Non-linear non-local Cosmology

TL;DR

This paper reviews the reformulation of non-local equations from string theory as diffusion-like equations with non-linear boundary conditions, providing a method to solve them numerically and analyze inflationary solutions.

Contribution

It introduces a novel approach to solving non-local equations by rewriting them as diffusion-like equations with boundary conditions and demonstrates its application to inflationary cosmology.

Findings

Successfully reformulated non-local equations as diffusion-like equations.

Developed a numerical method to solve these equations as initial value problems.

Identified conditions under which inflation occurs in the model.

Abstract

Non-local equations of motion contain an infinite number of derivatives and commonly appear in a number of string theory models. We review how these equations can be rewritten in the form of a diffusion-like equation with non-linear boundary conditions. Moreover, we show that this equation can be solved as an initial value problem once a set of non-trivial initial conditions that satisfy the boundary conditions is found. We find these initial conditions by looking at the linear approximation to the boundary conditions. We then numerically solve the diffusion-like equation, and hence the non-local equations, as an initial value problem for the full non-linear potential and subsequently identify the cases when inflation is attained.