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abstract: 'We study the time evolution of freely rolling moduli in the context of M-theory on a $G_2$ manifold. This free evolution approximates the correct dynamics of the system at sufficiently large values of the moduli when effects from non-perturbative potentials and flux are negligible. Moduli fall into two classes, namely bulk moduli and blow-up moduli. We obtain a number of non-trivial solutions for the time-evolution of these moduli. As a generic feature, we find the blow-up moduli always expand asymptotically at early and late time.'
author:
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    [André Lukas[^1] and Stephen Morris[^2]]{}\
    [*Department of Physics and Astronomy, University of Sussex*]{}\
    [Falmer, Brighton BN1 9QH, UK]{}\
title: '\'
---

Introduction
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Seven-dimensional manifolds with holonomy $G_2$ provide the general framework for relating M-theory to four-dimensional models with $N=1$ supersymmetry. Moreover, it has been shown that for certain singular limits of such manifolds four-dimensional models with phenomenologically interesting properties can be obtained [@Papadopoulos:1995da]–[@Acharya:2002kv].

Given these promising features it seems worthwhile to have a closer look at the cosmology of such $G_2$ compactifications of M-theory. More specifically, in this paper, we will be interested in the time-evolution of $G_2$ moduli. This problem has not previously been analysed since the four-dimensional 