Dynamical Systematics on Time Delay Lenses and the Impact on the Hubble Constant

While time-delay lenses can be an independent probe of $H_0$ 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 the 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 contributing to eight time-delay lenses used in cosmological analyses. We find that: (1) The characterization of the PSF both in absolute scale and in shape is important, with biases in $\Delta\sigma^2/\sigma^2$ up to 1-6%, depending on the lens system. Small slit 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 $\Delta\sigma^2/\sigma^2 \sim$ 3-8%. (3) The choice of anisotropy models is important with maximum changes of $\Delta\sigma^2/\sigma^2 \sim$ 5-18% and the frequently used Osipkov-Merritt models do not produce $h_4$ velocity moments typical of early-type galaxies. (4) Small differences between the true stellar mass distribution and the model light profile matter ($\Delta\sigma^2/\sigma^2 \sim$ 5-40%), with radial color gradients further complicating the problem. Finally, we discuss how 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.