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The data on spectroscopic galaxy clustering collected by the Dark EnergySpectroscopic Instrument (DESI) will allow the significant detection of subtlefeatures in the galaxy two-point correlation in redshift space, beyond the"standard" redshift-space distortions. Here we present an independentassessment of the detectability of the relativistic dipole in thecross-correlation of two populations of galaxies if they would be selected fromthe Bright Galaxy Survey (BGS) of DESI. We build synthetic galaxy catalogueswith the characteristics of the BGS using the light cone of a relativistic$N$-body simulation. Exploring different ways of splitting the populations ofgalaxies we find that with an unequal split with more bright galaxies thanfaint galaxies the detectability is significantly boosted, reaching 19 $\sigma$in the redshift bin $0.2 \lesssim z \lesssim 0.3$ and expected to be evenhigher at lower redshift. Moreover, we find that the measured dipole agreesvery well with the prediction of relativistic effects from linear theory downto separations of $\sim$ 30 Mpc/$h$.

A case study for measuring the relativistic dipole of a galaxy cross-correlation with the Dark Energy Spectroscopic Instrument