An optimal quadratic estimator for window-free cosmic shear power spectra

Abstract

The pseudo-$C_\ell$ estimator recovers the true cosmic shear power spectrum by correcting for the survey window convolution while employing inverse-variance weighting based on intrinsic shape noise of source galaxies. However, this weighting scheme is optimal only on small angular scales where shape noise dominates. In this paper, we derive a quadratic estimator for the unwindowed cosmic shear power spectrum by maximizing the Gaussian likelihood of the pixelized galaxy-shape field using the full covariance matrix, which accounts for both sample variance and shape noise. By combining FFTs in the flat-sky approximation, the conjugate-gradient method, and Monte Carlo realizations of Gaussian ancillary fields, we substantially reduce the computational cost of estimating the Fisher matrix, a key ingredient of the estimator that requires repeated inverse-covariance matrix operations. Using Gaussian simulations of shape fields, we validate the method and demonstrate that it can recover the input $E$-mode power spectrum with statistically optimal precision across all angular scales. We then apply the method to shape fields generated from ray-tracing simulations for a $Λ$CDM cosmology and show that, compared with the pseudo-$C_\ell$ method, it reduces the statistical uncertainties in the $E$-mode power spectrum by 5--15\% at multipoles of $\ell \lesssim 500$. We further demonstrate that the method significantly suppresses $E$- to $B$-mode leakage across the full multipole range. Our estimator therefore provides a statistically optimal approach for measuring cosmic shear power spectra from wide-area galaxy survey data.