Fundamental Physics from Future Weak-Lensing Calibrated
  Sunyaev-Zel'dovich Galaxy Cluster Counts
  Future high-resolution measurements of the cosmic microwave background (CMB)
will produce catalogs of tens of thousands of galaxy clusters through the
thermal Sunyaev-Zel'dovich (tSZ) effect. We forecast how well different
configurations of a CMB Stage-4 experiment can constrain cosmological
parameters, in particular the amplitude of structure as a function of redshift
$\sigma_8(z)$, the sum of neutrino masses $\Sigma m_{\nu}$, and the dark energy
equation of state $w(z)$. A key element of this effort is calibrating the tSZ
scaling relation by measuring the lensing signal around clusters. We examine
how the mass calibration from future optical surveys like the Large Synoptic
Survey (LSST) compares with a purely internal calibration using lensing of the
CMB itself. We find that, due to its high-redshift leverage, internal
calibration gives constraints on cosmological parameters comparable to the
optical calibration, and can be used as a cross-check of systematics in the
optical measurement. We also show that in contrast to the constraints using the
CMB lensing power spectrum, lensing-calibrated tSZ cluster counts can detect a
minimal $\Sigma m_{\nu}$ at the 3-5$\sigma$ level even when the dark energy
equation of state is freed up.
