Future dynamics of FLRW for the massless-scalar field system with positive cosmological constant

TL;DR

This paper proves the global existence and detailed asymptotic behavior of solutions to the Einstein-massless-scalar field system with positive cosmological constant, showing deviations from FLRW behavior due to scalar field gradient changes.

Contribution

It establishes the future global existence and precise asymptotics of near-FLRW solutions without symmetry assumptions, revealing breakdown of the stiff fluid interpretation at late times.

Findings

Solutions exist globally in the future direction.

Scalar field gradient becomes null and spacelike at infinity.

Perturbed solutions differ from FLRW in asymptotic behavior.

Abstract

We consider solutions to the Einstein-massless-scalar field system with a positive cosmological constant, arising from sufficiently regular, near-FLRW, initial data. We establish global existence in the future direction and derive their precise asymptotic behavior towards infinity. As a corollary, we infer that, unlike the FLRW background, the perturbed solutions do not describe a regular irrotational stiff fluid with linear equation of state $p=\rho$, for general asymptotic data at infinity. The reason for the breakdown of this interpretation is that the gradient of the scalar field stops being timelike at large times, eventually becoming null and then spacelike near infinity. Our results hold for open sets of initial data in Sobolev spaces without symmetries.