The stellar velocity anisotropy of strong lensing massive elliptical galaxies and its role in the inference of the Hubble parameter $H_0$ using spatially resolved kinematics
One of the biggest challenges in cosmology, the Hubble Tension, requires independent measurements of $H_0$, and strong lensing with time-delay cosmography is a promising avenue. The inclusion of spatially resolved kinematic data helps break the mass--sheet degeneracy, a key limitation in strong lensing. Kinematics, however, suffers from its own degeneracy due to unknown stellar velocity anisotropy, which can bias galaxy mass profile inferences. We investigate the bias in $H_0$ using a sample of ten massive elliptical galaxies at $z=0.2$ from the Illustris $TNG100$ simulations. We generate mock line-of-sight velocity-dispersion maps resembling JWST NIRSpec observations and test four anisotropy models: Osipkov--Merritt (OM), Mamon--Lokas (ML), constant $\beta$, and a generalized--OM (gOM) profile, under both kinematics-only and joint kinematics plus strong lensing analyses. We find a sub-percent average bias in $H_{0}$ across ten galaxies with joint modeling for three models: $+0.2 \pm 1.6\%$ (ML), $-0.9 \pm 1.9\%$ (constant) and $-0.9 \pm 1.6\%$ (gOM), with $\sim 5\%$ scatter. Joint modeling reduces bias, improves precision, and mitigates outlier results. Overall, the gOM model best recovers galaxy parameters and delivers the most accurate $H_{0}$ relative to posterior uncertainties considering both analyses. However, the single-parameter OM model produces large systematic biases: with kinematics only data, $H_{0}$ errors can exceed $20\%$, and even with joint modeling, produces an overall bias of $+11.5 \pm 1.3\%$ (OM). The higher bias in OM is unlikely to average out across an ensemble of galaxies. Our findings highlight the impact of anisotropy assumptions on $H_{0}$ inference and, more broadly, in galaxy dynamics.