PhD Completion Seminar: Louie Bernhardt

The PhD Completion Seminar of Louie Bernhardt will take place

When: Tuesday, February 10, 2026, 10AM
Where: Peter Hall 162 – Alison Harcourt Seminar Room

followed by tea/coffee in the Staff Tea Room. All welcome!

Title: Stability and scattering problems for expanding cosmologies in general relativity

Abstract: The phenomenon of spacetime expansion is of great importance in general relativity and cosmology. Indeed, our own universe is expanding, and current scientific understanding says the rate of this expansion has changed over time. In this thesis, we carry out a rigorous mathematical study of expanding solutions to the Einstein equations. This study is divided into two separate, but related parts.

In the first part, we investigate the stability of cosmological models in the critical regime of slow expansion. The Friedmann-Lemaitre-Robertson-Walker (FLRW) family of spacetimes, which are spatially homogeneous and isotropic, serve as fundamental models for the universe. Depending on the matter model one prescribes, members of the FLRW family of spacetimes can exhibit many different rates of expansion. We prove that slowly expanding FLRW spacetimes, which are solutions to the Einstein equations coupled to a particular nonlinear scalar field, are nonlinearly stable under small perturbations. We do this for several different matter models, including a relativistic perfect fluid satisfying the Euler equations. In particular, these are the first stability results for the Einstein equations in the regime of decelerated expansion.

In the second part, we study wave equations on expanding black hole spacetimes. Examples of such spacetimes include the Schwarzschild-de Sitter and Kerr-de Sitter families of spacetimes, which respectively model a spherically symmetric black hole in an expanding universe, and a rotating black hole in an expanding universe. We establish a scattering theory for the linear wave equation on Schwarzschild-de Sitter and Kerr-de Sitter spacetimes, with data prescribed at “infinite time”. Then we demonstrate that this scattering theory also holds for the Einstein equations themselves, and construct scattering solutions to the Einstein equations that behave asymptotically like Kerr-de Sitter, from data prescribed at infinity. These scattering solutions possess long-time asymptotics that are consistent with perturbations of Kerr-de Sitter arising from Cauchy data.