Linear power spectra in cold plus hot dark matter models: Analytical approximations and applications

This paper presents simple analytic approximations to the linear power spectra, linear growth rates, and rms mass fluctuations for both components in a family of cold + hot dark matter (CDM + HDM) models that are of current cosmological interest. The formulas are valid for a wide range of wavenumbers, neutrino fractions, redshifts, and Hubble constants: k less than or similar to 10 h Mpc(-1), 0.05 less than or similar to Omega(v) less than or similar to 0.3, 0 less than or equal to z less than or similar to 15, and 0.5 less than or similar to h less than or similar to 0.8. A new, redshift-dependent shape parameter, Gamma(v) = a(1/2)Omega(v)h(2), is introduced to simplify the multidimensional parameter space and to characterize the effect of massive neutrinos on the power spectrum. The physical origin of Gamma(v) lies in the neutrino free-streaming process, and the analytic approximations can be simplified to depend only on this variable and Omega(v). Linear calculations with these power spectra as input are performed to compare the predictions of Omega(v) less than or similar to 0.3 models with observational constraints from the reconstructed linear power spectrum and cluster abundance. The usual assumption of an exact scale-invariant primordial power spectrum is relaxed to allow a spectral index of 0.8 less than or similar to n less than or equal to 1. It is found that a slight tilt of n = 0.9 (no tensor mode) or n = 0.95 (with tensor mode) in Omega(v) similar to 0.1-0.2 CDM + HDM models gives a power spectrum similar to that of an open CDM model with a shape parameter Gamma = 0.25, providing good agreement with the power spectrum reconstructed by Peacock & Dodds and the observed cluster abundance at low redshifts. Late galaxy formation at high redshifts, however, will be a more severe problem in tilted models.