Karhunen‐Loeve Eigenvalue Problems in Cosmology: How Should We Tackle Large Data Sets?

Since cosmology is no longer "the data-starved science", the problem of howto best analyze large data sets has recently received considerable attention,and Karhunen-Loeve eigenvalue methods have been applied to both galaxy redshiftsurveys and Cosmic Microwave Background (CMB) maps. We present a comprehensivediscussion of methods for estimating cosmological parameters from large datasets, which includes the previously published techniques as special cases. Weshow that both the problem of estimating several parameters jointly and theproblem of not knowing the parameters a priori can be readily solved by addingan extra singular value decomposition step. It has recently been argued that the information content in a sky map from anext generation CMB satellite is sufficient to measure key cosmologicalparameters (h, Omega, Lambda, etc) to an accuracy of a few percent or better -in principle. In practice, the data set is so large that both a brute forcelikelihood analysis and a direct expansion in signal-to-noise eigenmodes willbe computationally unfeasible. We argue that it is likely that a Karhunen-Loeveapproach can nonetheless measure the parameters with close to maximal accuracy,if preceded by an appropriate form of quadratic "pre-compression". We also discuss practical issues regarding parameter estimation from presentand future galaxy redshift surveys, and illustrate this with a generalizedeigenmode analysis of the IRAS 1.2 Jy survey optimized for measuringbeta=Omega^{0.6}/b using redshift space distortions.