Sparsity covariance: a source of uncertainty when estimating correlation functions with a discrete sample of observations in the sky

TL;DR

This paper introduces the concept of sparsity covariance, a source of uncertainty in estimating correlation functions from finite, discrete sky samples, and provides a method to compute it, highlighting its impact on cosmic shear measurements.

Contribution

It defines sparsity covariance mathematically and presents a generic computation method, applying it to cosmic shear to assess its effect on shape noise.

Findings

Sparsity covariance increases shape noise in cosmic shear.

It is significant when the signal-to-noise ratio per measurement is high.

The method quantifies the sample dependence of correlation function estimates.

Abstract

Cosmological observables rely heavily on summary statistics such as two-point correlation functions. In many practical cases (e.g. the weak-lensing cosmic shear), those correlation functions are estimated from a finite, discrete sample of measurements that are randomly distributed in the sky. The result then inevitably depends on the sample at hand, regardless of any experimental noise. This sample dependence is a source of uncertainty for cosmological observables which I call sparsity covariance. This article proposes a mathematical definition and a generic method to compute sparsity covariance. It is then applied to the concrete case of cosmic shear, showing that sparsity covariance mostly enhances shape noise, whose amplitude is determined by the apparent ellipticity of galaxies rather than their intrinsic ellipticity. In general, sparsity covariance is non-negligible when the signal-to-noise ratio of individual measurements in the sample is comparable to, or larger than, unity.