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, $σ$, can be used to break the degeneracy, with uncertainties $ΔH/H_0 \propto Δσ^2/σ^2$ ultimately limited by the systematic uncertainties in the kinematic measurements - measuring $H_0$ to 2% requires $Δσ^2/σ^2$ < 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 $Δσ^2/σ^2$ 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 $Δσ^2/σ^2 \sim$ 3-8%. (3) The choice of anisotropy models is important with maximum changes of $Δσ^2/σ^2 \sim$ 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 ($Δσ^2/σ^2 \sim$ 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.

💡 Research Summary

The paper investigates the systematic uncertainties that affect the use of time‑delay gravitational lenses for measuring the Hubble constant (H₀). While the time‑delay distance provides a direct probe of H₀, the inference is degenerate with the convergence (κ_E) at the Einstein radius. Stellar dynamics, specifically the central velocity dispersion (σ), can break this degeneracy, but the required precision (Δσ²/σ² < 2 % to achieve a 2 % H₀ measurement) is extremely demanding.

The authors analyse eight well‑studied lenses (DES J0408‑5354, HE 0435‑1223, PG 1115+080, RX J1131‑1231, SDSS J1206+4332, B1608+656, WFI 2033‑4723, WGD 2038‑4008) and quantify four major sources of systematic error in σ:

1. Point‑Spread Function (PSF) modelling – Errors in the absolute scale and shape of the PSF bias σ² by 1–6 % depending on the lens. Small slit mis‑centering is comparatively negligible.

2. Difference between observed σ and the mean‑square velocity needed for the Jeans equation – Imperfect correction for the line‑spread function, template‑star mismatches, and spectral resolution uncertainties lead to a 3–8 % bias in σ².

3. Orbital anisotropy modelling – The widely used Osipkov‑Merritt (OM) anisotropy model fails to reproduce the typical fourth‑order Gauss‑Hermite moment (h₄≈0.01–0.04) of early‑type galaxies, causing σ² biases of 5–18 %. More flexible anisotropy prescriptions are required.

4. Assumed stellar mass distribution – Most analyses adopt a spherical Hernquist profile for the stars, whereas the actual light profiles are better described by double‑Sérsic or pseudo‑Jaffe models with radial colour gradients. This mismatch introduces σ² biases ranging from 5 % up to 40 %.

The authors introduce a sensitivity parameter ψ = d ln H₀ / d ln σ², which quantifies how a fractional change in σ² propagates to H₀. For the eight lenses ψ spans –0.63 to 3.68, with a median ≈ 0.93, indicating that a 1 % error in σ² typically translates into a ~1 % error in H₀. Consequently, to keep the total H₀ uncertainty below 2 % the combined systematic error budget on σ² must be ≤ 2 %.

A further key insight is that early‑type lens galaxies form a relatively homogeneous population in both dynamics and stellar populations. Treating each lens as an independent source of systematic error therefore underestimates the true uncertainty. Instead, the authors advocate marginalising over shared dynamical parameters (anisotropy, M/L gradients, colour‑gradient effects) at the sample level. This hierarchical approach preserves statistical power while correctly propagating systematic biases.

In summary, the paper demonstrates that current dynamical systematics—PSF characterization, Jeans‑equation velocity conversion, anisotropy modelling, and stellar mass‑light profile assumptions—can each contribute several percent bias to σ², which directly contaminates H₀ estimates. Achieving sub‑2 % precision on H₀ with time‑delay lenses will require (i) high‑fidelity PSF reconstruction, (ii) rigorous treatment of the σ–⟨v²⟩ conversion, (iii) adoption of flexible, data‑driven anisotropy models, (iv) realistic stellar mass models that incorporate colour gradients, and (v) a hierarchical Bayesian framework that marginalises shared systematics across the lens sample. Only by addressing these issues can time‑delay lenses fulfill their promise as an independent, high‑precision probe of the Hubble constant.