Geometric biases in power-spectrum measurements

The observed distribution of galaxies has local transverse isotropy around the line of sight (LOS) with respect to the observer. The difference in the statistical clustering signal along and across the LOS encodes important information about the geometry of the Universe, its expansion rate and the rate of growth of structure within it. Because the LOS varies across a survey, the standard fast Fourier transform (FFT) based methods of measuring the anisotropic power spectrum (APS) cannot be used for surveys with wide observational footprint, other than to measure the monopole moment. We derive a simple analytic formula to quantify the bias for higher order Legendre moments, and we demonstrate that it is scale independent for a simple survey model, and depends only on the observed area. We derive a similar numerical correction formula for recently proposed alternative estimators of the APS that are based on summing over galaxies rather than using an FFT, and can therefore incorporate a varying LOS. We demonstrate that their bias depends on scale but not on the observed area. For a quadrupole the bias is always less than 1 per cent for k > 0.01 h Mpc^{-1} at z > 0.32. For a hexadecapole the bias is below 5 per cent for k > 0.05 h Mpc^{-1} at z > 0.32.