In this talk, I consider, for the first time, cross-correlating galaxy peculiar velocities with the lensing of the cosmic microwave background. By looking at the statistical distribution of galaxies in the sky we can learn something about the underlying matter density field, i.e. the distribution of dark matter. However, our ignorance about the fundamental nature of dark matter implies a certain ignorance regarding the relationship between these distributions. At linear order in perturbation theory, it is reasonable to assume that the galaxy and dark matter densities are proportional, with the proportionality parametrised by the linear galaxy bias b(z). As a result, cosmological inference based on the number counting of galaxies is, to some degree, degenerate with the value of b(z). To avoid this problem, one could consider alternative tracers of the DM distribution with no dependence on b(z) and try to break these degeneracies. Two important examples are the peculiar velocities of galaxies and the lensing convergence of the MB photons, both sourced by the gradient of the dark matter distribution. I will show that their cross-correlation is a powerful cosmological probe of the underlying dark matter distribution on large angular scales, despite their kernels peaking at very different redshifts. An exciting feature of this cross-correlation is that most of its constraining power comes from the edge of the peculiar velocity survey and is even sensitive to the density beyond the edge.
I will discuss the detectability of this cross-correlation with existing and forthcoming surveys and the challenges involved with its detection. Whilst it appears that using direct peculiar velocity measurements the signal might be only marginally detectable, a reconstructed peculiar velocity field drastically improves the cumulative signal-to-noise
