The power spectrum of luminosity distance fluctuations in General Relativity

At low redshift, it is possible to combine spectroscopic information of galaxies with their luminosity or angular diameter distance to directly measure the projection of peculiar velocities (PV) along the line-of-sight. A PV survey probing a large fraction of the sky is subject to so-called wide-angle effects, arising from the variation of the line-of-sight across the sky, and other sub-leading projection effects due to the propagation of the photons in a perturbed cosmological background. In this work, for the first time, we provide a complete description, within linear theory and General Relativity, of the power spectrum of luminosity distance fluctuations, clarifying its relation to the observables in a PV survey. We find that wide-angle effects will be detected at high significance by future observations and will have to be included in the cosmological analysis. Other relativistic projections effects could also be detected provided accurate, per object, distances are available.