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(v) = Omega(v)/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(v), w, Omega(DE)) approximate to [1 - A(k)Omega(DE) f(v) + B(k)f(v)(2) - C(k) f(v)(3)] Omega(alpha)(m) (z) , where 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.