Mathematical issues in eternal inflation
Abstract
In this paper, we consider the problem of the existence and uniqueness of solutions to the Einstein field equations for a spatially flat FriedmannLemaîtreRobertsonWalker universe in the context of stochastic eternal inflation, where the stochastic mechanism is modelled by adding a stochastic forcing term representing Gaussian white noise to the KleinGordon equation. We show that under these considerations, the KleinGordon equation actually becomes a stochastic differential equation. Therefore, the existence and uniqueness of solutions to Einstein’s equations depend on whether the coefficients of this stochastic differential equation obey Lipschitz continuity conditions. We show that for any choice of V(φ ), the Einstein field equations are not globally wellposed, hence, any solution found to these equations is not guaranteed to be unique. Instead, the coefficients are at best locally Lipschitz continuous in the physical state space of the dynamical variables, which only exist up to a finite explosion time. We further perform Feller’s explosion test for an arbitrary powerlaw inflaton potential and prove that all solutions to the Einstein field equations explode in a finite time with probability one. This implies that the mechanism of stochastic inflation thus considered cannot be described to be eternal, since the very concept of eternal inflation implies that the process continues indefinitely. We therefore argue that stochastic inflation based on a stochastic forcing term would not produce an infinite number of universes in some multiverse ensemble. In general, since the Einstein field equations in both situations are not wellposed, we further conclude that the existence of a multiverse via the stochastic eternal inflation mechanism considered in this paper is still very much an open question that will require much deeper investigation.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 April 2015
 DOI:
 10.1088/02649381/32/7/075001
 arXiv:
 arXiv:1408.2249
 Bibcode:
 2015CQGra..32g5001S
 Keywords:

 Mathematical Physics;
 General Relativity and Quantum Cosmology;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Dynamical Systems
 EPrint:
 Accepted for publication in Classical and Quantum Gravity / This version: updated minor typos