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

TL;DR

The paper analyzes dynamical-systematic uncertainties in time-delay lens cosmography across eight lenses, focusing on how PSF modeling, aperture effects, anisotropy choices, and photometric mass modeling propagate into $H_0$ inferences via Jeans modeling. It demonstrates that $H_0$ constraints are sensitive to $\Delta\sigma^2/\sigma^2$ values, with PSF and aperture effects typically producing 1–6% changes, while anisotropy and photometric-profile choices can yield larger shifts up to 18% and 40% respectively, potentially biasing $H_0$ by several percent if not properly marginalized. The analysis also highlights that the common Osipkov-Merritt anisotropy models underproduce $h_4$ compared to observed early-type galaxies, suggesting the need for more physically motivated anisotropy profiles. A key conceptual takeaway is that early-type galaxies can be treated as a dynamically homogeneous population in joint lens analyses, which affects how uncertainties scale with the number of lenses. Overall, the work emphasizes that achieving a 2% precision on $H_0$ with time-delay lenses will require improved 2D kinematic data, non-Gaussian PSF modeling, and population-level marginalization of dynamical parameters.

Abstract

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.

Figures (13)

• Figure 1: Observed central velocity dispersion $\sigma^2/\sigma_{SIS}^2$ (top) and surface density at the Einstein radius $\kappa_E/\kappa_{SIS}$ normalized by the values for an SIS lens model. The heavier lines indicate the changes that will increase $H_0$ by 8% from the SIS model. The curves are for the eight time-delay lenses with the parameter values from Table \ref{['tab:data']}. The velocity dispersion includes a Gaussian PSF model and the extraction aperture. The estimated Hubble constant increases as $-\kappa/2\kappa_{SIS}$ as the surface density decreases.
• Figure 2: Absolute values of the fractional changes, $\left|\Delta\sigma^2_{\pm}/\sigma^2_0\right|$, due to misestimates of the seeing FWHM as a function of anisotropy, $\beta$. The curves with circles (squares) are the fractional changes if the reported FWHM is increased (decreased) by 10% assuming a Gaussian PSF. Each system is plotted in a different color and line style.
• Figure 3: Fractional changes, $\Delta\sigma^2_d/\sigma^2_d$, for apertures miscentered by half a pixel from the center of the galaxy as a function of anisotropy, $\beta$. Each system is plotted in a different color and line style, and we assumed a Gaussian PSF for all the cases.
• Figure 4: Fractional changes of squared velocity dispersion relative to the value for a Gaussian PSF with the same FWHM for single (circles) and double (squares) Moffat profiles as a function of anisotropy, $\beta$. Each system is plotted in a different color and linestyle.
• Figure 5: The differences $[P(v)-G(\sigma_{\ast})]/P(v=0)$ between the los velocity distribution ($P(v)$) and the best fit Gaussian model ($G(\sigma_{\ast})$) normalized by the peak of the los velocity distribution at zero velocity, $P(v=0)$, for SDSS J1206$+$4332. The anisotropy radii considered are $r_a/s\rightarrow \infty$ (isotropic), $10$, $3$, and $1$. The velocity distribution accounts for Gaussian PSF effects and the rectangular extraction aperture from Table \ref{['tab:data']} is applied. There is some noise due to the Monte Carlo integration and sampling.