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

TL;DR

The paper investigates how quantum fluctuations during inflation in Brans–Dicke cosmology generate spatial variations in the effective gravitational constant G(φ). Using a two-field, stochastic inflation framework, it analyzes the evolution of the physical-volume distribution P_p(σ,φ;t) and its stationary or nonstationary behavior across a range of inflaton potentials, under two time parametrizations. It shows that stationary distributions can arise for many models but their form depends on the time variable and boundary conditions, and it discusses possible anthropic implications for determining G. The work highlights the rich, environment-dependent structure of fundamental constants in an inflationary multiverse and the interpretive challenges posed by time-parametrization ambiguities. The results offer a framework for connecting Planck-scale physics, scalar-tensor gravity, and anthropic considerations in early-universe 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 φat the end of inflation. However, due to quantum fluctuations of the scalar fields produced during inflation, the gravitational constant G(φ) 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 σwhich 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.

Figures (14)

• Figure 1: Diffusion of the distribution $P_p$ along the Planck boundary in the simplest theory with $V(\sigma) = const$, in the time $t$ parametrization. Figs. 1a, 1b, 1c show different steps towards stationarity at the Planck boundary. Fig. 1d shows the same stage of the diffusion as Fig. 1c, but in a different perspective.
• Figure 2: Stationary probability distribution $P_p$ in the plane $(\sigma, \phi)$ for the theory $V(\sigma) = {\lambda\over4} \sigma^4$, in the time $t$ parametrization. The line OA corresponds to the end of inflation, while the line OB to the Planck boundary. Inflation occurs between these two lines. The line above OA shows the probability distribution ${\cal P}_p$.
• Figure 3: Same as in Fig. 2 for the potential $V(\sigma) = {\lambda \sigma^4\over 4}\, \exp{\sigma^4\over\sigma_0^4}$. The distribution ${\cal P}_p$ is shown on the left side of the box.
• Figure 4: Same as in Fig. 1 for the potential $V(\sigma) = {m^2\over 2}\sigma^2$.
• Figure 5: Same as in Fig. 3 for the potential $V(\sigma) = {\lambda \sigma^4\over 4} \, \exp\Bigl(-{\sigma^4\over\sigma_0^4}\Bigr)$.