The 2dF Galaxy Redshift Survey: The power spectrum and the matter content of the universe

TL;DR

The paper analyzes the 2dF Galaxy Redshift Survey power spectrum using a direct FFT-based estimator with a detailed window-function model to connect galaxy clustering to the linear matter power spectrum. It employs Eisenstein & Hu transfer functions and explores cosmologies parameterized by $\Omega_m h$ and $\Omega_b/\Omega_m$, incorporating mock catalogs to calibrate the covariance and test robustness. The main results favor a low-density universe with $\Omega_m h \approx 0.20$ and $\Omega_b/\Omega_m \approx 0.15$, with mild evidence for baryon oscillations and good consistency with CMB and BBN constraints. This work demonstrates the power of combining large redshift surveys with precise window-convolution modeling to constrain cosmological parameters from the shape of $P(k)$.

Abstract

The 2dF Galaxy Redshift Survey has now measured in excess of 160000 galaxy redshifts. This paper presents the power spectrum of the galaxy distribution, calculated using a direct FFT-based technique. We argue that, within the k-space region 0.02<k<0.15 h Mpc^-1, the shape of this spectrum should be close to that of the linear density perturbations convolved with the window function of the survey. This window function and its convolving effect on the power spectrum estimate are analyzed in detail. By convolving model spectra, we are able to fit the power-spectrum data and provide a measure of the matter content of the universe. Our results show that models containing baryon oscillations are mildly preferred over featureless power spectra. Analysis of the data yields 68% confidence limits on the total matter density times the Hubble parameter Ω_m h = 0.20 +/- 0.03, and the baryon fraction Ω_b/Ω_m = 0.15 +/- 0.07, assuming scale-invariant primordial fluctuations.

Figures (8)

• Figure 1: The 2dFGRS window function in Fourier space. The fitting formula of Equation \ref{['eq:win_fit']} is plotted (solid line) compared to the spherically averaged power obtained by Fourier transforming a random catalogue created to match the 2dFGRS window function (solid circles). This catalogue, containing 5 times as many points as the true galaxy catalogue was placed on a $512\times512\times256$ grid that encompassed all the data. Values from individual grid points are also plotted (dots), and show the anisotropy of the window function.
• Figure 2: The effect of convolving a linear power spectrum with the 2dFGRS window function. Two model power spectra are compared: The top panel shows a power spectrum including a baryonic component $P(k,\Omega_mh=0.2,\Omega_b/\Omega_m=0.16)$ while the bottom panel shows a power spectrum $P(k,\Omega_mh=0.2,\Omega_b/\Omega_m=0)$ with no baryonic component. These spectra are divided by a smooth reference model, $P(k,\Omega_mh=0.2,\Omega_b/\Omega_m=0)$. The upper solid lines show the unconvolved power spectra, and the lower solid lines after numerically convolving with the fit to the window function given by Equation \ref{['eq:win_fit']}. The dashed lines show the average power spectrum expected from data which has had the average over-density artificially set to zero (see Section \ref{['sec:window']} for details). Solid symbols show the shape of the average recovered power spectrum from 1000 Gaussian realizations of the density field (with different phases), placed on a $256\times256\times128$ grid at locations covered by the 2dFGRS. Due to the relatively coarse grid used, aliasing swamps the signal at $k>0.15\,h\,{\rm Mpc}^{-1}$, and data are only shown for the range of $k$ over which the 2dFGRS power spectrum is fitted (Section \ref{['sec:fit']}). This region is delineated by the vertical dotted lines.
• Figure 3: The 2dFGRS estimate of the redshift-space galaxy power spectrum, expressed as the ratio to a linear-theory CDM $P(k,\Omega_mh=0.2,\Omega_b/\Omega_m=0)$ power spectrum with $n=1$ & $\sigma_8=1$. These data do not estimate the true power spectrum, but give the power spectrum convolved with the window function (see Section \ref{['sec:window']}). Error bars are determined from the diagonal elements of the covariance matrix (calculated in Section \ref{['sec:cov_mat']}), for the $0.02\,h\,{\rm Mpc}^{-1}<k<0.15\,h\,{\rm Mpc}^{-1}$ data constrained by the vertical dotted lines. This is the region fitted in Section \ref{['sec:fit_mocks']}.
• Figure 4: A comparison of power spectra determined from the results of the $\Lambda$CDM and $\tau$CDM Hubble volume simulations divided by a $P(k,\Omega_mh=0.2,\Omega_b/\Omega_m=0)$ reference model. (a) the average spectrum recovered from the $z=0$ distribution of mass in 8 cubes each of side $1500\,h^{-1}\,{\rm Mpc}$ ($\Lambda$CDM) or $1000\,h^{-1}\,{\rm Mpc}$ ($\tau$CDM) (solid circles with 1$\sigma$ errors: the extremal data bound the shaded area), compared to the scaled input power spectrum (solid line). Here, the difference between the two is dominated by the non-linear collapse of small-scale structures at $k>0.15\,h\,{\rm Mpc}^{-1}$. The $\tau$CDM mass power spectrum is compared to a reference model with $\sigma_8=0.6$ (rather than $\sigma_8=1.0$) in order that it has approximately the same normalization as the other data. (b) the average recovered power spectrum from 35 ($\Lambda$CDM) and 30 ($\tau$CDM) real-space galaxy catalogues sampled from the simulations to match the 2dFGRS window function (solid circles with 1$\sigma$ errors: the extremal data bound the shaded area). These are compared to the linear spectrum convolved with the analytic approximation to the window function given by Equation \ref{['eq:win_fit']} (solid line). The difference between these two is dominated by both bias (used to create the galaxy catalogue) and non-linear effects. (c) the average recovered power spectra from 35 ($\Lambda$CDM) and 30 ($\tau$CDM) redshift-space galaxy catalogues designed to fully mimic the 2dFGRS redshift-space sample (solid circles with 1$\sigma$ errors: the extremal data bound the shaded area), and the linear spectra convolved with the window function (solid line). Here, the differences are due to all of the factors listed in Section \ref{['sec:FKP_method']}. The dashed lines in panels (b) & (c) show the power spectrum expected from data in which the average over-density is artificially set to zero (see Section \ref{['sec:window']} for details). The $0.02\,h\,{\rm Mpc}^{-1}<k<0.15\,h\,{\rm Mpc}^{-1}$ region fitted in Section \ref{['sec:fit_mocks']} is delineated by the vertical dotted lines. In calculating the error in the average power spectra measured from the simulations, we have assumed that the samples are independent. This is not precisely true, and the errors plotted therefore underestimate the true error.
• Figure 5: Likelihood surfaces for the best fit linear power spectrum over the region $0.02\,h\,{\rm Mpc}^{-1}<k<0.15\,h\,{\rm Mpc}^{-1}$. The normalization is a free parameter to account for the unknown large-scale biasing. Contours are plotted at the usual positions for one-parameter confidence of 68%, and two-parameter confidence of 68%, 95% and 99% (i.e. $-2\ln({\cal L}/{\cal L_{\rm max}}) = 1, 2.3, 6.0, 9.2$). We have marginalized over the missing free parameters ($h$ and the power spectrum amplitude) by integrating under the likelihood surface.