The problem of a self-gravitating scalar field with positive cosmological constant

TL;DR

This paper proves the nonlinear stability of de Sitter spacetime under small scalar field perturbations, demonstrating exponential decay and geodesic completeness, thus supporting the cosmic no-hair conjecture.

Contribution

It establishes well-posedness, global existence, and exponential decay for the Einstein-scalar field system with positive cosmological constant in spherical symmetry.

Findings

Solutions decay exponentially to de Sitter spacetime

Initial data close to de Sitter evolve to a complete spacetime

Supports the cosmic no-hair conjecture in this setting

Abstract

We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data given on a truncated null cone. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves to a causally geodesically complete spacetime (with boundary), which approaches a region of de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration, as well as a realization of the cosmic no-hair conjecture.