Stability of cosmological scaling solutions

We study the stability of cosmological scaling solutions within the class of spatially homogeneous cosmological models with a perfect fluid subject to the equation of state $p_{\gamma }=\left(\gamma -1\right){\rho }_{\gamma }$ (where γ is a constant satisfying $0<\mathrm{}\gamma <2)$ and a scalar field with an exponential potential. The scaling solutions, which are spatially flat isotropic models in which the scalar field energy density tracks that of the perfect fluid, are of physical interest. For example, in these models a significant fraction of the current energy density of the Universe may be contained in the scalar field whose dynamical effects mimic cold dark matter. It is known that the scaling solutions are late-time attractors (i.e., stable) in the subclass of flat isotropic models. We find that the scaling solutions are stable (to shear and curvature perturbations) in generic anisotropic Bianchi models when $\gamma <2/3.\mathrm{}$ However, when $\gamma >2/3,$ and particularly for realistic matter with $\gamma >~1,$ the scaling solutions are unstable; essentially they are unstable to curvature perturbations, although they are stable to shear perturbations. We briefly discuss the physical consequences of these results.