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: $k\lo 10\,h$ Mpc$^{-1}$, $0.05\lo \onu\lo 0.3$, $0\le z\lo 15$, and $0.5\lo h \lo 0.8$. A new, redshift-dependent shape parameter $\Gamma_\nu=a^{1/2}\onu h^2$ 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 $\Gamma_\nu$ lies in the neutrino free-streaming process, and the analytic approximations can be simplified to depend only on this variable and $\onu$. Linear calculations with these power spectra as input are performed to compare the predictions of $\onu\lo 0.3$ 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 $0.8\lo n\le 1$. It is found that a slight tilt of $n=0.9$ (no tensor mode) or $n=0.95$ (with tensor mode) in $\onu\sim 0.1-0.2$ 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 and Dodds (1994) and the observed cluster abundance.