Squared torsion fTTgravity and its cosmological implications Simran Arora 1Aaqid Bhat1 and P .K. Sahoo1 1Department of Mathematics Birla Institute of Technology and Science-Pilani

2025-05-03 0 0 626.41KB 8 页 10玖币

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Squared torsion f(T,T)gravity and its cosmological implications

Simran Arora ,1, ∗Aaqid Bhat,1, †and P.K. Sahoo 1, ‡

1Department of Mathematics, Birla Institute of Technology and Science-Pilani,

Hyderabad Campus, Hyderabad-500078, India.

(Dated: November 28, 2022)

We present the coupling of the torsion scalar Tand the trace of energy-momentum tensor

T, which produces new modiﬁed f(T,T)gravity. Moreover, we consider the functional form

f(T,T) = αT+βT2where αand βare free parameters. As an alternative to a cosmological

constant, the f(T,T)theory may offer a theoretical explanation of the late-time acceleration. The

recent observational data to the considered model especially the bounds on model parameters is

applied in detail. Furthermore, we analyze 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.

Keywords: f(T,T)gravity; acceleration; observational constraints; equation of state

I. INTRODUCTION

The in-depth veriﬁcation of late time acceleration has

led to immense research towards its explanation. It is

commonly known by the observations of type Ia Super-

novae [1,2], BAO [3,4], CMB [5], and H(z)measure-

ments [6]. The dark energy, which tried to explain the

late-time acceleration as the outcome of a type of en-

ergy connected to the cosmological constant, is one of

the successful primary models. In order to navigate

the path beyond the typical dark energy models, one

can go beyond the general theory of relativity by mod-

ifying the geometry. Alternative theories such as f(R)

gravity [7–9], a coupling between matter and curvature

through f(R,T)gravity [10,11], where Tis the trace

of energy momentum tensor, f(R,G)[12,13] (Gis the

Gauss-Bonnet) have all attempted to explain the dark

energy phenomenon in the context of curvature.

As a result, more general geometries than the Rieman-

nian, which may be valid at solar system level, may pro-

vide an explanation for the behavior of matter at large

scales in the universe. There has been a rising interest in

teleparallel gravity, a different type of modiﬁed gravity

that uses torsion instead of curvature. The basic idea be-

hind the teleparallel approach is to replace the metric of

spacetime by a set of tetrad vectors which is the physical

variable describing the gravitational properties. More-

over, this mathemcatical development employs a differ-

ent connection known as the Weitzen¨

ock connection.

∗dawrasimran27@gmail.com

†aaqid555@gmail.com

‡pksahoo@hyderabad.bits-pilani.ac.in

When one extends the action of the modiﬁed gravity

based on torsion, a separate and intriguing class of mod-

iﬁed gravity arises named as the teleparallel equivalent

of general relativity or f(T)gravity. However, a number

of analyses in f(T)gravity such as cosmological solu-

tions [14], late time acceleration [15,16], thermodynam-

ics [17], cosmological perturbations [18], cosmography

[19] have been applied in the literature. For a thorough

analysis of f(T)gravity, one can check [20].

Another new suggestion in modiﬁed gravity is to em-

ploy the coupling between the torsion and trace of

energy-momentum tensor known as f(T,T)theory, in

a similar fashion as f(R,T)gravity. The f(T,T)gravity

has been proposed in [21], and its consistency with cos-

mological data and the necessary physical conditions for

a coherent cosmological theory still has to be validated.

The coupling of torsion and matter expands the possi-

bilities for describing the characteristics of dark energy

or, more speciﬁcally, what is driving the observed accel-

eration. This theory has been investigated in the context

of reconstruction and stability [22,23], late-time acceler-

ation and inﬂationary phases [21], growth factor of sub-

horizon modes [24], quark stars [25].

The goal of the current study is to construct the extended

coupled-matter modiﬁed gravity by starting with TEGR

rather than GR. The construction of f(T,T)gravity,

which allows for arbitrary functions of the torsion scalar

Tand the trace of the energy-momentum tensor T, is the

focus of the current effort. In this paper, we investigate a

squared-torsion f(T,T)model, that raises a question on

the viability of such a theory as a candidate to account

for late-time acceleration. Further, the parameters are

constrained using the set of observational datasets and

in particular, we check the late-time accelerating behav-

arXiv:2210.01552v2 [gr-qc] 25 Nov 2022

ior holds true for f(T,T)using the cosmological param-

eters.

The plan of the work is the following: Starting from

the background of f(T), we introduce the framework

of f(T,T)gravity in section II. Section III is devoted

to the cosmological framework and the solutions to the

ﬁeld equations. Speciﬁcally, in section IV, we deal with

the observational data and methodology used to con-

strain the parameters involved. The late-time acceler-

ated phase is examined in section Vthrough cosmolog-

ical evolution. Finally, a conclusion is given in section

VI.

II. FIELD EQUATIONS

The fundamental preliminaries for the reconstruction

of the f(T)and f(T,T)theories of gravity are presented

in this section.

One needs a new connection, the Weitzenb ¨

ock connec-

tion [26] to obtain the torsion-based theory deﬁned as

Γα

µν =eα

a∂νea

µ, where ea

µand eα

aare tetrads (or vier-

beins). These vierbeins relate to the metric tensor gµν at

each point x of the spacetime manifold as

gµν(x) = ea

µ(x)eb

ν(x)ηab. (1)

Here, ηab =diag(1, −1, −1, −1)is the Minkowski metric

tensor. Hence, the torsion tensor describing the gravita-

tional ﬁeld is

Tα

µν =Γα

νµ −Γα

µν =eα

a∂µea

ν−∂νea

µ. (2)

We deﬁne the contortion and the superpotential tensor

through the components of the torsion tensor

Kµν

α=−1

2Tµν

α−Tνµ

α−Tµν

α, (3)

Sµν

α=1

2Kµν

α+δµ

αTλµ

λ−δν

αTλµ

λ. (4)

Using equations (2) and (4), one obtain the torsion scalar

[20,21,27]

T=Sµν

αTα

µν =1

2TαµνTαµν +1

2TαµνTνµα −Tα

αµ Tνµ

ν. (5)

One can deﬁne the gravitational action for teleparallel

gravity by

S=Zd4x e [T+Lm], (6)

where e=det(ea

µ) = √−gand Lmis the matter La-

grangian. In fact, one can extend Tto T+f(T), the so

called f(T)gravity. Moreover, it can be generalized to

become a general function of both the torsion scalar and

the trace of the energy-momentum tensor T, which re-

sults in the f(T,T)gravity.

The gravitational action for f(T,T)gravity is given by

S=1

16πGZd4x e[T+f(T,T)] + Zd4x e Lm(7)

Varying the action with respect to the vierbeins yields

the ﬁeld equations

(1+fT)he−1∂µ(ee α

aSλµ

α)−eα

aTµ

ναSνλ

µi+eλ

af+T

4+

fTT∂µT+fTT∂µTeα

aSλµ

α−fT





eα

α+pmeλ

2



=

4πGe α

α. (8)

where fT=∂f/∂T,fTT=∂2f/∂T∂T, and em

αis the

energy-momentum tensor.

We incorporate the ﬂat FRW metric as usual to apply

the aforementioned theory in a cosmological framework

to obtain modiﬁed Friedman equations. The FRW met-

ric read

ds2=dt2−a(t)2δijdxidxj, (9)

where a(t)is the scale factor. Further, (8) give rise to

modiﬁed Friedmann equations:

H2=8πG

3ρm−1

6f+12H2fT+fTρm+pm

3,

(10)

H=−4πG(ρm+pm)−˙

H(fT−12H2fTT)

−H(˙

ρm−3˙

pm)fTT−fTρm+pm

2. (11)

Here, T=ρm−3pmin the above equation is true for

the perfect matter ﬂuid.

Comparing the modiﬁed Friedmann equations (10)

and (11) to General Relativity equations

H2=8πG

3ρm+ρe f f , (12)

H=−4πGρm+pm+ρe f f +pe f f . (13)

we obtain

ρe f f =1

16πG[f+12 fTH2−2fT(ρm+pm)] (14)

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Squaredtorsionf(T,T)gravityanditscosmologicalimplicationsSimranArora,1,AaqidBhat,1,andP.K.Sahoo1,1DepartmentofMathematics,BirlaInstituteofTechnologyandScience-Pilani,HyderabadCampus,Hyderabad-500078,India.(Dated:November28,2022)WepresentthecouplingofthetorsionscalarTandthetraceofenergy-momentumte...

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Squared torsion fTTgravity and its cosmological implications Simran Arora 1Aaqid Bhat1 and P .K. Sahoo1 1Department of Mathematics Birla Institute of Technology and Science-Pilani

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