4\pi\xi\sinh
x\big{(}(\beta^{2}+120\beta-648)\pi^{2}+2\beta\xi^{2}\times$
$\displaystyle\\!\times(7\beta-24)\big{)}+1296\pi^{4}-4(19\beta^{2}-312\beta+432)\xi^{2}\pi^{2}+16\beta^{2}\xi^{4}\Big{)}+\cosh^{2}x\Big{(}4\pi\xi\sinh
x\big{(}(\beta^{2}-78\beta+648\big{)}\pi^{2}-4\beta(5\beta-$
$\displaystyle\\!-24)\xi^{2})+(24\beta-1008)\pi^{4}+32\xi^{2}(\beta^{2}-32\beta+54)\pi^{2}-32\beta^{2}\xi^{4}\Big{)}+24\xi\pi\sinh
x\Big{(}4(\beta-9)\pi^{2}+\xi^{2}\beta(\beta-8)\Big{)}+288\pi^{4}-$
$\displaystyle\\!-8\xi^{2}(\beta^{2}-36\beta+72)\pi^{2}+16\beta^{2}\xi^{4}\Bigg{\\}},$
(19)

where $x\equiv\sqrt{\frac{\pi}{\xi\Gamma}}\varphi.$

The spectral index of the mode:

$\displaystyle\\!n_{s}=\frac{1}{\pi\big{(}(\beta-12)^{2}N^{2}+9\Gamma^{2}\big{)}^{2}\big{(}(\beta-12)N+3\Gamma\big{)}^{2}\big{(}(\beta-12)N-3\Gamma\big{)}^{2}\Big{(}\pi(\beta-12)^{2}N^{2}-\xi\beta\Gamma
N(\beta-12)-9\pi\Gamma^{2}\Big{)}^{2}}$
$\displaystyle\\!\times\Bigg{\\{}\pi^{3}(\beta-12)^{12}N^{12}-2(\beta-12)^{11}\big{(}(\Gamma\xi+\pi\big{)}\beta-12\pi)\pi^{2}N^{11}-\Big{(}\xi(-\Gamma\xi+\pi)\beta^{2}+24\pi\xi\beta+18\pi(\pi\Gamma+8\xi)\Big{)}(\beta-12)^{10}$
$\displaystyle\\!\times\Gamma\pi
N^{10}-18(\beta-12)^{9}\Gamma^{2}\Big{(}(-\pi\xi\Gamma+7\pi^{2}-16\xi^{2})\beta+60\pi^{2}\Big{)}\pi
N^{9}-9\Big{(}(-4\pi^{2}\xi+16\xi^{3})\beta^{2}-192\pi^{2}\xi\beta+9\pi^{2}(\pi\Gamma-64\xi)\Big{)}$
$\displaystyle\\!\times(\beta-12)^{8}\Gamma^{3}N^{8}-108(\beta-12)^{7}\Gamma^{4}\Big{(}8\beta^{2}\xi^{2}+(-3\pi\xi\Gamma+9\pi^{2}+72\xi^{2})\beta-180\pi^{2}\Big{)}\pi
N^{7}+162(\beta-12)^{6}\Gamma^{5}\Big{(}(-\pi\xi^{2}\Gamma+5\pi^{2}\xi$
$\displaystyle\\!+16\xi^{3})\beta^{2}-168\pi^{2}\xi\beta+18\pi^{2}(\pi\Gamma-24\xi)\Big{)}N^{6}+972(\beta-12)^{5}\Gamma^{6}\Big{(}8\beta^{2}\xi^{2}+(-3\pi\xi\Gamma+9\pi^{2}+72\xi^{2})\beta-180\pi^{2}\Big{)}\pi
N^{5}-729$
$\displaystyle\\!\times\Big{(}(-4\pi^{2}\xi+16\xi^{3})\beta^{2}-192\pi^{2}\xi\beta+9\pi^{2}(\pi\Gamma-64\xi)\Big{)}(\beta-12)^{4}\Gamma^{7}N^{4}+13122(\beta-12)^{3}\Gamma^{8}\Big{(}(-\pi\xi\Gamma+7\pi^{2}-16\xi^{2})\beta+60\pi^{2}\Big{)}$
$\displaystyle\\!\times\pi
N^{3}-6561\Big{(}\xi(-\Gamma\xi+\pi)\beta^{2}+24\pi\xi\beta+18\pi(\pi\Gamma+8\xi)\Big{)}(\beta-12)^{2}\Gamma^{9}\pi
N^{2}+118098(\beta-12)\Gamma^{10}\big{(}(\Gamma\xi+\pi)\beta-12\pi\big{)}\pi^{2}N$
$\displaystyle\\!+531441\pi^{3}\Gamma^{12}\Bigg{\\}}$ (20)

The tensor-to-scalar ratio of the model:

$r=\frac{8N^{2}(\beta-12)^{4}\Gamma\xi\Big{(}\pi(\beta-12)^{4}N^{4}+12\xi\beta\Gamma(\beta-12)^{2}N^{3}+\big{(}18\pi\beta^{2}\Gamma^{2}-2592\pi\Gamma^{2}\big{)}N^{2}-108N\beta\Gamma^{3}\xi+81\pi\Gamma^{4}\Big{)}^{2}}{\pi\big{(}(\beta-12)^{2}N^{2}+9\Gamma^{2}\big{)}^{2}\big{(}(\beta-12)N+3\Gamma\big{)}^{2}\big{(}(\beta-12)N-3\Gamma\big{)}^{2}\Big{(}\pi(\beta-12)^{2}N^{2}-\xi\beta\Gamma
N(\beta-12)-9\pi\Gamma^{2}\Big{)}^{2}}$ (21)