Exponential potentials and cosmological scaling solutions

We present a phase-plane analysis of cosmologies containing a baryotropic fluid with an equation of state ${p}_{\ensuremath{\gamma}}=(\ensuremath{\gamma}\ensuremath{-}1){\ensuremath{\rho}}_{\ensuremath{\gamma}},$ plus a scalar field $\ensuremath{\varphi}$ with an exponential potential $V\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\ensuremath{\lambda}\ensuremath{\kappa}\ensuremath{\varphi})$ where ${\ensuremath{\kappa}}^{2}=8\ensuremath{\pi}G$. In addition to the well-known inflationary solutions for ${\ensuremath{\lambda}}^{2}<2$, there exist scaling solutions when ${\ensuremath{\lambda}}^{2}>3\ensuremath{\gamma}$ 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 $V(\ensuremath{\varphi})/{\ensuremath{\rho}}_{\ensuremath{\gamma}}\ensuremath{\rightarrow}0$ at late times, are always unstable (except for the cosmological constant case $\ensuremath{\gamma}=0)$. The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to ${\ensuremath{\lambda}}^{2}>20$. We show that standard inflation models are unable to solve this ``relic density'' problem.