Geometrically Consistent Approach to Stochastic DBI Inflation

TL;DR

This paper develops a geometrically consistent stochastic framework for DBI inflation, incorporating extra-dimensional constraints into the inflaton's probability distribution, and analyzes how these boundaries influence inflationary dynamics.

Contribution

It introduces a novel method to include geometric restrictions in the stochastic description of DBI inflation using absorbing walls and solves for the inflaton's PDF with these constraints.

Findings

The PDF vanishes at geometrically forbidden field values.

Walls significantly influence inflaton trajectories.

The method applies to arbitrary potential and warp factor functions.

Abstract

Stochastic effects during inflation can be addressed by averaging the quantum inflaton field over Hubble-patch sized domains. The averaged field then obeys a Langevin-type equation into which short-scale fluctuations enter as a noise term. We solve the Langevin equation for a inflaton field with Dirac Born Infeld (DBI) kinetic term perturbatively in the noise and use the result to determine the field value's Probability Density Function (PDF). In this calculation, both the shape of the potential and the warp factor are arbitrary functions, and the PDF is obtained with and without volume effects due to the finite size of the averaging domain. DBI kinetic terms typically arise in string-inspired inflationary scenarios in which the scalar field is associated with some distance within the (compact) extra dimensions. The inflaton's accessible range of field values therefore is limited because of the extra dimensions' finite size. We argue that in a consistent stochastic approach the distance-inflaton's PDF must vanish for geometrically forbidden field values. We propose to implement these extra-dimensional spatial restrictions into the PDF by installing absorbing (or reflecting) walls at the respective boundaries in field space. As a toy model, we consider a DBI inflaton between two absorbing walls and use the method of images to determine its most general PDF. The resulting PDF is studied in detail for the example of a quartic warp factor and a chaotic inflaton potential. The presence of the walls is shown to affect the inflaton trajectory for a given set of parameters.