Measurements of H(z) and D_A(z) from the Two-Dimensional Two-Point Correlation Function of Sloan Digital Sky Survey Luminous Red Galaxies

TL;DR

The paper develops and validates a method to extract the expansion rate $H(z)$ and angular diameter distance $D_A(z)$ from the two-dimensional two-point correlation function $\xi(\sigma,\,\pi)$ of galaxies, without assuming a specific dark energy model or spatial curvature. By combining a dewiggled nonlinear power spectrum, a Kaiser redshift-space distortion model, and a velocity-dispersion convolution, the authors fit $H(z)$ and $D_A(z)$ through a scaling transformation that maps theoretical predictions to fiducial coordinates, enabling a likelihood analysis with CosmoMC. Applying the method to SDSS DR7 LRGs (and validating with LasDamas mocks) yields $H(0.35)=82.1^{+4.8}_{-4.9}$ km s$^{-1}$ Mpc$^{-1}$ and $D_A(0.35)=1048^{+60}_{-58}$ Mpc, with derived quantities $H(0.35)r_s(z_d)/c=0.0434\pm0.0018$ and $D_A(0.35)/r_s(z_d)=6.60\pm0.26$, which are nearly uncorrelated and highly informative when combined with CMB data. The results demonstrate the feasibility and robustness of measuring both BAO radial and transverse scales from galaxy clustering, offering valuable inputs for dark energy studies and future surveys.

Abstract

We present a method for measuring the Hubble parameter, H(z), and angular diameter distance, D_A(z), from the two-dimensional two-point correlation function, and validate it using LasDamas mock galaxy catalogs. Applying our method to the sample of luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7), we measure H(z=0.35)=82.1_{-4.9}^{+4.8} km s^{-1}Mpc^{-1}, D_A(z=0.35)=1048_{-58}^{+60} Mpc without assuming a dark energy model or a flat Universe. We find that the derived measurements of H(0.35)r_s(z_d)/c and D_A(0.35)/r_s(z_d) (where r_s(z_d) is the sound horizon at the drag epoch) are nearly uncorrelated, have tighter constraints and are more robust with respect to possible systematic effects. Our galaxy clustering measurements of {H(0.35)r_s(z_d)/c, D_A(0.35)/r_s(z_d)}={0.0434\pm 0.0018,6.60\pm 0.26} (with the correlation coefficient r = 0.0604) can be used to combine with cosmic microwave background and any other cosmological data sets to constrain dark energy. Our results represent the first measurements of H(z) and D_A(z) (or H(z)r_s(z_d)/c and D_A(0.35)/r_s(z_d)) from galaxy clustering data. Our work has significant implications for future surveys in establishing the feasibility of measuring both H(z) and D_A(z) from galaxy clustering data.

Figures (7)

• Figure 1: \ref{['fig:sdssfull_twod']} The two-dimensional two-point correlation function (2D 2PCF) measured from SDSS DR7 LRGs in a redshift range $0.16 < z < 0.44$ (solid black contours), compared to a theoretical correlation function with parameters close to the best fit values in the likelihood analysis (dashed red contours). The theoretical model has $H(z=0.35)=81.8\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$, $D_A(z=0.35)=1042 \,{\rm Mpc}$, $\beta=0.35$, $\Omega_mh^2=0.117$, $\Omega_bh^2=0.022$, $n_s=0.96$, $\sigma_v=300 {\rm km\ s^{-1}}$, and $k_{\star}=0.11$. \ref{['fig:lasdamas_twod-single']} The 2D 2PCF measured from a single mock catalog, compared to a theoretical model with the input parameters of the LasDamas simulations and $\{\beta, \sigma_v, k_\star\}$ are set to $\{0.35, 300$km s$^{-1}, 0.11h$Mpc$^{-1}\}$ (dashed red contours). In both figures, the shaded disk indicates the scale range considered ($s=40-120\ h^{-1}$Mpc ) in this study. The thick dashed blue circle denotes the baryon acoustic oscillation scale. The observed 2D 2PCF has been smoothed by a Gaussian filter with rms variance of $2 h^{-1}$Mpc for illustration in these figures only; smoothing is not used in our likelihood analysis. The contour levels are $\xi=0.5, 0.1, 0.025, 0.01, 0.005, 0$. The $\xi=0$ contours are denoted with dotted lines for clarity.
• 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 $\{\beta, \sigma_v, k_\star\}$ are set to $\{0.35, 300$km s$^{-1}, 0.11h$Mpc$^{-1}\}$ (dashed red contours). The gray area is the scale range considered ($s=40-120\ h^{-1}$Mpc ) in this study. The thick dashed blue circle denotes the baryon acoustic oscillation scale. The contour levels are apparent in the 2D 2PCF measured from mock catalogs, even though no smoothing is used. The contour levels are $\xi=0.5, 0.1, 0.025, 0.01, 0.005, 0$. The $\xi=0$ contours are denoted with dotted lines for clarity.
• Figure 3: 2D marginalized contours ($68\%$ and $95\%$ C.L.) for $\{H(0.35)$, $D_A(0.35)$, $\Omega_m h^2$, $H(0.35) \,r_s(z_d)/c$, $D_A(0.35)/r_s(z_d)$, $D_V(0.35)/r_s(z_d)$, $A(0.35)\}$. The diagonal panels represent the marginalized probabilities. The unit of $H$ is $\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. The unit of $D_A$, $D_V$, and $r_s(z_d)$ is $\rm Mpc$.
• Figure 4: Measurements of the means of $H(0.35) \,r_s(z_d)/c$ from 80 individual mock catalogs (indexed as 01a to 40a and 01b to 40b). The black solid line shows the mean of these 80 measurements and the blue dashed lines show the range of $\pm \sigma$. The red dotted line shows the theoretical value computed with the input parameters of the simulations.
• Figure 5: Measurements of the means of $D_A(0.35)/r_s(z_d)$ from 80 individual mock catalogs (indexed as 01a to 40a and 01b to 40b). The black solid line shows the mean of these 80 measurements and the blue dashed lines show the range of $\pm \sigma$. The red dotted line shows the theoretical value computed with the input parameters of the simulations.