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Register a new userStatistical Analysis of Galaxy Surveys-II. The 3-point galaxy correlation function measured from the 2dFGRS
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E.Gaztanaga
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We present new results for the 3-point correlation function, zeta, measured as a function of scale, luminosity and colour from the final version of the two-degree field galaxy redshift survey (2dFGRS). The reduced three point correlation function, Q_3 is estimated for different triangle shapes and sizes, employing a full covariance analysis. The form of Q_3 is consistent with the expectations for the Lambda-cold dark matter model, confirming that the primary influence shaping the distribution of galaxies is gravitational instability acting on Gaussian primordial fluctuations. However, we find a clear offset in amplitude between Q_3 for galaxies and the predictions for the dark matter. We are able to rule out the scenario in which galaxies are unbiased tracers of the mass at the 9-sigma level. On weakly non-linear scales, we can interpret our results in terms of galaxy bias parameters. We find a linear bias term that is consistent with unity, b_1 = 0.93^{+0.10}_{-0.08} and a quadratic bias c_2 = b_2 /b_1 = -0.34^{+0.11}_{-0.08}. This is the first significant detection of a non-zero quadratic bias, indicating a small but important non-gravitational contribution to the three point function. Our estimate of the linear bias from the three point function is independent of the normalisation of underlying density fluctuations, so we can combine this with the measurement of the power spectrum of 2dFGRS galaxies to constrain the amplitude of matter fluctuations. We find that the rms linear theory variance in spheres of radius 8Mpc/h is sigma_8 = 0.88^{+0.12}_{-0.10}, providing an independent confirmation of values derived from other techniques. On non-linear scales, where xi>1, we find that Q_3 has a strong dependence on scale, colour and luminosity.
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We present galaxy-galaxy lensing measurements over scales 0.025 to 10 Mpc/h in the Sloan Digital Sky Survey. Using a flux-limited sample of 127,001 lens galaxies with spectroscopic redshifts and mean luminosity <L> = L_* and 9,020,388 source galaxies with photometric redshifts, we invert the lensing signal to obtain the galaxy-mass correlation function xi_{gm}. We find xi_{gm} is consistent with a power-law, xi_{gm} = (r/r_0)^{-gamma}, with best-fit parameters gamma = 1.79 +/- 0.06 and r_0 = (5.4+/-0.7)(0.27/Omega_m)^{1/gamma} Mpc/h. At fixed separation, the ratio xi_{gg}/xi_{gm} = b/r where b is the bias and r is the correlation coefficient. Comparing to the galaxy auto-correlation function for a similarly selected sample of SDSS galaxies, we find that b/r is approximately scale independent over scales 0.2-6.7 Mpc/h, with mean <b/r> = (1.3+/-0.2)(Omega_m/0.27). We also find no scale dependence in b/r for a volume limited sample of luminous galaxies (-23.0 < M_r < -21.5). The mean b/r for this sample is <b/r>_{Vlim} = (2.0+/-0.7)(Omega_m/0.27). We split the lens galaxy sample into subsets based on luminosity, color, spectral type, and velocity dispersion, and see clear trends of the lensing signal with each of these parameters. The amplitude and logarithmic slope of xi_{gm} increases with galaxy luminosity. For high luminosities (L ~5 L_*), xi_{gm} deviates significantly from a power law. These trends with luminosity also appear in the subsample of red galaxies, which are more strongly clustered than blue galaxies.
Measuring the two-point correlation function of the galaxies in the Universe gives access to the underlying dark matter distribution, which is related to cosmological parameters and to the physics of the primordial Universe. The estimation of the correlation function for current galaxy surveys makes use of the Landy-Szalay estimator, which is supposed to reach minimal variance. This is only true, however, for a vanishing correlation function. We study the Landy-Szalay estimator when these conditions are not fulfilled and propose a new estimator that provides the smallest variance for a given survey geometry. Our estimator is a linear combination of ratios between paircounts of data and/or random catalogues (DD, RR and DR). The optimal combination for a given geometry is determined by using lognormal mock catalogues. The resulting estimator is biased in a model-dependent way, but we propose a simple iterative procedure for obtaining an unbiased model- independent estimator.Our method can be easily applied to any dataset and requires few extra mock catalogues compared to the standard Landy-Szalay analysis. Using various sets of simulated data (lognormal, second-order LPT and N-Body), we obtain a 20-25% gain on the error bars on the two-point correlation function for the SDSS geometry and $Lambda$CDM correlation function. When applied to SDSS data (DR7 and DR9), we achieve a similar gain on the correlation functions, which translates into a 10-15% improvement over the estimation of the densities of matter $Omega_m$ and dark energy $Omega_Lambda$ in an open $Lambda$CDM model. The constraints derived from DR7 data with our estimator are similar to those obtained with the DR9 data and the Landy-Szalay estimator, which covers a volume twice as large and has a density that is three times higher.
Third-order statistics of the cosmic density field provides a powerful cosmological probe containing synergistic information to the more commonly explored second-order statistics. Here, we exploit a spectroscopic catalog of 72,563 clusters of galaxies extracted from the Sloan Digital Sky Survey, providing the first detection of the baryon acoustic oscillations (BAO) peak in the three-point correlation function (3PCF) of galaxy clusters. We measure and analyze both the connected and the reduced 3PCF of SDSS clusters from intermediate ($rsim10$ Mpc/h) up to large ($rsim140$ Mpc/h) scales, exploring a variety of different configurations. From the analysis of reduced 3PCF at intermediate scales, in combination with the analysis of the two-point correlation function, we constrain both the cluster linear and non-linear bias parameters, $b_1=2.75pm0.03$ and $b_2=1.2pm0.5$. We analyze the measurements of the 3PCF at larger scales, comparing them with theoretical models. The data show clear evidence of the BAO peak in different configurations, which appears more visible in the reduced 3PCF rather than in the connected one. From the comparison between theoretical models considering or not the BAO peak, we obtain a quantitative estimate of this evidence, with a $Delta chi^2$ between 2 and 75, depending on the considered configuration. Finally, we set up a generic framework to estimate the expected signal-to-noise ratio of the BAO peak in the 3PCF exploring different possible definitions, that can be used to forecast the most favorable configurations to be explored also in different future surveys, and applied it to the case of the Euclid mission.
We present a test of different error estimators for 2-point clustering statistics, appropriate for present and future large galaxy redshift surveys. Using an ensemble of very large dark matter LambdaCDM N-body simulations, we compare internal error estimators (jackknife and bootstrap) to external ones (Monte-Carlo realizations). For 3-dimensional clustering statistics, we find that none of the internal error methods investigated are able to reproduce neither accurately nor robustly the errors of external estimators on 1 to 25 Mpc/h scales. The standard bootstrap overestimates the variance of xi(s) by ~40% on all scales probed, but recovers, in a robust fashion, the principal eigenvectors of the underlying covariance matrix. The jackknife returns the correct variance on large scales, but significantly overestimates it on smaller scales. This scale dependence in the jackknife affects the recovered eigenvectors, which tend to disagree on small scales with the external estimates. Our results have important implications for the use of galaxy clustering in placing constraints on cosmological parameters. For example, in a 2-parameter fit to the projected correlation function, we find that the standard bootstrap systematically overestimates the 95% confidence interval, while the jackknife method remains biased, but to a lesser extent. The scatter we find between realizations, for Gaussian statistics, implies that a 2-sigma confidence interval, as inferred from an internal estimator, could correspond in practice to anything from 1-sigma to 3-sigma. Finally, by an oversampling of sub-volumes, it is possible to obtain bootstrap variances and confidence intervals that agree with external error estimates, but it is not clear if this prescription will work for a general case.
We perform theoretical and numerical studies of the full relativistic two-point galaxy correlation function, considering the linear-order scalar and tensor perturbation contributions and the wide-angle effects. Using the gauge-invariant relativistic description of galaxy clustering and accounting for the contributions at the observer position, we demonstrate that the complete theoretical expression is devoid of any long-mode contributions from scalar or tensor perturbations and it lacks the infrared divergences in agreement with the equivalence principle. By showing that the gravitational potential contribution to the correlation function converges in the infrared, our study justifies an IR cut-off $(k_{text{IR}} leq H_0)$ in computing the gravitational potential contribution. Using the full gauge-invariant expression, we numerically compute the galaxy two-point correlation function and study the individual contributions in the conformal Newtonian gauge. We find that the terms at the observer position such as the coordinate lapses and the observer velocity (missing in the standard formalism) dominate over the other relativistic contributions in the conformal Newtonian gauge such as the source velocity, the gravitational potential, the integrated Sachs-Wolf effect, the Shapiro time-delay and the lensing convergence. Compared to the standard Newtonian theoretical predictions that consider only the density fluctuation and redshift-space distortions, the relativistic effects in galaxy clustering result in a few percent-level systematic errors beyond the scale of the baryonic acoustic oscillation. Our theoretical and numerical study provides a comprehensive understanding of the relativistic effects in the galaxy two-point correlation function, as it proves the validity of the theoretical prediction and accounts for effects that are often neglected in its numerical evaluation.
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