Modeling the Anisotropic Two-Point Galaxy Correlation Function on Small Scales and Improved Measurements of H(z), D_A(z), and f(z)sigma_8(z) from the Sloan Digital Sky Survey DR7 Luminous Red Galaxies

TL;DR

This work develops a simple, efficient phenomenological model for the anisotropic 2D two-point galaxy correlation function that remains accurate from large scales down to $s\approx25\,h^{-1}\mathrm{Mpc}$ by incorporating nonlinear effects, a scale-dependent small-scale bias, and scale- and direction-dependent redshift-space distortions via a dewiggled power spectrum and velocity-dispersion modeling. The method combines large-scale linear theory with a small-scale bias kernel and a velocity distribution, implemented with fast one-dimensional convolutions and multipole expansions, enabling robust MCMC analyses. Validation with LasDamas mocks demonstrates accurate recovery of $H(0.35)$, $D_A(0.35)$, $\Omega_m h^2$, and growth-related quantities, and application to SDSS DR7 LRGs yields $H(0.35)r_s(z_d)/c=0.0433\pm0.0042$, $D_A(0.35)/r_s(z_d)=6.59\pm0.46$, and $f(0.35)\sigma_8(0.35)=0.429\pm0.089$ at $z=0.35$ over $25<s<120\,h^{-1}\mathrm{Mpc}$. The results, along with the provided covariance, offer a practical path to tighter dark energy and gravity constraints from current and future galaxy clustering data.

Abstract

We present a simple and efficient phenomenological model for the two-dimensional two-point galaxy correlation function that works well over a wide range of scales, from large scales down to scales as small as 25Mpc/h. Our model incorporates nonlinear effects, a scale-dependent galaxy bias on small scales, and allows the redshift-space distortions to be scale and direction dependent. We validate our model using LasDamas mock catalogs, and apply it to the Sloan Digital Sky Survey (SDSS) DR7 Luminous Red Galaxies (LRGs). Using only the monopole and quadrupole of the correlation function measured from the SDSS DR7 LRGs, we obtain improved measurements H(z)r_s(z_d)/c=0.0433\pm 0.0042, D_A(z)/r_s(z_d)=6.59\pm 0.46, and f(z)sigma_8(z)=0.429\pm 0.089 at z=0.35, using the scale range of 25<s<120Mpc/h. We expect our results and model to be useful in tightening dark energy and gravity constraints from the full analysis of current and future galaxy clustering data.

Figures (5)

• Figure 1: The normalized quadrupoles from the correlation functions computed with FFT and our convolution method. One can see that the results from FFT are converging to the result of the convolution method. In addition, FFT method just reaches reasonable convergence with box size = $1024^3$ (Mpc/h)$^3$ and grid size = $1024^3$ for the scales considered in this study. One would need to increase the box size or grid size if one want to include other scales. We also plot the $Q(s)$ from one-dimensional dewiggle model for comparison. It is a constant since the only redshift distortion effect comes from the Kaiser boost.
• Figure 2: The average two-dimensional two-point correlation function (2D 2PCF) measured from 160 LasDamas SDSS LRGfull mock catalogs (solid black contours), compared to a theoretical model with the input parameters of the LasDamas simulations and $\{\beta$, $b_A$, $\sigma_{v,0}$, $c_\mu$, $c_\sigma\}$ are set to $\{0.316$, $-0.0385$, $50$km s$^{-1}$, $10$, $4\}$ (dashed red contours). The thick dashed blue circle denotes the baryon acoustic oscillation scale. The contour levels are $\xi=2, 0.5, 0.1, 0.025, 0.01, 0.005, 0$. The $\xi=0$ contours are denoted with dotted lines for clarity.
• Figure 3: The averaged monopole of the correlation functions of the mock catalogs (black squares) comparing to the fitting model of this study (red dots) and a simpler model (linear model + 1D dewiggle damping + constant velocity dispersion,blue crosses). The error bars are taken as $1/\sqrt{160}$ of the square roots of the diagonal elements of the covariance matrix.
• Figure 4: The averaged quadrupole of the correlation functions of the mock catalogs (black squares) comparing to the fitting model of this study (red dots) and a simpler model (linear model + 1D dewiggle damping + constant velocity dispersion,blue crosses). The error bars are taken as $1/\sqrt{160}$ of the square roots of the diagonal elements of the covariance matrix.
• Figure 5: The averaged hexadecapole of the correlation functions of the mock catalogs (black squares) comparing to the fitting model of this study (red dots) and a simpler model (linear model + 1D dewiggle damping + constant velocity dispersion,blue crosses). The error bars are taken as $1/\sqrt{160}$ of the square roots of the diagonal elements of the covariance matrix.