  Compton scattering of the cosmic microwave background (CMB) from hot ionized
gas produces a range of effects, and the leading order effects are the kinetic
and thermal Sunyaev Zel'dovich (kSZ and tSZ) effects. In the near future, CMB
surveys will provide the precision to probe beyond the leading order effects.
In this work we study the cosmological information content of the next order
term which combines the tSZ and kSZ effects, hereafter called the
thermal-kinetic Sunyaev Zel'dovich (tkSZ) effect. As the tkSZ effect has the
same velocity dependence as the kSZ effect, it will also have many of the
useful properties of the kSZ effect. However, it also has its own, unique
spectral dependence, which allows it to be isolated from all other CMB signals.
We show that with currently-envisioned CMB missions the tkSZ effect can be
detected and can be used to reconstruct large scale velocity fields, with no
appreciable bias from either the kSZ effect or other extragalactic foregrounds.
Furthermore, since the relativistic corrections arise from the well-studied
pressure of ionized gas, rather than the gas number density as in the kSZ
effect, the degeneracy due to uncertain gas physics will be significantly
reduced. Finally, for a very low-noise experiment the tkSZ effect will be
measurable at higher precision than the kSZ.
