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 $β$, 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.

💡 Research Summary

The paper addresses one of the most pressing issues in contemporary cosmology—the Hubble tension—by investigating how assumptions about stellar velocity anisotropy affect the inference of the Hubble constant (H₀) from strong‑lens time‑delay cosmography when spatially resolved stellar kinematics are included. Using ten massive elliptical galaxies at redshift z≈0.2 drawn from the Illustris TNG100 simulation, the authors generate mock integral‑field spectroscopy (IFS) data that mimic JWST NIRSpec observations, producing line‑of‑sight velocity‑dispersion maps with realistic spatial resolution and noise characteristics.

Four anisotropy parameterizations are tested: (i) the classic Osipkov‑Merritt (OM) model, which uses a single scale radius to transition from isotropic to radial orbits; (ii) the Mamon‑Lokas (ML) model, which introduces separate central (β₀) and outer (β∞) anisotropy values together with a transition radius; (iii) a constant‑β model; and (iv) a generalized OM (gOM) model that adds a power‑law index to control the sharpness of the transition, thereby providing three free parameters.

Two analysis pipelines are employed. First, a “kinematics‑only” approach fits the mock IFS data using spherical Jeans modeling to recover the total mass profile and the anisotropy parameters. Second, a joint analysis combines the kinematic likelihood with strong‑lens constraints (image positions, magnifications, and time delays) within a Bayesian framework using nested sampling. The authors adopt a fiducial H₀=70 km s⁻¹ Mpc⁻¹ for all simulations and quantify the relative bias (ΔH₀/H₀) introduced by each anisotropy model under both pipelines.

Results show that the OM model performs poorly. In the kinematics‑only case, H₀ biases can exceed 20 %, reflecting the model’s inability to capture the complex radial variation of β seen in the simulated galaxies. Even when combined with lensing data, OM still yields a systematic overestimate of H₀ by +11.5 % ± 1.3 %, a bias that does not average out across the sample and would therefore be a serious systematic in real surveys.

In contrast, the more flexible models dramatically reduce bias. The ML model, with three free parameters, achieves an average bias of +0.2 % ± 1.6 % in the joint analysis, with a scatter of roughly 5 % among the ten galaxies. The constant‑β model, despite its simplicity, also performs well when high‑quality IFS data are available, giving a bias of –0.9 % ± 1.9 % under joint modeling. The gOM model, which balances flexibility and parsimony, consistently recovers the true mass, radius, and anisotropy parameters across all galaxies and yields the smallest systematic offset, –0.9 % ± 1.6 % in H₀, with uncertainties that faithfully reflect the posterior spread.

The authors further test the robustness of these conclusions by degrading the mock data (lower signal‑to‑noise, reduced radial coverage). Even under poorer data quality, the gOM and constant‑β models remain the most reliable, whereas OM’s bias worsens dramatically. This demonstrates that the choice of anisotropy parameterization is a dominant source of systematic error in H₀ inference from strong‑lens time‑delay cosmography with spatially resolved kinematics.

Overall, the study provides a clear prescription for future observational programs: (1) incorporate spatially resolved stellar kinematics to break the mass‑sheet degeneracy; (2) avoid overly restrictive anisotropy models such as the single‑parameter OM; (3) adopt flexible yet well‑constrained models like gOM (or ML when sufficient data support three parameters); and (4) perform joint lens‑dynamics Bayesian analyses to achieve sub‑percent accuracy in H₀. With upcoming JWST, ELT, Euclid, and LSST datasets delivering high‑quality IFS for thousands of strong lenses, the methodology outlined here offers a realistic pathway to independent, percent‑level measurements of the Hubble constant, thereby contributing crucial evidence toward resolving the Hubble tension.