SOME EXACT ANISOTROPIC SOLUTIONS VIA NOETHER SYMMETRY IN f(R) GRAVITY
M. Sharif* I. Nawazish**
Department of Mathematics, University of the Punjab Lahore-54590, Pakistan
Received June 16, 2014
We attempt to find exact solutions of the Bianchi I model in f(R) gravity using the Noether symmetry approach. For this purpose, we take a perfect fluid and formulate conserved quantities for the power-law f(R) model. We discuss some cosmological parameters for the resulting solution which are responsible for expanding behavior of the universe. We also explore Noether gauge symmetry and the corresponding conserved quantity. It is concluded that symmetry generators as well as conserved quantities exist in all cases and the behavior of cosmological parameters shows consistency with recent observational data.
DOI: 10.7868/S0044451015010046
1. INTRODUCTION
There is interest in investigating nonlinear higher-order differential equations whose exact solutions cannot be determined from well-known methods. This problem is resolved by Lie's theory, which helps not only to find exact solutions but also to explore new solutions by applying different transformations. The most interesting aspect of this theory is Noether symmetry, which is used to obtain analytical solutions as well as the corresponding conserved quantities. Different methods have been introduced to establish conservation laws, like the multiplier approach fl] and the partial Noether theorem for variational and 11011-variational problems [2]. Some authors [3] proposed computer packages to construct conserved quantities. Cheviakov [4] introduced Maple code to formulate conservation laws by using the multiplier approach.
The accelerated expansion of the universe is widely-discussed by modified theories of gravity such as the f(R) gravity, f(T) gravity (T is the torsion), modified Gauss-Bonnet gravity, f(R,T) gravity (T is trace of the energy momentum tensor), scalar-telisor theories, etc. The f(R) gravity is the simplest modification of general relativity, where the Ricci scalar R. is reE-mail: msharif.math'fflpu.edu.pk E-mail: iqranawazishOT'fflgmail.com
placed by an arbitrary function f(R). The field equations of f(R) gravity are fourth-order nonlinear partial differential equations whose exact solution can be found via the Noether symmetry approach. Different authors [5] elaborate a comprehensive review of f(R) gravity. Starobinsky [6] discussed the stability criteria of some f(R) models.
Observations of the CMBR and experimental data such as WMAP and Planck satellites indicate that the present universe is isotropic and largely homogeneous. This stage of the universe is described by the FRW model, which ignores all structure of the universe and observed anisotropy in the CMB temperature. However, the early stages of the universe are found to be spatially homogeneous as well as anisotropic. The anisotropy is still found in the present universe as the CMB temperature and to discuss this anisotropy, we consider the simplest anisotropic model, i.e., Bianchi type cosmological homogeneous models. These models describe the anisotropy effect in the early universe on present-day observations. Many authors have discussed these models from different standpoints. Akarsu and Kilinc [7] explored the Bianchi type I (BI) model which yields de Sitter volumetric expansion due to a constant effective energy density for anisotropic fluid along with an anisotropic equation of state (EoS) parameter. Yadav and Salia [8] investigated a locally rota-tionally symmetric (LRS) BI anisotropic cosmological model with dominance of dark energy for the condition ,4 = Bm. They found that the anisotropic distribution
of dark energy leads to the present accelerated expansion of the universe.
The symmetry approximation is extensively used to study exact solutions under different scenarios, which are then used to discuss cosmic aspects. Sharif and Waheed studied the Bardeen model [9] and stringy charged black holes [10] via approximate symmetry. The same authors [11] introduced curvature correction terms in a scalar tensor theory to explore Noether symmetries for FRW and LRS BI universe models. Aslam et al. [12] found maximum Noether symmetries for the BI universe in f(T) gravity. Ivucukakca [13] formulated exact solutions for a flat FRW universe in the scalar tensor theory nonminimally coupled to torsion scalar via Noether symmetry. Aslam et al. [14] investigated Noether gauge symmetry in f(T) gravity minimally coupled to a scalar field. Jamil et al. [15] explored Noether gauge symmetry in the Saez Ballester scalar tensor theory for the BI model. Ivucukakca et al. [16] found exact solutions by using the Noether symmetry approach for the LRS BI universe.
Many authors explored Noether symmetry in f(R) gravity. Capozziello et al. [17] investigated some exact spherically symmetric solutions with the help of Noether symmetry in f(R) gravity. Vakili [18] studied Noether symmetry for a flat FRW universe and discussed the effective EoS parameter for the quintessence phase. Jamil et al. [19] found Noether symmetry of the tachyon model for a flat FRW metric and discussed cosmic evolution via a power-law model. Hussain et al. [20] explored Noether gauge symmetry for a flat FRW spacetime which yields zero gauge term. They also checked the stability criteria for the power-law f(R) model. Shamir et al. [21] calculated a nonva-nishing gauge term for the same model and also discussed Noether gauge symmetry for the static spherically symmetric model. Ivucukakca and Camci [22] explored Noether gauge symmetry for the FRW universe in the Palatini f(R) gravity.
In this paper, we explore Noether and Noether gauge symmetries for an LRS BI universe model in f(R) gravity. We discuss some cosmological parameters to elaborate accelerated expansion of the universe. The paper is organized as follows. In Sec. 2, we provide a general formalism of Noether and Noether gauge symmetries and field equations of f(R) gravity. Section 3 is devoted to exact solutions, the Noether symmetry generator and corresponding conserved quantities, while Sec. 4 formulates symmetry generator via Noether gauge symmetry. In the last section, we summarize the results.
2. NOETHER SYMMETRY AND f(R) GRAVITY
In this section, we briefly discuss Noether and Noether gauge symmetry and f(R) field equations for the LRS BI universe model. Noether symmetry is obtained when a Lie derivative of the Lagrangian vanishes, while Noether gauge symmetry is its generalized form with a nonzero gauge term. The Noether theorem describes a strong connection between symmetries and conservation laws. We consider a point transformation that depends on an infinitesimal parameter A, i.e., Q* = Ql{qi, A) and generates a one-parameter Lie group. The vector field
A' = /?*(</
d_
dq*
A
dq* '
is referred to as Noether symmetry if the Lagrangian remains invariant, i.e., L_\-£ = 0. For Noether gauge symmetry, the vector field is defined as
where a:, 3*, and 7J are unknown functions and the dot represents the time derivative. This field yields Noether gauge symmetry if the Lagrangian satisfies the equation
A (£>«)£ = DG(t,q').
Here, G(t, q' ) represents the gauge term, D and A'W are the total derivative operator and the first-order prolongation given by
¡A
dq*
xw = x
(7i
VJ
1'
dq*
The conserved quantity corresponding to A' is defined
I = G — aC — (y3 — q3 a)
ÔC
dqi
For Noether symmetry, the gauge term vanishes and the conserved quantity takes the form
I = —y
.¡de
dqi
For a dynamical system, the Euler Lagrange equation and the associated energy function are defined as
- !L (= q V" (-j dC- £ = E
Oq* (It \dq* ' ft"*
dq*
The action of f(R) gravity is given by
A = hl ^^(/W + ^Gw^K (i)
where / is an arbitrary differentiable function of the scalar curvature and £m is the matter Lagrangian. Using metric variation of this action, we obtain the corresponding field equations as
/ii-ft/if - ^fStii' ~ ^i^i'fn + iJi-ti'Dfn = KTtll,. (2)
Here, fji = df/dR, represents the covariant derivative, and □ = V^V. In terms of the Einstein tensor, Eq. (2) takes the form
(3)
where the effective stress energy tensor is
G<x¿ — —(Ta¡:i +
T^ = ± (^-^.9^+VaV^/fl-D/^ ) . (4)
This contains such ingredients that are required to deal with accelerated expansion of the universe, referred to as dark source terms.
The LRS BI universe model is given as
d.s2 = dt'2 - A2(t) dx2 - B2(t)(dy2 + dz
(5)
where the scale factors ,4 and B are functions of time. In order to calculate the Lagrangian, we can write the action as
A = / [AD2i - \(R - R) + Cm]dt,
(6)
where the dynamical constraint R, the Lagrangian multiplier \. and the matter part of the Lagrangian Cm (taking perfect fluid in a matter-dominated universe) are
R =
AD■■
■(ÂD2+2ADD+2DÀÈ+AD2), (7)
\ = AD2f, Cm= PoÍAD2)-1. (8)
The Lagrangian corrcsponding to tlie action becomes
a.A.RH.Á.B.Ín = AD2(f - Rf)+4BÁBf+ +2AB2f'+2B2ÁRf"+4ABBRf"+po(AB2)~1, (9)
wliere tlie prime represents tlie derivativo witli respect to R.
3. EXACT SOLUTIONS AND CONSERVED QUANTITIES
Here, wo attempt to find exact solutions through Noethor symmetry and the corresponding conserved quantities. We also check the behavior of some cos-mological parameters for the resulting model to study the accelerated expansion of the universe. We assume ,4 = Bw, in 0,1, which is obtained from the constant ratio of shear and expansion scalars [23]. With this condition, Lagrangian (9) takes the form
C(D, R, B, R) = Bm+2(f—Rf')+2(2m+l)BmB2f'+ + 2 (tn + 2)D'r,+1 DRf" + PQ. (10)
The corresponding vector field for Noethor symmetry
9 , 9 ■ 9 ; 9 x = + + V+ n
'dB sai? 'as dR
where and £ are unknown functions that depend on the canonical variables B and R. The derivatives of these unknowns with respect to time are
ñ 9l l " = DdB
R
di/ OR'
t = D
dç •
dB
R
OR'
(12)
Using Eqs. (10) (12) in the condition for the existence of symmetry (L_\-£ = 0), we ob
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