Role of Gravity in Bianchi Type-V Dark Energy Model with Electromagnetic Field based on Lyra Geometry

In this paper, a Bianchi type V Dark energy cosmological model scenario is studied with the electromagnetic field in Lyra based on gravity. The source of the magnetic field is along the axis. Field equations found in Lyra geometry with electromagnetic field in gravity, by choosing ( ̃ ) ( ̃) with ( ̃) ̃ and . A hybrid scale factor discusses the physical and dynamical aspects of gravity.


Introduction
Bianchi type-V universes are natural generalizations of open FRW models that eventually become isotropic and homogeneous. They play an essential role in understanding galaxy formation in the early Universe [1]. There are substantial theoretical arguments for an anisotropic phase in the evolution of the Universe.
Numerous high-redshift supernovae discoveries demonstrate that the early Universe is expanding faster than previously anticipated [2][3][4][5]. In addition to these tests, observations such as cosmic microwave background (CMB) radiations and large-scale structure [6][7][8] suggest that the Universe is expanding quickly. Because of their capacity to explain the observed faster expansion of the Universe, modified gravity theories have sparked much attention in recent years. The essential ideas among them are And gravity [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The most prevailing significant theory in the current cosmological model of the Universe, examined with the FRW model and Bianchi type models, is The Lagrangian is a function of Ricci scalar R. The trace of the stress-energy tensor T. Many authors [23][24][25][26][27][28][29][30][31][32][33] have recently studied the dark energy model in gravity based on different contexts so far. Recently, Basumatary and Dewri [34] have reviewed the dark energy model with a particular form of scale factor in Sen-Dunn's theory of gravitation based on Bianchi type . The magnetic field [35][36][37] found in both galactic and intergalactic space has played a significant role in understanding the distribution of energy in the Universe. Numerous authors have explored the influence of magnetic fields on cosmological development in diverse circumstances, both theoretically and practically [38][39][40][41][42][43][44][45][46]; Melvin [47] has also revealed that matter was strongly ionized and seamlessly associated with the field during the development of the Universe.
Einstein introduced the cosmological constant in his general relativity field equations in 1917. However, his field equations do not acknowledge static solutions without the cosmological factor. After discovering and explaining the red-shift of galaxies, Einstein regretted the introduction of the cosmological constant [48]. Weyl [49] anticipated a further general theory that geometrically described gravitation and electromagnetism. Lyra [50] used a gauge function to introduce a structureless manifold to geometrize gravitation and electromagnetism. Similarly, Sen and his group [51,52] proposed a new scalar-tensor theory of gravitation and developed an analog of the EFE using Lyra geometry. Further, Halford [53] has pointed out that under the standard general relativistic interpretation, the constant vector displacement field ( ) in Lyra's geometry serves as a cosmological constant. As a result, Halford exhibits that the scalar-tensor approach based on Lyra's geometry predicts the same effects as Einstein's theory within the observable constraints.
Recently, Bhardwaj et al. [54] have explored dark energy models in f (R, T) gravity. Also, Singh and Devi [55] investigate the locally rotationally symmetric (LRS) Bianchi type-I cosmological models in gravity with Hybrid Expansion Law.

Overview of Gravity and the Field Equations
Bianchi type-V space-time is (1) where are functions of time alone and is a constant. The action of gravity using Lyra geometry [56,57] is given by in which the function of Ricci scalar , the trace of the energy-momentum tensor, and Lagrangian matter density are indicated by ̃ and Respectively. Also, the function of Ricci scalar together with the Riemannian curvature [57] is ̃ and the stress-energy tensor of the matter is where its trace Here, the matter Lagrangian depends only on the metric tensor component ( rather than its derivatives. So that it reduces to (5) Now, by varying the action S in eq. (2) for the metric tensor , the gravitational field equations of ̃ gravity takes the form as indicates covariant derivative in ̃ gravity model. It is essential to reveal the physical nature of the matter field through is used to form the field equations of ̃ gravity. The critical point of the case is that the model depicts diverse natures by considering different frames of Harko et al. [56]. So, in this study, it is assumed as The energy-momentum tensor with the electromagnetic field considered as (9) Here, and denote the energy density and thermodynamic pressure of the matter, where ( ) represent the electromagnetic fields of the source and is given as such that the Maxwell equation satisfies the relation In the co-moving coordinate system, the magnetic field is in the direction of axis so that is the only non-vanishing component of the model i.e. constant. Also is the four-velocity vector in a co-moving coordinate system satisfies the condition . The non-vanishing components of the source of for the given line element are as follows: Now by contracting eq. (6), it reduces to For a perfect fluid matter source, the field equations of ( ̃ ) gravity obtained by choosing ( ̃) ̃ and , where is an arbitrary constant, as given by Now, applying eq. (3) in (14), the field equations for Lyra Geometry [57] are obtained as where is a time like displacement field vector and ( ) , and the symbols have their usual meaning as in Riemannian Geometry. For the line element (1), to eq. (9), the Einstein Field Equations (15), reduces to The energy conservation equation takes the form

Solutions and the Physical Behavior of the Model in ̃ Gravity
Case-I: Presence of magnetic field: In solving the above-filed equations (16) -(20), the following physical parameters are vital, and these parameters are defined as follows: The spatial volume and the scale factor is given by (22) The generalized mean Hubble parameter defined as (23) where ̇ ̇ ̇ are the directional Hubble's parameters along and axes, respectively. Eqs. (22) and (23) give The scalar expansion , shear expansion and the anisotropy parameter are defined as Eq. (20) gives (28) In the Einstein field equations (16) - (20), there are five highly non-linear differential equations with six unknown variables, namely . Thus, to establish these six variables, let us consider the power-law relation (29) where is a positive constant that preserves the anisotropy of the space-time [58]. Now, a hybrid form of scale factor [59] is (30) Here, are positive constants such that lies between 2 and 3 where, for , the eq. (30) reduces to the hybrid scale factor, and it is crucial to construct cosmic transit from the early age of deceleration to late time acceleration [58]. Eqs. (22), (28), (29), and (30) together yields (31) (32) and, Then the line element (1) reduces to From eq. (22), the volume of the model reduces to (35) The other dynamical parameters are (Eqs. (24) -(27)) as follows: The Deceleration Parameter (DP) ( Adding Eqs. (16) -(18), the displacement field vector ( ) is obtain as Where and are both positive constant and depict the variation of the graph by considering the choice of , being .  41) and (42), the trace and the function of Ricci Scalar tensor ̃ and the Riemannian curvature of ̃ gravity with electromagnetic field based on Lyra geometry are obtained as And, where is a positive constant and keeping to describe the model's behavior, provided and are given by Here, the dark energy model using the frame of Harko et al. [56] as ̃ ̃ , is considered. Eqs. (44) and (46) give Case-II: In the absence of a magnetic field, i.e.
In this case, the pressure , density and the displacement vector is and, In the same way, the trace , the function of the Ricci Scalar Tensor ̃ , Riemannian curvature and the ̃ gravity obtained as follows: and, Gravity in Bianchi Type-V Dark Energy Model and, where, and The state finder parameters defined as and, which are obtained from (31), (36), and (40) as follows:

Conclusion
In this paper, the field equations in Lyra based ̃ gravity solved with by using . The absence of an electromagnetic field does not affect much in the present model. Also, at the initial moment , all the parameters like ̃ and ̃ tends to infinity. The beginning is the initial singularity with infinite pressure and density. When t ̃ and ( ̃ ) tends to be a constant value. From the eq. (35), it is seen that the universe approaches an infinitely large volume as time increases in both scenarios of the ̃ gravity model. The behavior of DP is for . The orientation of the energy density and pressure are positive and negative directions in each case with an increase in time . As per these observations of and , the present model has dark energy with the accelerated expansion of the Universe with an electromagnetic field. This result fits the recent observational data (LSS, CMB, SNe Ia). The displacement field vector positively gradually decreases with an increase in cosmic time . The CDM model does not evolve in this model. Eqs. (38) and (39) show that this model of ̃ gravity is not free from shear scalar, and the model's behavior is totally anisotropic provided .