A Simple Analytic Treatment of Neutrino Mass Impact on the Full Power Spectrum Shape via a Two-Fluid Approximation

Abstract

We present a new closed-form formula for the matter power spectrum in the presence of massive neutrinos that gives an accuracy of better than 5\% on all scales. It is the first closed-form result valid on all scales. To calculate this closed-form solution, we iteratively solve the fluid equations for cold dark matter+baryons, and neutrinos in terms of the neutrino mass fraction, $f_{\nu} \ll 1$, using variation of parameters to construct the response of the matter to the neutrino perturbation, which in turn will source the higher-order neutrino perturbations. This analytic solution accelerates calculations of the matter power spectrum with neutrinos. Also, it enables one to calculate the cross power spectra of matter and neutrinos which in turn can be used in computing the higher-order corrections to the power spectrum. To demonstrate our formula's accuracy and utility, we perform a Fisher forecast with it and show we reproduce the forecast from the Boltzmann solver class.