Anisotropies in the electromagnetic emission produced by dark matter annihilation or decay in the extragalactic sky are a recent tool in the quest for a particle dark matter evidence. We review the formalism to compute the two-point angular power spectrum in the halo-model approach and discuss the features and the relative size of the various auto- and cross-correlation signals that can be envisaged for anisotropy studies. From the side of particle dark matter signals, we consider the full multi-wavelength spectrum, from the radio emission to X-ray and gamma-ray productions. We discuss the angular power spectra of the auto-correlation of each of these signals and of the cross-correlation between any pair of them. We then extend the search to comprise specific gravitational tracers of dark matter distribution in the Universe: weak-lensing cosmic shear, large-scale-structure matter distribution and CMB-lensing. We have shown that cross-correlating a multi-wavelength dark matter signal (which is a direct manifestation of its particle physics nature) with a gravitational tracer (which is a manifestation of the presence of large amounts of unseen matter in the Universe) may offer a promising tool to demonstrate that what we call DM is indeed formed by elementary particles.
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