Cosmological Analysis of Three-Dimensional BOSS Galaxy Clustering and Planck CMB Lensing Cross Correlations via Lagrangian Perturbation Theory

We present a formalism for jointly fitting pre- and post-reconstruction redshift-space clustering (RSD) and baryon acoustic oscillations (BAO) plus gravitational lensing (of the CMB) that works directly with the observed 2-point statistics. The formalism is based upon (effective) Lagrangian perturbation theory and a Lagrangian bias expansion, which models RSD, BAO and galaxy-lensing cross correlations within a consistent dynamical framework. As an example we present an analysis of clustering measured by the Baryon Oscillation Spectroscopic Survey in combination with CMB lensing measured by Planck. The post-reconstruction BAO strongly constrains the distance-redshift relation, the full-shape redshift-space clustering constrains the matter density and growth rate, and CMB lensing constrains the clustering amplitude. Using only the redshift space data we obtain $\Omega_$\mathrm{m}$ = 0.303\pm 0.008$,$H_0 = 69.21\pm 0.78$and$\sigma_8 = 0.743\pm 0.043$. The addition of lensing information, even when restricted to the Northern Galactic Cap, improves constraints to ${\mathrm{\Omega }}_m=0.300\pm 0.008$, $H_0=69.21\pm 0.77$ and ${\sigma }_8=0.707\pm 0.035$, in tension with CMB and cosmic shear constraints. The combination of ${\mathrm{\Omega }}_m$ and $H_0$ are consistent with Planck, though their constraints derive mostly from redshift-space clustering. The low ${\sigma }_8$ value are driven by cross correlations with CMB lensing in the low redshift bin ($z\simeq 0.38$) and at large angular scales, which show a $20\mathrm{\%}$ deficit compared to expectations from galaxy clustering alone. We conduct several systematics tests on the data and find none that could fully explain these tensions.