Euclidean quantum gravity and stochastic inflation

TL;DR

This paper compares Euclidean quantum gravity and stochastic inflation by analyzing scalar field dispersions, revealing their correspondence through complex instantons, universe size, and stability limits, thus linking two different quantum cosmology approaches.

Contribution

It demonstrates the equivalence of Euclidean instantons and stochastic distributions in quantum gravity, especially for small mass and field cases, and explores their stability and classicality.

Findings

Complex instantons correspond to specific probability distributions.

Universe size matches the smoothing scale in stochastic inflation.

Both approaches break down at the same critical mass value.

Abstract

In this paper, we compare dispersions of a scalar field in Euclidean quantum gravity with stochastic inflation. We use Einstein gravity and a minimally coupled scalar field with a quadratic potential. We restrict our attention to small mass and small field cases. In the Euclidean approach, we introduce the ground state wave function which is approximated by instantons. We used a numerical technique to find instantons that satisfy classicality. In the stochastic approach, we introduce the probability distribution of Hubble patches that can be approximated by locally homogeneous universes down to a smoothing scale. We assume that the ground state wave function should correspond to the stationary state of the probability distribution of the stochastic universe. By comparing the dispersion of both approaches, we conclude three main results. (1) For a statistical distribution with a certain value, we can find a corresponding instanton in the Euclidean side, and it should be a complex-valued instanton. (2) The size of the universe of the Euclidean approach corresponds to the smoothing scale of the stochastic side; the universe is homogeneous up to the Euclidean instanton. (3) In addition, as the mass increases up to a critical value, both approaches break at the same time. Hence, generation of classical inhomogeneity in the stochastic approach and the instability of classicality in the Euclidean approach are related.