Cosmology with positive and negative exponential potentials

Abstract

We present a phase-plane analysis of cosmologies containing a scalar field ϕ with an exponential potential V ∝ exp(−λ κ ϕ) where κ2 = 8πG and V may be positive or negative. We show that power-law kinetic–potential scaling solutions only exist for sufficiently flat (λ2 < 6) positive potentials or steep (λ2 > 6) negative potentials. The latter correspond to a class of ever-expanding cosmologies with negative potential. However, we show that these expanding solutions with a negative potential are unstable in the presence of ordinary matter, spatial curvature or anisotropic shear, and generic solutions always recollapse to a singularity. Power-law kinetic–potential scaling solutions are the late-time attractor in a collapsing universe for steep negative potentials (the ekpyrotic scenario) and stable against matter, curvature or shear perturbations. Otherwise kinetic-dominated solutions are the attractor during collapse (the pre-big-bang scenario) and are only marginally stable with respect to anisotropic shear.