Hyperboloidal approach for linear and non-linear wave equations in FLRW spacetimes

TL;DR

This paper develops a hyperboloidal foliation method to numerically analyze wave propagation in FLRW spacetimes, providing insights into decay rates and global existence of solutions for linear and semi-linear wave equations in expanding cosmological backgrounds.

Contribution

It introduces a hyperboloidal approach adapted to FLRW spacetimes, offering new numerical evidence for decay estimates and conditions for global existence of solutions.

Findings

Decay rates for linear waves are sharp in expanding FLRW spacetimes.

Small data solutions exist globally under a null condition in decelerated expansions.

Solutions diverge in finite time without the null condition when expansion is slow.

Abstract

In this numerical work, we deal with two distinct problems concerning the propagation of waves in cosmological backgrounds. In both cases, we employ a spacetime foliation given in terms of compactified hyperboloidal slices. These slices intersect future null infinity, so our method is well-suited to study the long-time behaviour of waves. Moreover, our construction is adapted to the presence of the time--dependent scale factor that describes the underlying spacetime expansion. First, we investigate decay rates for solutions to the linear wave equation in a large class of expanding FLRW spacetimes, whose non--compact spatial sections have either zero or negative curvature. By means of a hyperboloidal foliation, we provide new numerical evidence for the sharpness of decay--in--time estimates for linear waves propagating in such spacetimes. Then, in the spatially-flat case, we present numerical results in support of small data global existence of solutions to semi-linear wave equations in FLRW spacetimes having a decelerated expansion, provided that a generalized null condition holds. In absence of this null condition and in the specific case of $ \square_g \phi = (\partial_t \phi)^2 $ (Fritz John's choice), the results we obtain suggest that, when the spacetime expansion is sufficiently slow, solutions diverge in finite time for every choice of initial data.