Scaling solutions on a brane

TL;DR

This paper explores the dynamics of a brane universe with matter and scalar fields, analyzing scaling solutions and their stability, especially focusing on the effects of different potentials and the nonstandard brane regime.

Contribution

It provides a detailed analysis of scaling solutions in brane cosmology with various scalar field potentials, including stability and existence conditions, highlighting differences from standard cosmology.

Findings

Scaling solutions depend on the potential's power-law index.

In the nonstandard brane regime, solutions differ significantly from standard cosmology.

Stability regions for solutions are characterized and analyzed.

Abstract

We investigate the dynamics of a flat isotropic brane Universe with two-component matter source: perfect fluid with the equation of state $p=(\gamma-1) \rho$ and a scalar field with a power-law potential $V \sim \phi^{\alpha}$. The index $\alpha$ can be either positive or negative. We describe solutions for which the scalar field energy density scales as a power-law of the scale factor (so called scaling solutions). In the nonstandard brane regime when the brane is driven by energy density square term these solutions are rather different from their analogs in the standard cosmology. A particular attention is paid to the inverse square potential. Its dynamical properties in the nonstandard brane regime are in some sense analogous to those of the exponential potential in the standard cosmology. Stability analysis of the scaling solutions are provided. We also describe solutions existing in regions of the parameter space where the scaling solutions are unstable or do not exist.