Noether Symmetry Approach in Scalar-Torsion fTϕGravity L.K. Duchaniya 1B. Mishra 1 and Jackson Levi Said2 3 1Department of Mathematics Birla Institute of Technology and Science-Pilani Hyderabad Campus Hyderabad-500078 India.

2025-05-02 0 0 357.52KB 13 页 10玖币

Noether Symmetry Approach in Scalar-Torsion f(T,ϕ)Gravity

L.K. Duchaniya ,1, ∗B. Mishra ,1, †and Jackson Levi Said 2, 3, ‡

1Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India.

2Institute of Space Sciences and Astronomy, University of Malta, Malta, MSD 2080.

3Department of Physics, University of Malta, Malta.

The Noether Symmetry approach is applied to study an extended teleparallel f(T,ϕ)gravity that

contains the torsion scalar Tand the scalar ﬁeld ϕin the context of an Friedmann-Lemaˆ

ıtre-Robertson-

Walker space-time. We investigate the Noether symmetry approach in f(T,ϕ)gravity formalism with

the speciﬁc form of f(T,ϕ)and analyze how to demonstrate a nontrivial Noether vector. The Noether

symmetry method is a helpful resource for generating models and ﬁnding out the exact solution of the

Lagrangian. In this article, we go through how the Noether symmetry approach enables us to deﬁne

the form of the function f(T,ϕ)and obtain exact cosmological solutions. We also ﬁnd the analytical

cosmological solutions to the ﬁeld equations, that is consistent with the Noether symmetry. Our

results demonstrate that the obtained solutions enable an accelerated expansion of the Universe. We

have also obtained the present value of the Hubble parameter, deceleration parameter, and effective

equation of state parameter, which is ﬁt in the range of current cosmological observations.

I. INTRODUCTION

General Relativity (GR) has gone through over a century of successfully describing the evolutionary processes

of the Universe in the form of the ΛCDM model [1–3], which is supported by overwhelming observational and

fundamental precision tests. This scenario predicts a Universe that drives the big bang through an inﬂationary

epoch and the well-known early Universe dynamics to eventually produce an accelerating late-time cosmology that

is sourced by dark energy [4,5]. ΛCDM describes dark energy through a cosmological constant Λwhich continues

to have fundamental problems associated with it [6–8] despite its observational successes. The next leading-order

contribution to this late-time cosmology is cold dark matter (CDM), which primarily acts on galactic scales. Despite

numerous decades-long efforts, this remains observational and undetected [9,10]. In the last few years, this has

become all the more dire with a new challenge coming from the observational sector, which is the suggestion of

tension in the value of the Hubble constant [11–13] as measured from local [14,15], early Universe sources [3,16].

This continues to seemingly increase as an observational tension in the data [17–19], and may permeate into other

sectors of cosmology [20,21].

One possible way to confront this problem is to consider even further modiﬁcations to the matter sector, which

would produce effective differences at particular epochs of the Universe, similar to inﬂation. However, another

approach is to reconsider the concordance model description of gravity through modiﬁcations to GR [2,22–24].

Recently, considerable work has gone into a new setting in which to consider gravitational interactions, namely

teleparallel gravity (TG). Here, the curvature associated with the Levi-Civita connection (˚

Γρ

µν, over-circles denote

any quantities calculated with the Levi-Civita connection) is exchanged with the torsion produced by the teleparallel

connection (Γρ

µν) [25–28]. This is a curvature-less connection that satisﬁes metricity. This means that all measures of

curvature will turn out to be identically zero, such as the Ricci scalar R(Γρ

µν) = 0. Saying that the regular Ricci scalar

remains nonzero in general ( ˚

R(˚

Γρ

µν)̸=0). TG can be used with regular GR to produce a torsion scalar T, equal to

the curvature-based Ricci scalar (up to a boundary term). Naturally, an action based on the torsion scalar will then

be dynamically equivalent to GR, and it is thus called the Teleparallel Equivalent of General Relativity (TEGR) since it

produces the same dynamical equations as that of the Einstein-Hilbert action.

Curvature-based modiﬁcations of GR have taken various forms over the years, with the most popular being f(˚

gravity [22,29,30]. Similarly, TEGR can be directly generalized to f(T)gravity [31–37]. f(T)gravity has the added

∗duchaniya98@gmail.com

†bivu@hyderabad.bits-pilani.ac.in

‡jackson.said@um.edu.mt

arXiv:2210.11944v3 [gr-qc] 18 Jul 2023

advantage that it is a second-order theory in terms of the derivatives that appears in the equations of motion. In

this context, it might also be interesting to add a scalar ﬁeld ϕto this general function since the resulting f(T,ϕ)

Lagrangian will continue to be second order in these derivatives [38–44]. This is the TG analog of f(R,ϕ)gravity

[45] with the important distinction that here all equations of motion are second-order in nature. This setting of

gravitational models has already been studied somewhat in works such as Refs. [46–49]. However, the Noether

symmetry considerations remain an open question for such classes of models.

In this work, we consider the Noether symmetry approach detailed in Refs. [50–55]. Through this approach,

we will study potential cosmological evolution scenarios produced by particular models of this class of theories.

Noether symmetries offer a tool to solve dynamical equations within cosmology, but more than that, it permits a

way to produce models that have some motivation from the fundamental sector. This provides better motivation

to study complex systems of equations of motion. Recently, Ref. [56] studied the full classiﬁcation of teleparallel

Horndeski scalar-tensor theories of cosmology stemming from Refs. [57,58]. This motivates us to analyze further

this particular subclass of models in which a simpler form of the scalar ﬁeld contribution is assumed.

We organize the work as follows; in Sec. II, we brieﬂy discuss the technical details of TG and its formulation of

f(T,ϕ)gravity, together with the formulation of the Friedmann equations for this setting. In Sec. III, we obtain

the point-like Lagrangian and derive the Noether equations using the Euler-Lagrangian equations in conﬁguration

space Q= (a,T,ϕ), leading to the cosmological equations of motion Eqs. (7–9). In Sec. IV, we introduce the concept

of Noether symmetries, leading to Sec. V, where these symmetries are studied for the present case. By determining

the Noether vector for a speciﬁc form of f(T,ϕ), we also determine the exact solutions of the cosmological ﬁeld

equations in Sec. VI. Finally, we conclude with a summary of the main results in Sec. VII.

II. SCALAR-TORSION f(T,ϕ)GRAVITY

Replacing the metric tensor as the fundamental variable with tetrad eAµ(inverse represented by Eµ

A) ﬁelds to-

gether with a spin connection ωabµas the dynamical variable, GR can be reformulated in the context of TG. The

tetrad ﬁeld eAµ(where Latin indices take on the values A=0, 1, 2, 3 refer to coordinates on the tangent space) relates

local Lorentz frames with the general spacetime manifold coordinates, which are denoted by Greek indices. The

metric can then be built as

gµν =ηABeAµeBν, (1)

where ηAB represents the Minkowski metric. The tetrad must also meet the requirements of orthogonality

Eµ

AeBµ=δB

A. Using the tetrad, the Levi-Civita connection can be substituted by the torsion-ful teleparallel con-

nection, given by [59]

Γσνµ :=Eσ

A∂µeAν+ωABµeB

ν, (2)

where the spin connection acts to retain the local Lorentz invariance of the ensuing ﬁeld equations, for a particular

frame, called the Weitzenb¨

ock gauge, these components vanish identically. Using this connection, an analog of the

Riemann tensor, which vanishes for the teleparallel connection, can be deﬁned as an anti-symmetric operator on this

connection through [60]

Tσµν :=2Γσ[νµ]. (3)

Using this torsion tensor, the torsion scalar can be deﬁned as [25–28]

T:=1

4Tαµν Tµν

α+1

2Tαµν Tνµ

α−Tαµα Tβµ

β, (4)

which is derived in such a way to be equivalent to the regular curvature-based Ricci scalar (up to a boundary

term). This means that TEGR will be deﬁned by an action based on the linear form of T.

TEGR can be directly modiﬁed to our scalar-tensor form by generalizing it to the action [44]

S=Zd4xe[f(T,ϕ) + P(ϕ)X] + Sm, (5)

where f(T,ϕ)is an arbitrary function of the torsion scalar Tand the scalar ﬁeld ϕ, and X=−∂µϕ∂µϕ/2. This

broad action includes non-minimally coupled scalar-torsion gravity models with f(T,ϕ)coupling function, f(T)

gravity, and a minimally coupled scalar ﬁeld. Here we assume geometric units, and write the tetrad determinant as

e=det[eAµ] = √−g

we consider the homogeneous and isotropic ﬂat Friedmann-Lemaˆ

ıtre-Robertson-Walker (FLRW) geometry in or-

der to proceed to the cosmological application of f(T,ϕ)gravity.

ds2=−dt2+a2(t)δµνdxµdxν, (6)

where a(t)is the scale factor that represents the expansion in the spatial directions. The tetrad, eAµ=diag(1, a(t),

a(t),a(t)). From Eq. (4), the torsion scalar becomes, T=6H2. Varying the action in Eq. (5) with respect to the

tetrad ﬁeld and the scalar ﬁeld ϕ, the ﬁeld equations of f(T,ϕ)gravity can be obtained along with the Klein-Gordon

equation as,

f(T,ϕ)−P(ϕ)X−2T f,T=ρm, (7)

f(T,ϕ) + P(ϕ)X−2T f,T−4˙

H f,T−4H f,T=−pm, (8)

−P,ϕX−3P(ϕ)H˙

ϕ−P(ϕ)¨

ϕ+f,ϕ=0 , (9)

where H=˙

ais the Hubble rate, and an over dot denotes the derivative with respect to cosmic time t. A comma

indicates the derivative for Tor ϕ. The functions pmand ρmrepresent the pressure and energy density of matter

respectively. One can refer the Friedmann equations and Klein-Gordon equation of f(T,ϕ)gravity in Refs. [46–49].

III. LAGRANGIAN FORMALISM OF f(T,ϕ)THEORY

The Lagrangian formalism of f(T,ϕ)theory has been formulated in this section. The point-like Lagrangian is

useful in the analysis of Noether symmetry, which deals with the Friedmann equations, and can be derived from

Eq. (5) or followed from Ref. [53]. One can establish a Canonical Lagrangian L=L(a,˙

a,T,˙

T,ϕ,˙

ϕ)to deduce the

cosmological equations in the FLRW metric, whereas Q= (a,T,ϕ)is the conﬁguration space from which it is possible

to derive the tangent space denoted by T Q and can be obtained as T Q = (a,˙

a,T,˙

T,ϕ,˙

ϕ), the corresponding tangent

space on which Lis deﬁned as an application. Here, the scale factor a(t), torsion scalar T, and scalar ﬁeld ϕ(t)are

taken as independent dynamical variables of the FLRW metric. One can use the method of Lagrange multipliers to

set T−6˙

a2=0 as a constraint of the dynamics and integrating by parts, the Lagrangian Lbecomes analogous to

Ref. [53], and so we obtain

S=2π2Za3

f(T,ϕ) + P(ϕ)˙

ϕ2

2−λ T−6˙

a2!−ρm0

a3

dt , (10)

where λis a Lagrange multiplier and ρm0is the matter energy density at present time. By varying this action in

Eq. (10) with respect to T, we get

λ=f,T(T,ϕ). (11)

Thus, the action in Eq. (10) can be written as

S=2π2Za3

f(T,ϕ) + P(ϕ)˙

ϕ2

2−f,T T−6˙

a2!−ρm0

a3

dt , (12)

and the point-like Lagrangian is

L(a,˙

a,T,˙

T,ϕ,˙

ϕ) = a3 f(T,ϕ) + P(ϕ)˙

ϕ2

2−T f,T(T,ϕ)!+6a˙

a2f,T(T,ϕ)−ρm0. (13)

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NoetherSymmetryApproachinScalar-Torsionf(T,ϕ)GravityL.K.Duchaniya,1,∗B.Mishra,1,†andJacksonLeviSaid2,3,‡1DepartmentofMathematics,BirlaInstituteofTechnologyandScience-Pilani,HyderabadCampus,Hyderabad-500078,India.2InstituteofSpaceSciencesandAstronomy,UniversityofMalta,Malta,MSD2080.3DepartmentofPhysi...

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Noether Symmetry Approach in Scalar-Torsion fTϕGravity L.K. Duchaniya 1B. Mishra 1 and Jackson Levi Said2 3 1Department of Mathematics Birla Institute of Technology and Science-Pilani Hyderabad Campus Hyderabad-500078 India.

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