Reassessing the Statistical Necessity of Stellar Velocity Anisotropy in Strong-Lensing Cosmology with Lens-by-Lens Photometric Constraints

Abstract

The stellar orbital anisotropy parameter ($\beta_{\rm ani}$) is a persistent systematic uncertainty in galaxy-scale strong gravitational lensing (SGL) cosmology. Typically fixed to isotropy or a local prior, it frequently degenerates with the lens density profile. We demonstrate this apparent redundancy largely arises from incomplete photometric constraints. We cross-matched 130 SGL systems with the Pantheon+ SN~Ia compilation, constructing a strictly matched sample of 107 SGL-SN pairs using a 5\% comoving-distance tolerance. Assuming a flat universe ($\Omega_k = 0$), the distance ratio is derived from apparent magnitude differences between paired SNe~Ia, eliminating $H_0$ and absolute magnitude dependence without fitting explicit dark-energy models. To break the kinematic degeneracy, we incorporate lens-by-lens luminosity density slopes ($\delta_i$) from high-resolution imaging. Adopting the quasi-model-independent P2 redshift-evolutionary framework ($\gamma(z) = \gamma_0 + \gamma_z z$), we find very strong statistical evidence for a free $\beta_{\rm ani}$. Fixing $\beta_{\rm ani}$ to isotropy ($\beta=0$) or a local prior ($\beta=0.18$) is strongly disfavored ($\Delta\bic = 14.2$ and $48.9$) and artificially inflates intrinsic scatter. A complementary P3 framework ($\gamma(z,\tilde{\Sigma}) = \gamma_0 + \gamma_z z + \gamma_s \log_{10}\tilde{\Sigma}$) confirms these penalties ($\Delta\bic = 13.5$ and $49.1$). Across all P2 variants, we consistently detect a negative redshift evolution of the density slope ($\gamma_z \approx -0.42$ to $-0.46$; ${\sim}1.5{-}2.0\sigma$), indicating ETG density profiles become shallower at higher redshifts. We conclude that when individual photometric constraints are incorporated, $\beta_{\rm ani}$ is statistically required as a free parameter to prevent severe dynamical modeling biases.