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

Technical Summary: Dynamical Systematics for Time Delay Lenses and the Impact on the Hubble Constant

Problem Statement
Time-delay gravitational lenses offer a method to measure the Hubble constant ($H_0$) independently of the cosmic distance ladder. However, these measurements suffer from a fundamental degeneracy where $H_0 \propto 1 - \kappa_E$, with $\kappa_E$ being the convergence near the Einstein radius. While lensing data constrain the Einstein radius ($R_E$) and a dimensionless quantity $\xi$ related to the mass profile slope, they cannot determine $\kappa_E$ without additional assumptions. Stellar dynamical measurements, specifically central velocity dispersions ($\sigma$), are used to break this degeneracy. The accuracy of $H_0$ derived from this method is limited by the precision of the kinematic measurements; achieving a 2% precision in $H_0$ requires systematic uncertainties in the squared velocity dispersion ($\Delta\sigma^2/\sigma^2$) to be less than 2%. This paper investigates the range of potential systematic uncertainties affecting eight time-delay lenses used in cosmological analyses, arguing that current estimates of $H_0$ may be biased by unaccounted systematics in the dynamical modeling.

Methodology
The authors analyze eight time-delay lenses (DES J0408−5354, HE 0435−1223, PG 1115+080, RX J1131−1231, SDSS J1206+4332, B1608+656, WFI 2033−4723, and WGD 2038−4008) using a spherically symmetric dynamical model. The study employs a Hernquist stellar distribution embedded in a Singular Isothermal Sphere (SIS) mass model to isolate systematic effects. The analysis focuses on four primary categories of systematics:

1. Point Spread Function (PSF) Modeling: The authors test the sensitivity of velocity dispersion measurements to errors in the PSF Full Width at Half Maximum (FWHM), aperture miscentering, and the assumption of a Gaussian PSF versus more realistic Moffat profiles.
2. Measured vs. True Dispersion: The study distinguishes between the measured velocity dispersion ($\sigma_*$, derived from fitting a Gaussian to spectral lines) and the root-mean-square velocity ($v_{rms}$) required for the Jeans equations. The authors quantify the bias introduced by non-Gaussian velocity distributions, characterized by the Gauss-Hermite moment $h_4$.
3. Anisotropy Models: The authors compare constant anisotropy models, the Osipkov-Merritt (O-M) model, and alternative profiles (Mamon & Lokas; Cuddeford) to determine how the choice of orbital anisotropy affects the inferred velocity dispersion and $H_0$.
4. Photometric and Stellar Population Models: The analysis examines the impact of using different stellar density profiles (Sérsic vs. Hernquist vs. Jaffe), the choice of effective radius axis (major vs. intermediate), and the complications arising from color gradients and multiple stellar populations in early-type galaxies.

Additionally, the paper addresses the statistical treatment of the lens sample, arguing that early-type galaxies constitute a homogeneous population in terms of dynamics and stellar populations, implying that certain systematic errors should be marginalized over the sample as a whole rather than treated as independent random errors for each lens.

Key Contributions and Results
The authors find that systematic uncertainties in dynamical modeling can easily exceed the 2% threshold required for competitive $H_0$ measurements. The specific findings include:

• PSF Systematics: For ground-based observations, biases in the PSF width and shape can induce fractional changes in $\sigma^2$ of 1–5%. Underestimating the PSF width or assuming a Gaussian profile when the true PSF has extended wings (Moffat) leads to underestimated velocity dispersions, driving models toward less centrally concentrated mass distributions and lower $H_0$ values. Aperture miscentering has a smaller effect (typically <0.3%) but still biases results downward.
• Dispersion Definition: The difference between the measured $\sigma_*$ and the required $v_{rms}$ is significant. For isotropic models, this difference ranges from ~1.8% to 5.6% across the sample. If early-type galaxies are moderately radially anisotropic (as expected), the true $v_{rms}$ is larger than the measured $\sigma_*$, leading to underestimates of $H_0$ if uncorrected.
• Anisotropy Models: The choice of anisotropy model has a profound impact. While constant anisotropy models and O-M models with large break radii lie within a narrow envelope, models allowing for moderate radial anisotropy in the central regions (e.g., Cuddeford models with $\beta_0 > 0$) can increase the inferred velocity dispersion by 2–18%. This shifts the mass distribution to be less concentrated, potentially biasing $H_0$ upward if not properly constrained. Furthermore, models producing higher $h_4$ values (consistent with observations of early-type galaxies) require larger corrections between $\sigma_*$ and $v_{rms}$.
• Photometric Profiles: Mismatches between the assumed stellar mass profile and the true distribution (e.g., using a Sérsic profile with an incorrect index) can cause $\Delta\sigma^2/\sigma^2$ variations of 1–10%. Additionally, the presence of color gradients implies that the stellar number density $n$ in the Jeans equation should be weighted by the equivalent width of spectral lines rather than the broadband light profile, a factor currently unaccounted for in standard analyses.
• Population Homogeneity: The paper argues that treating parameters like the anisotropy radius or stellar template choice as independent random variables for each lens underestimates the total uncertainty. Because early-type galaxies are a homogeneous population, systematic errors in these parameters are correlated across the sample. Marginalizing over these parameters at the population level (rather than per lens) prevents the artificial reduction of uncertainty by a factor of $N^{-1/2}$.

Significance and Claims
The paper claims that the current precision of $H_0$ estimates from time-delay lenses is likely limited by systematic uncertainties in the dynamical modeling rather than statistical noise. The authors demonstrate that biases in $\sigma^2$ of 1–18% are possible depending on the modeling choices, which is comparable to or larger than the required 2% precision for $H_0$.

The authors emphasize that:

1. Absolute Calibration Matters: Unlike pure stellar dynamical studies which rely on ratios, lensing depends on the absolute calibration of velocity dispersions, making it highly sensitive to systematic errors in PSF modeling and dispersion definition.
2. Model Dependencies: The choice of anisotropy model and the treatment of the velocity distribution function (specifically the $h_4$ correction) are critical. Current models often fail to reproduce the $h_4$ values observed in early-type galaxies, suggesting that current corrections are underestimates.
3. Statistical Treatment: The assumption of a heterogeneous lens population leads to an underestimation of final uncertainties. A homogeneous population model requires marginalizing systematic parameters over the entire sample, which preserves the systematic error contribution regardless of sample size.

The paper concludes that achieving a 2% measurement of $H_0$ requires controlling these systematic uncertainties, particularly the absolute calibration of velocity dispersions and the correct modeling of orbital anisotropy and stellar populations. While spatially resolved kinematics (2D data) offer a path to breaking degeneracies, the fundamental systematic biases regarding the definition of velocity dispersion and the homogeneity of the population must still be addressed.