# Slow-roll inflation in $f(T,\mathcal{T})$ modified gravity

Aaqid Bhat 0000-0002-8644-0677<EMAIL_ADDRESS>Department of Mathematics,
Birla Institute of Technology and Science-Pilani, Hyderabad Campus,
Hyderabad-500078, India. Sanjay Mandal 0000-0003-2570-2335
<EMAIL_ADDRESS>Faculty of Mathematics and Computer Science,
Transilvania University, Iuliu Maniu Str. 50, 500091 Brasov, Romania P.K.
Sahoo 0000-0003-2130-8832<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad
Campus, Hyderabad-500078, India.

###### Abstract

In this article, we explore the concept of cosmological inflation within the
framework of the $f(T,\mathcal{T})$ theory of gravity, where $f$ is a general
function of the Torsion scalar $T$ and the trace, $\mathcal{T}$, of the
energy-momentum tensor. It is assumed that the conditions of slow-roll
inflation are applicable in$f(T,\mathcal{T})$ gravity. To determine different
observables related to inflation, such as the tensor-to-scalar ratio $r$,
scalar spectral index $n_{s}$, spectral index $\alpha_{s}$, and tensor
spectral index $n_{t}$, the Hubble slow-roll parameters are utilized for a
particular model of $f(T,\mathcal{T})$. Lastly, an assessment has been carried
out to determine the feasibility of the models by conducting a numerical
analysis of the parameters. The findings indicate that it is feasible to
achieve compatibility with the observational measurements of slow-roll
parameters by utilizing different values of the free parameters.

Keywords: $f(T,\mathcal{T})$ gravity; early-time cosmology; Dark Energy;
equation of state; slow-roll inflation.

##  I Introduction

Recent observations have provided clear evidence that our universe is
expanding at an accelerating rate and is primarily made up of dark matter and
dark energy [1, 2, 3, 4, 5]. To account for this phenomenon, alternative
theories of gravity are being explored that do not rely on unknown sources of
energy. Modified gravity models, including $f(R),f(G),f(R,G)$, and $f(R,T)$
[6, 7, 8, 9, 10], are able to explain the expanding acceleration of the
universe. The $f(R)$ model was the initial modified gravity theory, which was
introduced by Buchdahl in 1970. In this theory, the curvature scalar $R$ in
the standard Einstein-Hilbert action is replaced by an arbitrary function of
$f(R)$. Modified gravitational theories typically involve expanding upon the
Einstein-Hilbert action in General Relativity, which is based on the curvature
description of gravity. However, an intriguing alternative arises when we
extend the action of the equivalent formulation of GR that incorporates
torsion.As it is known, Einstein constructed also the ”Teleparrell Equivalent
of General Relativity” (TEGR) in which the gravitational field is described by
the torsion tensor and not by the curvature one [15, 16]. In this approach,
the Lagrangian of the modified gravity theory is derived from contractions of
the torsion tensor, similar to how the Einstein-Hilbert Lagrangian is derived
from contractions of the curvature (Riemann) tensor. Therefore, rather than
beginning with General Relativity, one can initiate the development of
modified gravity theories by starting with the Teleparallel Equivalent of
General Relativity (TEGR). By extending the torsion scalar T to a general
function within the Lagrangian, one can construct the f(T) modified gravity
theory[17, 18].
A new proposal in modified gravity is to use the coupling between torsion and
the trace of the energy-momentum tensor, which is known as the
$f(T,\mathcal{T})$ theory. The $f(T,\mathcal{T})$ gravity theory was
introduced in [11].This newly formulated f(T, T) theory represents a novel
approach to modified gravity, distinctly separate from existing constructions
relying on torsion or curvature.When applied within a cosmological context,
this theory yields intriguing observational outcomes. Notably, it allows for a
coherent explanation encompassing the early inflationary stage, the subsequent
matter-dominated expansion that doesn’t involve acceleration, and the eventual
shift into a late-time accelerating phase. Furthermore, the effective dark
energy sector within this framework can exhibit characteristics ranging from
quintessence or phantom-like behavior to potentially crossing the phantom-
divide during its evolution. Thus,$f(T,\mathcal{T})$ cosmology offers a
unified description of the universe evolution. Recently the cosmological
implications of $f(T,\mathcal{T})$ was proposed by [12] in which the authors
analyzed the cosmological behavior of the deceleration, effective equation of
state and total equation of state parameters. However, it is seen that the
deceleration parameter depicts the transition from deceleration to
acceleration and the effective dark sector shows a quintessence-like
evolution. Constraining $f(T,\mathcal{T})$ gravity with gravitational
baryogenesis was studied in [13]. Also $H_{0}$ Tension in $f(T,\mathcal{T})$
was studied by [14] in which the authors gave a new way to alleviate $H_{0}$
Tension in torsion based modified gravity. The consistency of
$f(T,\mathcal{T})$ theory with cosmological data and the required physical
conditions for a consistent cosmological theory are yet to be confirmed.
However, the coupling of torsion and matter provides additional opportunities
for describing the properties of dark energy and identifying the source of the
observed acceleration. This theory has been evaluated regarding its
reconstruction and stability [19, 20], acceleration at late times and during
inflationary phases [11], the growth rate of sub-horizon modes[21], and also
in the context of quark stars[22].

Two valuable frameworks for characterizing the initial development of the
universe are the inflationary scenario [24, 23, 25] and bouncing cosmology[26,
27]. Inflationary cosmology is a widely accepted model that describes the very
early Universe, shortly after the Big Bang. The earliest inflationary models
were developed to address issues associated with primordial singularity and
emerged organically from the particles and entropy that existed in the
universe. However, these models were incomplete modifications of the big bang
theory as they assumed the universe was in a state of thermal equilibrium and
homogeneity on large scales prior to inflation. The chaotic inflation proposal
resolved this issue. The big bounce, also known as the phoenix universe, is a
cyclic cosmological model that oscillates between the big bang and big crunch
phases. This theory was widely accepted until the theory of inflation emerged
as a resolution to the horizon problem, which was prompted by observations
showing the large-scale structure of the universe. The authors of the study
[28] explored the future evolution of $f(R)$ gravity models, ensuring
consistency with local tests and unifying the history of universe expansion.
In [29], the authors examined the realization of inflation in the context of
unimodular $f(T)$ gravity, which is a modification of teleparallel gravity.
The study concluded that inflationary unimodular $f(T)$ gravity is a
successful candidate for describing the early universe.

The article aims to examine the generation of a viable inflationary era in
$f(T,\mathcal{T})$ gravity, which is a torsion matter coupling gravity. The
focus is on evaluating the viability of each inflationary scenario by
assessing the inflationary observational indices such as the tensor-to-scalar
ratio (r), the scalar spectral index ($n_{s}$), the running ($\alpha_{s}$) of
the spectral index, and the tensor spectral ($n_{T}$). The observational
indices will be assessed using conventional techniques, and we will examine
whether the resulting cosmologies comply with the most recent Planck and
BICEP2/Keck Array data [30, 31]. To conduct our research, we have examined a
well-known model $f(T,\mathcal{T})=\alpha\mathcal{T}+\beta T^{2}$ where
$\alpha$ and $\beta$ are free parameters. The letter is organized as follows:
Section II provides a concise overview of the field equations in
$f(T,\mathcal{T})$ gravity. Section III is dedicated to investigating the
potential for slow-roll inflation using $f(T,\mathcal{T})$ gravity. Our
findings are summarized in the concluding remarks in Section IV.

##  II  Brief review in $f(T,\mathcal{T})$ gravity

This section provides the basic principles necessary for rebuilding the
theories of gravity, specifically in $f(T)$ and $f(T,\mathcal{T})$.

To define the torsion-based theory, a new connection called the Weitzenböck
connection [32] is required. This connection is expressed as
$\tilde{\Gamma}^{\alpha}_{\mu\nu}=e_{a}^{\,\,\alpha}\partial_{\nu}e^{a}_{\,\,\mu}$.
These tetrads are related to the metric tensor $g_{\mu\nu}$ at every point x
on the spacetime manifold as

$g_{\mu\nu}(x)=e^{a}_{\,\,\mu}(x)e^{b}_{\,\,\nu}(x)\eta_{ab}.$ (1)

Here, $\eta_{ab}=diag(1,-1,-1,-1)$ is the Minkowski metric tensor. As a
result, the torsion tensor that characterizes the gravitational field can be
written as:

$T^{\alpha}_{\,\,\mu\nu}=\Gamma^{\alpha}_{\,\,\nu\mu}-\Gamma^{\,\,\alpha}_{\mu\nu}=e_{a}^{\,\,\alpha}\left(\partial_{\mu}e^{a}_{\,\,\nu}-\partial_{\nu}e^{a}_{\,\,\mu}\right).$
(2)

The components of the torsion tensor are used to define both the contortion
and superpotential tensor

$\displaystyle K^{\mu\nu}_{\,\,\alpha}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(T^{\mu\nu}_{\,\,\alpha}-T^{\nu\mu}_{\alpha}-T_{\alpha}^{\,\,\mu\nu}\right),$
(3) $\displaystyle S_{\alpha}^{\,\,\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(K^{\mu\nu}_{\,\,\alpha}+\delta^{\mu}_{\alpha}T^{\lambda\mu}_{\,\,\lambda}-\delta^{\nu}_{\alpha}T^{\lambda\mu}_{\,\,\lambda}\right).$
(4)

Using equations (2) and (4), one obtains the torsion scalar [11, 33, 34]

$T=S_{\alpha}^{\,\,\mu\nu}T^{\alpha}_{\,\,\mu\nu}=\frac{1}{2}T^{\alpha\mu\nu}T_{\alpha\mu\nu}+\frac{1}{2}T^{\alpha\mu\nu}T_{\nu\mu\alpha}-T_{\alpha\mu}^{\,\,\,\,\alpha}T^{\nu\mu}_{\,\,\,\,\nu}.$
(5)

The action for teleparallel gravity can be defined as follows:

$S=\int d^{4}x\,e\,[T+\mathcal{L}_{m}],$ (6)

where $e$ represents the determinant of the tetrad field $e^{a}_{\mu}$, which
is equal to the square root of the metric determinant $-g$, and
$\mathcal{L}{m}$ is the Lagrangian density of matter. Additionally, the $T$ in
teleparallel gravity can be extended to $T+f(T)$, which is known as $f(T)$
gravity. Furthermore, this can be further generalized to become a function of
both the torsion scalar and the trace of the energy-momentum tensor
$\mathcal{T}$, resulting in $f(T,\mathcal{T})$ gravity.

The action for $f(T,\mathcal{T})$ gravity is defined as follows

$S=\frac{1}{16\pi G}\int d^{4}x\,e[T+f(T,\mathcal{T})]+\int
d^{4}x\,e\,\mathcal{L}_{m}.$ (7)

$L_{m}=-P$ where $P$ is the total pressure.

The field equations for $f(T,\mathcal{T})$ gravity can be obtained by varying
the action with respect to the vierbeins.

$(1+f_{T})\left[e^{-1}\partial_{\mu}(ee_{a}^{\,\,\alpha}S_{\alpha}^{\,\,\lambda\mu})-e_{a}^{\,\,\alpha}T^{\mu}_{\nu\alpha}S_{\mu}^{\,\,\nu\lambda}\right]+e_{a}^{\,\,\lambda}\left(\frac{f+T}{4}\right)+\\\
\left(f_{TT}\partial_{\mu}T+f_{T\mathcal{T}}\partial_{\mu}\mathcal{T}\right)e_{a}^{\,\,\alpha}S_{\alpha}^{\,\,\lambda\mu}-f_{\mathcal{T}}\left(\frac{e_{a}^{\,\,\alpha}\overset{em}{T}_{\alpha}^{\,\,\lambda}+p_{m}e_{a}^{\,\,\lambda}}{2}\right)=\\\
4\pi Ge_{a}^{\,\,\alpha}\overset{em}{T}_{\alpha}^{\,\,\lambda}.$ (8)

where $f_{\mathcal{T}}={\partial f}/{\partial\mathcal{T}}$,
$f_{T\mathcal{T}}={\partial^{2}f}/{\partial T\partial\mathcal{T}}$, and
$\overset{em}{T}_{\alpha}^{\,\,\lambda}$ is the energy-momentum tensor.

To apply the aforementioned theory in a cosmological framework and obtain
modified Friedman equations, we can utilize the flat FRW metric as usual. The
FRW metric is given by

$ds^{2}=dt^{2}-a(t)^{2}\delta_{ij}dx^{i}dx^{j},$ (9)

where $a(t)$ is the scale factor. Further, (8) give rise to modified Friedmann
equations:

$H^{2}=\frac{8\pi
G}{3}\rho_{m}-\frac{1}{6}\left(f+12H^{2}f_{T}\right)+f_{\mathcal{T}}\left(\frac{\rho_{m}+p_{m}}{3}\right),$
(10) $\dot{H}=-4\pi
G(\rho_{m}+p_{m})-\dot{H}(f_{T}-12H^{2}f_{T\mathcal{T}})\\\
-H(\dot{\rho_{m}}-3\dot{p_{m}})f_{T\mathcal{T}}-f_{\mathcal{T}}\left(\frac{\rho_{m}+p_{m}}{2}\right).$
(11)

Here, (dot represents first order derivative with respect to $t$) and
$\mathcal{T}=\rho_{m}-3p_{m}$ in the above equation is true for the perfect
matter fluid.

Comparing the modified Friedmann equations (10) and (11) to General Relativity
equations

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3}\left(\rho_{m}+\rho_{eff}\right),$ (12) $\displaystyle\dot{H}$
$\displaystyle=$ $\displaystyle-4\pi
G\left(\rho_{m}+p_{m}+\rho_{eff}+p_{eff}\right).$ (13)

We obtain

$\rho_{eff}=-\frac{1}{16\pi
G}[f+12f_{T}H^{2}-2f_{\mathcal{T}}(\rho_{m}+p_{m})],$ (14)
$p_{eff}=\frac{1}{16\pi
G}[f+12f_{T}H^{2}-2f_{\mathcal{T}}(\rho_{m}+p_{m})]+\\\
(\rho_{m}+p_{m})+\left[\frac{(1+\frac{f_{T}}{8\pi
G})}{1+f_{T}12H^{2}f_{TT}+H(\frac{d\rho_{m}}{dH})(1-3{c_{s}}^{2})f_{T\mathcal{T}}}-1\right]$
(15)

Here, $c_{s}$ is the speed of light. The conservation equation involving the
effective energy and pressure reads as

$\dot{\rho}_{\text{eff}}+\dot{\rho}_{m}+3H(\rho_{m}+\rho_{\text{eff}}+p_{m}+p_{\text{eff}})=0$
(16)

. Hence, in the current model, the conservation of effective dark energy alone
does not hold, and there exists an effective interaction between dark energy
and regular matter, allowing for the potential exchange of energy between the
two components.

##  III Slow-Roll Inflation with $f(T,\mathcal{T})$ gravity

Slow-roll conditions are a set of requirements that inflationary models in
cosmology must satisfy to produce a period of exponential expansion in the
early universe. Slow-roll conditions have important implications for theories
of gravity in cosmology. In particular, they can be used to test and constrain
alternative theories of gravity, such as modified gravity models that attempt
to explain the observed accelerated expansion of the universe without the need
for dark energy. By studying the properties of the cosmic microwave background
radiation and large-scale structure, cosmologists can place constraints on the
parameters of these theories and rule out those that violate the slow-roll
conditions. The slow-roll conditions are typically expressed in terms of the
potential energy and the kinetic energy of the inflaton field, as well as its
derivatives with respect to time. In particular, for a scalar field $\phi$
with potential energy $V(\phi)$, the slow-roll conditions are.

In every inflationary scenario, one needs to calculate the values of various
inflation-related observables,such as the tensor-to-scalar ratio $r$, the
scalar spectral index $n_{S}$, the running of the spectral index $\alpha_{s}$,
and the tensor spectral $n_{T}$ .Tensor-to-scalar ratio is defined as the
ratio of the amplitude of tensor perturbations (primordial gravitational
waves) to the amplitude of scalar perturbations.The scalar spectral index
$n_{s}$ describes how the clumpiness of stuff varied on various scales just
after cosmic inflation.It is an important parameter describing the nature of
primordial density perturbations. In principle, the calculation of the above
observables requires a detailed and lengthy perturbation analysis. However,
one can bypass this procedure by transforming the given scenario to the
Einstein frame, where all the inflation information is encoded in the
(effective) scalar potential $V(\phi)$, defining the slow-roll parameters
$\epsilon$, $\eta$ and $\xi$ in terms of this potential and its derivatives
[35, 36].

$\epsilon\equiv\left(\frac{M}{2}\right)^{2}\left(\frac{1}{V}\frac{dV}{d\phi}\right)^{2}$
(17) $\eta\equiv\frac{M^{2}}{pV}\left(\frac{d^{2}V}{d\phi^{2}}\right)$ (18)
$\xi^{2}\equiv\frac{M^{4}}{pV^{2}}\left(\frac{dV}{d\phi}\right)\left(\frac{d^{3}V}{d\phi^{3}}\right)$
(19)

The slow-roll parameter $\epsilon$ must be much smaller than unity:

$\epsilon=\frac{1}{2}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2}\ll 1,$

where $V^{\prime}(\phi)$ is the derivative of the potential energy with
respect to $\phi$.
The second slow-roll parameter $\eta$ must also be much smaller than unity:

$\eta=\frac{V^{\prime\prime}(\phi)}{V(\phi)}\ll 1,$

where $V^{\prime\prime}(\phi)$ is the second derivative of the potential
energy with respect to $\phi$.

In this part, we are considering that the conditions for slow-roll inflation
are met within the $f(T,\mathcal{T})$ framework, and these conditions are
stated in relation to the parameter $H$ as

$\displaystyle\frac{\dot{H}}{H^{2}}\leq 1,$ (20)
$\displaystyle\frac{\ddot{H}}{H\dot{H}}\leq 1.$ (21)

To discuss the possibility of inflation in a theory of gravity that includes a
coupling between torsion and trace, let’s consider the following Lagrangian as
an example.

We have a function$f(T,\mathcal{T)}=\alpha\mathcal{T}+\beta
T^{2}=\alpha\rho_{m}+\beta T^{2}=\alpha\rho_{m}+\gamma H^{4}$, where $\alpha$
and $\gamma=36\beta$ are constants. To simplify, we use $8\pi G=c=1$. This
model is a deviation from General Relativity within the framework of
$f(T,\mathcal{T)}$. When $\alpha=0$ , the model behaves as a power-law
cosmology in $f(T)$ theory [37]. In this case, we get $f_{T}=\frac{\gamma
T}{18}$, $f_{TT}=\frac{\gamma}{18}$ ,$f_{\mathcal{T}}=\alpha$,
$f_{T\mathcal{T}}=0$.
Using these equations along with equations (10) & (11), we can derive the
following:

$\displaystyle\rho_{m}$ $\displaystyle=$
$\displaystyle\frac{3\left(1-\frac{\gamma
H^{2}}{2}\right)}{1+\frac{\alpha}{2}}H^{2},$ (22) $\displaystyle\dot{H}$
$\displaystyle=$ $\displaystyle-\frac{3(1+\alpha)\left(1-\frac{\gamma
H^{2}}{2}\right)}{(\alpha+2)(1-\gamma H^{2})}H^{2}.$ (23)

The deceleration parameter, denoted as $q$, is defined as
$q=-\frac{\dot{H}}{H^{2}}-1$. This progression aligns with the recent behavior
of the universe, characterized by three distinct stages: an initial
decelerating phase, a subsequent phase of accelerating expansion, and a late-
time acceleration phase. For our model, the deceleration parameter $q$ is
given as:

$q=\frac{3(1+\alpha)\left(1-\frac{\gamma H^{2}}{2}\right)}{(\alpha+2)(1-\gamma
H^{2})}-1.$ (24)

Moreover, the effective dark energy density and pressure from equations (14)
and (13) can be obtained as

$\displaystyle\rho_{eff}$ $\displaystyle=$
$\displaystyle\frac{3H^{2}(\alpha+\gamma H^{2})}{\alpha+2},$ (25)
$\displaystyle p_{eff}$ $\displaystyle=$
$\displaystyle-\frac{3H^{2}(\alpha+\gamma H^{2})}{(\alpha+2)(\gamma
H^{2}-1)}.$ (26)

which gives

$\omega_{eff}=\frac{1}{1-\gamma H^{2}}.$ (27)

Applying the slow roll conditions and using the above equation in equation
(16), we have

$\displaystyle 2\dot{H}\gamma+3k_{1}(\gamma*H^{2}-2)=0,$ (28) $\displaystyle
H(t)=p\text{tan}(q_{1}t),$ (29) $\displaystyle
H(t)=p\text{tan}\mathcal{T}_{1},$ (30)

where

$\displaystyle
p=\frac{\sqrt{2}}{\sqrt{-\gamma}},\,\,\,k_{1}=\frac{(\alpha+1)}{(\alpha+2)},$
(31) $\displaystyle
q_{1}=\sqrt{2}\sqrt{-\gamma},\,\,\,q_{1}\,t=\mathcal{T}_{1}$ (32)

In the slow roll regime, (27) yields

$\omega_{e}ff=\frac{1}{1-\gamma*p^{2}*\text{tan}^{2}q_{1}t}\\\ $ (33)

The inflationary model offers a coherent and effective explanation for the
rapid expansion of the universe and the resulting cosmological perturbations
that cause its anisotropy. For this, we verified the profile of the
deceleration parameter $q(t)$ and found that at a very early evolution
process, it presents a rapid expansion and then converges to the de Sitter
expansion in the late-time evolution. According to the Planck results, the
value of $n_{s}$ is estimated to be $0.968\pm 0.006$ (with $68\%$ confidence
level), the value of $r$ is less than 0.11 (with $95\%$ confidence level), and
the value of $\alpha_{s}$ is estimated to be $-0.003\pm 0.007$ (with $68\%$
confidence level). These parameters are derived from the slow-roll parameters
[38, 29].

$\displaystyle\epsilon_{1}\equiv-\frac{\dot{H}}{H^{2}},$ (34)
$\displaystyle\epsilon_{2}\equiv\frac{\ddot{H}}{H\dot{H}}-\frac{2\dot{H}}{H^{2}},$
(35)
$\displaystyle\epsilon_{3}\equiv\left(H\ddot{H}-2\dot{H}^{2}\right)^{-1},$
(36)

and rewritten as

$\displaystyle r\approx 16\epsilon_{1},$ (37) $\displaystyle n_{s}\approx
1-2\epsilon_{1}-2\epsilon_{2},$ (38) $\displaystyle
n_{s}\approx-2\epsilon_{1}\epsilon_{2}-\epsilon_{2}\epsilon_{3},$ (39)
$\displaystyle n_{T}\approx-2\epsilon_{1}.$ (40)

Now, we can rewrite the slow-roll parameters as follow;

$\displaystyle r=-\frac{16q_{1}\csc^{2}{\mathcal{T}_{1}}}{p},$ (41)
$\displaystyle n_{s}=1+\frac{2q_{1}(-1+\cot^{2}{\mathcal{T}_{1}})}{p},$ (42)
$\displaystyle\alpha_{s}=-\frac{2q_{1}^{2}(1-2pq_{1}+\cos{2\mathcal{T}_{1}})\cot^{2}{\mathcal{T}_{1}}\csc^{4}{\mathcal{T}_{1}}}{p^{2}(pq_{1}-\cot^{2}{\mathcal{T}_{1}})},$
(43) $\displaystyle n_{T}=-\frac{2q_{1}}{p}\csc^{2}{\mathcal{T}_{1}}.$ (44)

We can estimate the number of e-folds

$N=\ln\frac{a_{f}}{a_{i}}=\int_{t_{i}}^{t_{f}}H(t)dt.$ (45)

Hence where $a_{i}=a(t=t_{i})$ is the initial value of the scale factor $a$ at
the beginning of inflation $t_{i}$, and $a_{f}=a(t=t_{f})$ is its final value
at the end of inflation $t_{f}$.

$N=\frac{p}{q_{1}}\ln\left[\frac{\cos(q_{1}t_{f})}{\cos(q_{1}t_{i})}\right].$
(46)

Assuming that the Hubble parameter $H(t)$ can be expanded as a series around
$\mathcal{T}_{1}=0$, we can obtain the second-order approximation.

$H(t)\approx\ \ p\mathcal{T}_{1}+\mathcal{O}(\mathcal{T}^{2}_{1})$ (47)

and the slow-roll parameters become

$\displaystyle\varepsilon_{1}\approx\ \ -\frac{1}{\mathcal{T}^{2}_{1}},$ (48)
$\displaystyle\varepsilon_{2}\approx\ \ -2\frac{1}{p\mathcal{T}^{2}_{1}},$
(49) $\displaystyle\varepsilon_{3}\approx\ \
-2\frac{1}{p\mathcal{T}^{2}_{1}}.$ (50)

Hence in this scenario, we have

$\displaystyle r\approx\ \ -16\frac{1}{p\mathcal{T}_{1}},$ (51) $\displaystyle
n_{s}=1,$ (52) $\displaystyle\alpha_{s}\approx\ \
-\frac{8}{p^{2}\mathcal{T}^{2}_{1}},$ (53) $\displaystyle n_{T}\approx\ \
-2\frac{1}{p\mathcal{T}^{2}_{1}}.$ (54)

According to Hubble parameter $h(t)$, the e-folding number is given by

$N=\frac{pq_{1}(t_{f}^{2}-t_{i}^{2})}{2}.$ (55)

These statements indicate that when the Hubble parameter $H(t)$ can be
expanded in a series around $\mathcal{T}=0$, which occurs for low values of
$\mathcal{T}$, the inflationary parameters have large values when the slow-
roll conditions are satisfied when the slow-roll conditions are met.
In order to evaluate the feasibility of our model, we provide the numerical
outcomes for the different inflation parameters given by (41), (42), (43) by
comparing theory results with observational data coming from PLANK 2015 AND
BICEP2/LECK-Array-data [30, 31].

Figure 1: Profiles of tensor-to-scalar ratio $(r)$, scalar spectral index
$(n_{s})$, and running spectral index $(\alpha_{s})$ with varying model
parameters $\alpha$ and $\gamma$.

The given graphs illustrate how the inflation parameters of model
$f(T,\mathcal{T})$ change over time $t=0.1$s. The first graph shows the
tensor-to-scalar ratio $r$ for $\alpha$ values ranging from $0.0$ to $0.2$,
while the second graph displays the scalar spectral index $ns$ for $\alpha$
values between $0.1$ to $0.5$. The final graph demonstrates the trend of the
running of the spectral index $\alpha_{s}$. The evaluation of the curves in
the parametric diagram suggests that the tensor-to-scalar ratio $(r)$, scalar
spectral index $(ns)$, and running of the spectral index $(\alpha_{s})$ values
are consistent with the observed data.

##  IV Conclusion

This article investigates the inflationary scenario in torsion-trace coupling
gravity, which involves a Lagrangian density derived from a function called
$f(T,\mathcal{T})$, which makes use of the torsion $T$ and the trace of the
energy-momentum tensor $(\mathcal{T})$.We found evolution of deceleration
parameter, explicitly experiencing a change from a deceleration to
acceleration, capable of explaining the late-time universe. The study focuses
on a universe containing ordinary matter and dark energy, and a primary
differential equation in the Hubble parameter is derived. Slow-roll conditions
are applied for a specific $f(T,\mathcal{T})$ model, and various measurements
related to inflationary phenomena are evaluated, including the ratio of tensor
to scalar perturbations $(r)$, the spectral index of scalar perturbations
$(ns)$, the running of the spectral index $(\alpha_{s})$, the tensor spectral
$n_{T}$, and the number of e-folds parameter. The numerical findings are
consistent with observational data. So far, many studies have been done in the
background of the torsion-based modified theory focusing on the current
acceleration scenario of the universe, and its’ late-time acceleration [11,
12, 39]. Whereas in this letter, we aimed to study the early-inflationary
scenario and successfully presented it. {We have not only successfully
presented the slow-roll inflationary scenario but also presented a way to
constrain the $f(T,\mathcal{T})$ cosmological model’s parameters. We also
constrained the model parameters for the desirable results and discussed them
in the last of Section III. We hope this study will shed some light on a new
direction in torsion-based modified gravity cosmology.

In the near future, one can study more generalized models to put constraints
on the parameters and also may extend this study to explore some possible
cosmological scenarios. Upcoming surveys are expected to offer a more precise
depiction of the universe, leading to a reduction in the number of potential
candidates that could provide a more comprehensive explanation for the
inflationary period.

Data availability There are no new data associated with this article.

###### Acknowledgements.

Aaqid Bhat expresses gratitude to the BITS-Pilani, Hyderabad campus, India,
for granting him a Junior Research Fellowship. SM acknowledges Transilvania
University of Brasov for providing Postdoctoral fellowship. PKS acknowledges
Science and Engineering Research Board, Department of Science and Technology,
Government of India for financial support to carry out Research project No.:
CRG/2022/001847 and IUCAA, Pune, India for providing support through the
visiting Associateship program. We are very much grateful to the honorable
referees and to the editor for the illuminating suggestions that have
significantly improved our work in terms of research quality, and
presentation.

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