Qualitative and numerical study of the matter-radiation interaction in Kantowski-Sachs cosmologies

We examine, from both a qualitative and a numerical point of view, the evolution of Kantowski-Sachs cosmological models whose source is a mixture of a gas of weakly interacting massive particles (WIMP’s) and a radiative gas made up of a “tightly coupled” mixture of electrons, baryons and photons. Our analysis is valid from the end of nucleosynthesis up to the duration of radiative interactions ${(10\mathrm{}}^6\mathrm{K}>T>\mathrm{}4\mathrm{}\times {10}^3\mathrm{K}).\mathrm{}$ In this cosmic era annihilation processes are negligible, while the WIMP’s only interact gravitationally with the radiative gas and the latter behaves as a single dissipative fluid that can be studied within a hydrodynamical framework. Applying the full transport equations of extended irreversible thermodynamics, coupled with the field and balance equations, we obtain a set of governing equations that becomes an autonomous system of ordinary differential equations once the shear viscosity relaxation time ${\tau }_{\mathrm{rel}}$ is specified. Assuming that ${\tau }_{\mathrm{rel}}$ is proportional to the Hubble time, the qualitative analysis indicates that models begin in the radiation-dominated epoch close to an isotropic equilibrium point (saddle). We show how the form of ${\tau }_{\mathrm{rel}}$ governs the relaxation time scale of the models towards an equilibrium photon entropy, leading to “near-Eckart” and transient regimes associated with “abrupt” and “smooth” relaxation processes, respectively. Assuming the WIMP particle to be a supersymmetric neutralino with a mass $m_{\mathrm{w}}\sim 100\mathrm{GeV},$ the numerical analysis reveals that a physically plausible evolution, compatible with a stable equilibrium state and with observed bounds on CMB anisotropies and neutralino abundance, is only possible for models characterized by initial conditions associated with nearly zero spatial curvature and total initial energy density close to unity. An expression for the relaxation time, complying with physical requirements, is obtained in terms of the dynamical equations. It is also shown that the “truncated” transport equation does not give rise to acceptable physics.