Linear Power Spectra in Cold+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 wavenumber, neutrino fraction, redshift, and Hubble constant: k <∼ 10h Mpc −1, 0.05 <∼ Ων <∼ 0.3, 0 ≤ z <∼ 15, and 0.5 <∼ h <∼ 0.8. A new, redshift-dependent shape parameter Γν = a Ωνh 2 is introduced to simplify the multi-dimensional parameter space and to characterize the effect of massive neutrinos on the power spectrum. The physical origin of Γν lies in the neutrino free-streaming process, and the analytic approximations can be simplified to depend only on this variable and Ων . Linear calculations with these power spectra as input are performed to compare the predictions of Ων <∼ 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 <∼ n ≤ 1. It is found that a slight tilt of n = 0.9 (no tensor mode) or n = 0.95 (with tensor mode) in Ων ∼ 0.1 − 0.2 CDM+HDM models gives a power spectrum similar to that of an open CDM model with a shape parameter Γ = 0.25, providing good agreement with the power spectrum reconstructed by Peacock and Dodds (1994) and the observed cluster abundance. Subject headings: cosmology : theory – dark matter – elementary particles – large-scale structure of universe