Eigenmode analysis of galaxy redshift surveys .1. Theory and methods

We describe a method for estimating the power spectrum of density fluctuations from galaxy redshift surveys that yields improvement in both accuracy and resolution over direct Fourier analysis. The key feature of this analysis is expansion of the observed density held in the unique set of statistically orthogonal spatial functions that obtains for a given survey's geometry and selection function and the known properties of galaxy clustering (the Karhunen-Loeve transform). Each of these eigenmodes of the observed density held optimally weights the data to yield the cleanest (highest signal-to-noise ratio) possible measure of clustering power as a function of wavelength scale for any survey. Using Bayesian methods, we simultaneously estimate the mean density, power spectrum of density fluctuations, and redshift distortion parameters that best fit the observed data. This method is particularly important for analysis of surveys with small sky coverage, which are comprised of disjoint regions (e.g., an ensemble of pencil beams or slices), or that have large fluctuations in sampling density. We present algorithms for practical application of this technique to galaxy survey data.