# Perturbation Spectra of Warm Inflation in $f(Q,T)$ Gravity

Maryam Shiravand1<EMAIL_ADDRESS>Mehrdad Farhoudi1
<EMAIL_ADDRESS>Parviz Goodarzi2<EMAIL_ADDRESS>1Department
of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran
2Department of Physics, Ayatollah Boroujerdi University, Boroujerd, Iran

(February 6, 2024)

###### Abstract

We investigate the warm inflationary scenario within the context of the linear
version of $f(Q,T)$ gravity, coupled with both the inflaton scalar field and
the radiation field, under the conditions of the strong dissipation regime.
First, we calculate the modified Friedmann equations and the modified slow-
roll parameters. Subsequently, we apply the slow-roll approximations to derive
the scalar power spectrum and the tensor power spectrum. Also, we develop
formulations of the scalar and tensor perturbations for the $f(Q,T)$ gravity
with warm inflation scenario. Furthermore, we scrutinize two different forms
of the dissipation coefficient, a constant and a function of the inflaton
field to determine the scalar spectral index, the tensor-to-scalar ratio and
the temperature for the power-law potential case. By imposing some constraints
on the free parameters of the model, we attain results in good agreement with
both the Planck $2018$ data and the joint Planck, BK$15$ and BAO data for the
tensor-to-scalar ratio, and consistent results aligned with the Planck $2018$
data for the scalar spectral index. Consequently, we are able to revive the
power-law potential that was previously ruled out by observational data.
Moreover, for the variable dissipation coefficient, the model leads to the
scalar spectral index with the blue and red tilts in agreement with the WMAP
three years data.

Warm Inflation; Modified Gravity; Scalar Power Spectrum; Tensor Power
Spectrum; Cosmological Inflation; Strong Dissipation Regime

###### pacs:

98.80.Cq; 04.50.Kd; 98.80.Es; 04.25.Nx; 11.25.Db

##  I Introduction

In recent years, numerous scientific investigations, both theoretical and
observational, have been carried out to comprehend the dynamics of the
Universe. The outcomes of these studies align closely with the widely accepted
cosmological model known as $\Lambda$CDM, which is based on general relativity
(GR). Concurrently, surveys examining the cosmic microwave background (CMB)
radiation have provided crucial insights into the formation and development of
the Universe Hinshaw2013 ; Aghanim2021 . Despite the overall agreement,
certain perplexing concepts like the flatness and horizon problems persist
within the standard cosmological model. The concept of inflation has proven to
be a highly successful framework for addressing the issues in the standard
cosmology. It also serves as a foundation for explaining the generation and
evolution of the primordial seeds that give rise to the large-scale structure
of the Universe starobinsky1980 ; Guth1981 ; Albrecht1982 ; linde1982 ;
linde1990 .

The conventional inflation model consists of two distinct evolutionary phases.
The first phase is characterized by an accelerated expansion driven by an
inflaton field, which has negligible kinetic energy and potential energy, a
condition known as slow-roll. Notably, the quantum fluctuations of the
inflaton can lead to a period of inflation and provide an explanation for the
origin of large-scale structures Barrow1990 ; Barrow2006 . In simpler terms,
cosmic inflation produces density perturbations, which can be observed through
the measurement of temperature variations in the CMB wmap2003 ; Tegmark2004 .
On the other hand, a broad range of potential functions capable of describing
an inflationary period have been thoroughly investigated and constrained using
measurements of CMB anisotropies martin2014 . In the subsequent stage, known
as reheating, the inflaton undergoes decay, and giving rise to matter and
radiation fields similar to a hot big bang scenario Bassett2006 . The major
challenge lies in successfully bridging the evolution of the Universe towards
the end of this era. One intriguing approach to address this issue involves
investigating inflation in the context of the warm inflation scenario. Warm
inflation takes into account thermal effects and interactions between the
inflaton field and other particles in the Universe Bererafang1995 ; Berera1995
; Berera1997 ; Berera2023 . These interactions facilitate the continuous
transfer of energy from the inflaton to the surrounding particles and create a
significant thermal environment. The inclusion of thermal effects introduces
dissipation processes and has the potential to produce noticeable deviations
from the expected outcomes of cold inflation Berera1999 . This is achieved by
introducing a dissipation coefficient Berera2023 ; Oliveira1998 ; Berera2000 ;
Hall2004 . The dissipation term can be derived using quantum field theory
methods within a two-stage mechanism for the field interaction Bastero2011 ;
Bastero2013 . Consequently, warm inflation offers an alternative explanation
for the emergence of cosmic structure. Thus, the advantages of warm inflation
encompass the automatic reheating process at the end of inflation, where the
thermal bath begins to surpass the vacuum energy, as well as the suppression
of contributions to the scalar-tensor ratio Bellini1998 . This feature can
revive many inflationary models from the perspective of observational data.

A consequence of the friction term, which characterizes the decay of the
inflaton field into the radiation field during inflation, is that the density
perturbations are now mainly influenced by larger thermal fluctuations rather
than caused by quantum fluctuations Taylor2000 . For scalar perturbations, the
interaction between the inflaton field and the radiation field leads to
entropy perturbations along with adiabatic ones. This arises from the
characterization of warm inflation as an inflationary model with two
fundamental fields. In this framework, dissipative effects have the capacity
to produce a diverse range of spectra Herrera . In the large-scale limit of
slow-roll conditions, curvature perturbations remain constant and entropy
perturbations become negligible Hall2004 . In the literature, there are many
works in cosmological perturbations, see, e.g., Refs. Hall2004 ; Taylor2000 ;
Mukhanov1992 ; Oliveira2001 ; Brandenberger ; Bartolo2004 ; Weinberg ;
Bari2019 ; Antonio2022 .

On the other hand, while GR has been successful in predicting cosmological
phenomena Einestein1916 ; Will:2005va , it is not ideal for explaining the
existence of dark sectors (dark matter and dark energy) and their impacts on
the dynamics of the Universe, which align well with observational data
Huterer1999 ; Alam2004 ; Alam2007 ; Ishak2019 ; Ferreira:2019xrr . Such
shortcomings have prompted researchers to explore alternative theories of
gravity, see, e.g., Refs. Farhoudi2006 ; farb ; Sotiriou2010 ; Felice2010 ;
zare1 ; zare3 ; Haghani2018 ; zare5 . Various modified theories of gravity
have also been investigated in the context of inflationary cosmology, with the
aim of providing better alignment between their predictions and observational
data Starobinsky1980 ; Vilenkin1985 ; Majic1986 ; Fuji2003 ; Carrol2005 ;
Amendola2007 ; Nojiri2011 ; Shabani2014 ; Saba2018 ; Rasouli2018 ;
Odintsov2023 .

Alternative approaches in modifying GR involves exploring geometries beyond
the Riemannian framework, such as torsion and nonmetricity, which provide more
geometric descriptions of gravity. Early explorations in this direction were
conducted by Weyl, whose goal was to establish a geometric interpretation for
electromagnetism and a unified theory encompassing gravitation and
electromagnetism Weyl1918 ; Wheeler2018 . To incorporate this extension of
geometry, Weyl introduced a compensating vector field and also proposed that
during a parallel transformation of a vector, not only its direction but also
its magnitude undergoes a change. Within this concept, the connection of
system can be divided into two parts. Such that one part is a connection that
accounts for the variation in the magnitude of vector, while the other part is
the usual Levi-Civita connection that represents changes in the direction of
vector under a parallel transport. An important point is that both connections
are torsionless within the framework of the Weyl geometry. Weyl attempted to
interpret the introduced vector field as the electromagnetic potential, but
his theory ultimately proved unsuccessful and was even rejected by himself
Weyl1950 .

Nevertheless, Dirac made an attempt to revive the Weyl geometry. He
demonstrated Dirac1973  that the distinction between the Weyl and Riemannian
geometries lies in the expression of the partial derivative. In fact, through
the Weyl gauge transformation, the Weyl space can be reduced to a Riemannian
space with a metric that is conformally related to the original metric.
However, it has been argued Dirac1973  that the Weyl theory fails to describe
the electromagnetic interaction because the introduced vector field does not
interact with the spinor unlike the electromagnetic potential. While the Weyl
theory had limited physical success, it nevertheless presented some intriguing
aspects. Notably, one of the significant features of the Weyl geometry is the
presence of a nonzero covariant derivative of the metric tensor, which
geometrically leads to the introduction of a quantitative term known as
nonmetricity tensor. To simplify the geometric formalism of gravitation, an
alternative theory called the teleparallel equivalent of GR (TEGR) was
proposed. TEGR utilizes the Weitzenböck connection, which has zero curvature
and nonmetricity tensors, but non-zero torsion Maluf2013 .

Another alternative framework called symmetric teleparallel gravity has also
been developed, utilizing a connection with zero curvature and torsion
tensors, while incorporating a nonmetricity tensor to describe gravitational
interactions. This symmetric teleparallel gravity was further modified to
coincide with both GR and $f(Q)$ gravity,111The $Q$ term is the nonmetricity
scalar described in Sec. III. specifically in a gauge known as the coincident
gauge where the connection vanishes Naster1999 ; Jimenez2018 . A significant
aspect of this theory is its ability to separate gravitational and inertial
effects, something not achievable in GR. The symmetric teleparallel gravity
offers an alternative geometrical description of gravity that is dynamically
equivalent to GR. The theory is governed by the Einstein-Hilbert action in the
absence of boundary terms and naturally gives rise to self-accelerating
cosmological solutions in the early and late stages of the Universe, see,
e.g., Refs. Lu2019 ; Jimenez2020 ; Dimakis2022 . For a review on the
teleparallel gravity, see Ref. Bahamonde .

An alternative extension of the symmetric teleparallel gravity theory involves
the coupling between the geometry and the trace of the energy-momentum tensor
in action as an arbitrary function $f(Q,T)$. This construction enables a
straightforward description of gravitational interactions in the presence of
geometry-matter coupling. The obtained cosmological solutions can account for
both the accelerating and decelerating evolutionary phases of the Universe.
Consequently, $f(Q,T)$ gravity shows promise in providing valuable insights
into describing both the early and late stages of the Universe Xu2019 . To
date, numerous works and several cosmological aspects of $f(Q,T)$ gravity have
been investigated Arora:2020tuk ; Xu2020 ; Arora2020 ; Arora:2021jik ;
Gadbail:2021fjf ; Gadbail2021 ; Arora:2021met ; Godani2021 ; Pradhan2021 ;
Agrawal:2021rur ; Pati2021 ; 2022Arora ; Pati:2021zew ; Shiravand2022 ; El
Bourakadi2023 ; Bhagat2023 ; Tayde2023 ; Pati2022 ; Mandal2022 ; Narawada2023
; Das2023 . In this research, we investigate the warm inflation in $f(Q,T)$
gravity, warm inflation has already been considered in different areas and in
alternative gravitational theories, see, e.g., Refs. Herrera ; Nozari ;
Setare2012 ; Jamil2015 ; Sharif2016 ; Sharif2017 ; Amaek2022 ; Shahid2023 ;
Yeasmin2023 . Our focus lies on the context of linear version of $f(Q,T)$
gravity, where in our previous paper Shiravand2022 , we showed that the
evolution of general linear version of $f(Q,T)$ gravity is not fixed on a de
Sitter expansion and nor on the Minkowski metric. Although, it is also claimed
that the linear version of $f(Q,T)$ gravity is dynamically equivalent to the
linear version of $f(R,T)$ gravity that is claimed to be GR with a little
change in the content of matter Fisher2019 ; HMoraes ; Fisher2020 . However,
this equivalence is a matter of debate, and besides it provides another
geometrical formalism. In addition, we justified Shiravand2022  the
utilization of the linear version of $f(Q,T)$ gravity by providing a
comparison showing possible differences with what is obtained from the GR
case. Indeed, we indicated that the parameters in the linear version of
$f(Q,T)$ gravity provide better fittings to the observational data than GR.
Furthermore, through this version of gravity and based on the joint Planck,
BK$15$ and BAO data, we revived Shiravand2022  the previously excluded natural
potential.

The work structure is as follows. In Sec. II, we present some essential
relations on the conventional warm inflation and the related slow-roll
conditions and parameters. Then, in Sec. III, we give an overview on the
$f(Q,T)$ gravity in the framework of Friedmann-Lemaître-Robertson-Walker
(FLRW). Sec. IV is devoted to the investigation of warm inflation within the
context of linear version of $f(Q,T)$ gravity. In Sec. V, we investigate the
cosmological perturbations originating from $f(Q,T)$ gravity during warm
inflation. Thereafter, in Sec. VI, an analysis is conducted regarding warm
inflation featuring a power-law potential in the strong dissipation regime,
where constant and variable dissipation coefficients are considered. At the
end of this section, we discuss the weak dissipation regime very briefly.
Finally, in Sec. VII, we summarize the concluding results obtained.

##  II Cosmological Warm Inflation

Let us first start with the action of GR in the presence of a matter field as

$S=\int\sqrt{-g}\left(\dfrac{R}{2\kappa}+L_{\rm m}\right){\rm d}^{4}x.$ (1)

Here, $g$ is the metric determinant, $R$ stands for the Ricci scalar,
$\kappa=1/M^{2}_{\rm Pl}=8\pi G$ where $M_{\rm Pl}$ signifies the reduced
Planck mass in the natural units with $\hbar=1=c$, and $L_{\rm m}$ denotes the
Lagrangian of a matter field. By varying action (1) with respect to the
metric, one obtains

$R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}=\kappa\,T_{\mu\nu},$ (2)

where $T_{\mu\nu}$ represents the energy-momentum tensor generally defined as

$T_{\mu\nu}\equiv-\dfrac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{\rm m})}{\delta
g^{\mu\nu}}.$ (3)

We also consider the spatially flat FLRW geometry

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

where $a(t)$ represents the cosmological scale factor at the cosmic time $t$.

Assume that the matter is the inflaton scalar field with the Lagrangian

$L_{\rm
m}^{[\phi]}=-\dfrac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi-V(\phi)=\dfrac{1}{2}\dot{\phi}^{2}-V(\phi),$
(5)

where we have used $\phi\equiv\phi(t)$ as a homogeneous scalar field called
inflaton, $V(\phi)$ is its potential, and the lowercase Greek indices run from
zero to three. Then, upon inserting expression (5) into definition (3), it
gives

$\displaystyle T_{\mu\nu}^{[\phi]}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\phi\,\partial_{\nu}\phi-
g_{\mu\nu}\left[\dfrac{1}{2}\partial_{\sigma}\phi\partial^{\sigma}\phi+V(\phi)\right]$
(6) $\displaystyle=$
$\displaystyle\\!$$\dfrac{1}{2}\dot{\phi}^{2}+V(\phi)\\!$$\\!\delta^{0}_{\mu}\delta^{0}_{\nu}+a^{2}\\!$$\dfrac{1}{2}\dot{\phi}^{2}-V(\phi)\\!$$\\!\delta^{i}_{\mu}\delta_{\nu
i},$ (7)

where the lowercase Latin indices run from one to three. The energy-momentum
tensor associated with the inflaton field (which is conserved in this case)
can be characterized as equivalent to a perfect fluid with a linear barotropic
equation of state $p^{[\phi]}=w^{[\phi]}\rho^{[\phi]}$. Here, $\rho^{[\phi]}$
and $p^{[\phi]}$ respectively are the energy density and the pressure density
of the inflaton field as

$\rho^{[\phi]}=\dfrac{1}{2}\dot{\phi}^{2}+V(\phi)\qquad{\rm and}\qquad
p^{[\phi]}=\dfrac{1}{2}\dot{\phi}^{2}-V(\phi),$ (8)

where dot represents the derivative with respect to the cosmic time. Also, the
trace of the energy-momentum tensor is

$T^{[\phi]}=\dot{\phi}^{2}-4V.$ (9)

The variation of action (1) with respect to the inflaton field gives its
corresponding field equation, and hence by utilizing all field equations, one
obtains the corresponding Friedmann equations.

Now, in the warm inflationary paradigm, the inflaton field interacts with
other fields during inflationary era Berera1995 ; Berera2023 , and this leads
to the dissipation energy of inflaton as radiation. That is, the inflaton
field decays into radiation, and radiation is produced continuously. Under
these considerations, by employing Eq. (2) and relations (4), (6) and (8),
while assuming the radiation field is a perfect fluid, we readily derive the
new Friedmann equations, which are amended in the case of warm inflation, as
Berera1995

$3H^{2}=\kappa$\rho^{[\phi]}+\rho^{[\gamma]}$,$ (10)
$2\dot{H}+3H^{2}=-\kappa$p^{[\phi]}+p^{[\gamma]}$.$ (11)

Here, $H(t)\equiv\dot{a}(t)/a(t)$ is the Hubble parameter, and
$\rho^{[\gamma]}$ and $p^{[\gamma]}$ represent the energy density and the
pressure density associated with the radiation, respectively. Actually, in the
warm inflation, the background source is

$\rho=\rho^{[\phi]}+\rho^{[\gamma]}\qquad{\rm and}\qquad
p=p^{[\phi]}+p^{[\gamma]},$ (12)

which is now conserved.

Analogous to damped harmonic oscillators, where the damping term introduces
dissipation effects and prevents energy conservation with the rate of energy
loss, it is common BereraB2009 ; Arya2018 ; BereraB2018  to consider the
continuity equation

$\dot{\rho}^{[\phi]}+3H$\rho^{[\phi]}+p^{[\phi]}$=-\Upsilon\dot{\phi}^{2}$
(13)

for the inflaton field in the warm inflation. Here, $\Upsilon$ represents the
dissipation coefficient and, as a positive value in accordance with the second
law of thermodynamics, governs the transition of energy from the inflaton into
the radiation during the inflationary era. The dissipation of inflaton into
radiation leads to the generation of entropy. With a thermodynamic potential
as $V(\phi,\tau)$, the thermodynamic relation defines the entropy density of
radiation, denoted by $s(\phi,\tau)$, in terms of the thermodynamic potential
as Hall2004

$s(\phi,\tau)=-\dfrac{\partial V(\phi,\tau)}{\partial\,\tau},$ (14)

where $\tau$ is the temperature of the thermal bath. Obviously, if we consider
the potential only as $V(\phi)$, the corresponding entropy will vanish. Also,
since the dissipation leads to the generation of radiation hence, knowing that
its pressure density is $p^{[\gamma]}=\rho^{[\gamma]}/3$, its energy density
changes over time as

$\dot{\rho}^{[\gamma]}+4H\rho^{[\gamma]}=\Upsilon\dot{\phi}^{2}.$ (15)

Furthermore, by substituting relations (8) into Eq. (13), it gives the
corresponding Klein-Gordon equation as

$\ddot{\phi}+\dot{\phi}$3H+\Upsilon$+V^{\prime}=0,$ (16)

where the prime denotes derivative with respect to the inflaton field. Then,
by defining the dissipation rate, which quantifies the rate of dissipation
coefficient relative to the rate of expansion as

$\chi\equiv\dfrac{\Upsilon}{3H},$ (17)

we can rewrite Eq. (16) as

$\ddot{\phi}+3H\dot{\phi}$1+\chi$+V^{\prime}=0.$ (18)

The $\chi\gg 1$ corresponds to strong dissipation regime, while $\chi\ll 1$
corresponds to the domain of weak dissipation. Additionally, various forms for
the dissipation coefficient can be considered, such as a constant
$\Upsilon_{0}$, a function of the inflaton field $\Upsilon(\phi)$, a function
of temperature of thermal bath $\Upsilon(\tau)$, and/or a combination of both
$\Upsilon(\phi,\tau)$ Berera1995 ; Berera1997 ; Zhang2009 . In the following,
we investigate the necessary conditions for warm inflation.

Slow-Roll Conditions and Parameters

While considering relations (8) and Eqs. (10), (15) and (18), for warm
inflation to occur and last long enough, the conditions of slow-roll must be
satisfied as

$\displaystyle\dot{\phi}^{2}\ll V(\phi),\qquad\ddot{\phi}\ll
3H\dot{\phi}(1+\chi)$ (19)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rm
and}$
$\displaystyle\rho^{[\gamma]}\ll\rho^{[\phi]},\qquad\dot{\rho}^{[\gamma]}\ll
4H\rho^{[\gamma]}\ \,{\rm and}\ \,\Upsilon\dot{\phi}^{2}.$ (20)

Here, the last two terms indicate the dominance of the inflaton energy density
over the radiation energy density during the warm inflationary epoch and also
imply the state of relative stability in radiation generation Berera1995 ;
Berera2000 ; Hall2004 .

Under these considerations, Eqs. (10), (18) and (15) are reduced respectively
to

$\displaystyle 3H^{2}\approx\kappa\,V,$ (21) $\displaystyle
3H\dot{\phi}(1+\chi)\approx-V^{\prime},$ (22)
$\displaystyle\rho^{[\gamma]}\approx\dfrac{\Upsilon\dot{\phi}^{2}}{4H}=\dfrac{3}{4}\chi\dot{\phi}^{2}\mathrel{\mathop{=}^{{\rm
also}}}C_{\gamma}\tau^{4},$ (23)

where $C_{\gamma}=g_{\ast}\pi^{2}/30$ represents the Stefan-Boltzmann constant
for the black-body radiation BereraB2009 . Also, the constant $g_{\ast}$
denotes the effective number of relativistic degrees of freedom of the
radiation energy density in the thermal bath at the temperature $\tau$, whose
value is approximately $106.75$ for the standard model particles Correa2022 .
In addition, relations (8) read

$\rho^{[\phi]}\approx
V(\phi)\approx-p^{[\phi]}\quad\Rightarrow\quad\rho^{[\phi]}+p^{[\phi]}\approx
0$ (24)

during inflation. Also, by differentiating Eq. (21) and employing the Klein-
Gordon Eq. (22), we obtain the equation for $\dot{H}$ as

$\dot{H}\approx-\kappa$1+\chi$\dfrac{\dot{\phi}^{2}}{2}.$ (25)

Following Refs. liddle1994 ; Berera2009 ; Visinelli2011 ; Setare2013 ; myrzak
, there are several parameters that can be used to generally parameterize the
slow-roll approximations, namely

$\displaystyle\epsilon\equiv-\dfrac{\dot{H}}{H^{2}},$ (26)
$\displaystyle\eta\equiv-\dfrac{\dot{H}}{H^{2}}-\dfrac{\ddot{H}}{2\dot{H}H},$
(27)
$\displaystyle\lambda\equiv-\dfrac{\dot{\rho}^{[\gamma]}}{H\rho^{[\gamma]}}.$
(28)

Warm inflation occurs when these slow-roll parameters meet the conditions

$\epsilon\ll 1,\qquad\mid\eta\mid\ll 1,\qquad\mid\lambda\mid\ll 1.$ (29)

However, when the inflation ends, the value of the slow-roll parameters
becomes one. By utilizing Eqs. (21), (22), (23) and (25) plus conditions (29),
we can rephrase the slow-roll parameters in terms of the potential at leading
order as

$\displaystyle\epsilon\approx\dfrac{1}{2\kappa$1+\chi$}$\dfrac{V^{\prime}}{V}$^{2}\equiv\dfrac{\epsilon_{V}}{1+\chi},$
(30)
$\displaystyle\eta\approx\dfrac{1}{\kappa$1+\chi$}$\dfrac{V^{\prime\prime}}{V}$\equiv\dfrac{\eta_{V}}{1+\chi},$
(31)
$\displaystyle\lambda\approx\dfrac{1}{\kappa$1+\chi$}$\dfrac{\Upsilon^{\prime}V^{\prime}}{\Upsilon
V}$\equiv\dfrac{\lambda_{V}}{1+\chi},$ (32)

where obviously $1+\chi>0$ because $\chi$ is positive by definition, and we
have assumed that $\Upsilon$ is, in general, an explicit function of the
inflaton field. Therefore, for warm inflation to occur, the conditions
$\epsilon_{V}\ll 1+\chi$, $\eta_{V}\ll 1+\chi$ and $\lambda_{V}\ll 1+\chi$
must hold.

Moreover, using Eqs. (21) and (22), we can write the radiation energy density
in terms of the potential as

$\rho^{[\rm\gamma]}\approx\dfrac{\chi}{4\kappa$1+\chi$^{2}}\dfrac{$V^{\prime}$^{2}}{V}.$
(33)

Subsequently, by employing respectively relations (30), (19) and (8), we
obtain the relation between the radiation energy density and the inflaton
energy density during warm inflation as

$\rho^{[\rm\gamma]}\approx\dfrac{\chi}{2$1+\chi$}\epsilon
V\approx\dfrac{\chi}{2$1+\chi$}\epsilon\rho^{[\phi]}.$ (34)

This relation is self-consistent with the dominance of the inflaton energy
density over the radiation energy density during the warm inflationary epoch,
relation (20). At the end of inflation, in the strong dissipative regime
$\rho^{[\rm\gamma]}\approx\rho^{[\phi]}/2$, and in the weak dissipative regime
still $\rho^{[\gamma]}\ll\rho^{[\phi]}$.

Another significant parameter during inflation is the e-folding number. This
parameter quantifies the extent of expansion of the Universe throughout the
inflationary epoch and is generally defined as Liddle1999 ; Baumann2007

$N\equiv\ln$\dfrac{a_{\rm end}}{a}$=\int_{t}^{t_{\rm end}}H\rm{dt}.$ (35)

Here, the subscript ‘${\rm end}$’ denotes the value of the quantities at the
end of inflation. By utilizing Eqs. (21) and (22), the e-folding number in the
case of warm inflation can be reformulated in terms of the potential as

$N=\int_{\phi}^{\phi_{\rm
end}}\dfrac{H}{\dot{\phi}}\rm{d\phi}\approx\kappa\int_{\phi_{\rm
end}}^{\phi}$1+\chi$\dfrac{V}{V^{\prime}}\rm{d\phi}.$ (36)

In the subsequent sections, we first provide a concise overview of the
theoretical underpinnings of $f(Q,T)$ gravity222For more details, see, e.g.,
Refs. Xu2019 ; Shiravand2022  and references therein. and then investigate its
cosmological consequences.

##  III $f(Q,T)$ Gravity: a brief review

At first, it is known that any general connection can be decomposed into three
independent components, namely

$\Gamma^{\alpha}{}_{\mu\nu}=\\{^{\alpha}{}_{\mu\nu}\\}+K^{\alpha}{}_{\mu\nu}+L^{\alpha}{}_{\mu\nu},$
(37)

where $\\{^{\alpha}{}_{\mu\nu}\\}$, $K^{\alpha}{}_{\mu\nu}$ and
$L^{\alpha}{}_{\mu\nu}$ represent the Christoffel symbol, the contorsion
tensor and the disformation tensor, respectively. These terms are

$\displaystyle\\{^{\alpha}{}_{\mu\nu}\\}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}g^{\alpha\beta}(\partial_{\mu}g_{\beta\nu}+\partial_{\nu}g_{\beta\mu}-\partial_{\beta}g_{\mu\nu}),$
(38) $\displaystyle K^{\alpha}{}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\mathbb{T}^{\alpha}{}_{\mu\nu}+\mathbb{T}_{(\mu\nu)}{}^{\alpha},$
(39) $\displaystyle L^{\alpha}{}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-\dfrac{1}{2}g^{\alpha\beta}\left(Q_{\mu\beta\nu}+Q_{\nu\beta\mu}-Q_{\beta\mu\nu}\right),$
(40)

where $\mathbb{T}^{\alpha}{}_{\mu\nu}=2\Gamma^{\alpha}{}_{[\mu\nu]}$
represents the torsion tensor, and
$Q_{\alpha\mu\nu}\equiv\nabla_{\alpha}g_{\mu\nu}\neq 0$ stands for the
nonmetricity tensor.

Then, for later use, we define the traces of nonmetricity tensor, the
superpotential tensor and the nonmetricity scalar respectively as

${Q}_{\alpha}\equiv Q_{\alpha}{}^{\mu}{}_{\mu}\qquad{\rm
and}\qquad\tilde{Q}_{\alpha}\equiv Q^{\mu}{}_{\alpha\mu},$ (41)
$P^{\alpha}{}_{\mu\nu}\equiv-\dfrac{1}{2}L^{\alpha}{}_{\mu\nu}+\dfrac{1}{4}\left(Q^{\alpha}-\tilde{Q}^{\alpha}\right)g_{\mu\nu}-\dfrac{1}{4}\delta^{\alpha}_{(\mu}Q_{\nu)},$
(42)
$Q\equiv-g^{\mu\nu}\left(L^{\alpha}{}_{\beta\mu}L^{\beta}{}_{\nu\alpha}-L^{\alpha}{}_{\beta\alpha}L^{\beta}{}_{\mu\nu}\right).$
(43)

However, in the case of vanishing the Riemann and torsion tensors, i.e. a flat
manifold case, there is always an adapted coordinate system (i.e., the
coincident gauge) where
$\nabla_{\mu}\stackrel{{\scriptstyle*}}{{\rightarrow}}\partial_{\mu}$, that is
the connection effectively vanishes. Consequently, in this particular system,
relations (37) and (43) read

$L^{\alpha}{}_{\mu\nu}\mathrel{\mathop{=}^{*}}-\\{^{\alpha}{}_{\mu\nu}\\},$
(44)
$Q\mathrel{\mathop{=}^{*}}-g^{\mu\nu}\Big{(}\\{^{\alpha}{}_{\beta\mu}\\}\\{^{\beta}{}_{\nu\alpha}\\}-\\{^{\alpha}{}_{\beta\alpha}\\}\\{^{\beta}{}_{\mu\nu}\\}\Big{)},$
(45)

where the second result is equal to the minus of the effective Einstein-
Hilbert Lagrangian term. Moreover, relation (45) can be easily derived with
metric (4) and obtain

$Q=6H^{2},$ (46)

which is valid as a scalar in any frame.

Now, let us investigate the action for $f(Q,T)$ gravity in the presence of the
inflaton field as

$S=\int\sqrt{-g}\left[\dfrac{f(Q,T)}{2\kappa}+L_{\rm m}^{[\phi]}\right]{\rm
d}^{4}x,$ (47)

where its dynamical variables are the inflaton field, the metric and the
connection, which we assume to be in the case of vanishing the Riemann and
torsion tensors. Under these assumptions, varying action (47) with respect to
the metric and the connection, respectively, gives the field equations

$\begin{array}[]{l}\kappa\,T_{\mu\nu}^{[\phi]}=-\dfrac{2}{\sqrt{-g}}\bigtriangledown_{\alpha}\left(f_{\rm
Q}\sqrt{-g}P^{\alpha}{}_{\mu\nu}\right)-\dfrac{f\,g_{\mu\nu}}{2}\\\ \\\ \kern
16.79993pt+f_{\rm T}\left(T_{\mu\nu}^{[\phi]}+\Theta_{\mu\nu}\right)-f_{\rm
Q}\left(P_{\mu\alpha\beta}Q_{\nu}{}^{\alpha\beta}-2\,Q^{\alpha\beta}{}_{\mu}P_{\alpha\beta\nu}\right)\end{array}$
(48)

and

$\bigtriangledown_{\mu}\bigtriangledown_{\nu}$2\sqrt{-g}f_{\rm
Q}P^{\mu\nu}{}_{\alpha}+\kappa\,{\cal H}_{\alpha}{}^{\mu\nu}$=0,$ (49)

where

$\Theta_{\mu\nu}\equiv g^{\alpha\beta}\dfrac{\delta
T_{\alpha\beta}^{[\phi]}}{\delta g^{\mu\nu}},~{}~{}f_{\rm
Q}\equiv\dfrac{\partial f(Q,T)}{\partial Q},~{}~{}f_{\rm
T}\equiv\dfrac{\partial f(Q,T)}{\partial T}$ (50)

and

${\cal H}_{\alpha}{}^{\mu\nu}\equiv\dfrac{f_{\rm
T}\sqrt{-g}}{2\kappa}\dfrac{\delta\,T^{[\phi]}}{\delta\Gamma^{\alpha}{}_{\mu\nu}}+\dfrac{\delta$\sqrt{-g}\,L_{\rm
m}^{[\phi]}$}{\delta\Gamma^{\alpha}{}_{\mu\nu}}.$ (51)

Considering all the assumptions and after some manipulations, these field
equations lead to the modified Friedmann equations

$\displaystyle\kappa\,\rho^{[\phi]}=\dfrac{f}{2}-6FH^{2}-\dfrac{2\tilde{G}}{1+\tilde{G}}\left(\dot{F}H+F\dot{H}\right),$
(52)
$\displaystyle\kappa\,p^{[\phi]}=-\dfrac{f}{2}+6FH^{2}+2\left(\dot{F}H+F\dot{H}\right),$
(53)

where $F\equiv f_{\rm Q}$ and $\kappa\tilde{G}\equiv f_{\rm T}$. By
introducing the effective energy and pressure densities, as a conserved
perfect fluid, we rewrite the above equations as

$\displaystyle
3H^{2}=\dfrac{f}{4F}-\dfrac{\kappa}{2F}\left[\left(1+\tilde{G}\right)\rho^{[\phi]}+\tilde{G}p^{[\phi]}\right]\equiv\kappa\rho^{\rm[eff]},$
(54) $\displaystyle
2\dot{H}+3H^{2}=\dfrac{\kappa}{2F}\left[\left(1+\tilde{G}\right)\rho^{[\phi]}+\left(2+\tilde{G}\right)p^{[\phi]}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dfrac{f}{4F}-\dfrac{2\dot{F}H}{F}\equiv-\kappa
p^{\rm[eff]},$ (55)

Also, the evolution of the Hubble parameter is

$\dot{H}+\frac{\dot{F}}{F}H=\frac{\kappa}{2F}\left(1+\tilde{G}\right)\left(\rho^{[\phi]}+p^{[\phi]}\right).$
(56)

Furthermore, we remind that the energy-momentum tensor of the source,
$T_{\mu\nu}^{[\phi]}$, is not conserved in the context of the $f(Q,T)$
gravity. To illustrate this, let us first rewrite the field equation (48) as

$\displaystyle f_{\rm
T}\left(T^{[\phi]\mu}{}_{\nu}+\Theta^{\mu}{}_{\nu}\right)-\kappa
T^{[\phi]\mu}{}_{\nu}$ $\displaystyle=\dfrac{f}{2}\delta^{\mu}{}_{\nu}+f_{\rm
Q}Q_{\nu}{}^{\alpha\beta}P^{\mu}{}_{\alpha\beta}+\dfrac{2}{\sqrt{-g}}\nabla_{\alpha}\left(f_{\rm
Q}\sqrt{-g}P^{\alpha\mu}{}_{\nu}\right).$ (57)

Then, while using the field equation (49), the covariant divergence of
$T_{\mu\nu}^{[\phi]}$ can be obtained from the above equation, which becomes
Antonio2022 ; Xu2019

$\displaystyle\mathcal{D}_{\mu}T^{[\phi]\mu}{}_{\nu}-\dfrac{1}{f_{\rm
T}+\kappa}\left(f_{\rm T}\partial_{\nu}p^{[\phi]}-\dfrac{1}{2}f_{\rm
T}\partial_{\nu}T^{[\phi]}-B_{\nu}\right)=0,$ (58)

where $f_{\rm T}+\kappa\neq 0$, the symbol $\mathcal{D}_{\mu}$ signifies the
covariant derivative with respect to the Levi-Civita connection, and

$\displaystyle B_{\nu}\equiv$
$\displaystyle\dfrac{1}{\sqrt{-g}}\Big{[}Q_{\mu}\nabla_{\alpha}\Big{(}\sqrt{-g}f_{\rm
Q}P^{\alpha\mu}{}_{\nu}\Big{)}$
$\displaystyle+2\nabla_{\mu}\nabla_{\alpha}\Big{(}\sqrt{-g}f_{\rm
Q}P^{\alpha\mu}{}_{\nu}\Big{)}\Big{]}.$ (59)

Eq. (58) also indicates that in the case of metricity, i.e. when $B_{\nu}=0$,
plus lack of the matter-curvature coupling, i.e. when $f_{T}=0$, the energy-
momentum tensor of the source is conserved. However, finding a
$T^{\rm[eff]}_{\mu\nu}$ to write Eq. (58) in the form of its covariant
divergence equal to zero is not an easy task. That is why we resort to the
definition of $\rho^{\rm[eff]}$ and $p^{\rm[eff]}$, as given in Eqs. (54) and
(55), to define such an effective conserved term.

##  IV Warm Inflation in $f(Q,T)$ Gravity

In this section, we investigate the warm inflationary concept within the
framework of the linear version of $f(Q,T)$ gravity in the presence of the
inflaton scalar field $\phi$. In fact, we consider the specific functional
form of $f(Q,T)$ gravity as

$f(Q,T)=\alpha\,Q+\beta\,T^{[\phi]},$ (60)

where $\alpha$ and $\beta$ are constants within the model.

Analogous to Sec. II, the dynamics of the warm inflation scenario in the
framework of $f(Q,T)$ gravity can be described as

$\displaystyle 3H^{2}=\kappa\left(\rho^{[\rm eff]}+\rho^{[\gamma]}\right),$
(61) $\displaystyle 2\dot{H}+3H^{2}=-\kappa\left(p^{[\rm
eff]}+p^{[\gamma]}\right).$ (62)

Although, the inflaton field decays into radiation, due to Eqs. (54) and (55),
we have

$\displaystyle\dot{\rho}^{[\rm eff]}+3H$\rho^{[\rm eff]}+p^{[\rm
eff]}$=-\Upsilon\dot{\phi}^{2},$ (63)
$\displaystyle\dot{\rho}^{[\rm\gamma]}+4H\rho^{[\rm\gamma]}=\Upsilon\dot{\phi}^{2}.$
(64)

Furthermore, while considering the slow-roll condition
$\rho^{[\rm\gamma]}\ll\rho^{[\rm\phi]}$, upon utilizing relations (8), (9),
(46), (56) and (60) into Eqs. (61) and (62), we obtain

$\displaystyle
3H^{2}\approx-\dfrac{\left(\kappa+\beta\right)\dot{\phi}^{2}+2V(\phi)\left(\kappa+2\beta\right)}{2\alpha}\approx\kappa\rho^{[\rm
eff]},$ (65) $\displaystyle
2\dot{H}+3H^{2}\approx\dfrac{\left(\kappa+\beta\right)\dot{\phi}^{2}-2V(\phi)\left(\kappa+2\beta\right)}{2\alpha}\approx-\kappa
p^{[\rm eff]}.$ (66)

Then, by substituting Eqs. (65) and (66) into Eq. (63), we can derive the
corresponding modified Klein-Gordon equation

$\ddot{\phi}\left(\kappa+\beta\right)+3H\dot{\phi}\left(\kappa+\beta-\kappa\alpha\chi\right)+V^{\prime}\left(\kappa+2\beta\right)\approx
0.$ (67)

In addition, by differentiating Eq. (65) while using Eq. (67), we obtain the
modified equation governing $\dot{H}$ as

$\dot{H}\approx-$\frac{\kappa+\beta-\kappa\alpha\chi}{-\alpha}$\dfrac{\dot{\phi}^{2}}{2}$
(68)

that modifies Eq. (56) in this case for the warm inflation scenario. Actually,
the last term in the right hand side of Eq. (68) comes from Eq. (23).

Also, in this model, the slow-roll conditions (19) should be modified as

$\displaystyle$\kappa+\beta$\dot{\phi}^{2}\ll 2V$\kappa+2\beta$,$ (69)
$\displaystyle$\kappa+\beta$\ddot{\phi}\ll
3H\dot{\phi}$\kappa+\beta-\kappa\alpha\chi$.$ (70)

By employing these approximations, Eqs. (65) and (67) read

$\displaystyle 3H^{2}\approx\dfrac{-$\kappa+2\beta$}{\alpha}V,$ (71)
$\displaystyle
3H\dot{\phi}$\kappa+\beta-\kappa\alpha\chi$\approx-$\kappa+2\beta$V^{\prime}.$
(72)

Then, using Eqs. (23), (68), (71) and (72) plus conditions (29) into relations
(26), (27) and (28), we can derive the modified slow-roll parameters within
the linear version of $f(Q,T)$ gravity (60) in terms of the inflaton potential
as

$\displaystyle\epsilon\approx\dfrac{-\alpha}{2$\kappa+\beta-\kappa\alpha\chi$}$\dfrac{V^{\prime}}{V}$^{2},$
(73)
$\displaystyle\eta\approx\dfrac{-\alpha}{\kappa+\beta-\kappa\alpha\chi}$\dfrac{V^{\prime\prime}}{V}$,$
(74)
$\displaystyle\lambda\approx\dfrac{-\alpha}{\kappa+\beta-\kappa\alpha\chi}$\dfrac{\Upsilon^{\prime}V^{\prime}}{\Upsilon
V}$,$ (75)

where $\kappa+\beta-\kappa\alpha\chi\neq 0$. As is obvious, the comparison of
these parameters with the corresponding relations (30), (31), and (32) shows
that the slow-roll parameters have been modified by

$\dfrac{1}{\kappa$1+\chi$}\longrightarrow\dfrac{-\alpha}{\kappa+\beta-\kappa\alpha\chi}.$
(76)

Moreover, the obtained results dictate the constraint

$\dfrac{-\alpha}{\kappa+\beta-\kappa\alpha\chi}>0,$ (77)

which in turn, for negative values of $\alpha$, we must have

$\kappa+\beta-\kappa\alpha\chi>0.$ (78)

Additionally, by using Eqs. (71), (72) and (73) into Eq. (23), we obtain the
radiation energy density in this case as

$\rho^{[\rm\gamma]}\approx\dfrac{\chi$\kappa+2\beta$}{2$\kappa+\beta-\kappa\alpha\chi$}\epsilon
V\approx\dfrac{-\kappa\alpha\chi}{2$\kappa+\beta-\kappa\alpha\chi$}\epsilon\rho^{[\rm
eff]},$ (79)

where the last approximation is done employing Eq. (71) with approximation
(65). Relation (79) shows that $\rho^{[\rm\gamma]}\ll\rho^{[\rm eff]}$, which
is consistent with the condition $\rho^{[\rm\gamma]}\ll\rho^{[\rm\phi]}$.
Also, at the end of inflation, in the strong dissipative regime
$\rho^{[\rm\gamma]}\approx\rho^{[\rm eff]}/2$, and in the weak dissipative
regime still $\rho^{[\rm\gamma]}\ll\rho^{[\rm eff]}$. In addition, utilizing
Eqs. (71) and (72), the e-folding number in this case is

$N\approx\dfrac{1}{-\alpha}\int_{\phi_{\rm
end}}^{\phi}$\kappa+\beta-\kappa\alpha\chi$\dfrac{V}{V^{\prime}}d\phi.$ (80)

##  V COSMOLOGICAL PERTURBATIONS

The observations acquired from the CMB show that the power spectrum of
primordial perturbations aligns well with the power spectrum predictions in
the inflationary theory. In the cold inflation model, these perturbations stem
from the quantum fluctuations of the inflaton field. However, in the context
of warm inflation, its significance lies in the involvement of dissipative
effects throughout the inflationary period. This action leads to the
simultaneous occurrence of radiation production and inflationary expansion.
Dissipation is attributed to a friction term that characterizes the process of
dissipating a scalar field into a thermal bath through its interaction with
other fields. Moreover, warm inflation illustrates the influential role of
thermal fluctuations in generating initial perturbations during inflation. In
these models, density fluctuations stem primarily from thermal fluctuations
rather than quantum fluctuations, and these fluctuations constitute the
initial conditions for the formation of large-scale structures Berera1999 ;
Berera2009 . In the following subsections, we first prepare the necessary
equations for scalar and tensor perturbations in the context of warm inflation
within the $f(Q,T)$ gravity. Then, we probe the effects of these perturbations
on the linear version of $f(Q,T)$ gravity relation (60).

###  V.1 SCALAR PERTURBATIONS

In this subsection, we explore an equation that governs cosmological
perturbations within a system that encompasses both the inflaton field and
radiation. In the process of perturbation, we mainly perturb four quantities,
namely the metric, the four-velocity of the radiation fluid, the inflaton
field and the energy density of radiation field. Of course, these
perturbations in turn cause the other quantities involved to be perturbed.
Furthermore, our particular focus is up to the second-order of perturbations;
and to be consistent with the literature, from now on, we show unperturbed
quantities with a bar over them.

For perturbing the metric,, in the Einstein frame by adopting the
Newtonian/longitudinal gauge, the perturbation of the background (unperturbed)
metric (4), when the energy-momentum tensors involved are diagonal, is
expressed as Hall2004 ; Mukhanov1992 ; Bari2019

$ds^{2}=-(1+2\Psi)dt^{2}+a^{2}(t)(1-2\Psi)\delta_{ij}dx^{i}dx^{j}.$ (81)

Here, the metric perturbation variable is assumed to be $|\Psi(t,{\bf x})|\ll
1$ that parameterizes the scalar perturbations. For later use, from the
perturbed metric (81), we obtain

$\delta g^{\mu\nu}\approx{\rm
diag}\left[2\Psi,2\Psi/a^{2},2\Psi/a^{2},2\Psi/a^{2}\right].$ (82)

In the warm inflation scenario, it is possible to divide the energy-momentum
tensor into two components. That is, the effective component (which consists
of the inflaton part and the correction terms related to the $f(Q,T)$ theory,
i.e. Eqs. (54) and (55)) and the radiation part denoted by
$T_{\mu\nu}^{[\gamma]}$. That is, we have

$T_{\mu\nu}=T_{\mu\nu}^{[\gamma]}+T_{\mu\nu}^{[\rm eff]}.$ (83)

As mentioned, the radiation field is described as a perfect barotropic fluid,
i.e.

$T_{\mu\nu}^{[\gamma]}=\frac{\rho^{[\gamma]}}{3}$4u_{\mu}u_{\nu}+g_{\mu\nu}$,$
(84)

and the same relation for the corresponding unperturbed one. In this regard,
the components of the unperturbed four-velocity of the radiation fluid,
$\overline{u}_{\mu}$, are $\overline{u}_{0}=-1$ and $\overline{u}_{i}=0$ in
the rest frame. For perturbing the four-velocity of the radiation fluid, we
assume that the perturbed four-velocity of the radiation fluid is

$u_{\mu}(t,{\bf x})=\overline{u}_{\mu}(t)+\delta u_{\mu}(t,{\bf x})$ (85)

with the normalization constraint

$u^{\alpha}u_{\alpha}=g^{\mu\nu}u_{\mu}u_{\nu}=-1,$ (86)

where obviously $u^{\mu}\approx\overline{u}^{\mu}+\delta
u^{\mu}+\overline{u}_{\nu}\delta g^{\mu\nu}$. The variable $\delta u_{\mu}$
represents the perturbed part of the four-velocity of the radiation fluid,
where $\delta u^{\mu}\equiv\delta(u^{\mu})=\delta(g^{\mu\nu}u_{\nu})$ Weinberg
, and constraint (86) gives

$\delta u_{0}\approx-\Psi\qquad\Rightarrow\qquad\delta u^{0}\approx-\Psi.$
(87)

However, the velocity perturbation, $\delta u_{i}$, stands as an independent
dynamic variable that can333This is possible because for scalar perturbations
such a ‘flow’ is irrotational. be defined as

$\delta u_{i}\equiv
a(t)\partial_{i}(\delta{\mathfrak{u}})\qquad\Rightarrow\qquad\delta
u^{i}\approx a(t)\partial^{i}(\delta{\mathfrak{u}}),$ (88)

where $\delta{\mathfrak{u}}$ is a scalar velocity potential Weinberg ;
mukhanov .

For perturbing the inflaton field and the energy density of the radiation
field, we assume that these perturbed fields are

$\phi(t,{\bf x})=\overline{\phi}(t)+\delta\phi(t,{\bf x}),$ (89)
$\rho^{[\gamma]}(t,{\bf
x})=\overline{\rho^{[\gamma]}}(t)+\delta\rho^{[\gamma]}(t,{\bf x}),$ (90)

and in turn, we have $\delta p^{[\gamma]}=\delta\rho^{[\gamma]}/3$.

Next, the exchange of energy between the two components $\rho^{[\phi]}$ (and
as a result $\rho^{[\rm eff]}$) and $\rho^{[\gamma]}$ can be depicted through
a flux term as Moss2007

$\Lambda_{\mu}=-\Upsilon u^{\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi.$ (91)

From the same relation for its corresponding unperturbed one, we obtain
$\overline{\Lambda}_{0}=-(\dot{\overline{\phi}})^{2}\,\overline{\Upsilon}$ and
$\overline{\Lambda}_{i}=0$. Hence, Eqs. (63) and (64) read

$\displaystyle\dot{\overline{\rho^{[\rm eff]}}}+3H$\overline{\rho^{[\rm
eff]}}+\overline{p^{[\rm eff]}}$=-\overline{\Lambda}^{0},$ (92)
$\displaystyle\dot{\overline{\rho^{[\rm\gamma]}}}+4H\overline{\rho^{[\rm\gamma]}}=\overline{\Lambda}^{0},$
(93)

In the same merit, and in general, the conservation of the corresponding
energy-momentum tensors can be written as

$\displaystyle\overline{{\cal D}}_{\mu}\overline{T^{[\rm eff]}}^{\mu}{}_{\nu}$
$\displaystyle=$ $\displaystyle-\overline{\Lambda}_{\nu},$ (94)
$\displaystyle\overline{{\cal D}}_{\mu}\overline{T^{[\gamma]}}^{\mu}{}_{\nu}$
$\displaystyle=$ $\displaystyle\overline{\Lambda}_{\nu},$ (95)

and the same equations for the corresponding perturbed ones.

At this stage, we prepare some necessary equations for use in the next steps.
By writing the perturbed exchange of energy as
$\Lambda_{\mu}=\overline{\Lambda}_{\mu}+\delta\Lambda_{\mu}$ and substituting
all terms into its relation (91), we obtain the components of perturbation of
exchange energy as444Although relation $[\partial_{\mu},\delta]=0$ holds, we
do not use it to avoid mixing up functions, e.g., $\phi$ with $\delta\phi$,
where the former actually is $\phi(t)$ and the latter $\delta\phi(t,{\bf x})$.

$\displaystyle\delta\Lambda_{0}$ $\displaystyle=$
$\displaystyle-(\dot{\overline{\phi}})^{2}\,\delta\Upsilon+(\dot{\overline{\phi}})^{2}\,\Psi\overline{\Upsilon}-2\overline{\Upsilon}\,\dot{\overline{\phi}}\,\partial_{0}(\delta\phi),$
(96) $\displaystyle\delta\Lambda_{i}$ $\displaystyle=$
$\displaystyle-\overline{\Upsilon}\,\dot{\overline{\phi}}\,\partial_{i}(\delta\phi).$
(97)

Also, by perturbing Eq. (95) for the corresponding perturbed quantities, i.e.
$\delta${\cal D}_{\mu}T^{[\gamma]\mu}{}_{\nu}$=\delta\Lambda_{\nu}$, while
using metric (81) (to compute the Levi-Civita connection), relations (84),
(85) and (90), and Eqs. (93), (96) and (97), we obtain from its zeroth and
i-th components, respectively

$\displaystyle\partial_{0}(\delta\rho^{[\gamma]})+4H\delta\rho^{[\gamma]}+{4\over
3}\overline{\rho^{[\gamma]}}a\,\nabla^{2}(\delta{\mathfrak{u}})-4\dot{\Psi}\overline{\rho^{[\gamma]}}$
$\displaystyle\approx\delta\Upsilon(\dot{\overline{\phi}})^{2}-\Psi\overline{\Upsilon}\,(\dot{\overline{\phi}})^{2}+2\overline{\Upsilon}\,\dot{\overline{\phi}}\,\partial_{0}(\delta\phi)$
(98)

and

$\displaystyle
4\left[\overline{\rho^{[\gamma]}}\partial_{0}\partial_{i}(\delta{\mathfrak{u}})+{\dot{\overline{\rho^{[\gamma]}}}}\partial_{i}(\delta{\mathfrak{u}})+4\,H\overline{\rho^{[\gamma]}}\partial_{i}(\delta{\mathfrak{u}})\right]$
$\displaystyle\approx-a^{-1}\left[3\overline{\Upsilon}\,\dot{\overline{\phi}}\partial_{i}(\delta\phi)+\partial_{i}(\delta\rho^{[\gamma]})+4\,\overline{\rho^{[\gamma]}}\partial_{i}\Psi\right],$
(99)

where $\nabla^{2}=\partial_{i}\partial^{i}$.

Furthermore, by perturbing the field equation (48) while using all necessary
equations, one can obtain Antonio2022  from its $0-0$ component, trace, and
$0-i$ components, respectively

$\displaystyle\delta\rho^{[\phi]}\Big{[}\kappa+{3\over 2}\overline{f_{\rm
T}}-\overline{f_{\rm
TT}}\Big{(}\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}\Big{)}\Big{]}\\!$
$\displaystyle+\delta p^{[\phi]}\Big{[}\\!-{1\over 2}\overline{f_{\rm
T}}+3\overline{f_{\rm
TT}}$\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}$\\!\Big{]}$
$\displaystyle\approx$ $\displaystyle\,2\overline{f_{\rm
Q}}$3H^{2}\Psi+3H\dot{\Psi}-a^{-2}\nabla^{2}\Psi$$
$\displaystyle+72H^{3}\overline{f_{\rm QQ}}$H\Psi+\dot{\Psi}$,$ (100)
$\displaystyle{\dfrac{1}{4}}\overline{f_{\rm
T}}\delta\rho^{[\phi]}-\Big{(}\frac{\kappa}{2}+{3\over 4}\overline{f_{\rm
T}}\Big{)}\delta p^{[\phi]}$ $\displaystyle\approx$
$\displaystyle\overline{f_{\rm
Q}}$4H\dot{\Psi}+3H^{2}\Psi+\dot{H}\Psi+\ddot{\Psi}$$
$\displaystyle+12H\overline{f_{\rm
QQ}}$4H^{2}\dot{\Psi}+3H^{3}\Psi+4H\dot{H}\Psi+3\dot{H}\dot{\Psi}+\ddot{\Psi}$$
$\displaystyle+12H^{2}\dot{\overline{f_{\rm QQ}}}$H\Psi+\dot{\Psi}$,$ (101)
$\displaystyle$\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}$$\kappa+\overline{f_{\rm
T}}$a(t)\delta{\mathfrak{u}}$ $\displaystyle\approx 2\overline{f_{\rm
Q}}$H\Psi+\dot{\Psi}$+12H\overline{f_{\rm QQ}}$2\dot{H}\Psi+H\dot{\Psi}$,$
(102)

where $\overline{f_{\rm QQ}}=\partial\overline{f_{\rm
Q}}/\partial\overline{Q}$ and $\overline{f_{\rm TT}}=\partial\overline{f_{\rm
T}}/\partial\overline{T}$. Note that, Eqs. (V.1)-(V.1) are reduced to the
corresponding equations in GR Weinberg  when $\overline{f_{\rm Q}}=-1$,
$\overline{f_{\rm QQ}}=0$ and $\overline{f_{\rm T}}=0=\overline{f_{\rm TT}}$.

In addition, by substituting Eq. (58) into Eq. (94), for the case of warm
inflation, one obviously obtains

$\displaystyle-$\overline{f_{\rm T}}+\kappa$\overline{{\cal
D}}_{\mu}\overline{T^{[\phi]}}^{\mu}{}_{\nu}+\overline{f_{\rm
T}}\partial_{\nu}\overline{p^{[\phi]}}-\dfrac{1}{2}\overline{f_{\rm
T}}\partial_{\nu}\overline{T^{[\phi]}}$
$\displaystyle=\overline{B}_{\nu}+$\overline{f_{\rm
T}}+\kappa$\overline{\Lambda}_{\nu}\equiv\overline{U}_{\nu}.$ (103)

The perturbation of the right-hand side of this equation gives

$\displaystyle\delta U_{\mu}=\delta B_{\mu}+\delta f_{\rm
T}\overline{\Lambda}_{\mu}+$\overline{f_{\rm T}}+\kappa$\delta\Lambda_{\mu}$
$\displaystyle\hskip 17.64056pt=\delta B_{\mu}+\overline{f_{\rm
TT}}\,\overline{\Lambda}_{\mu}\delta T^{[\phi]}+$\overline{f_{\rm
T}}+\kappa$\delta\Lambda_{\mu}$ $\displaystyle\hskip 17.64056pt=\delta
B_{\mu}+\overline{f_{\rm
TT}}\,\overline{\Lambda}_{\mu}$\\!-\delta\rho^{[\phi]}+3\delta
p^{[\phi]}$\\!+$\overline{f_{\rm T}}+\kappa$\delta\Lambda_{\mu},$ (104)

where we have assumed that the function $f(Q,T)$ does not have mixed terms in
$Q$ and $T$, i.e. $f(Q,T)=g(Q)+h(T)$. Then, the 0-component of the
perturbation of Eq. (V.1) reads Antonio2022

$\displaystyle\Bigg{[}\kappa+\overline{f_{\rm T}}+{1\over 2}\overline{f_{\rm
T}}$1-{\partial_{0}\\!$\delta
p^{[\phi]}$\over\partial_{0}\\!$\delta\rho^{[\phi]}$}$\Bigg{]}\dot{\delta}$
$\displaystyle\\!-\\!\Bigg{\\{}\\!3H\\!$\kappa+\overline{f_{\rm
T}}$\\!\Bigg{[}\\!{$\kappa\\!+\\!\overline{f_{\rm T}}$w^{[\phi]}\\!-\\!{1\over
2}\overline{f_{\rm T}}$1\\!-\\!w^{[\phi]}$\over\kappa+\overline{f_{\rm
T}}+{1\over 2}\overline{f_{\rm T}}$1-w^{[\phi]}$}\\!-\\!{\delta
p^{[\phi]}\over\delta\rho^{[\phi]}}\\!\Bigg{]}$
$\displaystyle+{3H$\kappa+\overline{f_{\rm T}}$(w^{[\phi]}+1)\over
2\Big{[}\kappa+\overline{f_{\rm T}}+{1\over 2}\overline{f_{\rm
T}}$1-w^{[\phi]}$\Big{]}}\Bigg{[}\overline{f_{\rm
T}}$1-{\partial_{0}\\!$\delta
p^{[\phi]}$\over\partial_{0}\\!$\delta\rho^{[\phi]}$}$$
$\displaystyle+\overline{f_{\rm TT}}\overline{\rho^{[\phi]}}$1-3{\delta
p^{[\phi]}\over\delta\rho^{[\phi]}}$$1-3w^{[\phi]}$\Bigg{]}+3H\overline{f_{\rm
TT}}\overline{\rho^{[\phi]}}$ $\displaystyle\times\Big{(}1-3{\delta
p^{[\phi]}\over\delta\rho^{[\phi]}}\Big{)}(w^{[\phi]}+1)\Bigg{[}{\overline{f_{\rm
T}}$1-w^{[\phi]}$\over 2\kappa+\overline{f_{\rm
T}}$3-w^{[\phi]}$}\Bigg{]}\Bigg{\\}}\delta$
$\displaystyle+$\kappa+\overline{f_{\rm
T}}$\frac{\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}}{\overline{\rho^{[\phi]}}}\bigg{[}a^{-1}\nabla^{2}(\delta{\mathfrak{u}})-3\dot{\Psi}\bigg{]}$
$\displaystyle\approx\dfrac{-\nabla^{2}\left[8H\overline{f_{\rm
Q}}\Psi-12H^{2}\overline{f_{\rm
QQ}}$\dot{\Psi}+H\Psi$\right]}{a^{2}\overline{\rho^{[\phi]}}}$
$\displaystyle\hskip 14.22636pt+\dfrac{\overline{f_{\rm
TT}}$-\delta\rho^{[\phi]}+3\delta
p^{[\phi]}$\overline{\Lambda}_{0}+$\overline{f_{\rm
T}}+\kappa$\delta\Lambda_{0}}{\overline{\rho^{[\phi]}}},$ (105)

where
$\left(\rho^{[\phi]}-\overline{\rho^{[\phi]}}\right)/\overline{\rho^{[\phi]}}=\delta\rho^{[\phi]}/\overline{\rho^{[\phi]}}\equiv\delta$
is the density contrast parameter,
$w^{[\phi]}={\overline{p^{[\phi]}}/\overline{\rho^{[\phi]}}}$ represents the
equation of state parameter, and as $w^{[\phi]}$ is a constant, we have
used555According to the Newton-Laplace equation, the speed of sound in fluids,
in general, is $c_{s}^{2}=$\partial p/\partial\rho$_{S}$, where the last $S$
stands for isentropically, i.e. constant entropy. Hence, in the literature,
$c_{s}^{2}=\dot{p}/\dot{\rho}$ have been used, e.g., Refs. Antonio2022 ;
Pati2022 ; Bhagat2023 .
$\dot{\overline{p^{[\phi]}}}/\dot{\overline{\rho^{[\phi]}}}=w^{[\phi]}$. We
remind that the right-hand side of this equation comes from $\delta U_{\mu}$
which is due to nonmetricity and warm inflation. Hence, in the case of
metricity and without warm inflation, in the slow-roll regime by taking
$w^{[\phi]}\approx-1$, Eq. (V.1) gives

$\displaystyle\Bigg{[}\kappa+\overline{f_{\rm T}}+{1\over 2}\overline{f_{\rm
T}}$1-{\partial_{0}$\delta
p^{[\phi]}$\over\partial_{0}$\delta\rho^{[\phi]}$}$\Bigg{]}\dot{\delta}$
$\displaystyle+3H$\kappa+\overline{f_{\rm T}}$\frac{\delta\rho^{[\phi]}+\delta
p^{[\phi]}}{\overline{\rho^{[\phi]}}}$ $\displaystyle+$\kappa+\overline{f_{\rm
T}}$\frac{\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}}{\overline{\rho^{[\phi]}}}\bigg{[}a^{-1}\nabla^{2}(\delta{\mathfrak{u}})-3\dot{\Psi}\bigg{]}\approx
0.$ (106)

This equation reminds the corresponding equation in GR Weinberg  when
$\overline{f_{\rm T}}=0$.

On the other hand, by using relations (8), (82) and $\delta
V=\overline{V}^{\prime}\delta\phi$, while paying attention to relation (5) to
know where $(\dot{\overline{\phi}})^{2}$ comes from, the perturbation of the
energy density and pressure density of the scalar field is obtained as

$\displaystyle\delta\rho^{[\phi]}\approx\dot{\overline{\phi}}\,\partial_{0}(\delta\phi)+\overline{V}^{\prime}$\overline{\phi}$\delta\phi-(\dot{\overline{\phi}})^{2}\Psi,$
(107) $\displaystyle\delta
p^{[\phi]}\approx\dot{\overline{\phi}}\,\partial_{0}(\delta\phi)-\overline{V}^{\prime}$\overline{\phi}$\delta\phi-(\dot{\overline{\phi}})^{2}\Psi.$
(108)

The sum of these equations clearly gives

$\delta\rho^{[\phi]}+\delta p^{[\phi]}\approx
2$$\dot{\overline{\phi}}\,\partial_{0}(\delta\phi)-(\dot{\overline{\phi}})^{2}\Psi$$.$
(109)

Up to here, the main unknowns of perturbation are $\Psi=\Psi(t,{\bf x})$,
$\delta{\mathfrak{u}}=\delta{\mathfrak{u}}(t,{\bf x})$,
$\delta\phi=\delta\phi(t,{\bf x})$,
$\delta\rho^{[\gamma]}=\delta\rho^{[\gamma]}(t,{\bf x})$,
$\delta\rho^{[\phi]}=\delta\rho^{[\phi]}(t,{\bf x})$ and $\delta
p^{[\phi]}=\delta p^{[\phi]}(t,{\bf x})$, which are supposed to be determined
through the model equations. Meanwhile, as mentioned, the other quantities
involved will be perturbed subsequently, e.g.
$V\equiv\overline{V}(\phi)=\overline{V}$\overline{\phi}+\delta\phi$\approx\overline{V}$\overline{\phi}$+\overline{V}^{\prime}$\overline{\phi}$\delta\phi$,
and in the same way
$\rho^{[\phi]}\approx\overline{\rho^{[\phi]}}+\overline{\rho^{[\phi]}}^{\prime}\delta\phi$
and
$\Upsilon\approx\overline{\Upsilon}+\overline{\Upsilon}^{\prime}\delta\phi$
when $\Upsilon$ is an explicit function of $\phi$. However, later we will
consider special forms for the potential and the dissipation coefficient.
Accordingly, we have a system of six coupled differential equations (V.1-V.1)
and (V.1) for the unknown quantities. In general, these equations enable us to
explore the evolution of perturbed quantities. However, up to here, mentioning
the approximate equations is due to their writing up to the second-order of
perturbations. Whereas, since each quantity in the background and perturbation
varies slowly during warm inflation, in the continuation we rewrite all the
relevant equations according to the corresponding slow-roll approximations.

To proceed, we first transfer all the perturbed quantities into Fourier space,
which results in the spatial components of the quantities taking the form
$e^{i{\bf k}.{\bf x}}$, where ${\bf k}$ represents the wavenumber vector of
the corresponding modes. Accordingly, in this process, we introduce the
following definitions666In the literature, $\delta{\mathfrak{u}}(t)$ has been
introduced as proportional to $v(t)/k$, however we prefer to keep it
consistent with the other relevant quantities.

$\displaystyle\Psi(t,{\bf x})=\Psi(t)e^{i{\bf k}.{\bf
x}},\qquad\delta{\mathfrak{u}}(t,{\bf x})=\delta{\mathfrak{u}}(t)e^{i{\bf
k}.{\bf x}},$ (110) $\displaystyle\delta\phi(t,{\bf x})=\delta\phi(t)e^{i{\bf
k}.{\bf x}},\qquad\delta\rho^{[\gamma]}(t,{\bf
x})=\delta\rho^{[\gamma]}(t)e^{i{\bf k}.{\bf x}},$ (111)
$\displaystyle\delta\rho^{[\phi]}(t,{\bf x})=\delta\rho^{[\phi]}(t)e^{i{\bf
k}.{\bf x}},\qquad\delta p^{[\phi]}(t,{\bf x})=\delta p^{[\phi]}(t)e^{i{\bf
k}.{\bf x}}.$ (112)

Therefore, we can substitute $\partial_{j}\rightarrow ik_{j}$ and
$\nabla^{2}\rightarrow-k^{2}$ wherever necessary in the system equations.

Then, for the purpose of slow-roll approximations, we eliminate any quantity
with a higher time derivative and assume the plausible approximations
$\dot{\Psi}\ll H\Psi$, $\dot{H}\ll H^{2}$, $\ddot{\Psi}\ll H\dot{\Psi}$,
$\dot{\Psi}\ll H\Psi$, $\dot{H}\dot{\Psi}\ll H^{3}\Psi$ and
$\dot{\overline{f_{\rm QQ}}}\ll H\overline{f_{\rm QQ}}$. Also, we consider
modes whose longest wavelength satisfies ${k/a(t)}\ll H$. Hence, under these
considerations, Eqs. (V.1) and (V.1) are reduced respectively to

$\displaystyle\delta\rho^{[\phi]}(t)\left[\kappa+\dfrac{3}{2}\overline{f_{\rm
T}}-\overline{f_{\rm
TT}}\left(\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}\right)\right]$
$\displaystyle+\delta p^{[\phi]}(t)\left[-\dfrac{1}{2}\overline{f_{\rm
T}}+3\overline{f_{\rm
TT}}\left(\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}\right)\right]$
$\displaystyle\approx 6H^{2}\Psi(t)\left(\overline{f_{\rm
Q}}+12\overline{f_{\rm QQ}}H^{2}\right),$ (113)
$\displaystyle\dfrac{1}{2}\overline{f_{\rm
T}}\delta\rho^{[\phi]}(t)-\left(\\!\kappa+\dfrac{3}{2}\overline{f_{\rm
T}}\right)\delta p^{[\phi]}(t)$ $\displaystyle\approx
6H^{2}\Psi(t)\left(\overline{f_{\rm Q}}+12\overline{f_{\rm QQ}}H^{2}\right).$
(114)

We also assume777Of course, this is a restrictive assumption. that the order
of $\overline{f_{\rm T}}$ is the same as the order of
$\overline{T^{[\phi]}}\,\overline{f_{\rm TT}}$. Then, using relation (9) and
the slow-roll approximation $(\dot{\overline{\phi}})^{2}\ll\overline{V}$, we
get $\overline{f_{\rm
TT}}$\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}$\ll\overline{f_{\rm T}}$.
Hence, Eq. (V.1) reads

$\displaystyle\left(\kappa+\dfrac{3}{2}\overline{f_{\rm
T}}\right)\delta\rho^{[\phi]}(t)-\dfrac{1}{2}\overline{f_{\rm T}}\delta
p^{[\phi]}(t)$ $\displaystyle\approx 6H^{2}\Psi(t)\left(\overline{f_{\rm
Q}}+12\overline{f_{\rm QQ}}H^{2}\right).$ (115)

Thereafter, subtracting Eq. (V.1) from Eq. (V.1) gives

$\delta\rho^{[\phi]}(t)+\delta p^{[\phi]}(t)\approx 0,$ (116)

and in turn, $w^{[\phi]}\approx-1$, as expected during inflation. Accordingly,
by first inserting relations (110-112) into Eq. (109) and then substituting
its result into Eq. (116) yields

$\Psi(t)\approx\dfrac{\partial_{0}[\delta\phi(t)]}{\dot{\overline{\phi}}},$
(117)

and in turn, Eqs. (107) and (108) give

$\delta\rho^{[\phi]}(t)\approx\overline{V}^{\prime}$\overline{\phi}$\delta\phi(t)\approx-\delta
p^{[\phi]}(t).$ (118)

Similarly, we assume that the order of $\overline{f_{\rm Q}}$ is the same as
the order of $Q\,\overline{f_{\rm QQ}}$. Then, using relation (46) and the
slow-roll approximation $\dot{H}\ll H^{2}$, we get $\overline{f_{\rm
QQ}}\dot{H}\ll\overline{f_{\rm Q}}$. Hence, while also using the slow-roll
condition $\dot{\Psi}\ll H\Psi$, Eq. (V.1) is reduced to

$\delta{\mathfrak{u}}(t)\approx\dfrac{2\overline{f_{\rm
Q}}H\Psi(t)}{a(t)$\overline{\rho^{[\phi]}}+\overline{p^{[\phi]}}$$\kappa+\overline{f_{\rm
T}}$}.$ (119)

Furthermore, by using the slow-roll approximations
$\partial_{0}(\delta\rho^{[\gamma]})\ll H\delta\rho^{[\gamma]}$,
$\dot{\Psi}\ll H\Psi$ and $k/a(t)\ll H$,888Actually, for adequately slow
evolution, we ignore terms involving long wavelength perturbations. and by
employing Eq. (23) and relation
$\delta\Upsilon=\overline{\Upsilon}^{\prime}\delta\phi$, Eq. (V.1) is reduced
to

$\dfrac{\delta\rho^{[\gamma]}(t)}{\overline{\rho^{[\gamma]}}}\\!\approx\\!\dfrac{\overline{\Upsilon}^{\prime}}{\overline{\Upsilon}}\delta\phi(t)-\Psi(t)+2\dfrac{\partial_{0}$$\delta\phi(t)$$}{\dot{\overline{\phi}}}\\!\approx\\!\dfrac{\overline{\Upsilon}^{\prime}}{\overline{\Upsilon}}\delta\phi(t)+\Psi(t),$
(120)

where we have used relation (117) in the last approximation. Also, by
transferring to momentum space, Eq. (V.1) reads

$\displaystyle\overline{\rho^{[\gamma]}}\partial_{0}$$\delta{\mathfrak{u}}(t)$$+\dot{\overline{\rho^{[\gamma]}}}\delta{\mathfrak{u}}(t)+4H\overline{\rho^{[\gamma]}}\delta{\mathfrak{u}}(t)$
$\displaystyle\approx-\frac{1}{a(t)}\left[{3\over
4}\overline{\Upsilon}\,\dot{\overline{\phi}}\delta\phi(t)+{\delta\rho^{[\gamma]}(t)\over
4}+\overline{\rho^{[\gamma]}}\Psi(t)\right].$ (121)

Then, by applying the slow-roll approximation
$\partial_{0}\left[\overline{\rho^{[\gamma]}}\delta{\mathfrak{u}}(t)\right]\ll
H\overline{\rho^{[\gamma]}}\delta{\mathfrak{u}}(t)$ while using Eq. (120), it
reads

$\delta{\mathfrak{u}}(t)\approx-\dfrac{1}{4aH}\left[\dfrac{3\overline{\Upsilon}\,\dot{\overline{\phi}}}{4\overline{\rho^{[\gamma]}}}\delta\phi(t)+\dfrac{\overline{\Upsilon}^{\prime}}{4\overline{\Upsilon}}\delta\phi(t)+\dfrac{5}{4}\Psi(t)\right].$
(122)

Next, by substituting $\delta{\mathfrak{u}}(t)$ from Eq. (119) into Eq. (122),
while using relations (8) and Eq. (23), we obtain

$\Psi(t)\approx-\dfrac{\dot{\overline{\phi}}$\kappa+\overline{f_{\rm
T}}$$\overline{\Upsilon}^{\prime}\,\dot{\overline{\phi}}+12H\overline{\Upsilon}$}{\overline{\Upsilon}\Big{[}5(\dot{\overline{\phi}})^{2}$\kappa+\overline{f_{\rm
T}}$+32H^{2}\overline{f_{\rm Q}}\Big{]}}\delta\phi(t).$ (123)

Finally, while considering Eq. (V.1), and using Eqs. (24) and (116), and the
slow-roll condition $\dot{\Psi}\ll H\Psi$, Eq. (V.1) is reduced to

$\displaystyle\frac{$\kappa+2\overline{f_{\rm
T}}$}{\overline{\rho^{[\phi]}}(t)}$$\partial_{0}\\!$\delta\rho^{[\phi]}(t)$-\frac{\dot{\overline{\rho^{[\phi]}}}(t)}{\overline{\rho^{[\phi]}}(t)}\delta\rho^{[\phi]}(t)$$$
$\displaystyle\approx\dfrac{4k^{2}H\Psi(t)}{a^{2}(t)\overline{\rho^{[\phi]}}(t)}$2\overline{f_{\rm
Q}}-3H^{2}\overline{f_{\rm QQ}}$-\dfrac{4\overline{f_{\rm
TT}}\delta\rho^{[\phi]}(t)}{\overline{\rho^{[\phi]}}(t)}\overline{\Lambda}_{0}$
$\displaystyle+\dfrac{$\overline{f_{\rm
T}}+\kappa$}{\overline{\rho^{[\phi]}}(t)}\delta\Lambda_{0}.$ (124)

At this stage, all the perturbed equations involve exclusively temporal
aspects.

In the continuation, we represent the system equations for the warm inflation
model in the linear version of $f(Q,T)$ gravity, relation (60). At first, by
applying Eqs. (71) and (72) into relation (123), it reads

$\displaystyle\Psi(t)\approx$
$\displaystyle-\dfrac{\overline{V}^{\prime}$\kappa+\beta$\left[\alpha\overline{V}^{\prime}\overline{\Upsilon}^{\prime}+12\overline{\Upsilon}\,\overline{V}$\kappa+\beta-\kappa\alpha\overline{\chi}$\right]}{\overline{\Upsilon}\left[5\alpha\overline{V}^{\prime
2}$\kappa+\beta$+32\overline{V}^{2}$\kappa+\beta-\kappa\alpha\overline{\chi}$^{2}\right]}$
$\displaystyle\times\delta\phi(t).$ (125)

Thereafter, by substituting relations (24), (96) (while using relations (117)
and $\delta\Upsilon=\overline{\Upsilon}^{\prime}\delta\phi$ into it) and (118)
into Eq. (V.1), and then rearranging its terms, we obtain

$\displaystyle\left(\kappa+2\beta\right)\left[\dfrac{\overline{V}^{\prime\prime}}{\overline{V}}-\dfrac{\overline{V}^{\prime
2}}{\overline{V}^{2}}+\dfrac{\left(\kappa+\beta\right)\overline{\Upsilon}^{\prime}\dot{\overline{\phi}}}{\overline{V}\left(\kappa+2\beta\right)}\right]\delta\phi(t)$
$\displaystyle\approx\\!\left[\\!-\dfrac{\left(\kappa+2\beta\right)\overline{V}^{\prime}}{\overline{V}\,\dot{\overline{\phi}}}\\!-\\!\left(\kappa+\beta\right)\dfrac{\overline{\Upsilon}}{\overline{V}}\right]\\!\partial_{0}\left[\delta\phi(t)\right]\\!+\\!\dfrac{8\alpha
k^{2}H\Psi(t)}{a^{2}\overline{V}\,\dot{\overline{\phi}}}.$ (126)

Now, by using first Eq. (72) and then Eq. (71) into the last term in the
right-hand side of this equation, it reads

$\dfrac{8\,\alpha
k^{2}H\Psi(t)}{a^{2}\overline{V}\,\dot{\overline{\phi}}}\approx\dfrac{8k^{2}\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}{a^{2}\overline{V}^{\prime}}\Psi(t).$
(127)

Thereafter, by substituting $\Psi(t)$ from relation (125) into it, while using
the slow-roll parameters (73) and (75), we obtain

$\displaystyle\dfrac{8\alpha
k^{2}H\Psi(t)}{a^{2}\overline{V}\,\dot{\overline{\phi}}}$
$\displaystyle\approx\dfrac{-8k^{2}$\kappa+\beta$\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)\left(12-\lambda\right)}{a^{2}\overline{V}\left[-10$\kappa+\beta$\epsilon+32\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)\right]}\delta\phi(t)$
$\displaystyle\approx\dfrac{-3k^{2}$\kappa+\beta$}{a^{2}\overline{V}}\delta\phi(t)\approx\frac{k^{2}}{a^{2}H^{2}}\dfrac{\left(\kappa+\beta\right)\left(\kappa+2\beta\right)}{\alpha}\delta\phi(t),$
(128)

where we have employed conditions $\epsilon\ll 1$ and $\lambda\ll 1$ in the
middle approximation and Eq. (71) in the last approximation. However, due to
long wavelength perturbations, i.e. with $k^{2}\ll a^{2}H^{2}$, we can ignore
this term. Therefore, by using definition (17) and Eq. (72) into Eq. (V.1), it
reads

$\displaystyle\left(\kappa+2\beta\right)\left[\dfrac{\overline{V}^{\prime\prime}}{\overline{V}}-\dfrac{\overline{V}^{\prime
2}}{\overline{V}^{2}}-\dfrac{\left(\kappa+\beta\right)\overline{\chi}}{\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}\dfrac{\overline{\Upsilon}^{\prime}\overline{V}^{\prime}}{\overline{\Upsilon}\,\overline{V}}\right]\delta\phi(t)$
$\displaystyle\approx\dfrac{3H}{\overline{V}}\Big{[}\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\Big{]}\partial_{0}\left[\delta\phi(t)\right]$
$\displaystyle\approx\frac{-\overline{V}^{\prime}\left(\kappa+2\beta\right)\Big{[}\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\Big{]}}{\overline{V}\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}\frac{d\left[\delta\phi(t)\right]}{d\phi},$
(129)

where $\kappa+2\beta\neq 0$ has been assumed, and Eq. (72) has been used in
the last approximation. At last, from Eq. (V.1), we obtain the solution

$\delta\phi(t)\approx C\exp\left[\zeta(\phi)\right],$ (130)

where $C$ is a constant and $\zeta(\phi)$ is

$\displaystyle\zeta(\phi)\equiv$
$\displaystyle\int\left\\{-\dfrac{\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}{\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}\left(\dfrac{\overline{V}^{\prime\prime}}{\overline{V}^{\prime}}-\dfrac{\overline{V}^{\prime}}{\overline{V}}\right)\right.$
$\displaystyle\left.+\dfrac{\left(\kappa+\beta\right)\overline{\chi}}{\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}\dfrac{\overline{\Upsilon}^{\prime}}{\overline{\Upsilon}}\right\\}{\rm
d}\phi.$ (131)

It has been shown Berera1995 ; Hall2004 ; Herrera  that the density
fluctuation derived from solution (130) transforms as

$\Delta_{\rm R}\approx C\approx\delta\phi(t)\exp[-\zeta(\phi)].$ (132)

Actually in this relation, $\delta\phi(t)$ represents the fluctuation of the
scalar field during the warm inflationary era, generated by the thermal
interaction with the radiation field as Berera2000 ; Herrera

$\left[\delta\phi(t)\right]^{2}=\dfrac{k_{\rm F}\,\tau}{2\pi^{2}},$ (133)

where $k_{\rm F}$ is the freeze-out scale. The freeze-out time, $t_{\rm F}$,
is determined when the damping rate (the coefficient
$\partial_{0}\left[\delta\phi(t)\right]$ in the middle line of relation (V.1))
falls below the expansion rate $H$ Hall2004 ; Taylor2000 . Hence, the freeze-
out scale is

$k_{\rm
F}\approx\sqrt{\dfrac{3H^{2}}{\overline{V}}\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}.$
(134)

Accordingly, relation (132) undergoes the modification

$\displaystyle\Delta_{\rm R}^{2}$ $\displaystyle\approx\dfrac{k_{\rm
F}\,\tau}{2\pi^{2}}\exp[-2\zeta(\phi)]$
$\displaystyle\approx\dfrac{\tau\exp[-2\zeta(\phi)]}{2\pi^{2}}\sqrt{\dfrac{3H^{2}}{\overline{V}}\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}.$
(135)

On the other hand, the scalar spectral index, $n_{\rm S}$, is Encyclo2014

$n_{\rm S}-1=\dfrac{{\rm d}\ln\Delta_{\rm R}^{2}}{{\rm d}\ln k},$ (136)

where the inflationary observables are calculated at the first horizon
crossing Encyclo2014 , i.e. when $k=aH$. Hence, we have

$\displaystyle n_{\rm S}-1$ $\displaystyle\approx\dfrac{1}{2}\dfrac{{\rm d}\ln
k_{\rm F}^{2}}{{\rm d}\ln(aH)}+\dfrac{1}{4}\dfrac{{\rm d}\ln\tau^{4}}{{\rm
d}\ln(aH)}-\dfrac{2{\rm d}\zeta(\phi)}{{\rm d}\ln(aH)}$
$\displaystyle=$\dfrac{1}{2}\dfrac{{\rm d}\ln k_{\rm F}^{2}}{{\rm
d}\phi}+\dfrac{1}{4}\dfrac{{\rm d}\ln\tau^{4}}{{\rm d}\phi}-\dfrac{2{\rm
d}\zeta(\phi)}{{\rm d}\phi}$\dot{\overline{\phi}}\dfrac{{\rm d}t}{{\rm
d}\ln(aH)}.$ (137)

Then, by employing relations (23), (26), (71), (72) and (134), we obtain

$\dfrac{{\rm d}\ln k_{\rm F}^{2}}{{\rm
d}\phi}\approx\dfrac{-\left(\kappa+\beta+\kappa\alpha\right)\overline{\chi}}{\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}\left(\dfrac{\overline{\Upsilon}^{\prime}}{\overline{\Upsilon}}-\dfrac{1}{2}\dfrac{\overline{V}^{\prime}}{\overline{V}}\right),$
(138) $\displaystyle\dfrac{{\rm d}\ln\tau^{4}}{{\rm
d}\phi}\\!\approx\\!\dfrac{\left(\kappa\\!+\\!\beta\\!+\\!\kappa\alpha\overline{\chi}\right)}{\left(\kappa\\!+\\!\beta\\!-\\!\kappa\alpha\overline{\chi}\right)}\dfrac{\overline{\Upsilon}^{\prime}}{\overline{\Upsilon}}\\!-\\!\dfrac{\left(3\kappa\\!+\\!3\beta\\!-\\!\kappa\alpha\overline{\chi}\right)}{2\left(\kappa\\!+\\!\beta\\!-\\!\kappa\alpha\overline{\chi}\right)}\dfrac{\overline{V}^{\prime}}{\overline{V}}\\!+\\!2\dfrac{\overline{V}^{\prime\prime}}{\overline{V}^{\prime}},$
(139) $\dfrac{1}{\dot{\overline{\phi}}}\dfrac{{\rm d}\ln(aH)}{{\rm
d}t}\approx\dfrac{\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)\overline{V}}{\alpha\overline{V}^{\prime}}\left(1-\epsilon\right).$
(140)

Now, utilizing these relations, while employing relation (V.1) and using the
modified slow-roll parameters (73), (74) and (75), relation (V.1) becomes

$\displaystyle n_{\rm S}-1\approx$
$\displaystyle\left\\{\dfrac{8\left(\kappa+\beta\right)-\left(\kappa+\beta+9\kappa\alpha\right)\overline{\chi}}{2\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}\right.$
$\displaystyle\left.+\dfrac{\left(3\kappa+3\beta-\kappa\alpha\overline{\chi}\right)}{4\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}\right\\}\epsilon$
$\displaystyle-\left\\{\dfrac{1}{2}+\dfrac{2\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}{\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}\right\\}\eta$
$\displaystyle+\left\\{\dfrac{\left(5\kappa+5\beta+\kappa\alpha\right)\overline{\chi}}{2\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}\right.$
$\displaystyle\left.-\dfrac{\left(\kappa+\beta+\kappa\alpha\overline{\chi}\right)}{4\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)}\right\\}\lambda,$
(141)

where condition $1+\epsilon\approx 1$ has been used.

###  V.2 TENSOR PERTURBATIONS

In this subsection, we consider the evolution of tensor perturbations. For
this purpose, we denote the related metric as

$ds^{2}=-dt^{2}+a^{2}(t)(\delta_{ij}+h_{ij})dx^{i}dx^{j},$ (142)

wherein tensor perturbations are characterized by the spatially symmetric,
transverse, traceless tensor $h_{ij}$, with properties such that
$\partial^{i}h_{ij}=0=h^{i}_{i}$. Accordingly, the result can be achieved by
perturbing the energy-momentum tensor up to the second-order within the
framework of linear perturbations as Antonio2022 ; Bhattacharyaa2006

$\ddot{h}_{ij}+\Big{\\{}3H-\dfrac{d}{dt}\Big{[}\ln(f_{Q})\Big{]}\Big{\\}}\dot{h}_{ij}+\dfrac{k^{2}}{a^{2}}h_{ij}=0.$
(143)

However, in the linear functional form of $f(Q,T)$ gravity, e.g. relation
(60), one has $\dfrac{d}{dt}[\ln(f_{Q})]=0$. Hence, Eq. (143) is reduced to an
evolution equation for tensor perturbations in minimal coupling models.

In Ref. Bhattacharya2006 , a method has been provided to compute the emergence
of tensor perturbations during warm inflation by introducing an additional
factor $\coth\left[k/(2\,\tau)\right]$ that modifies the power spectrum of the
tensor fluctuation modes as Herrera

$\Delta_{\rm T}^{2}={4H^{2}\over\pi}\coth$\dfrac{k}{2\,\tau}$.$ (144)

Utilizing Eq. (71), this relation becomes

$\Delta_{\rm
T}^{2}\approx{4(\kappa+2\beta)\overline{V}(\phi)\over-3\alpha\,\pi}\coth$\dfrac{k}{2\,\tau}$.$
(145)

However, due to the weak nature of gravitational interactions, the tensor
primordial power spectrum remains unaltered, similar to its state during cold
inflation without any modification.

Furthermore, the inflationary parameter, denoted as the tensor-to-scalar ratio
$r$, is

${\rm r}=\dfrac{\Delta_{\rm T}^{2}}{\Delta_{\rm R}^{2}}.$ (146)

By applying relations (V.1) (while using Eq. (71) into it) and (145), it reads

$r\approx\dfrac{\kappa\,\overline{V}(\phi)\sqrt{(\kappa+2\beta)}\exp$$2\zeta(\phi)$$}{3\,\tau\sqrt{-\alpha\left[\left(\kappa+\beta-\kappa\alpha\overline{\chi}\right)-\left(\kappa+\beta\right)\overline{\chi}\right]}}\coth$\dfrac{k}{2\,\tau}$,$
(147)

where dictates

$\kappa+2\beta>0$ (148)

and

$\kappa+\beta-\kappa\alpha\overline{\chi}>\left(\kappa+\beta\right)\overline{\chi}$
(149)

for negative values of $\alpha$.

In the warm inflationary paradigm, the primordial scalar power spectrum
undergoes an increase during the warm inflation era compared to the cold
inflation era. Whereas, the tensor power spectrum remains consistent with the
cold inflation period. Consequently, the reduction of the tensor-to-scalar
ratio, compared to the cold inflation scenario, has the potential to revive
models that have been disregarded due to discrepancies with observational
data. Subsequently, we investigate the inflationary characteristics within the
framework of warm inflation, considering the linear version of $f(Q,T)$
gravity, which is coupled with both the inflaton field and the radiation
field. This analysis encompasses a power-law potential conducted under the
context of the strong dissipation regime. Within this regime, it becomes
possible to employ the approximation where $\chi\gg 1$.

##  VI Strong dissipation regime

Henceforth, since the quantities involved are unperturbed ones, we will not
label those with a bar for simplicity. In the strong dissipation regime, we
make the plausible assumption

$\beta\ll-\kappa\alpha\chi$ (150)

that, for negative values of $\alpha$, while considering condition (78),
requires

$-\alpha\chi>1.$ (151)

However, we prefer to assume $|-\alpha\chi|\gg 1$. Under these assumptions, we
can write

$\kappa+\beta-\kappa\alpha\chi\approx-\kappa\alpha\chi.$ (152)

Hence, condition (77) is always satisfied, since we have

$\dfrac{-\alpha}{\kappa+\beta-\kappa\alpha\chi}>0\quad\longrightarrow\quad\dfrac{1}{\kappa\chi}>0,$
(153)

where $\kappa\chi$ is obviously a positive value. Also, for negative values of
$\alpha$, under these assumptions, condition (149) requires

$\kappa+\beta+\kappa\alpha<0.$ (154)

In addition, assumption (150) implies that when $\alpha$ takes on negative
values, the $\beta$ parameter encompasses both positive and negative values.
Nevertheless, when $\alpha$ is a positive value, a prerequisite emerges for
$\beta$ to hold negative values.

Furthermore, in the context of the strong dissipation regime and under these
assumptions, the modified slow-roll parameters (73), (74) and (75) become

$\displaystyle\epsilon\approx\dfrac{1}{2\kappa\chi}$\dfrac{V^{\prime}}{V}$^{2},\qquad\eta\approx\dfrac{1}{\kappa\chi}$\dfrac{V^{\prime\prime}}{V}$,$
$\displaystyle\lambda\approx\dfrac{1}{\kappa\chi}$\dfrac{\Upsilon^{\prime}V^{\prime}}{\Upsilon
V}$.$ (155)

Also, the e-folding number (80) reads

$N\approx\kappa\int_{\phi_{\rm end}}^{\phi}\chi\dfrac{V}{V^{\prime}}{\rm
d}\phi.$ (156)

Thus, the model parameters $\alpha$ and $\beta$ are not appeared in the slow-
roll parameters and the e-folding number. Moreover, relation (V.1) and the
scalar spectral index (V.1) are now

$\displaystyle\zeta(\phi)\approx$
$\displaystyle\int\Bigg{[}\dfrac{-\kappa\alpha}{\left(\kappa+\beta+\kappa\alpha\right)}\left(\dfrac{V^{\prime\prime}}{V^{\prime}}-\dfrac{V^{\prime}}{V}\right)$
$\displaystyle-\dfrac{\left(\kappa+\beta\right)}{\left(\kappa+\beta+\kappa\alpha\right)}\dfrac{\Upsilon^{\prime}}{\Upsilon}\Bigg{]}{\rm
d}\phi,$ (157) $\displaystyle n_{\rm S}-1\approx$
$\displaystyle\dfrac{\left(3\kappa+3\beta+19\kappa\alpha\right)}{4\left(\kappa+\beta+\kappa\alpha\right)}\epsilon-\dfrac{\left(\kappa+\beta+5\kappa\alpha\right)}{2\left(\kappa+\beta+\kappa\alpha\right)}\eta$
$\displaystyle-\dfrac{\left(9\kappa+9\beta+\kappa\alpha\right)}{4\left(\kappa+\beta+\kappa\alpha\right)}\lambda.$
(158)

As mentioned, the behavior of the dissipation coefficient has a crucial impact
on the evolution of the inflaton field and the resulting cosmological
observables such as the spectrum of the scalar and tensor perturbations. It is
a key parameter that distinguishes warm inflation from the standard cold
inflation scenario. In the subsequent analysis, we will investigate the power-
law potential under two variations of the dissipation coefficient. One of
those as a constant value and the other as a function of the inflaton scalar
field to the power of two, within the context of the strong dissipation
regime.

###  VI.1 POWER-LAW POTENTIAL WITH CONSTANT DISSIPATION COEFFICIENT

In this subsection, with the power-law potential, we assume a constant
dissipation coefficient, say $\Upsilon_{0}$. Power-law potentials represent a
class of inflaton potentials that exhibit a straightforward mathematical
structure. It contains the potential energy $V(\phi)$ which is directly
proportional to the inflaton field $\phi$ raised to a fixed power $n$, namely

$V(\phi)=\nu\phi^{n},$ (159)

where $n$ is a positive rational number and $\nu$ is a constant of the model
Linde1983 ; Pavluchenko2004 . In this case, the slow-roll parameters (VI)
become

$\epsilon\approx\dfrac{n^{2}}{2\kappa\chi_{0}\phi^{2}},\qquad\eta\approx\dfrac{n(n-1)}{\kappa\chi_{0}\phi^{2}},\qquad\lambda\approx
0.$ (160)

Also, Eq. (71) reads

$3H^{2}\approx\dfrac{\nu$\kappa+2\beta$}{-\alpha}\phi^{n},$ (161)

and in turn, the dissipation rate (17) is now

$\chi_{0}=\dfrac{\Upsilon_{0}}{3H}\approx\sqrt{\dfrac{-\alpha\Upsilon_{0}^{2}}{3\,\nu$\kappa+2\beta$}}\,\phi^{-\dfrac{n}{2}}.$
(162)

Furthermore, by considering the condition $\epsilon(\phi_{\rm end})=1$
alongside the above relations, we obtain the scalar field at the end of
inflation as

$\phi_{\rm end}\approx\left[\dfrac{3\,\nu
n^{4}$\kappa+2\beta$}{-4\alpha$\kappa\Upsilon_{0}$^{2}}\right]^{1/(4-n)},$
(163)

which dictates $n\neq 4$. Also, the e-folding number (156) reads

$N\approx\left[\phi^{(4-n)/2}-\phi_{\rm
end}^{(4-n)/2}\right]\left[\dfrac{-4\alpha$\kappa\Upsilon_{0}$^{2}}{3\nu
n^{2}$4-n$^{2}$\kappa+2\beta$}\right]^{1/2}.$ (164)

Subsequently, the scalar field at the first horizon crossing becomes

$\displaystyle\phi\approx\left\\{\dfrac{-4\alpha$\kappa\Upsilon_{0}$^{2}}{3\nu
n^{2}\left(\kappa+2\beta\right)\left[n^{2}+\left(4-n\right)^{2}N^{2}\right]}\right\\}^{1/\left(n-4\right)}.$
(165)

Then, by substituting relations (160), while using relation (162), into
relation (VI), we obtain the scalar spectral index in terms of the inflaton
field as

$\displaystyle n_{\rm S}-1\approx$
$\displaystyle\sqrt{\dfrac{3\,\nu\left(\kappa+2\beta\right)}{-\alpha\Upsilon_{0}^{2}}}\,n\,\phi^{\left(n-4\right)/2}$
$\displaystyle\times\left[\dfrac{4\left(\kappa+\beta+5\kappa\alpha\right)-n\left(\kappa+\beta+\kappa\alpha\right)}{8\kappa\left(\kappa+\beta+\kappa\alpha\right)}\right]$
$\displaystyle\approx$
$\displaystyle\dfrac{4\left(\kappa+\beta+5\kappa\alpha\right)-n\left(\kappa+\beta+\kappa\alpha\right)}{4\left(\kappa+\beta+\kappa\alpha\right)\sqrt{n^{2}+\left(4-n\right)^{2}N^{2}}},$
(166)

where we have used relation (165) for the last row. Hence, the scalar spectral
index is only a function of the parameters $\alpha$, $\beta$, $n$ and $N$.

Also, in this case, relation (VI) becomes

$\zeta$\phi$\approx\int\dfrac{-\kappa\alpha}{\kappa+\beta+\kappa\alpha}\left(\dfrac{-1}{\phi}\right){\rm
d}\phi=\dfrac{\kappa\alpha}{\kappa+\beta+\kappa\alpha}\ln{\phi},$ (167)

then utilizing relations (17), (152) and (167) in relation (147), the tensor-
to-scalar ratio reads

$\displaystyle r\approx$
$\displaystyle\dfrac{\kappa\,\nu}{\tau}\sqrt{\dfrac{H(\kappa+2\beta)}{3\,\alpha\Upsilon_{0}\left(\kappa+\beta+\kappa\alpha\right)}}\coth$\dfrac{k}{2\tau}$$
$\displaystyle\times\phi^{\left[\dfrac{n(\kappa+\beta+\kappa\alpha)+2\kappa\alpha}{\kappa+\beta+\kappa\alpha}\right]}.$
(168)

Later, when we want to calculate the inflationary observables, we use for $k$
its value at the pivot scale, i.e. $k_{0}=0.002\ {\rm Mpc^{-1}}$.

To calculate the temperature of the thermal bath, at first by using relations
(72), (152) and (161), we obtain

$\dot{\phi}^{2}\approx\dfrac{n^{2}\nu$\kappa+2\beta$}{-3\,\alpha\kappa^{2}\chi_{0}^{2}}\phi^{(n-2)}.$
(169)

Then, from relation (23), we have

$\tau=$\dfrac{3\,\chi_{0}}{4C_{\gamma}}\dot{\phi}^{2}$^{1/4},$ (170)

wherein by inserting relations (162) and (169), it reads

$\tau\approx\left[\dfrac{n^{2}}{4C_{\gamma}\kappa^{2}\Upsilon_{0}}\sqrt{\dfrac{3\,\nu^{3}$\kappa+2\beta$^{3}}{$-\alpha$^{3}}}\right]^{1/4}\phi^{\dfrac{3n-4}{8}}.$
(171)

Next, by substituting relations (165) and/or (163) into the above result, we
can obtain the temperature of the thermal bath at the first horizon crossing
and/or at the end of inflation, respectively. Also, by substituting relations
(165) and (171) into relation (VI.1), we obtain the inflationary observable
$r$ as a function of $\alpha$, $\beta$, $n$, $N$, $\Upsilon_{0}$ and $\nu$
parameters.

To evaluate the ability of any theory of gravitation to explain inflation, it
is essential that the predicted values of $n_{\rm S}$ and $r$ be in agreement
with the observational data. These observables offer valuable insights into
the underlying mechanisms driving inflation and provide a means to test and
refine theoretical models. Achieving a high degree of observational
consistency ensures that our understanding of inflation remains grounded in
empirical evidence and bolsters the credibility of our scientific endeavors.
In this regards, the most recent constraints established by the Planck
collaboration to the scalar spectral index and the tensor-to-scalar ratio are
Planck:2018jri

$\displaystyle n_{\rm S}=0.9649\pm 0.0042\quad$
$\displaystyle{}\qquad\left({\rm at}\ 68\%\,{\rm
CL,PlanckTT,TE,EE+lowE+lensing}\right),$ $\displaystyle r<0.10\quad\left({\rm
at}\ 95\%\,{\rm CL,PlanckTT+lowE+lensing}\right).$ (172)

Nevertheless, through the combined examination of the Planck, BK$15$ and BAO
data, additional restrictions have been imposed, further constricting the
upper limit of $r$ to a specific value of

$r<0.056\quad{\rm at}\ 95\%\,{\rm CL}.$ (173)

In FIG. 1, we have depicted the scalar spectral index, the tensor-to-scalar
ratio, and the temperature at the end of inflation with respect to the model
parameters $\alpha$ and $\beta$ for the power-law potential with $n=2$,
$N=70$, $\nu\sim{\cal O}(10^{-1})$ and $\Upsilon_{0}\sim{\cal O}(10)$.999Note
that, the values chosen to draw the shapes within the given range of $\alpha$
and $\beta$ satisfy conditions (150), (152) and (154). The results indicate
that, in the range of $-30<\alpha<-20$ and $-5<\beta<1$, the model derived
values are in good agreement for $n_{\rm S}$ with the Planck $2018$ data
(VI.1) and for $r$ with both the Planck $2018$ data and the joint Planck,
BK$15$ and BAO data (173). However, we have also specified (although we have
not shown the corresponding plots) that (while using the same other values),
for $\alpha\geq-8$ $r$ disagrees with the Planck $2018$ data and for
$\alpha\geq-6$, it disagrees with both the Planck $2018$ data and the joint
Planck, BK$15$ and BAO data, and for $\alpha\geq-2$, it extremely increases.
In addition, for $\nu\leq{\mathcal{O}}(10^{-2})$ and/or
$\Upsilon_{0}\geq{\mathcal{O}}(10^{2})$, the tensor-to-scalar ratio is
extremely reduced, i.e. $r\leq{\mathcal{O}}(10^{-11}-10^{-5})$. Furthermore,
for the positive values of $\alpha$, the model does not produce real results
or compatible results with the observational data. Moreover, the result of
FIG. 1 shows that the temperature at the end of inflation, with the chosen
values, is approximately ${\mathcal{O}}(10^{-3})$. Although, by inserting the
exact value of the $\kappa$, this value becomes approximately
${\mathcal{O}}(10^{30})$.

Figure 1:  [color online] The inflationary observables and the temperature as
functions of the model parameters $\alpha$ and $\beta$ for the power-law
potential with $n=2$, $N=70$, $\Upsilon_{0}=10$ and $\nu=10^{-1}$ with
$\kappa=8\pi$, wherein (a) shows the scalar spectral index $n_{\rm S}$, (b)
indicates the tensor-to-scalar ratio $r$, and (c) demonstrates the temperature
at the end of inflation.

###  VI.2 POWER-LAW POTENTIAL WITH VARIABLE DISSIPATION COEFFICIENT

In this subsection, we continue the calculations with the assumption that the
dissipative coefficient is

$\Upsilon=\Upsilon_{*}\left(\dfrac{\phi}{\phi_{0}}\right)^{2},$ (174)

where $\Upsilon_{*}$ and $\phi_{0}$ are constant. In this case, the slow-roll
parameters can be derived in the similar way as we obtained relations (160),
except for $\lambda$, which are now

$\epsilon\approx\dfrac{n^{2}}{2\kappa\chi\phi^{2}},\qquad\eta\approx\dfrac{n(n-1)}{\kappa\chi\phi^{2}},\qquad\lambda\approx\dfrac{2n}{\kappa\chi\phi^{2}}.$
(175)

Next, from the condition $\epsilon\left(\phi_{\rm end}\right)=1$, we can
determine the value of the scalar field at the end of the inflationary era as

$\phi_{\rm end}^{2}\approx\dfrac{n^{2}}{2\kappa\chi_{\rm end}}.$ (176)

Thus, we need to specify the dissipation rate at the end of the inflationary
era. For this purpose, by substituting the dissipation coefficient (174) into
relation (17), while utilizing Eq. (161), we obtain

$\chi\approx\sqrt{\dfrac{-\alpha\,\Upsilon_{*}^{2}}{3\,\nu\left(\kappa+2\beta\right)\phi_{0}^{4}}}\,\phi^{\dfrac{\left(4-n\right)}{2}}.$
(177)

Meanwhile, this case with the power of $n=4$ clearly has a constant
dissipation rate. In turn, relation (176) gives

$\phi_{\rm end}\approx\left[\dfrac{3\,\nu
n^{4}\left(\kappa+2\beta\right)\phi_{0}^{4}}{-4\alpha(\kappa\Upsilon_{*})^{2}}\right]^{1/\left(8-n\right)},$
(178)

which dictates $n\neq 8$.

As in the previous subsection, the e-folding number (156) for this case yields

$\displaystyle
N\\!\approx\\!\left[\phi^{\left(8-n\right)/2}\\!-\\!\phi^{\left(8-n\right)/2}_{\rm
end}\right]\\!\left[\\!\dfrac{-4\alpha\,(\kappa\Upsilon_{*})^{2}}{3\nu
n^{2}\left(8-n\right)^{2}\\!\left(\kappa+2\beta\right)\phi_{0}^{4}}\\!\right]^{1/2}\\!,$
(179)

and in turn, the scalar field at the first horizon crossing becomes

$\displaystyle\phi\approx\left\\{\dfrac{-4\alpha$\kappa\Upsilon_{*}$^{2}}{3\nu
n^{2}\left(\kappa+2\beta\right)\left[n^{2}+\left(8-n\right)^{2}N^{2}\right]\phi_{0}^{4}}\right\\}^{1/\left(n-8\right)}.$
(180)

Also, by substituting relations (175), while using relation (177), into
relation (VI), we get the scalar spectral index in terms of the inflaton field
as

$\displaystyle n_{\rm S}-1\approx$
$\displaystyle\sqrt{\dfrac{3\,\nu\left(\kappa+2\beta\right)\phi_{0}^{4}}{-\alpha\Upsilon_{*}^{2}}}\,n\,\phi^{\left(n-8\right)/2}$
$\displaystyle\times\left[\dfrac{-16\left(2\kappa+2\beta-\kappa\alpha\right)-n\left(\kappa+\beta+\kappa\alpha\right)}{8\kappa\left(\kappa+\beta+\kappa\alpha\right)}\right]$
$\displaystyle\approx$
$\displaystyle\dfrac{-16\left(2\kappa+2\beta-\kappa\alpha\right)-n\left(\kappa+\beta+\kappa\alpha\right)}{4\left(\kappa+\beta+\kappa\alpha\right)\sqrt{n^{2}+\left(8-n\right)^{2}N^{2}}},$
(181)

where we have used relation (180) for the last row. Hence again, the scalar
spectral index is only a function of the parameters $\alpha$, $\beta$, $n$ and
$N$.

In addition, in this case, relation (VI) becomes

$\zeta$\phi$\approx\left[\dfrac{\kappa\alpha-2\left(\kappa+\beta\right)}{\kappa+\beta+\kappa\alpha}\right]\ln\phi,$
(182)

then utilizing relations (17), (152) and (182) in relation (147), the tensor-
to-scalar ratio reads

$\displaystyle r\approx$
$\displaystyle\dfrac{\kappa\nu}{\tau}\sqrt{\dfrac{H\phi_{0}^{2}\left(\kappa+2\beta\right)}{3\,\alpha\Upsilon_{*}\left(\kappa+\beta+\kappa\alpha\right)}}\coth$\dfrac{k}{2\tau}$$
$\displaystyle\times\phi^{\left[\dfrac{n(\kappa+\beta+\kappa\alpha)-(5\kappa+5\beta-\kappa\alpha)}{\kappa+\beta+\kappa\alpha}\right]}.$
(183)

Also, by using relation (170) for this case while substituting relations (169)
and (177) into it, we obtain the radiation temperature as

$\tau\approx\left[\dfrac{n^{2}}{4C_{\gamma}\kappa^{2}\Upsilon_{*}}\sqrt{\dfrac{3\,\nu^{3}$\kappa+2\beta$^{3}\phi_{0}^{4}}{$-\alpha$^{3}}}\right]^{1/4}\phi^{\dfrac{3n-8}{8}}.$
(184)

Now, using relations (161), (180) and (184) in relation (VI.2), we obtain
again the inflationary observable $r$ as a function of $\alpha$, $\beta$, $n$,
$N$, $\phi_{0}$, $\Upsilon_{*}$ and $\nu$ parameters.

We have plotted the scalar spectral index, the tensor-to-scalar ratio, and the
temperature at the end of inflation with respect to the model parameters
$\alpha$ and $\beta$ for the power-law potential with $n=2$ (in FIG. 2) and
$n=4$ (in FIG. 3), $N=70$, $\nu\sim{\cal O}(10^{-1})$, $\Upsilon_{*}\sim{\cal
O}(10^{2})$ and $\phi_{0}=M_{\rm Pl}$. The result for $n_{\rm S}$ indicates
that, in this case, the linear version of $f(Q,T)$ gravity can generate
diverse spectra that extend across the spectrum range from the blue tilt,
namely $n_{\rm S}>1$ for $-30<\alpha<-3$ (in FIG. 2 and FIG. 3), to the red
tilt, i.e. $n_{\rm S}<1$ for $-0.5<\alpha<-0.1$ (in FIG. 2)
and$-0.2<\alpha<-0.05$ (in FIG. 3), which is in agreement with the WMAP three
years data WMAP2006 . Moreover, the result illustrates that the tensor-to-
scalar ratio is in the range of $r\sim{\mathcal{O}}(10^{-4}-10^{-3})$ (in FIG.
2) and $r\sim{\mathcal{O}}(10^{-7}-10^{-6})$ (in FIG. 3) compatible with both
the observational data (VI.1) and (173). Therefore, the model with $n=4$
imposes more restrictions on $r$ than the model with $n=2$. Furthermore, the
values of both $n_{\rm S}$ and $r$ are very sensitive to different values of
the lower bound of the $\alpha$ parameter, but not very sensitive to different
values of the parameter $\beta$. However, in this case, for positive values of
the parameter $\alpha$, no real and/or compatible results are attainable. In
addition, the results show that the temperature at the end of inflation, with
the chosen values, is approximately ${\mathcal{O}}(10^{-2})$ (in FIG. 2) and
${\mathcal{O}}(10^{-3})$ (in FIG. 3). Although, by inserting the exact value
of the $\kappa$, these values are approximately ${\mathcal{O}}(10^{16})$ and
${\mathcal{O}}(10^{15})$, respectively. Moreover, for the power-law potential
with $n=4$, the dissipation rate is constant and does not depend on the
inflaton scalar field.

Figure 2:  [color online] The inflationary observables and the temperature as
functions of the model parameters $\alpha$ and $\beta$ for the power-law
potential with $n=2$, $N=70$, $\phi_{0}=M_{\rm Pl}$, $\Upsilon_{*}=10^{2}$ and
$\nu=10^{-1}$ with $\kappa=8\pi$, wherein (a) and (b) show the scalar spectral
index $n_{\rm S}$, (c) indicates the tensor-to-scalar ratio $r$, and (d)
demonstrates the temperature at the end of inflation.

Figure 3:  [color online] The inflationary observables and the temperature as
functions of the model parameters $\alpha$ and $\beta$ for the power-law
potential with $n=4$, $N=70$, $\phi_{0}=M_{\rm Pl}$, $\Upsilon_{*}=10^{2}$ and
$\nu=10^{-1}$ with $\kappa=8\pi$, wherein (a) and (b) show the scalar spectral
index $n_{\rm S}$, (c) indicates the tensor-to-scalar ratio $r$, and (d)
demonstrates the temperature at the end of inflation.

In the weak dissipation regime, the scalar spectral index (V.1) becomes

$n_{\rm S}-1=\dfrac{19}{4}\epsilon-\dfrac{5}{2}\eta-\dfrac{1}{4}\lambda.$
(185)

Also, relations (V.1) and (147) are now

$\displaystyle\zeta(\phi)\approx\int\left(\dfrac{V^{\prime}}{V}-\dfrac{V^{\prime\prime}}{V^{\prime}}+\chi\dfrac{\Upsilon^{\prime}}{\Upsilon}\right){\rm
d}\phi,$ $\displaystyle
r\approx\dfrac{\kappa\sqrt{\left(\kappa+2\beta\right)}\,V}{3\tau\sqrt{-\alpha\left(\kappa+\beta\right)}}\exp\left[2\zeta(\phi)\right]\coth\left(\dfrac{k}{2\tau}\right).$
(186)

Furthermore, within this regime, the slow-roll parameters (73), (74) and (75)
become

$\displaystyle\epsilon\approx\dfrac{-\alpha}{2$\kappa+\beta$}$\dfrac{V^{\prime}}{V}$^{2},\qquad\eta\approx\dfrac{-\alpha}{$\kappa+\beta$}$\dfrac{V^{\prime\prime}}{V}$,$
$\displaystyle\lambda\approx\dfrac{-\alpha}{$\kappa+\beta$}$\dfrac{\Upsilon^{\prime}V^{\prime}}{\Upsilon
V}$.$ (187)

##  VII Conclusions

Based on observations, it has been evident that the Universe has experienced
two periods of accelerated expansion in its history. The first phase occurred
in the early universe, known as inflation, and the second phase is taking
place in the late universe, which is related to the dark energy sector.

The inflationary paradigm addresses the deficiencies of the standard
cosmological model by introducing a period of accelerated expansion. In
general, inflationary models can be categorized into two groups: cold
inflation and warm inflation. In the cold inflation model, the inflationary
scalar field does not interact with other fields or radiation. It slowly rolls
down along its relatively flat potential, and the Universe experiences a
quasi-exponential expansion during inflation, which comes to an end when the
inflaton reaches the minimum of the potential. In this case, the inflaton
begins to oscillate around this minimum, decays into the particles of the
standard model, and the Universe, which has been in a very cold phase due to
very rapid expansion, reheats and connects to the radiation-dominant era of
the standard model of cosmology. This phase is called reheating. However, in
the warm inflation model, the inflaton does interact with other fields,
leading to an energy transfer from the inflaton to radiation field during its
slow-roll. Towards the end of inflation, the inflaton undergoes complete decay
into radiation, preventing the Universe from having a very cold phase due to
inflation. As a consequence, the Universe is seamlessly transitioning into the
radiation-dominant phase without the need for a separate reheating phase.
While clear predictions have been successfully made by GR in describing
cosmological phenomena, these predictions are not sufficient to fully account
for the impacts of dark sectors on the dynamics of the Universe, which are
consistent with observational data. Therefore, the exploration of alternative
gravitational theories provides a compelling motivation to address this issue.

In this study, we have investigated warm inflation within the framework of the
linear version of $f(Q,T)$ gravity. To accomplish this, we initially outlined
the basic theoretical setup for warm inflation, starting with GR and
considering an isotropic and homogeneous scalar field, commonly referred to as
inflaton. Next, we have provided a concise explanation of the $f(Q,T)$ gravity
theory along with its mathematical aspects and interconnections. Subsequently,
we have applied this theory to a spatially flat FLRW metric, while taking into
account the inflaton scalar field and a dissipation term. As a result, we have
derived the modified Friedmann equations and the modified Klein-Gordon
equation specific to the linear version of $f(Q,T)$ gravity. Furthermore, we
have extracted the extended slow-roll parameters and extended slow-roll
conditions for this model. Following that, we have analyzed the scalar and
tensor perturbations, while have developed the corresponding formulations, for
the linear version of $f(Q,T)$ gravity with warm inflation scenario. By
employing the slow-roll conditions, we have calculated both the scalar power
spectrum and the power spectrum of the tensor perturbations. Subsequently, we
have determined the scalar spectral index and the tensor-to-scalar ratio based
on these power spectra. Then, in the strong dissipation regime, by imposing a
restriction on the parameters and assuming the power-law potential, we have
derived those inflationary observables as well as the temperature of the
thermal bath for two cases. One case involves a constant dissipation
coefficient, while the other one considers a more general case, a specific
function of the inflaton scalar field.

Under the assumption of the constant dissipation coefficient, we have obtained
that the scalar spectral index is a function of two parameters of the linear
model of $f(Q,T)$ gravity ($\alpha$ and $\beta$) plus the power value $n$ of
the power-law potential and the e-folding number. While, the tensor-to-scalar
ratio is a function of these four parameters plus the two constant
coefficients of the potential and the dissipation coefficient. Then, we have
plotted the scalar spectral index, the tensor-to-scalar ratio, and the
temperature at the end of inflation with respect to the model parameters for
the power-law potential with $n=2$ while keeping the other parameters fixed.
The obtained results for the scalar spectral index and the tensor-to-scalar
ratio in the selected range of negative values of the $\alpha$ parameter and
positive and negative values of the $\beta$ parameter are in good agreement
with the Planck $2018$ data and the joint Planck, BK$15$ and BAO data. Hence,
the power-law potential can make predictions consistent with the observational
data. However, these inflationary observables rely crucially on the model
parameters. For instance, when considering positive values of the parameter
$\alpha$, it is not possible to achieve results consistent with the
observational data. Also, the results show that the temperature of the
Universe at the end of inflation, with the chosen values, is approximately
${\mathcal{O}}(10^{-3})$, although by inserting the exact value of the
$\kappa$ it is approximately ${\mathcal{O}}(10^{30})$.

Next, assuming that the variable dissipation coefficient is a function of the
inflaton field to the power of two, we have again obtained that the scalar
spectral index is a function of two model parameters plus the parameter $n$
and the e-folding number. While, the tensor-to-scalar ratio is a function of
these four parameters plus the three constant coefficients of the potential
and the dissipation coefficient. Then, we have plotted the scalar spectral
index, the tensor-to-scalar ratio, and the temperature at the end of inflation
with respect to the model parameters for the power-law potential with $n=2$
and $n=4$ while keeping the other parameters fixed. The model produces a
spectrum for the scalar spectral index that transitions from the blue tilt to
the red tilt as the parameter $\alpha$ increases, which is in agreement with
the WMAP three years data. The findings show that for negative values of
$\alpha$ and positive and negative values of $\beta$, the results are
consistent with both the Planck $2018$ data and the joint Planck, BK$15$ and
BAO data. Once again, although these inflationary observables rely crucially
on the model parameters, the power-law potential can make predictions
consistent with the observational data. Besides, in this case, again, when
considering positive values of $\alpha$, it is not possible to obtain results
in agreement with the observational data. Also, the results show that the
temperature at the end of inflation, with the chosen values, is approximately
${\mathcal{O}}(10^{-2})$ (in FIG. 2) and ${\mathcal{O}}(10^{-3})$ (in FIG. 3),
although by inserting the exact value of the $\kappa$ these are approximately
${\mathcal{O}}(10^{16})$ and ${\mathcal{O}}(10^{15})$, respectively.

Finally, in contrast to the results obtained from GR in the same scenario,
since the $f(Q,T)$ gravity approach imposes tighter constraints on the value
of the tensor-to-scalar ratio, it effectively revives the power-law potential.
Such results can provide a justification for applying $f(Q,T)$ gravity. In
addition, the utilization of the linear version of $f(Q,T)$ gravity produces
distinct differences in the constraints on the parameter of the power-law
potential compared to the results obtained under the GR assumptions.
Therefore, by choosing the appropriate parameters of the model, the linear
version of $f(Q,T)$ gravity in the warm inflationary paradigm can revive the
power-law potential.

## ACKNOWLEDGMENTS

M.S. and M.F. thank the Research Council of Shahid Beheshti University.

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