Pure-mode correlation functions for cosmic shear and application to KiDS-1000

¹ Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
e-mail: peter@astro.uni-bonn.de
² Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
³ E. A. Milne Centre, University of Hull, Cottingham Road, Hull HU6 7RX, UK
⁴ Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany
⁵ School of Mathematics, Statistics and Physics, Newcastle University, Herschel Building, NE1 7RU Newcastle-upon-Tyne, UK
⁶ Leiden Observatory, Leiden University, PO Box 9513 2300 RA Leiden, The Netherlands
⁷ Shanghai Astronomical Observatory (SHAO), Nandan Road 80, Shanghai 200030, PR China
⁸ University of Chinese Academy of Sciences, Beijing 100049, PR China

Received: 19 October 2021
Accepted: 8 March 2022

Abstract

One probe for systematic effects in gravitational lensing surveys is the presence of so-called B modes in the cosmic shear two-point correlation functions, ξ_±(ϑ), since lensing is expected to produce only E-mode shear. Furthermore, there exist ambiguous modes that cannot uniquely be assigned to either E- or B-mode shear. In this paper we derive explicit equations for the pure-mode shear correlation functions, ξ_±^E/B(ϑ), and their ambiguous components, ξ_±^amb(ϑ), that can be derived from the measured ξ_±(ϑ) on a finite angular interval, ϑ_min ≤ ϑ ≤ ϑ_max, such that ξ_±(ϑ) can be decomposed uniquely into pure-mode functions as ξ₊ = ξ₊^E+ξ₊^B+ξ₊^amb and ξ₋ = ξ₋^E−ξ₋^B+ξ₋^amb. The derivation is obtained by defining a new set of Complete Orthogonal Sets of E and B mode-separating Integrals (COSEBIs), for which explicit relations are obtained and which yields a smaller covariance between COSEBI modes. We derive the relation between ξ_±^E/B/amb and the underlying E- and B-mode power spectra. The pure-mode correlation functions can provide a diagnostic of systematics in configuration space. We then apply our results to Scinet LIght Cone Simulations (SLICS) and the Kilo-Degree Survey (KiDS-1000) cosmic shear data, calculate the new COSEBIs and the pure-mode correlation functions, as well as the corresponding covariances, and show that the new statistics fit equally well to the best fitting cosmological model as the previous KiDS-1000 analysis and recover the same level of (insignificant) B modes. We also consider in some detail the ambiguous modes at the first- and second-order level, finding some surprising results. For example, the shear field of a point mass, when cut along a line through the center, cannot be ascribed uniquely to an E-mode shear and is thus ambiguous; additionally, the shear correlation functions resulting from a random ensemble of point masses, when measured over a finite angular range, correspond to an ambiguous mode.