Neutrino mass, dark energy, and the linear growth factor

We study the degeneracies between neutrino mass and dark energy as they manifest themselves in cosmological observations. In contradiction to a popular formula in the literature, the suppression of the matter power spectrum caused by massive neutrinos is not just a function of the ratio of neutrino to total mass densities ${f}_{\ensuremath{\nu}}={\ensuremath{\Omega}}_{\ensuremath{\nu}}/{\ensuremath{\Omega}}_{m}$, but also each of the densities independently. We also present a fitting formula for the logarithmic growth factor of perturbations in a flat universe, $f(z,k;{f}_{\ensuremath{\nu}},w,{\ensuremath{\Omega}}_{\mathrm{DE}})\ensuremath{\approx}[1\ensuremath{-}A(k){\ensuremath{\Omega}}_{\mathrm{DE}}{f}_{\ensuremath{\nu}}+B(k){f}_{\ensuremath{\nu}}^{2}\ensuremath{-}C(k){f}_{\ensuremath{\nu}}^{3}]{\ensuremath{\Omega}}_{m}^{\ensuremath{\alpha}}(z)$, where $\ensuremath{\alpha}$ depends on the dark energy equation of state parameter $w$. We then discuss cosmological probes where the $f$ factor directly appears: peculiar velocities, redshift distortion, and the integrated Sachs-Wolfe effect. We also modify the approximation of Eisenstein and Hu [Astrophys. J. 511, 5 (1999)] for the power spectrum of fluctuations in the presence of massive neutrinos and provide a revised code [http://www.star.ucl.ac.uk/~lahav/nu_matter_power.f].

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