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 $\Omega_m = 0.300 \pm 0.008$, $H_0 = 69.21 \pm 0.77$
and $\sigma_8 = 0.707 \pm 0.035$, in tension with CMB and cosmic shear
constraints. The combination of $\Omega_m$ 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
$20\%$ 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.
