Initial Conditions of Inflation in a Bianchi I Universe

TL;DR

This paper explores the initial conditions for inflation in a homogeneous but anisotropic Bianchi I universe, constructing a finite, invariant measure on initial states and finding that inflation favors finely tuned initial conditions near the potential minimum.

Contribution

It introduces a geometric approach using the Eisenhart lift to define a finite, invariant measure on initial conditions in a Bianchi I universe, highlighting the fine-tuning needed for inflation.

Findings

The phase-space volume for initial conditions is finite for many models.

The measure favors initial states near the potential minimum.

Inflation requires finely tuned initial conditions.

Abstract

We investigate the initial conditions of inflation in a Bianchi~I universe that is homogeneous but not isotropic. We use the Eisenhart lift to describe such a theory geometrically as geodesics on a field space manifold. We construct the phase-space manifold of the theory by considering the tangent bundle of the field space and equipping it with a natural metric. We find that the total volume of this manifold is finite for a wide class of inflationary models. We therefore take the initial conditions to be uniformly distributed over it in accordance with Laplace's principle of indifference. This results in a normalisable, reparametrisation invariant measure on the set of initial conditions of inflation in a Bianchi~I universe. We find that this measure favours an initial state in which the inflaton field is at or near its minimum, with a mild preference for some initial anisotropy. Since inflation requires an initial field value with a large displacement from its minimum, we therefore conclude that the theory of inflation requires finely tuned initial conditions.