Propagating data-driven galaxy redshift distribution uncertainties in 3$\times$2-pt analyses

Uncertainties in the radial distribution of galaxies, $\boldsymbol{n}(\boldsymbol{z})$, are one of the major contributions to the error budget of early Stage-IV galaxy survey analyses of weak gravitational lensing, galaxy clustering and galaxy-galaxy lensing (3$\times$2-pt). Based on ensembles of simulated $\boldsymbol{n}(\boldsymbol{z})$ including stochastic and systematic variations, we study the impact of four different $\boldsymbol{n}(\boldsymbol{z})$ uncertainty models: shifts, shifts & stretches, Gaussian processes (GP) and principal component analysis (PCA). Due to the high dimensionality of the latter models, we make use of state-of-the-art gradient-based inference methods as well as approximate analytical marginalisation schemes. Our results show that Stage-IV 3$\times$2-pt analyses must go beyond simple shift & stretch models. In particular, we advocate for the adoption of PCA models even in early Stage-IV surveys. Our results show that considering a five-parameters PCA model only degrades the constraint on the $S_{\rm 8}$ parameter by $5$ per cent with respect to the case when only a shift and a stretch parameter are included, while incurring half the bias in its constituents parameters, $\Omega_{\rm m}$ and $\sigma_{\rm 8}$. We demonstrate that all models considered can be safely marginalised analytically, with speed-ups of up to a factor of 25 depending on the dimensionality of the model. This will allow Stage-IV analyses to safely include higher-dimensional $\boldsymbol{n}(\boldsymbol{z})$ uncertainty models in their analysis at negligible additional computational cost.