Self-calibration and robust propagation of photometric redshift distribution uncertainties in weak gravitational lensing

TL;DR

This work tackles the challenge of propagating photometric redshift distribution uncertainties into weak-lensing cosmology by introducing a flexible redshift model based on a modified Gaussian mixture (a Gaussian comb) whose amplitudes are linear nuisance parameters. The authors derive an analytic marginalisation of these nuisance amplitudes within the weak-lensing likelihood, enabling a high-flexibility redshift model without inflating the sampling dimensionality. They implement an iterative self-calibration scheme that alternates between cosmology optimisation and nuisance parameter (comb-amplitude) refinement, and they apply the method to the KV450 data, obtaining results in good agreement with the fiducial KV450 analysis while accounting for more complex redshift distributions. The approach reduces biases from redshift uncertainties, yields slightly shifted IA and $S_8$ constraints due to degeneracies, and is ready to support future 6×2pt analyses with larger surveys where higher-order redshift-distribution features become important.

Abstract

We present a method that accurately propagates residual uncertainties in photometric redshift distributions into the cosmological inference from weak lensing measurements. The redshift distributions of tomographic redshift bins are parameterised using a flexible modified Gaussian mixture model. We fit this model to pre-calibrated redshift distributions and implement an analytic marginalisation over the potentially several hundred redshift nuisance parameters in the weak lensing likelihood, which is demonstrated to accurately recover the cosmological posterior. By iteratively fitting cosmological and nuisance parameters arising from the redshift distribution model, we perform a self-calibration of the redshift distributions via the tomographic cosmic shear measurements. Our method is applied to the third data release of the Kilo-Degree Survey combined with the VISTA Kilo-Degree Infrared Galaxy Survey (KV450). We find constraints on cosmological parameters that are in very good agreement with the fiducial KV450 cosmic shear analysis and investigate the effects of the more flexible model on the self-calibrated redshift distributions. We observe posterior shifts of the medians of the five tomographic redshift distributions of up to $Δz \approx 0.02$, which are however degenerate with an observed decrease of the amplitude of intrinsic galaxy alignments by about $10\%$.

Figures (10)

• Figure 1: Sketch of the iterative fitting method used to determine the best fit in the combined parameter space of cosmological and nuisance parameters. We alternate between optimising cosmological parameters (numerically; blue arrows), keeping nuisance parameters fixed, and optimising nuisance parameters (using Newton's method; red arrows), keeping cosmological parameters fixed. After several iterations we achieve convergence to the best fit in the combined parameter space. After optimising the likelihood, we set the amplitudes of the Gaussian comb to the best-fit parameters and proceed with sampling the likelihood in cosmological parameter space (dotted orange line) while analytically marginalising over nuisance parameters (green arrows).
• Figure 2: Fit results of a Gaussian mixture with 30 components to the redshift distribution in five tomographic redshift bins. Blue curves indicate redshift distributions fitted to the pre-calibrated DIR redshift histograms, shown in black. Shaded regions indicate the uncertainties on the redshift distributions derived from the diagonal elements of the correlation matrix of fit parameters, shown in Fig. \ref{['fig:correlation_matrix']}. Orange curves represent the redshift distributions after iterative optimisation of cosmological and nuisance parameters.
• Figure 3: Correlation matrix of best-fit comb amplitudes with 30 components per redshift bin.
• Figure 4: Posterior distribution of the median redshift of each tomographic redshift bin, inferred by drawing realisations of the Gaussian comb amplitudes from a multivariate Gaussian distribution. Black curves indicate the median redshift of the KV450 redshift histograms calibrated using the fiducial DIR method. The blue curves show the median redshift of the Gaussian comb that is fitted to the DIR histograms. The orange curves represent the median redshift of the Gaussian comb after iterative self-calibration with cosmic shear measurements.
• Figure 5: Marginalised posteriors for $A_{\rm IA}$ and $S_8$. The orange contours present the results from the KV450 likelihood with a self-calibrated Gaussian comb and analytical marginalisation over nuisance parameters, while the blue contours refer to the fiducial KV450 constraints. The star indicates the best-fit values from Table \ref{['tab:iterative_calibration']} for the KV450 likelihood with a Gaussian comb, and the cross indicates the best-fit values for the fiducial KV450 likelihood. The dashed contour shows the posterior distribution from the KV450 'gold' sample wright_som_kv450, which is constructed by removing photometric sources that are not directly represented by the overlapping spectroscopic reference samples using SOMs. Therefore, this contour is inferred from a different sample of galaxies with a different redshift distribution.