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 how to analyze large data sets best has recently received consider able attention, and Karhunen-Loeve ve eigenvalue methods have been app lied to both galaxy redshift surveys and cosmic microwave background ( CMB) maps. We present a comprehensive discussion of methods for estima ting cosmological parameters from large data sets, which includes the previously published techniques as special cases. We show that both th e problem of estimating several parameters jointly and the problem of not knowing the parameters a priori can be readily solved by adding an extra singular value decomposition step. It has recently been argued that the information content in a sky map from a next-generation CMB s atellite is sufficient to measure key cosmological parameters (h, Omeg a, Lambda, etc.) to an accuracy of a few percent or better-in principl e. In practice, the data set is so large that both a brute force likel ihood analysis and a direct expansion in signal-to-noise eigenmodes wi ll be computationally unfeasible, We argue that it is likely that a Ka rhunen-Loeve approach can nonetheless measure the parameters with clos e to maximal accuracy, if preceded by an appropriate form of quadratic ''precompression.'' We also discuss practical issues regarding parame ter estimation from present and future galaxy redshift surveys and ill ustrate this with a generalized eigenmode analysis of the IRAS 1.2 Jy survey optimized for measuring beta=Omega(0.6)/b using redshift space distortions.