Power Spectrum Analysis of Three-Dimensional Redshift Surveys

Abstract

We develop a general method for power spectrum analysis of three dimensional redshift surveys. We present rigorous analytical estimates for the statistical uncertainty in the power and we are able to derive a rigorous optimal weighting scheme under the reasonable (and largely empirically verified) assumption that the long wavelength Fourier components are Gaussian distributed. We apply the formalism to the updated 1-in-6 QDOT IRAS redshift survey, and compare our results to data from other probes: APM angular correlations; the CfA and the Berkeley 1.2Jy IRAS redshift surveys. Our results bear out and further quantify the impression from e.g.\ counts-in-cells analysis that there is extra power on large scales as compared to the standard CDM model with $Ωh\simeq 0.5$. We apply likelihood analysis using the CDM spectrum with $Ωh$ as a free parameter as a phenomenological family of models; we find the best fitting parameters in redshift space and transform the results to real space. Finally, we calculate the distribution of the estimated long wavelength power. This agrees remarkably well with the exponential distribution expected for Gaussian fluctuations, even out to powers of ten times the mean. Our results thus reveal no trace of periodicity or other non-Gaussian behavior.