[Paper Review] The 3D power spectrum of galaxies from the SDSS

Max Tegmark, Michael R. Blanton+33 more|CERN Bulletin|Oct 27, 2003

Galaxies: Formation, Evolution, PhenomenaPhysics and Astronomy943 citations

TL;DR

This paper presents a high-precision measurement of the three-dimensional real-space matter power spectrum $P(k)$ from the Sloan Digital Sky Survey (SDSS) using 205,443 galaxies. Employing a matrix-based pseudo-Karhunen-Loève eigenmode method, it achieves uncorrelated, minimum-variance estimates across 22 $k$-bands from $0.02 o0.3 ext{ }h/{ m Mpc}$, with robust corrections for systematics, nonlinear redshift distortions, and luminosity-dependent bias, yielding a curvature in $P(k)$ well-fit by a flat adiabatic model with $h ilde{ ho}_m = 0.213 \pm 0.023$ and $\sigma_8 = 0.89 \pm 0.02$ for $L_*$ galaxies.

ABSTRACT

We measure the large-scale real-space power spectrum P(k) using a sample of 205,443 galaxies from the Sloan Digital Sky Survey, covering 2417 square degrees with mean redshift z~0.1. We employ a matrix-based method using pseudo-Karhunen-Loeve eigenmodes, producing uncorrelated minimum-variance measurements in 22 k-bands of both the clustering power and its anisotropy due to redshift-space distortions, with narrow and well-behaved window functions in the range 0.02 h/Mpc < k < 0.3h/Mpc. We pay particular attention to modeling, quantifying and correcting for potential systematic errors, nonlinear redshift distortions and the artificial red-tilt caused by luminosity-dependent bias. Our final result is a measurement of the real-space matter power spectrum P(k) up to an unknown overall multiplicative bias factor. Our calculations suggest that this bias factor is independent of scale to better than a few percent for k<0.1h/Mpc, thereby making our results useful for precision measurements of cosmological parameters in conjunction with data from other experiments such as the WMAP satellite. As a simple characterization of the data, our measurements are well fit by a flat scale-invariant adiabatic cosmological model with h Omega_m =0.201+/- 0.017 and L* galaxy sigma_8=0.89 +/- 0.02 when fixing the baryon fraction Omega_b/Omega_m=0.17 and the Hubble parameter h=0.72; cosmological interpretation is given in a companion paper.

Motivation & Objective

• To measure the three-dimensional real-space matter power spectrum $P(k)$ from the Sloan Digital Sky Survey with high precision and minimal systematics.
• To develop and apply a matrix-based pseudo-Karhunen-Loève eigenmode method that produces uncorrelated, minimum-variance power spectrum estimates across multiple $k$-bands.
• To model, quantify, and correct for systematic errors including angular and radial density fluctuations, nonlinear redshift distortions, and luminosity-dependent bias.
• To assess the robustness of the power spectrum measurement across different sky regions and data subsets, ensuring consistency and reliability.
• To provide a cosmologically interpretable $P(k)$ measurement that can be combined with other datasets, such as WMAP, for precision cosmology.

Proposed method

• The analysis uses a matrix-based method based on pseudo-Karhunen-Loève (PKL) eigenmodes to decompose the galaxy power spectrum into uncorrelated, minimum-variance bandpower estimates across 22 $k$-bands.
• The method computes signal covariance matrices $\mathbf{S}$ and their derivatives $\mathbf{P}_i = \partial\mathbf{S}/\partial p_i$ using a truncated multipole expansion up to $\ell_{\rm cut} = 260$, ensuring numerical convergence of the $\mathbf{P}$-matrices.
• A prior power spectrum is iteratively refined using a BBKS model for $P_{\rm gg}(k)$ and analytic forms for $P_{\rm gv}(k)$ and $P_{\rm vv}(k)$ with $r=1$, $\beta=0.5$, to minimize error bars while maintaining unbiased estimates.
• The method disentangles the galaxy-galaxy ($gg$), galaxy-velocity ($gv$), and velocity-velocity ($vv$) power spectra, allowing for correction of redshift-space distortions and isolation of real-space clustering.
• Systematic effects such as angular and radial density fluctuations are tested by omitting different sky regions and subsets, confirming robustness of the final $P(k)$ result.
• The final $P(k)$ is calibrated to the real-space matter power spectrum up to an unknown multiplicative bias factor, which is shown to be scale-independent to better than a few percent for $k < 0.1\,h/{\rm Mpc}$.

Experimental results

Research questions

• RQ1 What is the shape and amplitude of the three-dimensional real-space matter power spectrum $P(k)$ on large scales from the SDSS galaxy survey?
• RQ2 How can a matrix-based pseudo-Karhunen-Loève method be used to produce uncorrelated, minimum-variance power spectrum estimates with well-behaved window functions?
• RQ3 To what extent do systematics such as density fluctuations, nonlinear redshift distortions, and luminosity-dependent bias affect the measured $P(k)$, and how can they be corrected?
• RQ4 Is the measured $P(k)$ consistent across different regions of the sky and robust to data subset variations?
• RQ5 Can the measured $P(k)$ be well-fit by a flat adiabatic cosmological model, and what are the resulting constraints on cosmological parameters like $h\Omega_m$ and $\sigma_8$?

Key findings

• The measured real-space matter power spectrum $P(k)$ is not well described by a single power law, showing clear curvature across the range $0.02\,h/{\rm Mpc} < k < 0.3\,h/{\rm Mpc}$.
• The power spectrum is robust to omission of angular and radial density fluctuations and shows consistent results across different regions of the sky.
• The overall bias factor relating the observed galaxy power to the true matter power spectrum is found to be independent of scale to better than a few percent for $k < 0.1\,h/{\rm Mpc}$, enabling precise cosmological interpretation.
• The data are well-fit by a flat adiabatic cosmological model with $h\Omega_m = 0.213 \pm 0.023$ and $\sigma_8 = 0.89 \pm 0.02$ for $L_*$ galaxies, assuming $\Omega_b/\Omega_m = 0.17$ and $h = 0.72$.
• The method's numerical convergence is confirmed with $\ell_{\rm cut} = 260$, and results are stable even at lower $\ell_{\rm cut}$ (e.g., 120), indicating robustness of the $P(k)$ estimates.
• The use of iterative priors and a matrix-based approach ensures that error bars are reliable and unbiased, even with imperfect initial assumptions, due to theoretical guarantees from previous work on similar methods.