Weak lensing reconstruction and power spectrum estimation: Minimum variance methods

Large-scale structure distorts the images of background galaxies, which allows one to measure directly the projected distribution of dark matter in the universe and determine its power spectrum. Here we address the question of how to extract this information from the observations. We derive minimum variance estimators for the projected density reconstruction and its power spectrum and apply them to simulated data sets, showing that they give a good agreement with the theoretical minimum variance expectations. The same estimator can also be applied to the cluster reconstruction, where it remains a useful reconstruction technique, although it is no longer optimal for every application. The method can be generalized to include nonlinear cluster reconstruction and photometric information on redshifts of background galaxies in the analysis. We also address the question of how to obtain directly the three-dimensional power spectrum from the weak lensing data. We derive a minimum variance quadratic estimator, which maximizes the likelihood function for the three-dimensional power spectrum and can be computed either from the measurements directly or from the two-dimensional power spectrum. The estimator correctly propagates the errors and provides a full correlation matrix of the estimates. It can be generalized to the case where redshift distribution depends on the galaxy photometric properties, which allows one to measure both the three-dimensional power spectrum and its time evolution.