A Generalized Bianchi Type I Cosmological Model with Time-Varying Gravitational and Cosmological Constants in f (R, T) Gravity

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Introduction

General Relativity (GR) was introduced by Einstein in 1915 and completely changed our view on gravitation and cosmology at large [1]. Gravity is understood by GR as curving space-time caused by the presence of matter and energy, GR says. General Relativity has been very successful at describing a variety of astrophysical and cosmological phenomena, but some observations are now becoming a serious problem for the standard cosmological model [2,3], notably the accelerating expansion of the universe.

A distant-type Ia supernova [2–3] and the Cosmic Microwave Background (CMB) radiation [4] observations have revealed that the present universe is expanding at an accelerated rate. As part of Einstein's theory, this acceleration can be understood by adding the cosmological constant Λ, a source of dark energy. The cosmological constant has, however, very severe theoretical problems, including the fine tuning problem and the coincidence problem. These problems have sparked cosmologists to look for alternate solutions, such as modified theories of gravity [5].

One of the best investigated modifications of GR is f(R) gravity. According to this theory, the Einstein-Hilbert action is generalized replacing the Ricci scalar R with an arbitrary function f(R) [6, 7]. These changes inevitably emerge in an attempt to quantum correct gravity, and it has been demonstrated that they can give a geometrical explanation for the early inflationary phase, as well as the late-time accelerated expansion of the universe, without the need of exotic dark energy components [6]. In modern theoretical cosmology, several cosmological models via f(R) gravity have been successfully used during various phases of evolution of the universe, and have gained great interest [7]. Another generalization of f(R) gravity was introduced by Harko et al. [8] in the form of the f(R,T) gravity, which involves the action depending on the scalar R and the trace T of the energy-momentum tensor. The explicit coupling to T, on the other hand, couples the matter to the geometry, resulting in changes to the gravitational field equations. The coupling of this matter with geometry can lead to further gravitational effects and to new understanding regarding the accelerated expansion of the Universe, dark energy effects and cosmological dynamics [8]. The richness of the mathematical structure and physical meaning of f(R,T) gravity has made it an interesting field of research in the modern cosmology.

Interestingly, it is possible in cosmology that the fundamental constants in nature can also change over time. Dirac proposed the Large Number Hypothesis in [9] which leads to the idea of a varying gravitational constant, G. Many investigations have dealt with the possible cosmological models involving variable G and Λ in order to comprehend their effect on cosmic evolution since then. A time-dependent cosmological constant (C) can provide a natural explanation for the present accelerated expansion, and a varying gravitational constant (C) can have an impact on the dynamics of the universe from the early epochs to the present day [10]. Anisotropic models of the cosmos are especially useful in the study of the early universe. While the large-scale universe is highly isotropic, anisotropies might have been important in the early universe. Bianchi Type I space-time is the most basic space-time model which is homogeneous and anisotropic and has been extensively used to study the impact of anisotropy on cosmic evolution [11]. These models are more abstract than the Friedmann-Lemaître-Robertson-Walker (FLRW) universe, and help to understand the transition between anisotropy and isotropy.

In the present work, the study of cosmological models based on the variable gravitational constant G and cosmological constant Λ in the framework of the General Relativity theory is continued. In f(R,T) gravity, we assume that we have a homogeneous and anisotropic Bianchi Type I universe in which a perfect fluid is present. The exact functional form is f(R,T) = R + 2f(T)

If is adopted, there are corrections to Einstein's field equations, due to the coupling between the energy-momentum tensor and the geometry of space-time [8]. By making the assumption that the cosmological constant is time dependent with a relationship, Exact solutions of the modified field equations can be obtained. Λ ∝ R⁻², where R is the mean scale factor of the universe. The resulting model is able to describe both the inflationary period of the early universe and the present-day accelerated expansion of the universe. A number of physical and geometrical parameters are explored, such as the Hubble parameter H, the gravitational constant G, the cosmological constant Λ, the expansion scalar θ, and the shear scalar σ². The analysis shows that the evolution of the universe is consistent and there is a coupling between the shear scalar, the gravitational constant, the cosmological constant and the Hubble parameter. The model has a singularity of the point type at the initial epoch where Λ has the value of infinity. The gravitational constant changes dynamically with cosmic time and the cosmological constant decreases as the universe expands. Moreover, as the universe evolves, the anisotropy decreases and at late times the universe tends to be isotropic (σ/θ → 0). No explicit equation of state or a pre-given functional form of the G has been imposed, and the solutions thus obtained are more general and physically realistic. Hence, the present model constitutes a viable idea for exploring cosmological evolution in anisotropic space-time in the framework of modified f(R,T) gravity and it also plays a role in understanding how the variation of the gravitational and cosmological parameters can affect the evolution of the universe.

Thus the universe will be decelerating phase for negative Λ and for positive Λ universe will be slow down the rate decrease, showing that the rate volume expansion decreases during time evaluation and presence of positive Λ, slows down the rate of this decreases whereas a Λ negative would promote it.

Summary and Conclusion

In this section, I summarize the results obtained in this research. Here, my work is extension of Cosmological Model With varying � and Λ in general relativity. A lot of work has been done by Saha (2005), in studying the anisotropic Bianch type I Cosmological Model in general relativity with varying � and Λ. In this article, I extend this work by converting cosmological Model with homogenous Bianchi type I space time with variable � and Λ containing matter in the form of perfect fluid inn the framework of �(??????,??????)gravity. In this article, I concentrated on the first class as �(??????,??????)= ??????+2�(??????). We considered the modified gravity which naturally unifies two expansion phases of the universe i.e., inflation at early time and cosmic acceleration at current approach. The model based on the exact solution of the �(??????,??????) gravity field equations for the anisotropic Biandi-I space-time filled with perfect fluid with time dependent Λ term. Solution of Einstein’s equations involves the conservation of energy-momentum tensor. Through physical process I detected the evolution of different parameters which imply that dissimilar parameters are coupled. In this manner the field equation provide trivial assumption implying a linear linkage among ??????2,�,Λ and �. In short, I have explored Bianchi type I space-time in variable � and Λ in presence of perfect fluid with Cosmological term Λ proportional to ??????−2 (where ?????? is scale factor). Initially the model has a point type singularity, gravitational constant �(�) is decreasing and Cosmological constant Λ is infinite at this time, when time increases Λ decreases. Here, I have neither assumed equation of state not particular form of G. The model approach isotropy for large value of t, the model is quasi-isotopic i.e., ?????? ??????=0.