The wave equation in cosmological spacetimes

We discuss the geometric wave equation in a class of spacetimes that are homogeneous and isotropic, i.e. that approximately describe, at large scales, the universe we live in. While this hyperbolic PDE can be seen as a damped wave equation in $\mathbb{R}^n$, where the damping term is time-dependent, its geometric interpretation in the Lorentzian setting allows to associate dispersive properties of solutions to physical properties of the spacetime, such as its rate of expansion. For specific universes of positive, zero or negative spatial curvature, we are able to provide explicit expressions of the solutions to this wave equation in terms of spherical means, by generalizing a method employed by S. Klainerman and P. Sarnak in the '80s. Numerical results regarding decay rates and the failure of Huygens' principle will also be presented. This is joint work with J. Natário and A. Vañó-Viñuales.