Euclid preparation. LIX. Angular power spectra from discrete observations

TL;DR

This paper develops a formalism for accurately measuring angular power spectra from discrete galaxy survey data, accounting for shot noise and enabling precise theoretical predictions for Euclid's observations.

Contribution

It introduces a novel framework for computing angular power spectra from discrete data without assuming a continuous underlying field, including exact shot noise calculations and an efficient convolution method.

Findings

Achieved less than 1% bias in simulated Euclid data

Derived exact expressions for shot noise in galaxy clustering and cosmic shear

Validated the methodology with simulations showing high accuracy

Abstract

We present the framework for measuring angular power spectra in the Euclid mission. The observables in galaxy surveys, such as galaxy clustering and cosmic shear, are not continuous fields, but discrete sets of data, obtained only at the positions of galaxies. We show how to compute the angular power spectra of such discrete data sets, without treating observations as maps of an underlying continuous field that is overlaid with a noise component. This formalism allows us to compute exact theoretical expectations for our measured spectra, under a number of assumptions that we track explicitly. In particular, we obtain exact expressions for the additive biases ("shot noise") in angular galaxy clustering and cosmic shear. For efficient practical computations, we introduce a spin-weighted spherical convolution with a well-defined convolution theorem, which allows us to apply exact theoretical predictions to finite-resolution maps, including HEALPix. When validating our methodology, we find that our measurements are biased by less than 1% of their statistical uncertainty in simulations of Euclid's first data release.