Cosmology with positive and negative exponential potentials

TL;DR

This work classifies the qualitative cosmological dynamics of a scalar field with an exponential potential, allowing both signs for V, using one- and two-dimensional phase-space analyses. The authors identify fixed points corresponding to kinetic domination and potential–kinetic scaling, detailing their existence and stability across regimes defined by λ^2 and the sign of V. In a single-field FRW setting, a flat positive potential yields a late-time scaling attractor leading to power-law inflation when λ^2<6, whereas negative potentials can lead to recollapse or, for steep negatives, an unstable scaling attractor in collapsing phases. When a barotropic fluid is included, the two-dimensional phase-space reveals fluid-dominated and fluid–scalar scaling solutions, with stability depending on γ and λ^2; in particular, the steep negative potential (ekpyrotic) regime produces a robust, ultra-stiff attracting solution in collapsing universes, stable against matter, curvature, and shear perturbations. The results have implications for ekpyrotic and pre-big-bang scenarios, brane-world cosmologies, and the general understanding of scalar-field driven dynamics near cosmological singularities.

Abstract

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

Figures (9)

• Figure 1: One-dimensional phase-space for flat positive potentials, $\lambda^2<6$. Arrows indicate evolution in cosmic time, $t$. Note that in the lower half-plane, $H<0$, this has the opposite sense to $N\equiv\ln(a)$.
• Figure 2: One-dimensional phase-space for steep positive potentials, $\lambda^2>6$.
• Figure 3: One-dimensional phase-space for flat negative potentials, $\lambda^2<6$.
• Figure 4: One-dimensional phase-space for steep negative potentials, $\lambda^2>6$.
• Figure 5: Two-dimensional phase-space for flat positive potentials, $\lambda^2<3\gamma$.