We present a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state p _ { \gamma } = ( \gamma - 1 ) \rho _ { \gamma } , plus a scalar field \phi with an exponential potential V \propto \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 barotropic 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 } \rightarrow 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 .