A Simple Way to Estimate the Variation of the Gravitational Constant as a Function of Redshift in the Framework of Brans-Dicke Theory

Abstract

The objective of the present study is to find theoretically the nature of evolution of the time-varying gravitational constant $(G)$ and its relative time-rate of change $(\dot{G}/G)$ with respect to the redshift parameter $(z)$. For this purpose, we have used the field equations of the Brans-Dicke (BD) theory of gravity for a flat universe of zero pressure, with a homogeneous and isotropic space-time. Our entire formulation is based on four mathematical models constructed with empirical expressions involving the scale factor, BD scalar field and their time derivatives. Substituting these expressions into the field equations, we have determined the values of the constants associated with these ansatzes. It is clearly evident from these values that the gravitational constant increases as the redshift $(z)$ decreases with time. We have also determined the nature of variation of the relative time-rate of change of the gravitational constant $(\dot{G}/G)$. It has been found to be increasing as $z$ decreases with time. The variation of the gravitational constant and its relative time-rate of change, as functions of the redshift parameter, have been depicted graphically on the basis of the four models discussed in the present article. Based on their characteristics of variation, we have proposed an empirical relation representing the evolution of the gravitational constant $(G)$ as a function of time. Using this relation, we have determined the nature of dependence of redshift $(z)$ upon time and represented it graphically. Similar findings have been obtained from studies based on various other methods. An important feature of the present study is that all its findings have been obtained without solving the field equations.