6 × 2 pt: Forecasting gains from joint weak lensing and galaxy clustering analyses with spectroscopic-photometric galaxy cross-correlations

¹ Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands
² Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, The Netherlands
³ Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Av. Complutense 40, E-28040 Madrid, Spain
⁴ Institute of Cosmology & Gravitation, Dennis Sciama Building, University of Portsmouth, Portsmouth PO1 3FX, UK
⁵ Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada
⁶ Department of Physics and Astronomy, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada
⁷ Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing, 44780 Bochum, Germany
⁸ The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden
⁹ Astrophysics Group and Imperial Centre for Inference and Cosmology (ICIC), Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, UK
¹⁰ Donostia International Physics Center, Manuel Lardizabal Ibilbidea, 4, 20018 Donostia, Gipuzkoa, Spain
¹¹ School of Mathematics, Statistics and Physics, Newcastle University, Herschel Building, NE1 7RU Newcastle-upon-Tyne, UK
¹² Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland
¹³ Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
¹⁴ Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK
¹⁵ Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
¹⁶ Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France
¹⁷ Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria
¹⁸ Shanghai Astronomical Observatory (SHAO), Nandan Road 80, Shanghai 200030, China
¹⁹ Key Laboratory of Radio Astronomy and Technology, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100101, P. R. China
²⁰ University of Chinese Academy of Sciences, Beijing 100049, China
²¹ Institute for Particle Physics and Astrophysics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
²² Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK
²³ Department of Physics, Centre for Extragalactic Astronomy, Durham University, South Road, Durham DH1 3LE, UK

^⋆ Corresponding authors: harrysj100@gmail.com; n.e.chisari@uu.nl; shahab.joudaki@ciemat.es

Received: 2 October 2024
Accepted: 23 April 2025

Abstract

Accurate knowledge of galaxy redshift distributions is crucial in the inference of cosmological parameters from large-scale structure data. We explore the potential for enhanced self-calibration of photometric galaxy redshift distributions, n(z), through the joint analysis of up to six two-point functions. Our 3 × 2 pt configuration comprises photometric shear, spectroscopic galaxy clustering, and spectroscopic-photometric galaxy-galaxy lensing (GGL). We expand this to include spectroscopic-photometric cross-clustering, photometric GGL, and photometric auto-clustering, using the photometric shear sample as an additional density tracer. We performed simulated likelihood forecasts of the cosmological and nuisance parameter constraints for stage-III- and stage-IV-like surveys. For the stage-III-like survey, we employed realistic redshift distributions with perturbations across the full shape of the n(z), and distinguished between ‘coherent’ shifting of the bulk distribution in one direction, versus more internal scattering and full-shape n(z) errors. For perfectly known n(z), a 6 × 2 pt analysis gains ∼40% in figure of merit (FoM) on the S₈ ≡ σ₈√Ω_m/0.3 and Ω_m plane relative to the 3 × 2 pt analysis. If untreated, coherent and incoherent redshift errors lead to inaccurate inferences of S₈ and Ω_m, respectively, and contaminate inferences of the amplitude of intrinsic galaxy alignments. Employing bin-wise scalar shifts, δz_i, in the tomographic mean redshifts reduces cosmological parameter biases, with a 6 × 2 pt analysis constraining the δz_i parameters with 2 − 4 times the precision of a photometric 3^ph × 2 pt analysis. For the stage-IV-like survey, a 6 × 2 pt analysis doubles the FoM (σ₈–Ω_m) compared to the 3 × 2 pt or 3^ph × 2 pt analyses, and is only 8% less constraining than if the n(z) were perfectly known. A Gaussian mixture model for the n(z) is able to reduce mean-redshift errors whilst preserving the n(z) shape, and thereby yields the most accurate and precise cosmological constraints for any given N × 2 pt configuration in the presence of n(z) biases.