Exponential potentials and cosmological scaling solutions

We present a phase-plane analysis of cosmologies containing a baryotropic fluid with an equation of state $p_{\gamma }=\left(\gamma -1\right){\rho }_{\gamma },$ plus a scalar field $\phi$ with an exponential potential $V\propto \exp \left(-\lambda \kappa \phi \right)$ where ${\kappa }^2=8\mathrm{}\pi G$. In addition to the well-known inflationary solutions for ${\lambda }^2<2\mathrm{}$, there exist scaling solutions when ${\lambda }^2>3\mathrm{}\gamma$ in which the scalar field energy density tracks that of the baryotropic fluid (which for example might be radiation or dust). We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where $V\left(\phi \right)/{\rho }_{\gamma }\to 0$ at late times, are always unstable (except for the cosmological constant case $\gamma =0)$. The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to ${\lambda }^2>20\mathrm{}$. We show that standard inflation models are unable to solve this “relic density” problem.