  By numerically solving the quantum kinetic equations we compute the range of
parameters where the $\nu_\mu \to \nu_s$ oscillation solution to the
atmospheric neutrino anomaly is consistent with a stringent big bang
nucleosynthesis (BBN) bound of $N_{eff}^{BBN} \stackrel{<}{\sim} 3.6$. We show
that this requires tau neutrino masses in the range $m_{\nu_\tau} \stackrel
{>}{\sim} 4 eV$ (for $|\delta m^2_{atm}| = 10^{-2.5} eV^2$). We discuss the
implications of this scenario for hot+cold dark matter, BBN, and the anisotropy
of the cosmic microwave background.
