Efficient calculation of cosmological neutrino clustering in the non-linear regime

We study in detail how neutrino perturbations can be followed in linear theory by using only terms up to l = 2 in the Boltzmann hierarchy. We provide a new approximation to the third moment and demonstrate that the neutrino power spectrum can be calculated to a precision of better than 5% for masses up to 1 eV and k less than or similar to 10 h/Mpc. The matter power spectrum can be calculated far more precisely and typically at least a factor of a few better than with existing approximations. We then proceed to study how the neutrino power spectrum can be reliably calculated even in the non-linear regime by using the non-linear gravitational potential, sourced by dark matter overdensities, as it is derived from semi analytic methods based on N-body simulations in the Boltzmann evolution hierarchy. Our results agree extremely well with results derived from N-body simulations that include cold dark matter and neutrinos as independent particles with different properties.