The optimal weighting function for cosmic magnification measurement through foreground galaxy-background galaxy (quasar) cross correlation

TL;DR

This paper derives the optimal weighting function for cosmic magnification measurements using foreground-background galaxy or quasar cross correlations, improving signal-to-noise ratios especially for dense background populations.

Contribution

It introduces a generalized optimal weighting function that accounts for intrinsic clustering, enhancing measurement accuracy over previous methods.

Findings

Optimal weighting improves S/N by ~20% for BigBOSS-like surveys.

For denser background populations, the improvement can reach a factor of ~2.

The method extends previous weighting schemes to include scale-dependent effects.

Abstract

Cosmic magnification has been detected through cross correlation between foreground and background populations (galaxies or quasars). It has been shown that weighing each background object by its $\alpha-1$ can significantly improve the cosmic magnification measurement \citep{Menard02,Scranton05}. Here, $\alpha$ is the logarithmic slope of the luminosity function of background populations. However, we find that this weighting function is optimal only for sparse background populations in which intrinsic clustering is negligible with respect to shot noise. We derive the optimal weighting function for general case including scale independent and scale dependent weights. The optimal weighting function improves the S/N (signal to noise ratio) by $\sim 20%$ for a BigBOSS-like survey and the improvement can reach a factor of $\sim 2$ for surveys with much denser background populations.