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) = (gamma-1)rho(gamma), plus a scalar field phi with an exponential potential V proportional to exp(-lambda kappa phi) where kappa(2)= 8 pi G. In addition to the well-known inflationary solutions for lambda(2)<2, there exist scaling solutions when lambda(2)>3 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(phi)/rho(gamma)-->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. We show that standard inflation models are unable to solve this "relic density" problem.