A METHOD FOR EXTRACTING MAXIMUM RESOLUTION POWER SPECTRA FROM GALAXY SURVEYS

摘要

The power spectrum estimated from a galaxy redshift survey is the real spectrum convolved with a window function, so when estimating the power on very large scales (for small k), it is important that this window function be as narrow as possible. A method that achieves this is presented. The optimal fluctuation estimate is found to be the Fourier transform of the number density fluctuations n — 1 weighted by a function ψ_0, and the optimal ψ_0 is found to be the ground-state solution of the Schroedinger equation, with the inverse selection function as the potential. This quantum mechanics analogy occurs basically because we want the weight function to be narrow both in Fourier space (to give a narrow window) and in real space (to minimize the variance from shot noise). An optimal method for averaging the estimates at different k is also presented, generalizing the standard procedure of averaging over shells in k-space. Finally, a discrete version of the method is presented, dividing space into "fuzzy pixels," which has the advantage of being able to handle red-shift distortions in a straightforward way.