Fluctuations of the Gravitational Constant in the Inflationary Brans-Dicke Cosmology

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 $\phi$ at the end of inflation. However, due to quantum fluctuations of the scalar fields produced during inflation, the gravitational constant $G(\phi)$ may take different values in different exponentially large parts of the Universe. We investigate the probability distribution $P_p$ 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 $\sigma$ which drives inflation, and with two different time parametrizations. Our work is based on the analytical study of the diffusion equations for $P_p$, 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 $P_p$ 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 $P_p$ 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.