Power Spectra for Cold Dark Matter and its Variants

TL;DR

This work presents an accurate fitting formula for the linear transfer function in adiabatic cosmologies containing cold dark matter, baryons, and massive neutrinos, extending to multiple degenerate neutrino species and nonzero curvature or cosmological constant. The method separates time evolution into scale-dependent growth factors and a time-independent master function that encodes the drag-epoch spectrum, with a small-scale suppression modeled by a suppression function and an optional correction for neutrino fractions. The authors achieve roughly 5% accuracy in the transfer function (about 10% in power) across a broad parameter range, and apply the framework to observational constraints from the power spectrum shape, cluster abundances, damped Ly$\alpha$ systems, and the Ly$\alpha$ forest. By enabling efficient exploration of parameter space, this approach facilitates joint constraints on baryon and neutrino content and, ultimately, on neutrino masses, by combining CMB, large-scale structure, and high-redshift data.

Abstract

The bulk of recent cosmological research has focused on the adiabatic cold dark matter model and its simple extensions. Here we present an accurate fitting formula that describes the matter transfer functions of all common variants, including mixed dark matter models. The result is a function of wavenumber, time, and six cosmological parameters: the massive neutrino density, number of neutrino species degenerate in mass, baryon density, Hubble constant, cosmological constant, and spatial curvature. We show how observational constraints---e.g. the shape of the power spectrum, the abundance of clusters and damped Lyman-alpha systems, and the properties of the Lyman-alpha forest--- can be extended to a wide range of cosmologies, including variations in the neutrino and baryon fractions in both high-density and low-density universes.

Figures (4)

• Figure 1: Comparison of the fitting formula to the numerical results of CMBfast (Sel96 1996). (a) Results at $z=0$. (b) $z=9$. Upper panels: Transfer functions divided by a fiducial pure-CDM transfer function, formed by using equation (\ref{['eq:Tzero']}) with $\alpha_\nu=\beta_c=1$ and $q_{\rm eff} = q$. Density-weighted CDM+baryon ( short-dashed) and CDM+baryon+neutrino ( long-dashed) transfer functions are shown. Lower panels: Fractional residuals. The cosmology is $\Omega_0=1$, $\Omega_\nu = 0.2$, $\Omega_b = 0.05$, and $h = 0.5$.
• Figure 2: Same as Figure \ref{['fig:stMDM']}, but for different choices of cosmology. (a) High-baryon model, with $\Omega_b = 0.3$, $\Omega_\nu = 0.2$, $\Omega_0 = 1$, and $h=0.5$. (b) Low-density, flat model, with $\Omega_0 = 0.3$, $\Omega_\Lambda = 0.7$, $\Omega_\nu = 0.06$, $\Omega_b = 0.03$, and $h=0.5$. (c) Model with two degenerate neutrino species ($N_\nu=2$). $\Omega_b = 0.05$, $\Omega_\nu = 0.2$, $\Omega_0 = 1$, and $h=0.5$. (d) Model with high $\Omega_0h^2$, beyond our range of $[0.06, 0.40]$. $\Omega_b = 0.2$, $\Omega_\nu = 0.2$, $\Omega_0 = 1$, and $h=0.8$.
• Figure 3: ( a--d) Ratio of fluctuations within $50h^{-1}{\rm\,Mpc}$ spheres to that within $8h^{-1}{\rm\,Mpc}$ spheres as a function of neutrino and baryon fractions for various cosmologies. The shaded region shows the preferred range of $\sigma_{50}/\sigma_8 = 0.151\pm0.016$. ( a) $\Omega_0 = 1$, $h = 0.5$, $n=1$, $N_\nu = 1$. ( b) As ( a), but with $n=0.95$ and tensors as per power-law inflation. ( c) As ( a), but with two degenerate neutrino species ($N_\nu = 2$). ( d) $\Omega_0 = 0.35$, $\Omega_\Lambda = 0.65$, $h = 0.7$, $n=1$, $N_\nu = 1$. All are COBE-normalized. ( e--h) Amplitude of fluctuations within $8h^{-1}{\rm\,Mpc}$ spheres for the cosmologies given in ( a--d), respectively. The shaded region is the preferred range of $\sigma_8 = (0.5\pm0.15)\Omega_0^{-0.65}$ (Pen97 1997) and $\sigma_8 > 0.59$ (Fan97 1997).
• Figure 4: ( a--d) Amplitude of fluctuations at $z=4$ within a Gaussian window of mass corresponding to halos of $50{\rm\,km\,\,s^{-1}\,\,Mpc^{-1}}$ circular velocity. Cosmologies are as per Figure \ref{['fig:lss']}. The shaded region indicates cosmologies where the neutral gas in halos of $v_c>50{\rm\,km\,\,s^{-1}\,\,Mpc^{-1}}$ [using the prescription of Kly95 (1995) and $f_{{\rm H\,\sc i}}=1$] exceeds that observed in damped Ly$\alpha$ systems. The region to the left of the dashed line is the allowed region for $f_{{\rm H\,\sc i}}=0.1$. ( e--h) Amplitude of fluctuations at $z=3$ for a Gaussian window of radius $0.0416(\Omega_0h^2)^{-1/2}{\rm\,Mpc}$, suggested by Gne97 (1997) as a indicator of the slope of the column-density distribution of the Ly$\alpha$ forest. The region $\sigma_{{\rm Ly}\alpha}>2$ is shaded. Cosmologies are as in Figure \ref{['fig:lss']}.