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

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