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

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.