Understanding the neutrino mass constraints achievable by combining CMB lensing and spectroscopic galaxy surveys

We perform a thorough examination of the neutrino mass ($M_\nu$) constraints achievable by combining future spectroscopic galaxy surveys with cosmic microwave background (CMB) experiments, focusing on the contribution of CMB lensing. CMB lensing can help by breaking the $M_\nu$-curvature degeneracy when combined with baryon acoustic oscillation (BAO)-only measurements, but we demonstrate this combination wastes a great deal of constraining power, as the broadband shape of the power spectrum contributes significantly to constraints. We also expand on previous work to demonstrate how cosmology-independent constraints on $M_\nu$ can be extracted by combining measurements of the scale-dependence in the power spectrum caused by neutrino free-streaming with the full power of future CMB surveys. These free-streaming constraints are independent of the optical depth to the CMB ($\tau$) and give stronger constraints alone on $M_\nu$ than are given by the combination of BAOs and CMB lensing. Finally, we demonstrate that the effect of including the galaxy-CMB lensing cross power spectrum is negligible.