# Scalar warm inflation in holographic cosmology

Zahra Bouabdallaoui1<EMAIL_ADDRESS>Ahmed Errahmani1
<EMAIL_ADDRESS>Mariam Bouhmadi-López2,3<EMAIL_ADDRESS>Taoufik Ouali1<EMAIL_ADDRESS>1Laboratory of Physics of Matter and
Radiation,
University of Mohammed first, BP 717, Oujda, Morocco
2Department of Physics University of the Basque Country UPV/EHU. P.O. Box 644,
48080 Bilbao, Spain
3IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain

###### Abstract

We consider warm inflation in the context of holographic cosmology. The weak
and the strong dissipative regime are analysed in the slow-roll approximation
and within what is known as intermediate inflation. For an appropriate choice
of the equation of state, the intermediate inflation is not only an exact
solution in general relativity, but also in the holographic setup considered
in this paper. Within this approach several dissipative and physically
relevant functions are considered. We constrain our model using the latest
Planck data. We conclude that three of the models analysed are consistent with
Planck data for some ranges of the model parameters. However, one of them is
ruled out by the observations.

##  I Introduction

The inflationary paradigm was proposed to sort out several shortcomings of the
standard Big Bang theory. 1 ; Albrecht ; Starobinsky ; Linde2 ; Sato ; Linde1
. In addition, this paradigm reproduces correctly the distribution of the
large scale structure 6 ; 7 ; 8 ; 9 ; 10  and the observed anisotropy of the
cosmic microwave background Bennett ; 11 ; 12 ; 2 ; Larson . The zoo of the
inflationary models is quite wild and many models are consistent with the
current observations. Therefore, it would be quite interesting to have some
approach that alleviate such a degeneracy Arxi17 ; Arxi18 ; Arxi19 ; Arxi20 ;
Arxi21 . Within the inflationary scenario with a slow roll approach, two roads
can be taken. On the one hand, in the cold inflationary scenario, the inflaton
does not interact with its environment and the reheating of the Universe takes
place at the end of inflation when the scalar field reach the bottom of the
potential and starts oscillating. However, this kind of approach might face
some fine-tunning problems and the reheating problem itself. Furthermore, the
latest Planck data ruled out many forms of inflaton potentials, including, for
instance, the quadratic and quartic potentials, both well motivated from a
particle physics perspective. On the other hand, there is the warm
inflationary (WI) scenario, where the radiation production occurs gradually as
the Universe inflates through the dissipation of the inflaton field into a
thermal radiation 18 ; 3 . In addition, initial density fluctuations (the
seeds of the large scale structure) is produced during the WI era i.e. the
fluctuation of the inflaton field is due to a thermal process rather than to a
quantum one as in cold inflation 4 ; 28 ; 29 ; 30 ; 31 . This thermal process
affect deeply the background and the perturbative results of the inflationary
dynamics Ber ; Bast .

Furthermore, cold and warm inflation have been well studied in the framework
of an effective approach to quantum gravity. Indeed, for example in extra-
dimensions models, motivated by superstring theory Stewart:1994ts ;
Dvali:1998pa  such as brane-world models where matter is considered to be
confined in a 3-brane while gravity can propagate in the bulk, have been a
natural arena for inflation. In addition, warm inflation has been analysed in
modified theory of gravity, like brane-worlds delCampo:2007cia ; Cid:2007fk ;
Deshamukhya:2009wc ; Nozari:2009th ; Romero:2009mg ; BasteroGil:2011mr ;
Herrera:2014nta ; Herrera:2015aja ; Kamali:2019xnt ; Cid  and in loop quantum
cosmology Herrera11 ; Zhang_2013 ; Herrera_2014 ; Graef ; Benetti_2019 . We
would to remember that Randall and Sundrum model Randall:1999vf  is extremely
well motivated and can be used as the main setup of the AdS/CFT
correspondence. We recall that the AdS/CFT correspondence, conjectured by
Maldacena Maldacena , consists in describing the 5-gravitational theory as a
conformal field theory on the 4-D boundary space-time. This duality can be
seen as an holographic approach. The effect of the holographic picture on cold
inflation has been analysed in Lidsey ; ZahraPRD  by a conformal field theory
on its 4-dimensional boundary space-time. Previous study of the effect of this
holographic picture on cold inflation scenario might be found in Refs. Lidsey
; ZahraPRD , and its extension by a non minimal coupling (NMC) to the induced
gravity by means of the dynamical system and the modifications of the
amplitude of the primordial perturbation in AatifaNPB ; AatifaIJMPD .
Recently, we have studied a NMC Higgs inflation in this context AatifaHiggs .

In this paper, we study warm inflation within an holographic view point in
order to complete our previous study ZahraPRD  and to see how well is warm
inflation supported within the holographic picture. In this context, we
consider a well known form of the scale factor and named as intermediate
inflation. This kind of model was already studied in general relativity and in
a modified theory of gravity Barrow1 ; Barrow2 ; Herrera3 ; Herrera4 ;
Herrera5 ; Herrera2 .

The outline of the paper is as follow. In Sec. II, we construct the main
formulas of warm inflation in the context of the holographic duality. In Sec.
III, we show how the intermediate inflation occurs from an equation according
to the holographic setup. In Sec. IV, we review the main formulas used in warm
inflation at the perturbative level. In Secs. V and VI we consider the case of
the weak and the strong dissipation regime respectively. In these sections, we
compute the main parameters of these regime in terms of the holographic
parameter and of the perturbation parameters. We conclude our paper in section
VII.

##  II HOLOGRAPHIC WARM INFLATION

In the AdS/CFT correspondence, the Friedmann equation is modified as Kiritsis
; Lidsey

$H^{2}=\frac{2\kappa\rho_{c}}{3}\Big{(}1+\epsilon\sqrt{1-\frac{\rho}{\rho_{c}}}\Big{)},$
(1)

where $\kappa=\hat{m}_{p}^{-2}=8\pi m_{p}^{-2}$ and $m_{p}$ is the
4-dimensional Planck mass, $\rho$ is the total energy density,
$\rho_{c}={3\hat{m}_{p}^{4}}/{8c}$ and $c={\hat{m}_{p}^{6}}/{M^{6}}$ is the
conformal anomaly coefficient which is defined as the ratio between the
4-dimensional Planck mass and the 5-dimensional mass $M$. At the low-energy
limit, $\rho\ll\rho_{c}$, and for the branch $\epsilon=-1$ the standard form
of the Friedmann equation can be recovered. In the following only this branch
will be considered.
In warm inflation, we will consider that the universe is filled with a self-
interacting scalar field with energy density $\rho_{\phi}$ and a radiation
energy density $\rho_{\gamma}$.

The conservation law of the total energy density
$\rho=\rho_{\phi}+\rho_{\gamma}$ reads

$\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})+\dot{\rho}_{\gamma}+4H\rho_{\gamma}=0$
(2)

The dynamical equations for the energies densities, $\rho_{\phi}$ and
$\rho_{\gamma}$, in warm inflation are described respectively by 3

$\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=-\Upsilon\dot{\phi}^{2},$ (3)
$\dot{\rho}_{\gamma}+4H\rho_{\gamma}=\Upsilon\dot{\phi}^{2},$ (4)

where a dot means derivative with respect to the cosmic time. The positive
dissipation factor, $\Upsilon$, is responsible for the reheating the universe
through the decay of the scalar field $\phi$. Several phenomenological
expression for the dissipation term $\Upsilon\dot{\phi}^{2}$ are given in the
literature 32 ; Gleiser ; 33 ; 34 ; Zhang ; Bastero ; Bastero1 .

Following Refs.Zhang ; Bastero , we consider the general form of the
dissipative coefficient, given by

$\Upsilon(\phi,T)=C_{\phi}\frac{T^{m}}{\phi^{m-1}}$ (5)

where $m$ is an integer and $C_{\phi}$ is associated to the dissipative
microscopic dynamics. Different expressions for the dissipation coefficient,
i.e. different choices of values of $m$, have been analyzed in Zhang ; Bastero
; Bastero1 . In this paper, we will discuss the following cases

  * •

$m=-1$ which corresponds to the dissipation rate of the non supersymmetric
model Gleiser ; 33 .

  * •

$m=0$ in which the dissipation coefficient represents an exponentially
decaying propagator in the high temperature regime. A dissipation coefficient
which depends only on the scalar field was first considered in warm inflation
in Oliveira .

  * •

$m=1$ corresponds to the high temperature regime Panotopoulos ; Ber ; Moss2 ;
Bastero2 .

  * •

$m=3$ is motivated by a supersymmetric model Ber ; BasteroB ; Bastero , a
minimal warm inflation Berghaus ; Laine ; Motaharfar , a quantum field theory
model of inflation Bastero-Gil  and through axion inflation DeRocco .
Furthermore, this case is used in other contexts such as in a hilltop model
Bueno , in a stochastic approach Ramos2 , in a potential with an inflection
point Cerezo  and in runway potentials Das_2020 . By using a sphaleron rate in
a non-abelian gauge fields Berghaus , a Chern-Simons diffusion rate in a
minimal warm inflation Laine  and in the Dirac-Born infeld Motaharfar , the
authors show that the dissipative coefficient in the case $m=3$ does not
depend on the inflaton fields as in Eq. (5).

The energy density, $\rho_{\phi}$, and the pressure, $p_{\phi}$, of the
standard scalar field can be written as

$\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V(\phi),\quad\quad
p_{\phi}=\frac{\dot{\phi}^{2}}{2}-V(\phi),$ (6)

where $V(\phi)$ represents the effective potential.

By introducing the dimensionless dissipation parameter $Q$, defined as

$Q\equiv\frac{\Upsilon}{3H},$ (7)

Eq. (3) can be rewritten as

$\dot{\rho}_{\phi}=-3H\dot{\phi}^{2}(1+Q).$ (8)

We will consider two dissipative regimes on this paper. The strong dissipative
one characterised by $Q\gg 1$ and the weak dissipative one in which $Q\ll 1$.

At the epoch of warm inflation it is safe to assume that the energy density of
the scalar field is the dominant component of the cosmic fluid
($\rho_{\phi}\gg\rho_{\gamma}$ ) 3 ; 4 ; 30 . Therefore, the effective
Friedmann Eq. (1) reduces to

$\displaystyle H^{2}$ $\displaystyle\approx$
$\displaystyle\frac{2\kappa\rho_{c}}{3}\Big{(}1-\sqrt{1-\frac{\rho_{\phi}}{\rho_{c}}}\Big{)}$
(9) $\displaystyle=$
$\displaystyle\frac{2\kappa\rho_{c}}{3}\Big{(}1-\sqrt{1-\frac{\frac{\dot{\phi}^{2}}{2}+V(\phi)}{\rho_{c}}}\Big{)},$

Furthermore, by combining Eq. (8) and the derivative of Eq. (9), one can show
that

$\dot{\phi}^{2}=[\dot{\phi}^{2}]_{\textrm{std}}G,$ (10)

where

$[\dot{\phi}^{2}]_{\textrm{std}}=-\frac{2\dot{H}}{\kappa(1+Q)},$ (11)

and

$G=1-\frac{3H^{2}}{2\kappa\rho_{c}},$ (12)

are the kinetic energy density of the scalar field in standard warm cosmology
RamonHerrera  and the correction term characterising the effect of AdS/CFT
correspondence, respectively.

We can notice that at the low energy limit, ($\rho\ll\rho_{c}$), the
correction term reduces to one and the standard expression of the scalar field
is recovered.

We suppose that during warm inflation the radiation production is quasi-
stable, i.e. $\dot{\rho}_{\gamma}\ll 4H\rho_{\gamma}$ and
$\dot{\rho}_{\gamma}\ll\Upsilon\dot{\phi}^{2}$ 3 ; 4 ; 30 . The combination of
Eqs. (4) and (10) yields

$\rho_{\gamma}=[\rho_{\gamma}]_{\textrm{std}}G,$ (13)

where $[\rho_{\gamma}]_{\textrm{std}}$ is the radiation energy density in
standard warm inflation RamonHerrera  expressed as

$[\rho_{\gamma}]_{\textrm{std}}=-\frac{\Upsilon\dot{H}}{2\kappa H(1+Q)}.$ (14)

Furthermore, the energy density of radiation is related to the temperature as
Kolb1

$\rho_{\gamma}=C_{\gamma}T^{4},$ (15)

where $C_{\gamma}=\pi^{2}g_{*}/30$ and $g_{*}$ characterise the number of
relativistic degrees of freedom. Substituting Eq. (15) into Eq.(13), the
temperature of the thermal bath $T$ can be written as

$T=[T]_{\textrm{std}}G^{\frac{1}{4}},$ (16)

where $[T]_{\textrm{std}}$ is the temperature in standard warm inflation
RamonHerrera

$[T]_{\textrm{std}}=\Big{[}-\frac{\Upsilon\dot{H}}{2\kappa
C_{\gamma}H(1+Q)}\Big{]}^{\frac{1}{4}}.$ (17)

Substituting Eq. (16) in (5), we get that

$\Upsilon^{\frac{4-m}{4}}=C_{\phi}\Big{[}\frac{1}{2\kappa
C_{\gamma}}\Big{]}^{\frac{m}{4}}\phi^{1-m}\Big{[}-\frac{\dot{H}}{H}\Big{]}^{\frac{m}{4}}\Big{(}1+Q\Big{)}^{-\frac{m}{4}}G^{\frac{m}{4}}.$
(18)

At low energy limit, ($\rho\ll\rho_{c}$), the standard expression RamonHerrera
of Eq. (18) is recovered.

Furthermore, by solving Eq. (9) for the scalar density and using
simultaneously Eq. (10), we find the effective potential

$V(\phi)=\frac{3H^{2}}{2\kappa}\Big{(}1+G\Big{)}+\frac{\dot{H}}{\kappa(1+Q)}\Big{(}1+\frac{3}{2}Q\Big{)}G.$
(19)

At low energy limit, Eq.(19) recovers the expression of standard warm
inflation RamonHerrera .

##  III Intermediate inflation

In order to illustrate our purpose, we will focus on the intermediate scenario
of inflation Barrow2 ; Barrow1 ; Herrera3 ; Herrera4 ; Herrera5 ; Bast  in
which an exact solution can be obtained. Nevertheless, exact solutions can
also be found for power law expansion of the universe 55  and in the de Sitter
inflationary scenario 1 . In the intermediate scenario of inflation, the scale
factor obeys the following expression Barrow2 ; Barrow1 ; Herrera3 ; Herrera4
; Herrera5 ; Bast

$a(t)=a_{I}\exp(At^{f}),$ (20)

where the two constants $f$ and $A$ satisfy the conditions $0<f<1$ and $0<A$
respectively. Intermediate inflation model may be derived from an effective
theory at low dimensions of a fundamental string theory. Therefore, the study
of the intermediate inflationary model is motivated by string/M-theory. In
this scenario, the cosmic expansion evolves slower than the standard de Sitter
model, $a(t)\propto\exp(H_{I}t)$ and faster than power law inflation
$(a\propto t^{p},p>1)$.

The intermediate scenario of inflation is still an exact solution in our
holographic setup. Indeed, an appropriate choice of the equation of state of
the form

$\rho+p=\gamma\rho_{c}\sqrt{1-\frac{\rho}{\rho_{c}}}\left[(1-\sqrt{1-\frac{\rho}{\rho_{c}}})\right]^{\lambda},$
(21)

for $\lambda>1$ and $\gamma>0$ reproduces this intermediate paradigm. For
$\lambda=1$ and $\rho\ll\rho_{c}$, we recover the standard Friedmann equation
and the barotropic equation of state $p=(\gamma-1)\rho$. The equation of state
(21) implies that while the weak energy conditions IJMPD  is satisfied, the
strong energy condition ($\rho+3p\geq 0$) is violated for a sufficiently small
energy density and for $\lambda>1$. The violation of the strong energy
condition assure that the acceleration of the Universe takes place.

With the help of Eq. (1) and the conservation of the total energy density

$\overset{\cdot}{\rho}+3H(\rho+p)=0,$ (22)

where $\rho=\rho_{\phi}+\rho_{\gamma}$ and $p=p_{\phi}+p_{\gamma}$ are the
energy density and the pressure respectively, a straightforward calculus leads
to the intermediate inflation given by Eq. (20).

On the other hand, the dimensionless slow-roll parameters are given by

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{1-f}{Aft^{f}}$ (23)
$\eta\equiv-\frac{\ddot{H}}{2H\dot{H}}=\frac{2-f}{2Aft^{f}}.$ (24)

In order to estimate the scale where the perturbations cross the Hubble
radius, we need to define the end of inflation, i.e. $\epsilon=1$. According
to the behaviour of the potential (see Eq. (19)), a description of how
inflation ends in our model requires a deeper analysis which is out of the
scope of this paper. However, to illustrate our purpose, we have plotted, in
Fig. 1 and 2, the potential $V$ and the first slow roll parameter $\epsilon$
for the couple of parameters ($A$, $f$). The weak dissipative regime is
illustrated in Fig. 1 for ($A=0.01$, $f=0.45$), ($A=0.08$, $f=0.27$) and
($A=0.1$, $f=0.13$) and the strong dissipative regime is illustrated in Fig. 2
for ($A=0.1$, $f=0.2$), ($A=0.1$, $f=0.12$) and ($A=0.1$, $f=0.06$). We notice
from Fig. 1, that the slow roll parameter is always less than one and its
violation cannot stop inflation, in the decreasing branch of the potential, in
the cases ($A=0.01$, $f=0.45$) and ($A=0.1$, $f=0.2$) for weak and strong
dissipative regime, respectively. In this case, a modification of the model by
adding a new a parameter $\phi_{end}$ may trigger the stop of the intermediate
inflation martin2013 . In the case of ($A=0.1$, $f=0.13$) and ($A=0.1$,
$f=0.0.06$), the slow roll parameter is always higher than one and the
intermediate inflation may occur between values of the scalar field where the
scalar field reaches the maximum of the potential and the scalar field value
at which $\epsilon=1$ (first solution) for weak and strong dissipative
regimes, respectively. In the last case, the intermediate inflation may also
occur between values of the scalar field at which $\epsilon=1$ (the second
solution) and infinite values of the scalar field.

|  |
---|---|---
Figure 1: Plot of the potential (left curve) and the slow roll parameter (right curve) in the weak dissipative regime for ($A=0.01$, $f=0.45$), ($A=0.08$, $f=0.27$) and ($A=0.1$, $f=0.13$). The horizontal dashed line (right curve) corresponds to $\epsilon=1$.  |  |
---|---|---
Figure 2: Plot of the potential (left curve) and the slow roll parameter
(right curve) in the strong dissipative regime for ($A=0.1$, $f=0.2$),
($A=0.1$, $f=0.12$) and ($A=0.1$, $f=0.06$). The horizontal dashed line (right
curve) corresponds to $\epsilon=1$.

Furthermore, the number e-folds $N$ is given by

$N=\int^{t_{end}}_{t_{k}}Hdt=A\Big{(}t_{end}^{f}-t_{k}^{f}\Big{)},$ (25)

where $t_{k}$ and $t_{end}$ are the cosmic time at the horizon crossing and at
the end of inflation, respectively. By equating Eq.(23) to unity, we obtain
$t_{end}=\Big{(}\frac{1-f}{Af}\Big{)}^{\frac{1}{f}}$, and Eq.(25) give us the
cosmic time at the horizon crossing

$t_{k}\dot{=}I(N)=\Big{(}\frac{1-f(N+1)}{Af}\Big{)}^{\frac{1}{f}}.$ (26)

##  IV PERTURBATIONS AND WARM INFLATION

Warm inflation not only affects the background parameters but can also modify
those at the perturbative level as the scalar perturbations; though the tensor
perturbations remains unchanged. The general idea for the perturbations in
warm inflation being of thermal origin were given in 3 ; 4 ; 28 ; 29 ; 30 .
However, the most relevant works with the modern notation in which the
fluctuations in the scalar field are mainly originated from thermal
fluctuation rather than quantum fluctuations during warm inflation can be
found in Graham_2009 ; Gleiser ; Bastero_Gil_2014 ; Bastero_2014 .

In order to study the cosmological perturbations, we consider the scalar
perturbations of a Friedmann-Lemaître-Robertson-Walker (FLRW) background in
the longitudinal gauge. Therefore, the perturbed metric reads

$ds^{2}=-(1+2\Phi)dt^{2}+a^{2}(t)(1-2\Phi)\delta_{ij}dx^{i}dx^{j},$ (27)

where $a(t)$ is the scale factor, $\Phi(t,x)$ is the scalar perturbation.
Furthermore, the dissipative coefficient depends explicitly, in general, on
the temperature and on the inflaton field as given in Eq (5). This dependence
modifies the set of the perturbative equations. In fact, the amplitude of the
scalar perturbation of the inflaton field will be enhanced by a correction
factor. These modifications appear, not only because of the scalar
perturbations of the background metric but also due to the existence of a
dissipative coefficient in the conservation of the energy momentum tensor
through

$\delta\Upsilon=\frac{\partial\Upsilon}{\partial T}\delta
T+\frac{\partial\Upsilon}{\partial\phi}\delta\phi.$ (28)

As we will notice in the next paragraph, this modification will affect
strongly the amplitude of the scalar spectrum. This treatment has been already
studied in literatures e.g. Bastero ; Bartrum_14 ; Bastero_Gil_2014 ;
Bastero_2014 ; Benetti_2017 . In the next paragraph, we introduce the
amplitude of the scalar perturbation of the inflaton field as evaluated at the
horizon crossing. Like we did previously, we assume that the perturbations
will not be highly affected by the holographic approach effect. Therefore, we
assume that the amplitude of the scalar perturbation is given by Benetti_2017
; Bartrum_14 ; Motaharfar_2019 ; Das_2020 ; Bastero_Gil_2014

$P_{{R}}=\left(\frac{H^{2}}{2\pi\dot{\phi}}\right)^{2}\left(1+2n+\frac{2\sqrt{3}\pi
Q}{\sqrt{3+4\pi Q}}\frac{T}{H}\right)F(Q)$ (29)

where the coefficient $n$ denotes the statistical distribution for the
inflaton field. By assuming that the inflaton is in thermal equilibrium with
the radiation bath, $n$ behaves like a Bose-Einstein distribution. The
function $F(Q)$ describes the coupling between the inflaton field and
radiation. Its expression is obtained numerically by resolving the systems of
perturbations equations for warm inflation Bastero ; Bastero2 ;
Bastero_Gil_2014 ; Graham_2009 . We will specify its numerical fit in the
strong dissipative regime for each value of the dissipative coefficient. From
Eqs. (10), (11) and (16), equation (29) can be rewritten as

$P_{R}=\frac{1}{4\pi^{2}}H^{4}\left[-\frac{2\dot{H}}{\kappa(1+Q)}\right]^{-1}G^{-1}\left(1+2n+\frac{2\sqrt{3}\pi
Q}{\sqrt{3+4\pi Q}}\frac{T}{H}\right)F(Q),$ (30)

where

$\frac{T}{H}=\left[-\frac{\Upsilon\dot{H}}{2\kappa
C_{\gamma}H(1+Q)}\right]^{\frac{1}{4}}H^{-1}G^{\frac{1}{4}},$ (31)

and the combination of Eqs. (7) and (18) give

$Q\left[1+Q\right]^{\frac{m}{4-m}}=\frac{1}{3H}C_{\phi}^{\frac{4}{4-m}}\left[\frac{1}{2\kappa
C_{\gamma}}\right]^{\frac{m}{4-m}}\phi^{4\frac{1-m}{4-m}}\left[-\frac{\dot{H}}{H}\right]^{\frac{m}{4-m}}G^{\frac{m}{4-m}},$
(32)

which can be rewritten in term of the parameters of the intermediate
inflation, Eq. (20), as

$P_{R}=\frac{A^{4}f^{4}}{4\pi^{2}}\left[\frac{2Af(1-f)}{\kappa(1+Q)}\right]^{-1}I(N)^{3f-2}(1-S(I(N))^{2f-2})^{-1}\left(1+2n+\frac{2\sqrt{3}\pi
Q}{\sqrt{3+4\pi Q}}\frac{T}{H}\right)F(Q),$ (33)
$\frac{T}{H}=\frac{1}{Af}\left[\frac{3Af(1-f)}{2\kappa
C_{\gamma}}\frac{Q}{1+Q}\right]^{\frac{1}{4}}I(N)^{\frac{2-3f}{4}}(1-S(I(N))^{2f-2})^{\frac{1}{4}},$
(34)
$Q\left[1+Q\right]^{\frac{m}{4-m}}=\frac{C_{\phi}^{\frac{4}{4-m}}}{3Af}\left[\frac{(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{m}{4-m}}\phi^{4\frac{1-m}{4-m}}I(N)^{-f+\frac{2m-4}{m-4}}\left[1-S(I(N))^{2f-2}\right]^{\frac{m}{4-m}}.$
(35)

where the quantities $S=3A^{2}f^{2}/2\kappa\rho_{c}$ and $I(N)$ are given by
Eq. (26).

In the next section, we will examine separately the weak dissipative regime
and the strong one. In the weak dissipative case, we will approximate the
function $F(Q)$ to be equal to one while for the strong dissipative regime we
will fix this function according to the values of the parameters $m$ found in
Benetti_2017 ; Motaharfar_2019

$\displaystyle F(Q)$ $\displaystyle\sim$ $\displaystyle
1+4.981Q^{1.946}+0.127Q^{4.330},$ (36) $\displaystyle F(Q)$
$\displaystyle\sim$ $\displaystyle 1+0.335Q^{1.364}+0.0185Q^{2.315},$ (37)
$\displaystyle F(Q)$ $\displaystyle\sim$
$\displaystyle\frac{1+0.4Q^{0.77}}{(1+0.15Q^{1.09})^{2}},$ (38) $\displaystyle
F(Q)$ $\displaystyle\sim 1,$ (39)

for $m=3$, $m=1$, $m=-1$ and $m=0$, respectively. We can notice, from Eqs.
(36)-(39), that in the strong dissipative regime, the scalar perturbations of
the inflaton field will be enhanced for the case $m\neq 0$ as compared with
the weak dissipative regime.

To complete the set of parameters required to constrain our model by the
observational data, we introduce the amplitude of the tensor perturbation,
$P_{T}$. As mentioned above, while the amplitude of the scalar perturbation in
warm inflation is given by Eq. (29), the amplitude of the tensor perturbation
remain unchanged and is given by Motaharfar_2019

$P_{T}=2\kappa\Big{(}\frac{H}{\pi}\Big{)}^{2},$ (40)

and from Eq. (29), we can express the tensor-to-scalar ratio,
$r={P_{T}}/{P_{R}}$, in terms of the number of e-folds N as

$r=\frac{8\kappa}{A^{2}f^{2}}\left[\frac{2Af(1-f)}{\kappa(1+Q)}\right]I(N)^{-f}(1-S(I(N))^{2f-2})\left(1+2n+\frac{2\sqrt{3}\pi
Q}{\sqrt{3+4\pi Q}}\frac{T}{H}\right)^{-1}F(Q)^{-1}.$ (41)

##  V WEAK DISSIPATIVE REGIME

Considering that the system evolves according to the weak dissipative regime
($\Upsilon\ll 3H$). Performing the integration of Eq. (10), the evolution of
the scalar field is given by

$\displaystyle\phi_{t}-\phi_{0}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{8A(1-f)}{\kappa f}}t^{\frac{f}{2}}H_{A,f,c}(t)$ (42)
$\displaystyle=$ $\displaystyle[\phi_{t}]_{\textrm{std}}H_{A,f,c}(t)$

where the integration constant, $\phi_{0}$, will be set to zero in the rest of
the paper and $H_{A,f,c}(t)$, a correction term as compared to standard warm
inflation RamonHerrera , can be expressed as

$H_{A,f,c}(t)={{}_{2}F_{1}}\Big{(}-\frac{1}{2},\frac{f}{4(f-1)};\frac{5f-4}{4(f-1)};St^{2(f-1)}\Big{)},$
(43)

where ${{}_{2}F_{1}}(a,b;c;d)$ is the hypergeometric function and
$S=\frac{3A^{2}f^{2}}{2\kappa\rho_{c}}$.

From Eqs. (20) and (18), as $Q=\Upsilon/3H\ll 1$, the dissipation coefficient
$\Upsilon$ can be rewritten in terms of the scalar field as

$\Upsilon=C_{\phi}^{\frac{4}{4-m}}\Big{[}\frac{1-f}{2\kappa
C_{\gamma}}\Big{]}^{\frac{m}{4-m}}\phi^{\frac{4(1-m)}{4-m}}t^{-\frac{m}{4-m}}\Big{(}1-St^{2f-2}\Big{)}^{\frac{m}{4-m}}.$
(44)

The evolution of the scalar field and of the dissipation coefficient are given
respectively, at the horizon crossing $t_{k}$, by

$\phi_{t_{k}}=\sqrt{\frac{8A(1-f)}{\kappa
f}}\Big{(}I(N)\Big{)}^{\frac{f}{2}}H_{A,f,c}(N),$ (45)

and

$\displaystyle\Upsilon$ $\displaystyle=$ $\displaystyle
C_{\phi}^{\frac{4}{4-m}}\Big{[}\frac{1-f}{2\kappa
C_{\gamma}}\Big{]}^{\frac{m}{4-m}}\phi_{t_{k}}^{\frac{4(1-m)}{4-m}}\Big{(}I(N)\Big{)}^{-\frac{m}{4-m}}$
(46) $\displaystyle\times$
$\displaystyle\Big{(}1-S\Big{(}I(N)\Big{)}^{2f-2}\Big{)}^{\frac{m}{4-m}}.$

If we compare the scalar field and the dissipation coefficient of our model at
the horizon crossing with that obtained in the case of standard warm inflation
RamonHerrera  we find, respectively

$\phi_{t_{k}}=[\phi_{t_{k}}]_{\textrm{std}}H_{A,f,c}(N),$ (47)

and

$\Upsilon=[\Upsilon]_{\textrm{std}}[\Upsilon]_{c},$ (48)

with

$[\Upsilon]_{\textrm{std}}=C_{\phi}^{\frac{4}{4-m}}\Big{[}\frac{1-f}{2\kappa
C_{\gamma}}\Big{]}^{\frac{m}{4-m}}\phi_{\textrm{std}}^{\frac{4(1-m)}{4-m}}\Big{(}I(N)\Big{)}^{-\frac{m}{4-m}},$
(49)

and

$[\Upsilon]_{c}=\Big{(}H_{A,f,c}(N)\Big{)}^{\frac{4(1-m)}{4-m}}\Big{(}1-S(I(N))^{2f-2}\Big{)}^{\frac{m}{4-m}}.$
(50)

Here $H_{A,f,c}(N)$ and $[\Upsilon]_{c}$ represent corrections term to the
scalar field and to the dissipation coefficient in the weak dissipation
regime, respectively. We can show that the correction term $H_{A,f,c}(N)$ does
not deviate too much from 1 i.e. the variation of the scalar field is not
strongly affected by the AdS/CFT correspondence.

The dissipation term introduced previously in Eq. (5) describes the thermal
radiation bath and its evolution versus the conformal anomaly coefficient tell
us about the effect of AdS/CFT correspondence on the thermal process. From
Fig. 3, we notice that the effect of the AdS/CFT correspondence is more
significative and increases with the conformal anomaly coefficient for $c\geq
10^{7}$. This effect increases the thermal process. However, we can see from
Eq. (46) that for specific values of $A$ and $f$ there must be a limiting
value $S(I(N))=1$ beyond which the dissipation coefficient no longer makes
sense.

|  |
---|---|---
|  |
Figure 3: Evolution of the correction term $[\Upsilon]^{W}_{c}$ versus the
conformal anomaly coefficient $c$ in the weak dissipative regime for $f=0.125$
and for three different values of the parameter $A$, for the case $m=0$ (upper
left panel), $m=1$ (upper right panel), $m=-1$ (down left panel) and $m=3$
(down right panel). We have used $\kappa=1$, $N=60$ and $C_{\gamma}=70$.

To incorporate the observed parameters in warm inflation at the perturbative
level, we rewrite the amplitude of the scalar perturbation Eqs. (33)-(35) and
the tensor-to-scalar ratio Eq. (41) in the weak dissipative regime (i.e.
$F(Q)\simeq 1$) as follow

$\displaystyle P_{R}$ $\displaystyle=$
$\displaystyle\frac{A^{4}f^{4}}{4\pi^{2}}\left[\frac{2Af(1-f)}{\kappa}\right]^{-1}I(N)^{3f-2}(1-S(I(N))^{2f-2})^{-1}\left(1+2n+2\pi
Q\frac{T}{H}\right),$ (51) $\displaystyle\frac{T}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{A.f}\left[\frac{3Af(1-f)}{2\kappa
C_{\gamma}}{Q}\right]^{\frac{1}{4}}I(N)^{\frac{2-3f}{4}}(1-S(I(N))^{2f-2})^{\frac{1}{4}},$
(52) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{C_{\phi}^{\frac{4}{4-m}}}{3Af}\left[\frac{(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{m}{4-m}}\phi^{4\frac{1-m}{4-m}}I(N)^{-f+\frac{2m-4}{m-4}}\left[1-S(I(N))^{2f-2}\right]^{\frac{m}{4-m}},$
(53) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{8\kappa}{A^{2}f^{2}}\left[\frac{2Af(1-f)}{\kappa}\right]I(N)^{-f}(1-S(I(N))^{2f-2})\left(1+2n+2\pi
Q\frac{T}{H}\right)^{-1},$ (54)

where at the crossing horizon time, the inflaton field is given by Eqs. (42)
and (62). Furthermore, by combining Eqs. (52) and (53), the above equations
read

$\displaystyle P_{R}$ $\displaystyle=$
$\displaystyle\frac{A^{4}f^{4}}{4\pi^{2}}\left[\frac{2Af(1-f)}{\kappa}\right]^{-1}\text{
}I(N)^{3f-2}(1-S(I(N))^{2f-2})^{-1}\left(1+2n+2{\pi}Q\frac{T}{H}\right),$ (55)
$\displaystyle\frac{T}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{Af}\left[\frac{C_{\phi}(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{1}{4-m}}I(N)^{-f+\frac{m-3}{m-4}}\left[1-S(I(N))^{2f-2}\right]^{\frac{1}{4-m}}\phi^{\frac{1-m}{4-m}},$
(56) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{C_{\phi}}{3Af}\left[\frac{C_{\phi}(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{m}{4-m}}\phi^{4\frac{1-m}{4-m}}I(N)^{-f+\frac{2m-4}{m-4}}\left[1-S(I(N))^{2f-2}\right]^{\frac{m}{4-m}},$
(57) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{8\kappa}{A^{2}f^{2}}\left[\frac{2Af(1-f)}{\kappa}\right]I(N)^{-f}(1-S(I(N))^{2f-2})\left(1+2n+2\pi
Q\frac{T}{H}\right)^{-1},$ (58)

and the scalar spectral index $n_{s}=1-\frac{d\ln(P_{R})}{dN}$ can be written
as

$\displaystyle n_{s}-1$ $\displaystyle=$ $\displaystyle-(3f-2)\frac{d\ln
I(N)}{dN}+\frac{d\ln(1-S(I(N))^{2f-2})}{dN}-\frac{d\ln\left(1+2n+2\pi
Q\frac{T}{H}\right)}{dN}$ (59) $\displaystyle=$
$\displaystyle-(3f-2)\frac{I(N)^{-f}}{Af}+\frac{2(1-f)SI(N)^{f-2}}{Af(1-S(I(N))^{2f-2})}+n_{4},$

where we have used Eq. (26) and we have defined $n_{4}$ as

$n_{4}=-\frac{d\ln\left(1+2n+2\pi Q\frac{T}{H}\right)}{dN}.$ (60)

The term $n_{4}$ is obtained numerically. The numerical analysis is done for
different values of the number of e-folds and the conformal anomaly
coefficient is set to be $c=10^{7}$. We have also assumed the warm condition
($\frac{T}{H}>1)$, the weak dissipation case $(Q\ll 1)$ and we have
constrained the spectral index in the observational range $0.9<n_{s}<1$.

The results show that the dissipative coefficients $\Upsilon_{1}=C_{\phi}T$
and $\Upsilon_{0}=C_{\phi}\phi$ are not supported by Planck data 12  for any
set of values of $C_{\phi}$, $A$ and $f$. However, for a convenient choice of
those parameters, namely $C_{\phi}$, $A$ and $f$, the dissipative coefficients
$\Upsilon_{-1}=C_{\phi}\phi^{2}/T$ and $\Upsilon_{3}=C_{\phi}T^{3}/\phi^{2}$
are well supported by data. Indeed,

  * •

for $m=-1$, we plot the dissipative parameter $Q$ and the tensor-to-scalar
ratio $r$ as parametric functions of $n_{s}$ for the following range of
parameters $C_{\phi}=10^{-25}$, $0.1<A<2$ and $f=0.125$. The parametric plot
is shown in Fig. 4.

  * •

for $m=3$, we plot the dissipative parameter $Q$ and the tensor-to-scalar
ratio $r$ as parametric functions of $n_{s}$ for the following range of
parameters $C_{\phi}=5.7\times 10^{7}$, $0.1<A<0.5$ and $f=0.5$. The
parametric plot is shown in Fig. 5.

We notice from the plots of Fig. 4 and 5, that we are, indeed, in the weak
dissipative regime. From the plots of the same Fig. 4 i.e. $m=-1$ and $m=3$ in
the dissipative coefficient, we notice that the tensor-to-scalar ratio lies in
the $1\sigma$ contour of Planck data for $N=55$ and $N=60$. This means that
the weak dissipative regime in warm inflation in the holographic context under
consideration may reproduce the observational data for the choice of
parameters considered above.

(a) |  | (b)
---|---|---
|  |
Figure 4: The plots show the variation of the dissipative parameter $Q$ (plot (a)) and the variation of the tensor-to-scalar ratio $r$ (plot (b)) versus the spectral index $n_{s}$ for different values of the e-fold number, the conformal anomaly coefficient $c=10^{7}$ and the parameter $m=-1$. These plots are obtained for the following choice of parameters $C_{\phi}=10^{-25}$, $0.1<A<2$ and $f=0.125$.  (a) |  | (b)
---|---|---
|  |
Figure 5: The plots show the variation of the dissipative parameter $Q$ (plot (a)) and the variation of the tensor-to-scalar ratio $r$ (plot (b)) versus the spectral index $n_{s}$ for different values of the e-fold number, the conformal anomaly coefficient $c=10^{7}$ and the parameter $m=3$. These plots are obtained for the following choice of parameters $C_{\phi}=5.7\times 10^{7}$, $0.1<A<0.5$ and $f=0.5$. (a) |  | (b)
---|---|---
|  |
Figure 6: The value of $T/H$ versus the spectral index $n_{s}$ for the
conformal anomaly coefficient $c=10^{7}$ for $m=-1$, $C_{\phi}=10^{-25}$ curve
(a) and for $m=3$, $C_{\phi}=5.7\times 10^{7}$ curve (b).

##  VI STRONG DISSIPATIVE REGIME $\Upsilon>>3H$

In this section, we analyse the strong dissipative regime, ($Q\gg 1$), by
considering exact solutions for the cases $m=3$ and $m\neq 3$. Integrating Eq.
(10) by using Eq. (20), the evolution of the scalar field is given by

$\phi_{t}-\phi_{0}=\left\\{\begin{array}[]{cc}\Big{[}\frac{(3-m)G_{m}(t)}{2K_{m}}\Big{]}^{\frac{2}{3-m}},&\quad\text{for}\;\;m\neq
3\\\
\exp\Big{[}\frac{G_{m}(t)}{K_{m}}\Big{]},&\quad\text{for}\;\;m=3\end{array}\right.$
(61)

with

$\displaystyle G_{m}(t)$ $\displaystyle=$ $\displaystyle
H_{A,f,c,m}^{s}(t)t^{\frac{(8-m)f+2(m-2)}{8}},$ (62) $\displaystyle
H_{A,f,c,m}^{s}(t)$ $\displaystyle=$
$\displaystyle{{}_{2}F_{1}}\Big{(}\frac{m-4}{8},\frac{(8-m)f+2(m-2)}{16(f-1)};1+\frac{(8-m)f+2(m-2)}{16(f-1)};St^{2(f-1)}\Big{)},$
(63) $\displaystyle K_{m}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{(8-m)f+2(m-2)}{8}\Big{)}\Big{(}\frac{4AfC_{\gamma}}{C_{\phi}}\Big{)}^{-\frac{1}{2}}\Big{(}\frac{3Af(1-f)}{2\kappa
C_{\gamma}}\Big{)}^{\frac{m-4}{8}},$ (64)

where $\phi_{0}$ is an integration constant which will be set to $\phi_{0}=0$
in the rest of the paper, $H_{A,f,c,m}^{s}(t)$, is a correction term to
standard warm inflation and $S=\frac{3A^{2}f^{2}}{2\kappa\rho_{c}}$.

Since $Q=\Upsilon/3H\gg 1$ and using Eqs. (18) and (20), the dissipation
coefficient $\Upsilon$ can be rewritten in terms of the scalar field as

$\Upsilon=C_{\phi}\Big{[}\frac{-3Af(f-1)}{2\kappa
C_{\gamma}}\Big{]}^{\frac{m}{4}}t^{\frac{m}{4}(f-2)}\phi^{1-m}\Big{(}1-St^{2(f-1)}\Big{)}^{\frac{m}{4}}.$
(65)

If we compare the amplitude of the scalar inflaton field Eq (61) and the
dissipation coefficient Eq. (65) at the horizon crossing and for $m\neq 3$
with that obtained in the case of standard warm inflation (see Ref
RamonHerrera ) we find respectively that

$\phi_{t_{k}}=[\phi_{t_{k}}]_{\textrm{std}}\Big{(}H_{A,f,c,m}^{s}(N)\Big{)}^{\frac{2}{3-m}},$
(66)

and

$\Upsilon=[\Upsilon]_{\textrm{std}}[\Upsilon]^{s}_{c},$ (67)

with

$[\Upsilon]^{s}_{c}=\Big{(}H_{A,f,c,m}^{s}(N)\Big{)}^{2\frac{(1-m)}{(3-m)}}\Big{(}1-St_{k}^{2f-2}\Big{)}^{\frac{m}{4}},$
(68)

is the correction term of the dissipation coefficient in the strong
dissipation regime.

Fig. 6 shows the plot of $T/H$ versus the spectral index in the weak
dissipation regime for $m=-1$ (curve (a)) and $m=3$ (curve (b)). The two
curves, (a) and (b), are obtained for the same choice of parameters as in
curve (a) and curve (b) in Fig. 5, respectively. These figures show clearly
that the results obtained in Fig. 5 satisfy the warm condition.

Fig. 7 shows the evolution of the correction term $[\Upsilon]^{s}_{c}$ versus
the conformal anomaly coefficient $c$ for three values of the parameter $m$
and for a fixed value of the parameter $f=0.25$ and for three values of the
parameter $A$. We notice that the AdS/CFT correspondence has no significant
effect and the thermal process is described by standard warm inflation for
$c<10^{7}$ for the three chosen values of $m$ parameters. Furthermore, while
for $m=0$ and $m=1$ the thermal process decreases, it increases strongly for
$m=-1$ with the increasing values of the conformal anomaly coefficient
$c>10^{7}$ for the range of $f$ and $A$ considered. However this range of the
conformal anomaly may change by changing the value of $f$ and $A$.

|  |
---|---|---

Figure 7: Evolution of the correction term $[\Upsilon]^{s}_{c}$ versus the
conformal anomaly coefficient $c$ for $f=0.125$ and for different values of
the parameter $A$ for the cases $m=0$ (upper left panel), $m=1$ (upper right
panel), and $m=-1$ (lower panel), we have used, $\kappa=1$, $N=60$ and
$C_{\gamma}=70$.

Furthermore, in the strong dissipative regime, $1\ll Q$, Eqs. (33)-(35) and
(41) take the following form

$\displaystyle P_{R}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3\pi}A^{2}f^{2}\kappa}{8\pi^{2}(1-f)}\left[\frac{3Af(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{1}{4}}I(N)^{\frac{3\left(3f-2\right)}{4}}(1-S(I(N))^{2f-2})^{-\frac{3}{4}}Q^{\frac{3}{2}}F(Q),$
(69) $\displaystyle\frac{T}{H}$ $\displaystyle=$
$\displaystyle\frac{1}{Af}\left[\frac{3Af(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{1}{4}}I(N)^{\frac{2-3f}{4}}(1-S(I(N))^{2f-2})^{\frac{1}{4}},$
(70) $\displaystyle Q$ $\displaystyle=$ $\displaystyle
C_{\phi}\left[3Af\right]^{\frac{4}{4-m}}\left[\frac{(1-f)}{2\kappa
C_{\gamma}}\right]^{\frac{m}{4}}I(N)^{f\frac{(m-4)}{4}+\frac{m-2}{2}}\left[1-S(I(N))^{2f-2}\right]^{\frac{m}{4}}\phi^{1-m},$
(71) $\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{16(1-f)}{\sqrt{3\pi}}\left[\frac{3Af(1-f)}{2\kappa
C_{\gamma}}\right]^{-\frac{1}{4}}I(N)^{-\frac{\left(f+2\right)}{4},}(1-S(I(N))^{2f-2})^{\frac{3}{4}}Q^{-\frac{3}{2}}F(Q)^{-1}.$
(72)

We notice here that $\frac{T}{H}$ in Eq. (70) is independent of the parameters
$Q$, $m$ and $C_{\phi}$. We notice also that according to the function $F(Q)$
which characterises the coupling between the inflaton field and radiation,
Eqs. (36)-(38), and Eqs. (69) and (71) that a strong effect is observed on the
amplitude of the scalar perturbation in the strong dissipative regime
particularly in the case of $m\neq 0$.

As in the weak dissipation case, a numerical approach to calculate the scalar
spectral index $n_{s}=1-\frac{d\ln(P_{R})}{dN}$ is mandatory. To this aim, we
set the value of the conformal anomaly coefficient to $c=10^{7}$, we consider
as well the warm condition ($\frac{T}{H}>1)$ and the strong dissipation case
$(Q\gg 1)$.

The results show that, for any range of parameters $C_{\phi}$, $A$ and $f$,
the dissipative coefficient $\Upsilon_{1}$ and $\Upsilon_{0}$ are excluded by
observational data. We plot the dissipative parameters and the tensor-to-
scalar ratio ($Q$ and $r$) as parametric functions of the spectral index
$n_{s}$. For the dissipative parameter $Q$, Fig. 8, the range of parameters
considered are $C_{\phi}=10^{10}$, $0.1<A<6$ and $f=0.0625$. While for the
dissipative parameter $Q$, Fig. 9, the range of parameters considered are
$C_{\phi}=1.5\times 10^{5}$, $0.1<A<4$, and $f=0.25$. We notice from the plots
of Figs. 8 and 9, that we are, indeed, in the strong dissipative regime. For
the plots of the same Figs. 8 and 9 i.e. $m=1$ and $m=3$ in the dissipative
coefficient, we notice that the tensor-to-scalar ratio and the spectral index
values agrees with the $1\sigma$ contour of Plank data for a large values of
$N$-efolds. In Fig. 10 we show the evolution of $T/H$ versus the spectral
index in the strong regime for $c=10^{7}$, $0.1<A<6$ and $f=0.25$. The curve
confirms that the result obtained in the strong dissipation regime satisfy the
conditions of warm inflation. This means that the strong dissipative regime of
warm inflation in the holographic context under consideration is in agreement
with the observational data for the choice of parameters mentioned above.

(a) |  | (b)
---|---|---
|  |
Figure 8: The plots show the variation of the dissipative parameter $Q$ (plot (a)) and the variation of the tensor-to-scalar ratio $r$ (plot (b)) versus the spectral index $n_{s}$ for different values of the e-fold number, the conformal anomaly coefficient $c=10^{7}$ and the parameter $m=1$. These plots are obtained for the following choice of parameters $C_{\phi}=10^{10}$, $0.1<A<6$ and $f=0.0625$. (a) |  | (b)
---|---|---
|  |
Figure 9: The plots show the variation of the dissipative parameter $Q$ (plot
(a)) and the variation of the tensor-to-scalar ratio $r$ (plot (b)) versus the
spectral index $n_{s}$ for different values of the e-fold number, the
conformal anomaly coefficient $c=10^{7}$ and the parameter $m=3$. These plots
are obtained for the following choice of parameters $C_{\phi}=1.5\times
10^{5}$, $0.1<A<4$, and $f=0.25$. Figure 10: The strong dissipation regime
value of $T/H$ versus the spectral index for the conformal anomaly coefficient
$c=10^{7}$, $0.1<A<6$ and $f=0.25$ .

##  VII Conclusions

In this work, we have studied warm inflation with a scalar field in the
context of the AdS/CFT correspondence and with a general form for the
dissipative coefficient. We have considered that the scale factor evolve with
time during intermediate inflation as $a(t)=a_{I}\exp{At^{f}}$, ($0<f<1$). In
this context, we have provided the basic equation in the weak and the strong
dissipation regime and we have shown that these equations are equal to the
standard one times some corrections term which at the low energy limit tends
to one. We have discussed our model in the frame of warm inflation condition
($T/H>1$) by determining the consistent range of $C_{\phi}$-parameter of the
dissipative coefficent, $A$ and $f$ parameters of the intermediate inflation
(see Figs. 4, 5, 8 and Fig. 9).

We have examined our theoretical prediction by plotting the evolution of
different inflationary parameters versus observational data. In both
dissipative regimes, while our model is well supported by Planck data for the
dissipative coefficient $\Upsilon_{3}$, it is not the case for the dissipative
coefficient $\Upsilon_{0}$. However, while the dissipative coefficient
$\Upsilon_{-1}$ is well supported by Planck data in the weak dissipative
regime, the dissipative coefficient $\Upsilon_{1}$ is well supported in the
strong dissipative regime.

We have shown that for a suitable interval of $C_{\phi}$-parameters of the
dissipative coefficient and for $A$ and $f$ parameters of the intermediate
inflation our predicted inflationary parameters are consistent with the
$1\sigma$ confidence level contours derived from Planck data.

Finally, we conclude that scalar warm inflation in the context of holographic
cosmology for the cases $m=-1$, $m=1$ and $m=3$ may describe the inflationary
era and predicts the appropriate inflationary parameters with respect to the
observational data within a slow roll approximation for the weak, strong and
both regime, respectively.

##  VIII Acknowledgment

We thank the referee for her/his suggestion concerning the perturbation issue.
The work of M. B. L. is supported by the Basque Foundation of Science
Ikerbasque. She also would like to acknowledge the support from the Basque
government Grant No. IT956-16 (Spain) and from the Grant No. PID2020–114035
GB-100, funded by MCIN/ 638AEI/10.13039/501100011033 and by “ERDF A way of
making Europe.” She is also partially supported by the grant FIS2017-85076-P,
funded by MCIN/ 638 AEI/10.13039/501100011033 and by “ERDF A way of making
Europe.”

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