On the asymptotics of 3+1D cosmologies with bounded scalar potential and isometry group forming 2-dimensional orbits

TL;DR

This paper demonstrates that in 3+1D cosmologies with bounded scalar potential and specific symmetry conditions, inflationary expansion occurs from inhomogeneous initial states, and analyzes the asymptotic behavior of related geometric and physical quantities.

Contribution

It introduces a mean curvature flow approach to prove inflation onset in symmetric 3+1D cosmologies with bounded potentials and studies the asymptotic dynamics using an inflationary time coordinate.

Findings

Inflation occurs with inhomogeneous initial conditions.

Volume growth is bounded between de Sitter slices with different cosmological constants.

Asymptotic analysis of metric, stress-energy tensor, and inflaton field dynamics.

Abstract

We study the onset of inflation in 3+1 dimensional cosmologies with an inflationary potential $U$ satisfying $0 < \Lambda_1 \leq U \leq \Lambda_2$, matter satisfying the dominant and strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are orbits under an isometry group. Assuming an initial Cauchy slice with positive mean curvature everywhere, we show, via mean curvature flow, that there exists a family of spatial slices parameterized by $\lambda$, whose volume grows between the flat slicings in de Sitter spaces with cosmological constants $\Lambda_1$ and $\Lambda_2$. In particular, inflationary expansion indeed occurs in this setting with inhomogeneous initial conditions. Finally, we apply this "inflationary time coordinate" $\lambda$ to study asymptotics of the variation in the metric, the average stress-energy tensor, and the dynamics of an inflaton field on a spatial slice.