Euclid preparation. Impact of redshift distribution uncertainties on the joint analysis of photometric galaxy clustering and weak gravitational lensing
One of the $\textit{Euclid}$ mission's key projects is the so-called 3$\times$2pt analysis, that is, the combination of cosmic shear, photometric galaxy clustering, and galaxy-galaxy lensing. Although $\textit{Euclid}$ has established quality requirements for the photo-$z$ accuracy needed for the weak lensing galaxy sample, no such requirements have been set for the photometric clustering sample. In this paper, we investigate the impact of redshift uncertainties on $\textit{Euclid}$'s photometric galaxy clustering analysis and its combination with weak gravitational lensing, focusing on data release 1 (DR1). In particular, we study whether having precise knowledge of the mean of the redshift distributions per bin is sufficient to avoid biases in the resulting cosmological constraints or whether accuracy in the higher-order moments of the distribution is required. We evaluate the results based on their constraining power on $w_{\mathrm{0}}$ and $w_{a}$ and define thresholds for the precision and accuracy of $\textit{Euclid}$'s redshift distribution of the photometric clustering sample. We find that the redshift distributions of the photometric clustering sample must be known at an accuracy of 0.004(1+$z$) in the mean in order to recover 80$\%$ of the constraining power in $\textit{Euclid}$'s DR1 $w_{\mathrm{0}}w_{a}$CDM 3$\times$2pt analysis. The impact of the uncertainty on the width is negligible, provided the mean redshift is constrained with sufficient accuracy. For most sources of redshift distribution error, attaining the requirement on the mean will also reduce uncertainty in the width well below the required level.