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

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.