On the Cosmic No-Hair Conjecture in T2-symmetric non-linear scalar field spacetimes

TL;DR

This paper proves the Cosmic No-Hair conjecture for T2-symmetric spacetimes with non-linear scalar fields, establishing global existence, uniqueness, and detailed asymptotic behavior of solutions under broad conditions.

Contribution

It demonstrates the Cosmic No-Hair conjecture in a new class of spacetimes with non-linear scalar fields, extending previous results to more general settings.

Findings

Global existence and uniqueness of solutions for all future times.

Detailed asymptotic estimates for spacetime metrics in the constant potential case.

Future causal geodesic completeness and validation of the Cosmic No-Hair conjecture.

Abstract

We consider spacetimes solving the Einstein non-linear scalar field equations with T2-symmetry and show that they admit an areal time foliation in the expanding direction. In particular, we prove global existence and uniqueness of solutions to the corresponding system of evolution equations for all future times. The only assumption we have to make is that the potential is a non-negative smooth function. In the special case of a constant potential, a setting which is equivalent to a linear scalar field on a background with a positive cosmological constant, we achieve detailed asymptotic estimates for the different components of the spacetime metric. This result holds for all T3-Gowdy symmetric metrics and extends to certain T2-symmetric ones satisfying an a priori decay property. Building upon these asymptotic estimates, we show future causal geodesic completeness and prove the Cosmic No-Hair conjecture.