Mathematical issues in eternal inflation

Abstract

Abstract. In this paper, we consider the problem of existence and uniqueness of solutions to the Einstein field equations for a spatially flat FLRW 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 Klein-Gordon 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 well-posed, 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 power-law inflaton potential and prove that all solutions to the EFE 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 well-posed, 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.

Eternal hilltop inflation

“…Some authors have argued that inflation is not truly eternal even in this case[53,54] 4. "Saddle-point" inflation models with odd exponent p can avoid eternal inflation through an appropriate choice of parameters[35], so we consider even powers of p.…”

Cosmic Background

“…Owing to hotter-colder CMB rains, decreasing mass and volume of PL will equal to increasing ones of universe that cosmic average mass density [17] ~ 4.5×10 −31 g/cm 3 also roughly was of PL, implying that here, however, disagrees with big bang that the density will be lower as time went on, as eternal inflation in fact cannot be eternal [18],…”