A Generalized Solution of Bianchi Type-V Models with Time-Dependent G and Λ

Abstract

Abstract:We study the homogeneous but anisotropic Bianchi type-V cosmological model with time-dependent gravitational and cosmological "constants". Exact solutions of the Einstein field equations (EFEs) are presented in terms of adjustable parameters of quantum field theory in a spatially curved and expanding background. It has been found that the general solution of the average scale factor a as a function of time involved the hypergeometric function. Two cosmological models are obtained from the general solution of the hypergeometric function and the Emden-Fowler equation.The analysis of the models shows that, for a particular choice of parameters in our first model, the cosmological "constant" decreases whereas the Newtonian gravitational "constant" increases with time, and for another choice of parameters, the opposite behaviour is observed. The models become isotropic at late times for all parameter choices of the first model. In the second model of the general solution, both the cosmological and gravitational "constants" decrease while the model becomes more anisotropic over time. The exact dynamical and kinematical quantities have been calculated analytically for each model.

Bianchi type–I Model with Time Varying Λ and G: The Generalized Solution

“…This equation is called the generalized Friedmann equation, which is a non-linear second order differential equation, and it is similar to the equation of Bianchi type V model that we have obtained in our previous paper (Alfedeel et al 2018), except the third term on the left-hand side here is different by a factor of 2 (i.e. α type1 = α type V + 2).…”

“…The volume expansion parameter θ, the average anisotropy parameter Ap, and shear modules σ are defined as in (Alfedeel et al 2018;Carvalho et al 1992) by:…”

“…This equation cannot be integrated as it is because of the unknown functions Λ, G, ρ and the parameter w. Hence, we use the same form of Λ as in Alfedeel et al (2018) and Tiwari (2008)…”

A Note on Varying G and Λ in Chern–Simons Modified Gravity

“…Sarfraz and Saddique [35] investigated G and Λ in the framework of CSMG taking Λ ∼ ( ȧ a ) 2 is naturally attractive in existence, whereas Λ ∼ ä a corresponds to a repulsive conditions and thus pertains to the present situation of the accelerating universe. It is also noticed that the universe is expanding with acceleration due to the simultaneous influence of time variables Λ and G. Alfedeel et al [36,37] studied time dependent gravitational constant G(t) and cosmological constants Λ(t) taking into account Bianchi type-I and V matrices and analyzed that the Λ(t) decreases with time while the G(t) increases, and the reverse tendency is found for another range of constraints. Motivated by these studies, we opted for two models named holographic dark energy (HDE) Λ ∼ (α Ḣ + βH 2 ) and modified holographic Ricci dark energy (MHRDE) Λ ∼ 2 α−β ( Ḣ + 3 2 αH 2 ) [12] to investigate their influence on universe expansion in the framework of the CSMG theory.…”

k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

“…Since then, Euler, too, had researched the topic on the hypergeometric series, but the first full systematic study was introduced by Gauss [12]. Some works and complete references concerning both the hypergeometric series and the certain equation 2can be found in Kummer [13], Riemann [14], Bailey [15,16], Chaundy [17], Srivastava [18], Whittaker [19], Beukers [20], Gasper [21], Olde Daalhuis [22,23], Dwork [24], Chu [25], Yilmazer et al [26], Morita et al [27], Abramov et al [28], Alfedeel et al [29], and the literature therein. However, in contrast to the extensive studies on Equation 2, other hypergeometric differential equations with k ∈ R + are very limited.…”