Impact of Weak Lensing Mass Calibration on eROSITA Galaxy Cluster
  Cosmological Studies -- a Forecast
  We forecast the impact of weak lensing (WL) cluster mass calibration on the
cosmological constraints from the X-ray selected galaxy cluster counts in the
upcoming eROSITA survey. We employ a prototype cosmology pipeline to analyze
mock cluster catalogs. Each cluster is sampled from the mass function in a
fiducial cosmology and given an eROSITA count rate and redshift, where count
rates are modeled using the eROSITA effective area, a typical exposure time,
Poisson noise and the scatter and form of the observed X-ray luminosity-- and
temperature--mass--redshift relations. A subset of clusters have mock shear
profiles to mimic either those from DES and HSC or from the future Euclid and
LSST surveys. Using a count rate selection, we generate a baseline cluster
cosmology catalog that contains 13k clusters over 14,892~deg$^2$ of
extragalactic sky. Low mass groups are excluded using raised count rate
thresholds at low redshift. Forecast parameter uncertainties for
$\Omega_\mathrm{M}$, $\sigma_8$ and $w$ are 0.023 (0.016; 0.014), 0.017 (0.012;
0.010), and 0.085 (0.074; 0.071), respectively, when adopting DES+HSC WL
(Euclid; LSST), while marginalizing over the sum of the neutrino masses. A
degeneracy between the distance--redshift relation and the parameters of the
observable--mass scaling relation limits the impact of the WL calibration on
the $w$ constraints, but with BAO measurements from DESI an improved
determination of $w$ to 0.043 becomes possible. With Planck CMB priors,
$\Omega_\text{M}$ ($\sigma_8$) can be determined to $0.005$ ($0.007$), and the
summed neutrino mass limited to $\sum m_\nu < 0.241$ eV (at 95\%). If
systematics on the group mass scale can be controlled, the eROSITA group and
cluster sample with 43k objects and LSST WL could constrain $\Omega_\mathrm{M}$
and $\sigma_8$ to 0.007 and $w$ to 0.050.
