# Logolinear series expansions with applications to primordial cosmology

## TL;DR

This paper introduces a method for deriving series solutions involving linear and logarithmic terms in differential equations, specifically applied to early universe cosmology models, with accompanying computational tools.

## Contribution

It presents a novel technique for generating logolinear series expansions in differential equations relevant to primordial cosmology, including implementation code.

## Key findings

- Successfully applied to polynomial and Starobinsky inflationary potentials
- Provides analytic and numerical tools for series computation
- Addresses singularity expansions in pre-inflationary universe models

## Abstract

We develop a method for computing series expansions for solutions to ordinary differential equations when the asymptotic form contains both linear and logarithmic terms. Such situations are common in primordial cosmology when considering series expansions out of a singularity in the equations arising from a pre-inflationary phase of the universe. We develop mathematical techniques for generating these series expansions, and apply them to polynomial and Starobinsky inflationary potentials with kinetic initial conditions. Code for analytic and numerical computation of logolinear series is provided on GitHub.