Blake Temple

University of California, Davis, April 11, 12:00 pm

Expanding spacetimes which characterize the radial instability of critical and under-dense Friedmann Spacetimes

We give a definitive characterization of the instability of the pressureless (p = 0) critical (k = 0) Friedmann spacetime to smooth radial perturbations. The analysis is based on a derivation of the Einstein field equations for smooth spherically symmetric spacetimes expressed in self-similar coordinates (t, ξ), ξ =rt < 1, conceived to realize the critical Friedmann spacetime as a stationary solution whose character as an unstable saddle rest point SM is determined via an expansion of smooth solutions in even powers of ξ. Using the expansion we characterize the local instability of non-critical Friedmann spacetimes within the unstable manifold of SM. With an eye toward Cosmology, we then identify a new maximal asymptotically stable family F of global smooth outwardly expanding solutions which characterizes the evolution of underdense perturbations of SM. Solutions in F align with a k < 0 Friedmann spacetime at early times and at leading order in ξ, generically introduce accelerations away from k < 0 Friedmann spacetimes at intermediate times, and then decay back to the same k < 0 Friedmann spacetime as t → ∞, uniformly at each fixed radius r > 0. The analysis establishes that the spatially homogeneous self-similar Big Bang of Friedmann spacetimes is universal at leading order in ξ, but surprisingly, the existence of a negative eigenvalue at SM at order n = 2 implies the Big Bang is generically not self-similar beyond leading order. We propose F as the maximal asymptotically stable family of solutions into which generic underdense radial perturbations of the unstable critical Friedmann spacetime will evolve and naturally admit accelerations away from Friedmann spacetimes within the dynamics of solutions of Einstein’s original field equations, that is, without recourse to a cosmological constant or dark energy. This is joint work with Christopher Alexander and Zeke Vogler.