  Weak gravitational lensing of the Cosmic Microwave Background (CMB) changes
CMB statistics in a nontrivial way, allowing for reconstruction of the lensing
potential and the use of these reconstructed maps in determining cosmological
parameters that affect the formation of intervening large-scale structures.
Although in principle there are correlations between the primary CMB and the
reconstructed lensing potential due to the lensing procedure itself, in
practice CMB analyses treat these as negligible when combining these band
powers in likelihoods. In this paper we quantify explicitly the impact on
parameter constraints due to these cross-covariances between the lensed CMB and
reconstructed lensing power, and we compare to the effect of including all
lensing-induced non-Gaussian covariances, which have previously shown to impact
parameter constraints on the order of 10%. We perform our analysis for a range
of experimental setups, scanning over instrumental noise levels of 0.5 to 10.0
$\mu$K-arcmin in temperature assuming fully polarized detectors, and using a
fixed beam size of 1.4 arcmin. When the correlations between the lensed CMB and
lensing power are neglected, we find that forecasted constraints shift by at
most 3% of the error bar for a 6-parameter $\Lambda$CDM model, and for the
noise levels considered in this paper. For some of the $\Lambda$CDM extensions
considered here, however, these correlations have a nontrivial impact, in some
cases more than 10% of the error bar, even for current experimental noise
levels.
