General Relativity as an Attractor in Scalar-Tensor Stochastic Inflation

TL;DR

In scalar-tensor models of gravity, quantum fluctuations during inflation diffuse the effective Planck mass $M_P$ and the Brans–Dicke parameter $ω$, raising the question of whether general relativity can be dynamically selected as an attractor. In the Brans–Dicke limit with constant $ω$, diffusion along the Planck boundary yields runaway, nonstationary distributions for $M_P$. The authors show that introducing a pole in $ω$ via a maximum of $f(φ)$ (i.e., a pole in $ω(f)$) yields a stationary distribution along the Planck boundary and drives $ω$ to exponentially large values by the end of inflation, effectively recovering GR. They illustrate this with a concrete model where $f(φ)=f_0 \, sin^2(a φ)$ and $ω(f)=ω_0/\cos^2(a φ)$, and demonstrate that diffusion is effectively one-dimensional along the Planck boundary, producing a peak at σ* and a large end-state $ω$. Combined with the subsequent classical evolution, this quantum diffusion selects general relativity as the late-time attractor, suggesting the mechanism may be generic in theories with Planck-mass upper bounds.

Abstract

Quantum fluctuations of scalar fields during inflation could determine the very large-scale structure of the universe. In the case of general scalar-tensor gravity theories these fluctuations lead to the diffusion of fundamental constants like the Planck mass and the effective Brans--Dicke parameter, $ω$. In the particular case of Brans--Dicke gravity, where $ω$ is constant, this leads to runaway solutions with infinitely large values of the Planck mass. However, in a theory with variable $ω$ we find stationary probability distributions with a finite value of the Planck mass peaked at exponentially large values of $ω$ after inflation. We conclude that general relativity is an attractor during the quantum diffusion of the fields.

Figures (2)

• Figure 1: The classical evolution of dilaton and inflaton fields during inflation in the $(x,y)$ plane, see Eq. (4.4), for the scalar-tensor theory defined by (4.1) is represented by the continuous curves, while the dashed curves correspond to the solutions in ordinary Brans--Dicke theory. Classical motion starts at Planck boundary ($z_p$) and ends at the end of inflation boundary ($z_e$), represented by the thick straight lines, while the dotted line corresponds to the self-reproduction boundary ($z_s$). The dot-dashed horizontal line corresponds to the general relativistic limit $y\to1$. Note that the classical motion in our theory very quickly approaches that limit.
• Figure 2: The probability distribution $\Psi(\sigma)$ along the Planck boundary, for $\sigma_b = 1$ and $\sigma_\ast = 0.812$. The WKB solutions (5.4) are good approximations to the numerical result away from the turning point $\sigma_\ast$, marked here with a vertical line.