ЖЭТФ, 2023, том 163, вып. 4, стр. 496-502

NOETHER SYMMETRIES AND SOME EXACT SOLUTIONS IN

f (R, T²) THEORY

M. Sharif^a*, M. Z. Gul^a**

^a Department of Mathematics and Statistics, The University of Lahore

54000, Lahore, Pakistan

Received September 30, 2022,

revised version October 09, 2022

Accepted for publication October 12, 2022

DOI: 10.31857/S0044451023040065

to mention here that this theory explains the complete

EDN: LRECZQ

cosmic history and the cosmic evolution. Roshan and

Shojai [4] examined that EMSG resolves the primor-

dial singularity as it has bounce in the early universe.

Abstract. The main objective of this article is to

Board and Barrow [5] used a specific model of this the-

examine some physically viable solutions through the

ory and discussed exact solution, singularities as well

Noether symmetry technique in f(R, T²) theory. In or-

as cosmic evolution with the isotropic configuration of

der to investigate Noether equations, symmetry gener-

matter in this theory. Bahamonde et al [6] studied

ators and conserved quantities, we use a specific model

various EMSG models and analyzed that these models

of this modified theory. We find exact solutions and ex-

manifest the current cosmic evolution and acceleration.

amine the behavior of various cosmological quantities.

We have examined some physically viable solutions [7]

It is found the behavior these quantities is consistent

and dynamics of celestial objects in this theory [8].

with current observations indicating that this theory

describes the cosmic accelerated expansion. We con-

The Noether symmetry (NS) strategy gives a fasci-

clude that generators of Noether symmetry and con-

nating method to develop new cosmic models and as-

served quantities exist in this theory.

sociated structures in modified theories of gravity. The

1. Introduction. The current cosmic expansion

NS approach is significant as it recovers symmetry gen-

has been the most stunning and dazzling result for the

erators as well as some conservation laws of the system

scientific community [1]. Although general relativity

[9]. This method does not deal only with the dynamical

(GR) is a widely accepted theory which explains the

solutions but it also provides some viable conditions to

cause of this expansion but it has some issues like coin-

select cosmic models based on recent observations [10].

cidence and fine tuning problems. To addresses these

Moreover, this method is an important and useful tech-

issues, several modifications of GR (modified gravita-

nique to examine exact solutions by using conserved

tional theories) have been formulated to unveil the cos-

values of the system. Conservation laws are the main

mic mysteries. The first modification of GR is f(R)

ingredients to analyze the distinct physical phenomena.

theory and significant literature [2] is available to un-

These are the particular cases of the Noether theorem,

derstand the physical features of this theory. Recently,

according to which every differentiable symmetry pro-

Katirci and Kavuk [3] modified f(R) theory by intro-

duces conservation laws. The conservation laws of lin-

ducing a non-linear term (T² = T_ξηT^ξη) in the func-

ear and angular momentum govern the translational

tional action referred to as f(R, T²) theory.

and rotational symmetry of any object. The Noether

charges are important in the literature as they are used

This proposal is also dubbed as energy-momentum

to examine various major cosmic problems in various

squared gravity (EMSG) and contains higher-order

considerations [12-21].

matter source terms which are helpful to analyze var-

ious interesting cosmological results. It is worthwhile

This manuscript investigates the NS for anisotropic

and homogenous cosmic models such as BT-I, BT-

^* E-mail: msharif.math@pu.edu.pk

III and Kantowski-Sachs (KS) in the background of

^** E-mail: mzeeshangul.math@gmail.com

EMSG. The manuscript is organized as follows. Section

496

ЖЭТФ, том 163, вып. 4, 2023

Noether Symmetries and Some Exact Solutions in f(R, T²) Theory

Fig. 1. Behavior of effective energy density for ϵ = -1 (green

line) and ϵ = 1 (red line)

2 studies the basic formalism of EMSG. Section 3 pro-

vides a detailed study of the NS approach and derives

exact cosmological solutions which are then discussed

through graphs. The summary of the consequences is

given in sect. 4.

2. Field Equations. We derive the field equations

of the homogeneous and anisotropic spacetime in this

section. The action of EMSG is expressed as [3]

∫ (

)

f (R, T²)

+L_m d⁴x√-g,

(1)

2κ²

where κ² = 1 and L_m manifest the coupling constant

and Lagrangian of matter, respectively. The corre-

sponding equations of motion are obtained as

R_ξηf_R + g_ξη□f_R - ∇_ξ∇_ηf_R -

g_ξηf =

Fig. 2. Behavior of effective pressure (upper pannel) and de-

=T_ξη -Θ_ξηf_T2,

(2)

celeration parameter (lower pannel) for ϵ = -1 (green line),

ϵ = 1 (red line) and ϵ = 0 (orange line)

where

∂f

□=∇_ξ∇^ξ, f_T2 =

f_R =

Rearranging Eq.(2), we have

∂T²

∂R

and

(D)

G_ξη =

+T_ξη)=T^effξη,

(3)

ξη

f_R

Θ_ξη = -2L_m(T_ξη -

g_ξηT) -

where

∂²Lm

-4

T^αβ - TT_ξη + 2T^αξT_ηα.

∂g^ξη∂g^αβ

T_ξη = (ρ + p)U_ξU_η + pg_ξη

497

ЖЭТФ, том 163, вып. 4, 2023

Noether Symmetries and Some Exact Solutions in f(R, T²) Theory

For ϵ = 0, -1, 1, the BT-I, BT-III and KS cosmic mod-

(2äa^-1 - 4bb^-1)f_RR + (3p² + ρ²)f_RT 2 -

els are obtained. The resulting equations of motion

- 2ϵb^-1f_RR - Rf_RR - T²f_RT2 -

become

- (4 ˙aba^-1b^-1 + 2b2b^-2

)f_RR = 0,

(13)

ρ^eff =

[ρ -

f + (3p² + ρ² + 4pρ)f_T2 +

f_R

(2äa^-1 - 4bb^-1)f_RT 2 + (3p² + ρ²)f_T 2_T 2 -

+ ϵb^-2f_R - (˙aa^-1 + 2bb^-1)(R˙ f_RR +T˙²f_RT2) +

- 2ϵb^-1f_RT2 - Rf_RT2 - T²f_T2_T2 -

+ (äa^-1 + 2bb^-1 + 2˙aba^-1b^-1 +

b²b^-2)f_R],

(6)

- (4 ˙aba^-1b^-1 + 2b2b^-2)f_RT 2 = 0.

(14)

We formulate the Hamiltonian to examine the total

p^eff =

[p +

f + 2bb^-1( Rf_RR +T˙²f_RT2) -

f_R

energy of the system as

− ϵb^-2f_R +

Rf_RR +

T²f_RT2 -

E = -ab²(f - Rf_R - T²f_T2 + (3p² + ρ²)f_T2) -

− (äa^-1 + 2bb^-1 + 2˙aba^-1b^-1 +

b²b^-2)f_R +

- (2 ˙abba^-1 +

b²) × 2af_R - ϵf_R - ab²p -

+R˙²f_RRR +

²f_RT2_T2 + 2

R TfRRT2],

(7)

- (2b² a + 4abb)f˙R.

(15)

p^eff =

[p+

f+(aa^-1+

bb^-1)(R˙ f_RR +T˙²f_RT2 )-

The dynamical equations (11)-(14) are extremely com-

f_R

plex due to multivariate functions and their derivatives.

- ϵb^-2f_R +

T²f_RT2 +

Rf_RR -

In the next section, we use NS technique to identify

− (äa^-1 + 2bb^-1 + 2˙aba^-1b^-1 +

b²b^-2)f_R +

exact solutions. Although this theory is not conserved

but one can obtain conserved values through NS ap-

+R˙²f_RRR +

²f_RT2_T2 + 2

R TfRRT2].

(8)

proach, which are then used to examine the mysterious

universe. As a result, this strategy is more intriguing

Now, we apply Lagrange multiplier method to formu-

and we adopt it in this article.

late the Lagrangian as

3. Noether Symmetries in EMSG. This section

L = ab²(f - Rf_R - T²f_T2 + (3p² + ρ²)f_T2 + p) -

formulates the Noether equations for the homogenous

and anisotropic universe model in EMSG. The symme-

- 2a(2˙abba^-1 +

b² - ϵ)f_R -

try generators are expressed as

− (2b² a + 4abb)R˙ f_RR - (2b² a + 4abb)T˙²f_RT 2 .

(9)

∂

Y = λ(t,qⁱ)

+ Υ^j(t,qⁱ)

i = 1,2,3,...,n,

The fundamental properties of the system can be ex-

∂t

∂q^j

plained using the Hamiltonian (E) and the dynamical

where λ(t, a, b, R, T²) and Υ^j (t, a, b, R, T²) are the un-

equations, determined as

known parameters. The Lagrangian must satisfy the

∂L

invariance constraint, expressed as

(

)=0, E=

qⁱ(

) - L,

(10)

∂qⁱ

qⁱ

∂

Y [1]L + (Dλ)L = DΩ, Y ^[1] = Y +

(16)

where generalized coordinates are denoted by qⁱ. The

qⁱ

resulting dynamical equations are

where Ω is the boundary term and

f - Rf_R - T²f_T2 + (3p² + ρ²)f_T2 + p +

∂

qⁱ

∂t

∂qⁱ

+4bb^-1(R˙ f_RR +f_RT 2 T˙²)+a(f_T 2 (6pp_,a +2ρρ_,a)+p_,a)+

+b^-2(2b2+4bb+2ϵ)f_R+2 Rf_RR+2

T²f_RT2 +2

R²f_RRR+

defines the total derivative. The corresponding integral

integral of motion is expressed as

²f_RT2_T2 + 4

RT²f_RRT2 = 0,

(11)

∂L

I =Υ

- λE - Ω.

(17)

qⁱ

f - Rf_R - T²f_T2 + (3p² + ρ²)f_T2 + (

Rf_RR +

+T˙²f_RT²)4˙aa^-1 + p + b(f_T2(6pp_,b + 2ρρ_,b) + p_,b) +

This is a crucial component of NS that is essential for

computing viable solutions and is also named as the

+ 2a^-1b^-1(äb + ˙ab +ba)f_R + 2

R +

conserved quantities.

+ 4bb^-1(R˙ f_RRT 2 + T˙²f_RT 2_T 2 ) = 0,

(12)

We take the vector field (Y ) with configuration

space Q = (t, a, b, R, T²) to examine the generators

499

M. Sharif, M. Z. Gul

ЖЭТФ, том 163, вып. 4, 2023

with corresponding first integrals of Lagrangian (9) un-

2bΥ¹f_RT2 + 2aΥ²f_RT2 + 2abΥ³f_RRT2 +

der invariance condition (16). By comparing the coef-

+2abΥ⁴f_RT2_T2 +b²Υ^1,bf_RT2 +2bΥ^1,T2fR+2abΥ,bfRT2 +

ficients of Eq.(16), we have

RT²

+ 2aΥ^2,T2fR + 2abΥ^,T2fRR + 2abΥ^,T2f

2b²Υ^1,tf_RT2 + 4abΥ^2,tf_RT2 + Ω_,T2 = 0,

(18)

- 2abλ_,tf_RT2 = 0,

(30)

2b²Υ^1,tf_RR + 4abΥ^2,tf_RR + Ω_,R = 0, λ_,af_R = 0, (19)

λ_,Rf_RT2 = 0, λ_,T2 f_RR = 0,

bΥ^1,T2 fRR + bΥ,RfRT2 + 2aΥ,T2fRR +

b²Υ¹[f - Rf_R - T²f_T2 + (3p² + ρ²)f_T2 + p +

+ 2aΥ^2,Rf_RT2 = 0,

(20)

+ a((6pp_,a + 2ρρ_,a)f_T2 + p_,a) + 2ϵf_R] +

4bΥ^1,tf_R + 4aΥ^2,tf_R + 4abΥ^3,tf_RR +

+ Υ²[2ab(f - Rf_R - T²f_T2 + (3p² + ρ²)f_T2 + p) +

+ 4abΥ^4,tf_RT2 + Ω_,b = 0,

(21)

+ ab²((6pp_,b + 2ρρ_,b)f_T2 + p_,b)] + (3p² + ρ²)f_RT2 ) +

+Υ³[-ab²(Rf_RR-T²f_RT2+(3p²+ρ²)f_T2_T2)+2aϵf_RR]+

4bΥ^2,tf_R + 2b²Υ^3,tf_RR + 2b²Υ^4,tf_RT2 + Ω_,a = 0,

(22)

+ Υ⁴[-ab²(Rf_RT2 - T²f_T2_T2 -

λ_,bf_R = 0,

- (3p² + ρ²)f_T 2_T 2 ) + 2aϵf_RT 2 ] +

bΥ^1,T2 fRR + 2abΥ^,T2fRT2 = 0,

(23)

+ λ_,t[ab²(f - Rf_R - T²f_T2 + 2ϵf_R +

λ_,Rf_RR = 0, λ_,T2 f_RT2 = 0,

+ (3p² + ρ²)f_T 2 + p)] - Ω_,t = 0.

(31)

bΥ^1,Rf_RR + 2abΥ^2,Rf_RR = 0,

(24)

These equations help to study the dark cosmos in the

2Υ^2,af_R + bΥ^3,af_RR + bΥ^4,af_RT2 = 0,

context of f(R, T²). We solve the above system to ob-

tain exact solutions for specific f(R, T²) model in the

Υ¹f_R + aΥ³f_RR + aΥ⁴f_RT2 + 2bΥ^1,bf_R +

following section.

+ 2abΥ^4,bf_RT2 + 2abΥ^3,bf_RR - aλ_,tf_R +

3.1. Exact Solutions. Here, we formulate the

+ 2aΥ^2,bf_R = 0,

(25)

generators of NS, conserved values of the system and

corresponding physical solutions. Due to the above sys-

λ_,af_RR = 0, λ_,af_RT2 = 0,

tem’s complexity and nonlinearity, we assume a partic-

ular EMSG model f(R, T²) = R + T² which minimizes

2Υ^2,Rf_R + bΥ^3,Rf_RR + bΥ^4,Rf_RT2 - bλ_,tf_RR +

the system complexity and help to examine the exact

solutions. Manipulating Eqs.(18)-(31), we obtain

+ 2Υ²f_RR + bΥ^1,af_RR + bΥ³f_RRR +

+ bΥ⁴f_RRT2 + 2aΥ^2,af_RR = 0,

(26)

1 aF₂(t)

F₄(t)

c₂

Υ¹ =

aF˙₁(t) - 2c₁a -

√

√ ,

λ_,bf_RR = 0,

F₂(t)

Υ² =

√

1(t) + c1)b, λ = F1(t),

2Υ²f_RT2 - bλ_,tf_RT2 + bΥ^3,T2 fRR +

√

Ψ=-

ab²F¨₁(t) - 4a

bF˙₂(t) + F₃(t) +

4(t)b² ,

+ bΥ^1,af_RT2 + bΥ^4,T2 f_RT2 + 2Υ^2,T2fR + bΥ⁴fRT2T2 +

√

+ 2aΥ^2,af_RT2 + bΥ³f_RRT2 = 0,

(27)

3c₁(-3c₁a²ϵ - 6c₁aϵ - 2c₃at)

ρ=

(32)

3c₁ba

λ_,bf_RT2 = 0,

where c_i(i = 1, ..., 5) are integration constants with

c₁ = 0. The corresponding symmetry generators be-

2Υ²f_R+2bΥ³f_RR +2bΥ⁴f_RT2 +2bΥ^1,af_R+2aΥ^2,af_R+

come

+ 2abΥ^3,af_RR + 2bΥ^2,T2fR + b²Υ^,bfRR + 2abΥ^,afRT2 +

∂

+ b²Υ^4,bf_RT2 - 2bλ_,tf_R = 0,

(28)

Y₁ = -3t

Y₂ = -3a

∂t

∂a

2bΥ¹f_RR + 2aΥ²f_RR + 2abΥ³f_RRR + 2abΥ⁴f_RRT2 +

Substituting the value of Lagrangian (9), Hamiltonian

(15) and above solutions (32) in Eq.(17), we obtain first

+ b²Υ^1,bf_RR + 2bΥ^1,Rf_R + 2abΥ^2,bf_RR + 2aΥ^2,Rf_R +

integral as

+ 2abΥ^3,Rf_RR + 2abΥ^4,Rf_RT2 - 2abλ_,tf_RR = 0,

(29)

[3c1a2ϵ + 6c1aϵ + 2c3at

λ_,Rf_R = 0, λ_,T2 f_R = 0,

I = 12abbc₁ + 3

3c₁a

500

ЖЭТФ, том 163, вып. 4, 2023

Noether Symmetries and Some Exact Solutions in f(R, T²) Theory

]

are negative for BT-III universe model which support

- 2ab2 - 4˙abb + 2

c₁t - c₃t².

b²

the current cosmic acceleration. Figure 3 determines

that r - s and EoS parameters describe quintessence

By comparing the coefficients of c₁ and c₃, we have

and phantom phases of DE which represent the cos-

(

2ϵ)

mic expansion. The obtained solutions for ϵ = -1 are

I₁ = t², I₂ = 12abb + 3t ϵa - 2ab2 - 4abb +

b²

consistent with recent observations which indicate that

this theory demonstrates expansion of the universe.

We substitute Eqs.(32) into dynamical equations (11)-

(15) and obtain the exact solution as

The total amount of energy density can be ex-

(

pressed as fractional energy density. The fractional

a(t) =

6c₅c₃(c₅ + t)³ - 15c₁ϵ(c₅ + t)³ -

density is defined as

)(

)_-1

- 4c₃t(c₅ + t)³ + 60c₄c₁ϵ

60c₁ϵ(c₅ + t)³

1+Ω_σ =Ω_m +Ω_Λ,

where

1√

b(t) =

-6ϵ(c₅ + t).

(33)

ρ_Λ

Ω_m =

Ω_Λ =

Ω_σ =

3H²

To analyze this solution, we study the graphical be-

The evaluation of fractional densities corresponding

havior of some important cosmological parameters like

to ordinary matter (Ω_m) and dark energy (Ω_Λ) plays

deceleration and r - s parameters that are the major

a vital role to measure the contribution of these ele-

factors in the field of cosmology. These cosmic param-

ments in the cosmos. The densities for isotropic uni-

eters for anisotropic and homogeneous universe model

verse model defined as Ω_m +Ω_Λ = 1 whereas expression

are defined as

equality becomes Ω_m +Ω_Λ = 1+Ω_σ for anisotropic uni-

verse model. We analyze the behavior of fractional den-

H =

(

q=-

- 1.

H²

sities corresponding to matter and dark energy graph-

-1

ically at redshift scale factor where a(t) = a₀(1 + z)

The pair of r - s parameters constructs a relationship

and z is the redshift parameter. From observations of

between formulated and standard models of the uni-

Planck data 2018, it is suggested that Ω_m = 0.3111 and

verse which is used to examine the characteristics of

Ω_Λ

= 0.6889. According to some recent observations,

DE, expressed as

there are some evidences in favor of closed universe

r-1

model with fractional density Ω_Λ

= 0.01. For ε = -1,

r = q + 2q² -

3(q -¹² )

the fractional density of matter indicates inconsistent

behavior and the trajectory of fractional density pro-

For (r, s) = (1, 0), the constructed model corresponds

vides Ω_m = 0.3 for ε = 1 as shown in Figure 4 (left

to ΛCDM model whereas quintessence and phantom

plot). In this regard, it implies consistent behavior with

DE eras are obtained for s > 0 and r < 1, respectively.

Planck data 2018. The right plot of Figure 4 reveals

The EoS parameter

the behavior of fractional density of dark energy which

shows consistent behavior with Plank data for ε = 1

eff

ω^eff =

and it exhibits inconsistent behavior for ε = -1.

ρ^eff

4. Final Remarks. Modified theories are as-

is a dimensionless quantity that determines the correla-

sumed as the most propitious and elegant proposals to

tion between state parameters. This parameter differ-

examine the dark universe due to the presence of extra

entiates the DE era into quintessence (-1 < ω ≤ -1/3)

higher-order geometric terms. In this paper, we have

and phantom (ω < -1) phases.

formulated exact solutions of anisotropic and homoge-

We have considered the values of integration con-

neous spacetimes in f(R, T²) gravity. For this reason,

stants as c₁ = -2, c₃ = -10, c₄ = 10 and c₅ = 5.7

we have considered the NS technique to examine the

to analyze the graphical behavior of physical quanti-

exact solutions. We have formulated the Lagrangian,

ties. Figure 1 shows that the effective energy density is

NS generators with conserved values in the background

positively increasing for ϵ = -1 which manifests that

of EMSG. The behavior of exact solutions have been

our universe is in the expansion phase. Figure 2 shows

investigated through different cosmological quantities.

that the effective pressure and deceleration parameter

501

M. Sharif, M. Z. Gul

ЖЭТФ, том 163, вып. 4, 2023

The main findings are summarized as follows.

S. Bahamonde, M. Marciu, and P. Rudra, Phys.

1. We have established two non-zero NS genera-

Rev. D 100, 083511 (2019).

tors and corresponding conserved quantities. We have

M. Sharif and M.Z. Gul, Phys. Scr. 96, 025002

obtained the exact solutions for BT-I, BT-III and KS

(2021); Phys. Scr. 96, 125007 (2021); Chin. J.

universe models.

Phys. 80, 58 (2022).

2. The effective energy density show accelerated

and constant expansion corresponding to BT-III, BT-I

M. Sharif and M.Z. Gul, Int. J. Mod. Phys. A 36,

and KS spacetimes, respectively (Figure 1).

2150004 (2021); Universe 7, 154 (2021); Int. J.

3. The value of effective pressure and deceleration

Geom. Methods Mod. Phys. 19, 2250012 (2021);

parameter remain negative for ϵ = -1 which support

Chin. J. Phys. 71, 365 (2021); Mod. Phys. Lett. A

the current cosmic acceleration (Figure 2).

37, 2250005 (2022).

4. The r-s and EoS parameters yield quintessence

and phantom DE phases which determine the rapid ex-

E. Noether, Tramp. Th. Stat, Phys 1, 189 (1918);

pansion of the universe (Figure 3).

T. Feroze, F.M. Mahomed, and A. Qadir, Nonlin-

5. In the background of BT-III universe models,

ear Dyn. 45, 65 (2006).

the analysis of fractional density parameter of matter

10.

S. Capozziello, M. De Laurentis, and S.D.

reveals that the EMSG is consistent with Plancks 2018

Odintsov, Eur. Phys. J. C 72, 1434 (2012).

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not preserved (Figure 4). We conclude that the EMSG

11.

S. Capozziello, R.D. Ritis, and A.A. Marino, Class.

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Quantum Gravity 14, 3259 (1997).

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6. We find that first integrals of motion are very

12.

S. Capozziello, G. Marmo, and C.P. Rubano, Int.

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13.

A.K. Sanyal, Phys. Lett. B 524, 177 (2002).

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