Non-Linear Effects on the Angular Correlation Function

Abstract

Extracting the three dimensional power spectrum from the 2D distribution of galaxies has become a standard tool of cosmology. This extraction requires some assumptions about the scaling of the power spectrum with redshift; all treatments to date assume a simple power law scaling. In reality, different scales grow at different rates, due to non-linearities. We show that angular surveys are sensitive to a weighted average of the power spectrum over a distribution of redshifts, where the weight function varies with wavenumber. We compute this weight function and show that it is fairly sharply peaked at \bar{z}, which is a function of k. As long as the extracted power spectrum is understood to be $P(k,\bar{z})$, the error introduced by non-linear scaling is quantifiable and small. We study these effects in the context of the APM and SDSS photometric surveys. In general the weight matrix is peaked at larger z and is broader for deeper surveys, leading to larger (but still quantifiable) errors due to non-linear scaling. The tools introduced here -- in particular the weight function and effective redshift $\bar{z}$ -- can also be profitably applied to plan surveys to study the evolution of the power spectrum.