Dust-radiation universes: stability analysis

In this paper we consider flat and open universe models containing a mixture of cold matter (dust) and radiation, interacting only through gravity, with the aim of studying their stability with respect to linear scalar perturbations. To this end, we consider the perturbed universe as a dynamical system, described by coupled differential equations for a gauge-invariant perturbation variable and a relevant background variable. The phase-space analysis of this dynamical system shows that flat dust-radiation models are unstable, and open models are marginally stable, with respect to adiabatic perturbations. For flat models, there are actually three different regimes of evolution for the perturbations, depending on their wavelength: (i) large-scale perturbations that grow unbounded (unstable modes); (ii) overdamped intermediate-scale perturbations that decay; (iii) damped small-scale perturbations that oscillate (wave modes). The transition scales from one regime to another are today of the order of 60 Mpc, and are determined by a critical wavenumber for the perturbations, k_EC (an invariant of the model), which in turn depends on the only scale entering the model, i.e. the Hubble radius H^-1_E at the equidensity of matter and radiation. We find that k_EC < k_JE (the Jeans wavenumber at equidensity), which implies that there are perturbations that decay even if their wavelength at equidensity is larger than the corresponding Jeans scale. We also briefly discuss metric and curvature perturbations: in flat models the large-scale components of a suitably defined metric perturbation Φ_n (which plays the role of the Newtonian potential) grow to a constant value, but those of the dimensionless scalar E/H² (E is the tidal field magnitude) grow unbounded as the corresponding density perturbations. In open models, both the curvature perturbation E/H² and Φ_n decay, irrespective of the scale, while the corresponding density perturbations are frozen-in to a constant value. We believe that the analysis given here gives a clearer idea of the stability properties of realistic universe models than the standard analysis based on the Jeans scale, despite our simplifying assumptions.