Asymptotics of solutions of the Einstein equations with positive cosmological constant

TL;DR

This paper analyzes the asymptotic behavior of solutions to Einstein's equations with a positive cosmological constant, providing formal series expansions and existence results for vacuum and fluid spacetimes in various dimensions.

Contribution

It offers a general mathematical framework for understanding the asymptotics of Einstein solutions with positive cosmological constant, including formal series and existence proofs without symmetry assumptions.

Findings

Vacuum solutions have full asymptotic expansions given by formal series.

Large gradients can develop in stiffer-than-radiation equations of state.

Existence of spacetimes with prescribed asymptotics depending on free functions.

Abstract

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.