We explore an alternative method to the usual shear correlation function approach for the estimation of aperture mass statistics in weak lensing survey data. Our approach builds on the direct estimator method. In this paper, to test and validate the methodology, we focus on the aperture mass dispersion. After computing the signal and noise for a weighted set of measured ellipticites we show how the direct estimator can be made into a linear order algorithm that enables a fast and efficient computation. We then investigate the applicability of the direct estimator approach in the presence of a real survey mask with holes and chip gaps. For this we use a large ensemble of full ray-tracing mock simulations. By using various weighting schemes for combining information from different apertures we find that inverse variance weighting the individual aperture estimates with an aperture completeness greater than 70% yields an answer that is in close agreement with the correlation function approach. We then apply this approach to the CFHTLenS as a pilot scheme and find that our method recovers to high accuracy the official result for the variance of both the E and B mode signal. We then explore the cosmological information content of the direct estimator using the Fisher information approach. We show that there is a only modest loss in cosmological information from the rejection of apertures that are of low completeness. This method unlocks the door to efficient methods for recovering higher order aperture mass statistics in linear order operations.

We explore an alternative method to the usual shear correlation function approach for the estimation of aperture mass statistics in weak lensing survey data. Our approach builds on the direct estimator method. In this paper, we extend our analysis to statistics of arbitrary order and to the multiscale aperture mass statistics. We show that there always exists a linear order algorithm to retrieve any of these generalised aperture mass statistics from shape catalogs when the direct estimator approach is adopted. We validate our approach through application to a large number of Gaussian mock lensing surveys where the true answer is known and we do this up to 10th order statistics. We then apply our estimators to an ensemble of real-world mock catalogs obtained from N-body simulations – the SLICS mocks, and show that one can expect to retrieve detections of higher order clustering up to fourth order in a KiDS-1000 like survey. We expect that these methods will be of most utility for future wide-field surveys like Euclid and the Rubin Telescope.

The power spectrum of the nonlinearly evolved large-scale mass distribution recovers only a minority of the information available on the mass fluctuation amplitude. We investigate the recovery of this information in 2D ‘slabs’ of the mass distribution averaged over ≈100 h−1Mpc along the line of sight, as might be obtained from photometric redshift surveys. We demonstrate a Hamiltonian Monte Carlo (HMC) method to reconstruct the non-Gaussian mass distribution in slabs, under the assumption that the projected field is a point-transformed Gaussian random field, Poisson-sampled by galaxies. When applied to the Quijote N-body suite at z = 0.5 and at a transverse resolution of 2 h−1Mpc, the method recovers ∼30 times more information than the 2D power spectrum in the well-sampled limit, recovering the Gaussian limit on information. At a more realistic galaxy sampling density of 0.01 h3Mpc−3, shot noise reduces the information gain to a factor of five improvement over the power spectrum at resolutions of 4 h−1Mpc or smaller.

The power spectrum of the nonlinearly evolved large-scale mass distribution recovers only a minority of the information available on the mass fluctuation amplitude. We investigate the recovery of this information in 2D "slabs" of the mass distribution averaged over ≈ 100 ℎ −1 Mpc along the line of sight, as might be obtained from photometric redshift surveys. We demonstrate a Hamiltonian Monte Carlo (HMC) method to reconstruct the non-Gaussian mass distribution in slabs, under the assumption that the projected field is a point-transformed Gaussian random field, Poisson-sampled by galaxies. When applied to the Quijote 𝑁-body suite at 𝑧 = 1 and at a transverse resolution of 2 ℎ −1 Mpc, the method recovers ∼ 30 times more information than the 2D power spectrum in the well-sampled limit, recovering the Gaussian limit on information. At a more realistic galaxy sampling density of 0.01 ℎ 3 Mpc −3 , shot noise reduces the information gain to a factor of five improvement over the power spectrum at resolutions of 4 ℎ −1 Mpc or smaller.