Fluctuations of the gravitational constant in the inflationary Brans-Dicke cosmology

Abstract

According to the Brans-Dicke theory, the value of the gravitational constant G which we measure at present is determined by the value of the Brans-Dicke scalar field \ensuremath{\varphi} at the end of inflation. However, because of quantum fluctuations of the scalar fields produced during inflation, the gravitational constant G(\ensuremath{\varphi}) may take different values in different exponentially large parts of the Universe. We investigate the probability distribution ${\mathit{P}}_{\mathit{p}}$ to find a domain of a given volume with a given value of the gravitational constant G at a given time. The investigation is performed for a wide class of effective potentials of the scalar field \ensuremath{\sigma} which drives inflation, and with two different time parametrizations. Our work is based on the analytical study of the diffusion equations for ${\mathit{P}}_{\mathit{p}}$, as well as on the computer simulation of stochastic processes in the inflationary universe. We have found that in some inflationary models the probability distribution ${\mathit{P}}_{\mathit{p}}$ rapidly approaches a stationary regime. The shape of the distribution depends, however, on the choice of the time parametrization. In some other models the distribution ${\mathit{P}}_{\mathit{p}}$ is not stationary. An interpretation of our results and of all ambiguities involved is outlined, and a possible role of anthropic considerations in determination of the gravitational constant is discussed.