Missing baryons, bulk flows and the E-mode polarization of the Cosmic Microwave Background

If the peculiar motion of galaxy groups and clusters indeed resembles that of the surrounding baryons, then the kinetic Sunyaev-Zel'dovich (kSZ) pattern of those massive halos should be closely correlated to the kSZ pattern of all surrounding electrons. Likewise, it should also be correlated to the CMB E-mode polarization field generated via Thomson scattering after reionization. We explore the cross-correlation of the kSZ generated in groups and clusters to the all sky E-mode polarization in the context of upcoming CMB experiments like Planck, ACT, SPT or APEX. We find that this cross-correlation is effectively probing redshifts below $z=3-4$ (where most of baryons cannot be seen), and that it arises in the very large scales ($l<10$). The significance with which this cross-correlation can be measured depends on the Poissonian uncertainty associated to the number of halos where the kSZ is measured and on the accuracy of the kSZ estimations themselves. Assuming that Planck can provide a cosmic variance limited E-mode polarization map at $l<20$ and S/N $\sim 1$ kSZ estimates can be gathered for all clusters more massive than $10^{14} M_{\odot}$, then this cross-correlation should be measured at the 2--3 $\sigma$ level. Further, if an all-sky ACT or SPT type CMB experiment provides similar kSZ measurements for all halos above $10^{13} M_{\odot}$, then the cross-correlation total signal to noise (S/N) ratio should be at the level of 4--5. A detection of this cross-correlation would provide direct and definite evidence of bulk flows and missing baryons simultaneously.