Dynamics with Infinitely Many Derivatives: Variable Coefficient Equations

TL;DR

This paper develops a mathematical framework for solving variable coefficient infinite order differential equations, especially relevant in nonlocal cosmology, and demonstrates its application to p-adic inflation perturbations.

Contribution

It introduces a new formalism for variable coefficient infinite order equations, extending previous methods for constant coefficients, with applications to cosmological perturbation theory.

Findings

Leading corrections to scalar perturbations are small on large scales.

The formalism effectively handles equations in nonlocal inflation models.

Application to p-adic inflation confirms the approach's utility.

Abstract

Infinite order differential equations have come to play an increasingly significant role in theoretical physics. Field theories with infinitely many derivatives are ubiquitous in string field theory and have attracted interest recently also from cosmologists. Crucial to any application is a firm understanding of the mathematical structure of infinite order partial differential equations. In our previous work we developed a formalism to study the initial value problem for linear infinite order equations with constant coefficients. Our approach relied on the use of a contour integral representation for the functions under consideration. In many applications, including the study of cosmological perturbations in nonlocal inflation, one must solve linearized partial differential equations about some time-dependent background. This typically leads to variable coefficient equations, in which case the contour integral methods employed previously become inappropriate. In this paper we develop the theory of a particular class of linear infinite order partial differential equations with variable coefficients. Our formalism is particularly well suited to the types of equations that arise in nonlocal cosmological perturbation theory. As an example to illustrate our formalism we compute the leading corrections to the scalar field perturbations in p-adic inflation and show explicitly that these are small on large scales.