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Register a new userInverting the Angular Correlation Function
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Scott Dodelson
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The two point angular correlation function is an excellent measure of structure in the universe. To extract from it the three dimensional power spectrum, one must invert Limbers Equation. Here we perform this inversion using a Bayesian prior constraining the smoothness of the power spectrum. Among other virtues, this technique allows for the possibility that the estimates of the angular correlation function are correlated from bin to bin. The output of this technique are estimators for the binned power spectrum and a full covariance matrix. Angular correlations mix small and large scales but after the inversion, small scale data can be trivially eliminated, thereby allowing for realistic constraints on theories of large scale structure. We analyze the APM catalogue as an example, comparing our results with previous results. As a byproduct of these tests, we find -- in rough agreement with previous work -- that APM places stringent constraints on Cold Dark Matter inspired models, with the shape parameter constrained to be $0.25pm 0.04$ (using data with wavenumber $k le 0.1 h{rm Mpc}^{-1}$). This range of allowed values use the full power spectrum covariance matrix, but assumes negligible covariance in the off-diagonal angular correlation error matrix, which is estimated with a large angular resolution of 0.5degrees (in the range 0.5 and 20 degrees).
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We estimate the two-point angular correlation function (CF) of quasars from SDSS DR3 using a special method of comparison random catalog generation. The best-fit value for the CF power-law index is found to be $alpha=0.78pm0.18$ on the $2<theta<250$ $arcmin$ interval. This is lower (though in marginal agreement) than earlier result of markcite{{it Myers et al.} (2005)} based on SDSS DR1 catalogue of photometrically-classified quasars.
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The 2-point angular correlation function $w(theta)$ (2PACF), where $theta$ is the angular separation between pairs of galaxies, provides the transversal Baryon Acoustic Oscillation (BAO) signal almost model-independently. In this paper we use 409,337 luminous red galaxies in the redshift range $z = [0.440,0.555]$ obtained from the tenth data release of the Sloan Digital Sky Survey (SDSS DR10) to estimate $theta_{rm{BAO}}(z)$ from the 2PACF at six redshift {shells}. Since noise and systematics can hide the BAO signature in the $w - theta$ plane, we also discuss some criteria to localize the acoustic bump. We identify two sources of model-dependence in the analysis, namely, the value of the acoustic scale from Cosmic Microwave Background (CMB) measurements and the correction in the $theta_{rm{BAO}}(z)$ position due to projection effects. Constraints on the dark energy equation-of-state parameter w$(z)$ from the $theta_{rm{BAO}}(z)$ diagram are derived, as well as from a joint analysis with current CMB measurements. We find that the standard $Lambda$CDM model as well as some of its extensions are in good agreement with these $theta_{rm{BAO}}(z)$ measurements.
We estimate the angular correlation function for the standard CDM, tilted $n=0.7$ CDM and hybrid (CHDM) models, and compare with observations. When compared with the APM observational results scaled to the Lick depth, there appears to be fair agreement with the estimate from the CHDM model. But a more detailed comparison using the unscaled APM data for the five magnitude slices with $Delta b_j=0.5$ shows that, in fact, none of the models can actually fit w(theta) for all the slices simultaneously. As $n=0.7$ tilted CDM and SCDM bracket all possible tilted models, we conclude that none of the tilted models is consistent with the APM results. With CHDM, a wtheta amplitude that is 30%--40% too high on scales $theta < 0.deg 5$ is predicted for the deepest slices ($b_japprox 20$). We find that no reasonable simple variation of the parameters for the luminosity function or for the evolution of the correlation function with redshift could change the situation. Thus, the discrepancy between the APM data and the CHDM model, though small, seems to be real.
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