  We present a method to obtain a high-significance detection of relativistic
effects on cosmological scales. Measurements of such effects would be
instrumental for our understanding of the Universe, as they would provide a
further confirmation of the validity of general relativity as the correct
description of the gravitational interaction, in a regime very far from that of
strong gravity, where it has been tested to exquisite accuracy. Despite its
relevance, the detection of relativistic effects has hitherto eluded us, mainly
because they are stronger on the largest cosmic scales, plagued by cosmic
variance. Our work focuses on the cosmological probe of galaxy clustering,
describing the excess probability of finding pairs of galaxies at a given
separation due to them being part of the same underlying cosmic large-scale
structure. We focus on the two-point correlation function of the distribution
of galaxies in Fourier space -- the power spectrum -- where relativistic
effects appear as an imaginary contribution to the real power spectrum. By
carefully tailoring cuts in magnitude/luminosity, we are able to obtain two
samples (bright and faint) of the same galaxy population, whose
cross-correlation power spectrum allows for a detection of the relativistic
contribution. In particular, we optimise the definition of the samples to
maximise the detection significance of the relativistic Doppler term for both a
low-$z$ Bright Galaxy Sample and a high-$z$ H$\alpha$ emission line galaxy
population.
