Projection and Galaxy Clustering Fourier Spectra

Abstract

Second order perturbation theory predicts a specific dependence of the bispectrum, or three-point correlation function in the Fourier transform domain, on the shape of the configuration of its three wave vector arguments, which can be taken as a signature of structure formed by gravitational instability. Comparing this known dependence on configuration shape with the weak shape dependence of the galaxy bispectrum has been suggested as an indication of bias in the galaxy distribution. However, to interpret results obtained from projected catalogs, we must first understand the effects of projection on this shape dependence. We present expressions for the projected power spectrum and bispectrum in both Cartesian and spherical geometries, and we examine the effects of projection on the predicted bispectrum with particular attention to the dependence on configuration shape. Except for an overall numerical factor, for Cartesian projection with characteristic depth $ \Dstar $ there is little effect on the shape dependence of the bispectrum for wavelengths small compared to $ \Dstar $ or projected wavenumbers $ q \Dstar \gg 1 $. For angular projection, a scaling law is found for spherical harmonic index $ \ell \gg 1 $, but there is always a mixing of scales over the range of the selection function. For large $ \ell $ it is sufficient to examine a small portion of the sky.