Cosmology with positive and negative exponential potentials

Abstract

We present a phase-plane analysis of cosmologies containing a scalar field $\varphi$ with an exponential potential $V\propto \exp (-\lambda \kappa \varphi )$ where ${\kappa }^2=8\pi G$ and $V$ may be positive or negative. We show that power-law kinetic-potential scaling solutions only exist for sufficiently flat (${\lambda }^2<6$) positive potentials or steep (${\lambda }^2>6$) 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 to 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.

8 pages, latex with revtex, 9 figures

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