Euclid: Photometric redshift calibration with the clustering redshifts technique

Aims: The precision of cosmological constraints from imaging surveys hinges on accurately estimating the redshift distribution $n(z)$ of tomographic bins, especially their mean redshifts. We assess the effectiveness of the clustering redshifts technique in constraining Euclid tomographic redshift bins to meet the target uncertainty of $\sigma ( \langle z \rangle ) < 0.002 (1 + z)$. In this work, these mean redshifts are inferred from the small-scale angular clustering of Euclid galaxies, which are distributed into bins with spectroscopic samples localised in narrow redshift slices. Methods: We generate spectroscopic mocks from the Flagship2 simulation for the Baryon Oscillation Spectroscopic Survey (BOSS), the Dark Energy Spectroscopic Instrument (DESI), and Euclid's Near-Infrared Spectrometer and Photometer (NISP) spectroscopic survey. We evaluate and optimise the clustering redshifts pipeline, introducing a new method for measuring photometric galaxy bias (clustering), which is the primary limitation of this technique. Results: We have successfully constrained the means and standard deviations of the redshift distributions for all of the tomographic bins (with a maximum photometric redshift of 1.6), achieving precision beyond the required thresholds. We have identified the main sources of bias, particularly the impact of the 1-halo galaxy distribution, which imposed a minimal separation scale of 1.5 Mpc for evaluating cross-correlations. These results demonstrate the potential of clustering redshifts to meet the precision requirements for Euclid, and we highlight several avenues for future improvements.