Scaling solutions and geodesics in moduli space

TL;DR

This paper explores the relationship between cosmological scaling solutions and geodesics in moduli space, providing methods to construct scalar potentials and metrics that enable such solutions in scalar-tensor theories of gravity.

Contribution

It establishes a direct link between scaling cosmological solutions and specific geodesics in moduli space, offering explicit construction techniques for potentials and metrics.

Findings

Scaling solutions correspond to geodesics traced by gradient integral curves.

Explicit construction of moduli metrics for given scalar potentials.

Method to derive scalar potentials from known moduli metrics.

Abstract

In this paper we consider cosmological scaling solutions in general relativity coupled to scalar fields with a non-trivial moduli space metric. We discover that the scaling property of the cosmology is synonymous with the scalar fields tracing out a particular class of geodesics in moduli space - those which are constructed as integral curves of the gradient of the log of the potential. Given a generic scalar potential we explicitly construct a moduli metric that allows scaling solutions, and we show the converse - how one can construct a potential that allows scaling once the moduli metric is known.