Linear Power Spectra in Cold+Hot Dark Matter Models: Analytical Approximations and Applications

TL;DR

The paper develops compact analytic approximations for the linear power spectra, growth rates, and rms mass fluctuations in CDM+HDM cosmologies with massive neutrinos, introducing the neutrino-free-streaming shape parameter $Γ_ν = a^{1/2}\,Ω_ν\,h^2$ to collapse parameter dependence. It presents explicit fits for the growth rate $f$, the CDM and HDM spectra, the density-weighted spectrum, and $σ$, achieving typical accuracy better than ~10% for $k \lesssim 20\,h\mathrm{Mpc}^{-1}$ and $0\le z \lesssim 15$. By comparing to reconstructed linear $P(k)$ from Peacock & Dodds and to observed cluster abundances, the work shows that small primordial tilts ($n \sim 0.9$–$0.95$) with modest neutrino content ($Ω_ν \sim 0.1$–$0.3$) yield good agreement, effectively linking neutrino physics to large-scale structure data. These analytic tools enable rapid, reliable linear calculations and serve as practical inputs for generating initial conditions in CDM+HDM simulations.

Abstract

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 $Γ_ν=a^{1/2}\onu h^2$ 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 $Γ_ν$ 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 $Γ=0.25$, providing good agreement with the power spectrum reconstructed by Peacock and Dodds (1994) and the observed cluster abundance.

Figures (9)

• Figure 1: Present-day linear power spectra for the standard CDM model (dotted) and four CDM+HDM models with $n=1$, $\Omega_b$, and $H_0=50$ km s$^{-1}$ Mpc$^{-1}$, computed from integrations of the Boltzmann equations. The four CDM+HDM models have $\Omega_\nu=0.05, 0.1, 0.2$, and 0.3, and the power in the cold (solid) and hot (dashed) components are shown separately. All are normalized to the 4-year COBE result $Q_{\rm rms-PS}=18\,\mu K$ (Gorski et al. 1996).
• Figure 2: Direct integration of the Boltzmann equations (solid) vs. analytic fitting results (dashed) for the present-day CDM and HDM power spectra and the growth rate in four CDM+HDM models ($n=1, H_0=50$) with $\Omega_\nu=0.05, 0.1, 0.2$, and 0.3. (a) Growth rate $f(k,a,\Omega_\nu)$ of the density field. The fitting formula is given by eq. (\ref{['f']}). (b) Ratio of the CDM power spectrum in CDM+HDM models to that in the pure CDM model, $g=P_c(\Omega_\nu)/P_c(\Omega_\nu=0)$. The fitting formula is given by eq. (\ref{['g']}). (c) Ratio of the HDM to CDM power spectrum, ${\cal H}=P_\nu(\Omega_\nu)/P_c(\Omega_\nu)$. The fitting formula is given by eq. (\ref{['h']}).
• Figure 3: Same as Figure \ref{['fig:fgh1']} but for $a=0.1$
• Figure 4: Scaling of $k$ with the Hubble constant for functions $f$, $g$ and ${\cal H}$ in the $\Omega_\nu=0.2$, $n=1$ CDM+HDM model at $a=1$. As discussed in the text, $f$ and $\cal H$ scale perfectly with $k/h^3$, while $k/h^2$ is a good approximation for $g$ for a large range of $k$.
• Figure 5: Root-mean-square of the linear mass fluctuations $\sigma(R,\Omega_\nu)$ in spheres of radius $R h^{-1}$ Mpc for $n=1$ and $H_0=50$ models with $\Omega_\nu=0.0, 0.05, 0.1, 0.2$, and 0.3 (top down). The solid and dashed curves are from numerical integration and the fitting formulas eqs. (\ref{['sigfitc']}) and (\ref{['sigfit']}), respectively. The dotted curves show the scaled $a^{-1}\sigma(R,\Omega_\nu\,,a)$ for $a=0.1$, which start to deviate from $\sigma(R,\Omega_\nu,a=1)$ only at small scale. All models are normalized to $Q_{\rm rms-PS}=18\,\mu K$ at large $R$.