The Physical Origin of Dark Energy Constraints from Rubin Observatory
  and CMB-S4 Lensing Tomography
  We seek to clarify the origin of constraints on the dark energy equation of
state parameter from CMB lensing tomography, that is the combination of galaxy
clustering and the cross-correlation of galaxies with CMB lensing in a number
of redshift bins. In particular, we consider the two-point correlation
functions which can be formed with a catalog of galaxy locations and
photometric redshifts from the Vera C. Rubin Observatory Legacy Survey of Space
and Time (LSST) and CMB lensing maps from the CMB-S4 experiment. We focus on
the analytic understanding of the origin of the constraints. Dark energy
information in these data arises from the influence of three primary
relationships: distance as a function of redshift (geometry), the amplitude of
the power spectrum as a function of redshift (growth), and the power spectrum
as a function of wavenumber (shape). We find that the effects from geometry and
growth play a significant role and partially cancel each other out, while the
shape effect is unimportant. We also show that Dark Energy Task Force (DETF)
Figure of Merit (FoM) forecasts from the combination of LSST galaxies and
CMB-S4 lensing are comparable to the forecasts from cosmic shear in the absence
of the CMB lensing map, thus providing an important independent check. Compared
to the forecasts with the LSST galaxies alone, combining CMB lensing and LSST
clustering information (together with the primary CMB spectra) increases the
FoM by roughly a factor of 3-4 in the optimistic scenario where systematics are
fully under control. We caution that achieving these forecasts will likely
require a full analysis of higher-order biasing, photometric redshift
uncertainties, and stringent control of other systematic limitations, which are
outside the scope of this work, whose primary purpose is to elucidate the
physical origin of the constraints.
