W., Lang, D., & Goodman, J

While time-delay lenses can independently probe $H_0$, the estimates are degenerate with the convergence of the lens near the Einstein radius. Velocity dispersions, $\sigma$, can be used to break the degeneracy, with uncertainties $\Delta H/H_0 \propto \Delta\sigma^2/\sigma^2$ ultimately limited by systematic uncertainties in the kinematic measurements - measuring $H_0$ to 2\% requires $\Delta\sigma^2/\sigma^2 < 2\%$. Here we explore a broad range of potential systematic uncertainties affecting eight time-delay lenses used in cosmological analyses. We find that: (1) The characterization of the PSF in both absolute scale and shape is important, with biases in $\Delta\sigma^2/\sigma^2$ up to $1$-$5\%$ for ground-based observations. Small miscenterings of the lens are less important. (2) The difference between the measured velocity dispersion and the mean square velocity needed for the Jeans equations is important, with up to $\Delta\sigma^2/\sigma^2 \sim 2$-$6\%$. (3) The choice of anisotropy models is important with maximum changes of $\Delta\sigma^2/\sigma^2 \sim2$-$18\%$. Biases may be minimized by using models that reproduce the $h_4$ velocity moments typical of early-type galaxies. (4) Small differences between the true stellar mass distribution and the model light profile matter ($\Delta\sigma^2/\sigma^2 \sim 1$-$10\%$), with radial color gradients further complicating the problem. The Jeans equations for mixed stellar populations imply that the correct profile is a population line equivalent width weighting corresponding to no broad band filter profile. Finally, the homogeneity of the early-type galaxy population means that many dynamically related parameters must be marginalized over the lens sample as a whole and not over individual lenses.