Theory and Statistics of Weak Lensing from Large-Scale Mass Inhomogeneities

TL;DR

The paper develops a rigorous theoretical and statistical framework for detecting weak gravitational lensing signals from large-scale mass inhomogeneities through ellipticity correlations. It formulates a gradient–curl decomposition of the shear field, shows that lensing produces a dominant gradient component with parity-zero curl, and ties the signal to the power spectrum P_{gg} while treating P_{cc} and P_{gc} as null tests. It then defines Fourier-domain estimators and a maximum-likelihood approach to infer the lensing amplitude, demonstrating significant detectability for FIRST-like surveys under cluster-abundance normalization and even larger signals for COBE normalization. The work also discusses systematic controls, the advantages of a Fourier-based analysis, and the broader implications for constraining the mass power spectrum and cosmological parameters from wide-area weak-lensing data.

Abstract

Weak lensing by large-scale mass inhomogeneities in the Universe induces correlations in the observed ellipticities of distant sources. We first review the harmonic analysis and statistics required of these correlations and discuss calculations for the predicted signal. We consider the ellipticity correlation function, the mean-square ellipticity, the ellipticity power spectrum, and a global maximum-likelihood analysis to isolate a weak-lensing signal from the data. Estimates for the sensitivity of a survey of a given area, surface density, and mean intrinsic source ellipticity are presented. We then apply our results to the FIRST radio-source survey. We predict an rms ellipticity of roughly 0.011 in $1^\circ \times 1^\circ$ pixels and 0.018 in $20' \times 20'$ pixels if the power spectrum is normalized to $σ_8 Ω^{0.53} = 0.6$, as indicated by the cluster abundance. The signal is significantly larger in some models if the power spectrum is normalized instead to the COBE anisotropy. The uncertainty in the predictions from imprecise knowledge of the FIRST redshift distribution is about 25% in the rms ellipticity. We show that FIRST should be able to make a statistically significant detection of a weak-lensing signal for cluster-abundance-normalized power spectra.