Phase--space Distribution of Volatile Dark Matter

Abstract

We discuss the phase--space distribution of $\mu$ neutrinos if $\tau$ neutrinos are unstable and decay into $\nu_\mu + scalar$. If this scalar is a familon or a Majoron, in the generic case the $\nu_\mu$ background is NOT the straightforward overlap of neutrinos of thermal and decay origins. A delay in $\nu_\tau$ decay, due to the Pauli exclusion principle, can modify it in a significant way. We provide the equations to calculate the $\nu_\mu$ distribution and show that, in some cases, there exists a good approximate solution to them. However, even when such solution is not admitted, the equations can be numerically solved following a precise pattern. We give such a solution for a number of typical cases. If $\nu_\mu$ has a mass $\sim 2$ eV and the see--saw argument holds, $\nu_\tau$ must be unstable and the decay into $\nu_\mu + scalar$ is a reasonable possibility. The picture leads to a delayed equivalence redshift, which could allow to reconcile COBE data with a bias parameter $b\ge 1$.