Phase--space Distribution of Volatile Dark Matter

We discuss the phase--space distribution of $μ$ neutrinos if $τ$ neutrinos are unstable and decay into $ν_μ+ scalar$. If this scalar is a familon or a Majoron, in the generic case the $ν_μ$ background is NOT the straightforward overlap of neutrinos of thermal and decay origins. A delay in $ν_τ$ decay, due to the Pauli exclusion principle, can modify it in a significant way. We provide the equations to calculate the $ν_μ$ 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 $ν_μ$ has a mass $\sim 2$ eV and the see--saw argument holds, $ν_τ$ must be unstable and the decay into $ν_μ+ scalar$ is a reasonable possibility. The picture leads to a delayed equivalence redshift, which could allow to reconcile COBE data with a bias parameter $b\ge 1$.

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