TDCOSMO IV: Hierarchical time-delay cosmography -- joint inference of the Hubble constant and galaxy density profiles

Abstract

The H0LiCOW collaboration inferred via gravitational lensing time delays a Hubble constant $H_0=73.3^{+1.7}_{-1.8}$ km s$^{-1}{\rm Mpc}^{-1}$, describing deflector mass density profiles by either a power-law or stars plus standard dark matter halos. The mass-sheet transform (MST) that leaves the lensing observables unchanged is considered the dominant source of residual uncertainty in $H_0$. We quantify any potential effect of the MST with flexible mass models that are maximally degenerate with H0. Our calculation is based on a new hierarchical approach in which the MST is only constrained by stellar kinematics. The approach is validated on hydrodynamically simulated lenses. We apply the method to the TDCOSMO sample of 7 lenses (6 from H0LiCOW) and measure $H_0=74.5^{+5.6}_{-6.1}$ km s$^{-1}{\rm Mpc}^{-1}$. In order to further constrain the deflector mass profiles, we then add imaging and spectroscopy for 33 strong gravitational lenses from the SLACS sample. For 9 of the SLAC lenses we use resolved kinematics to constrain the stellar anisotropy. From the joint analysis of the TDCOSMO+SLACS sample, we measure $H_0=67.4^{+4.1}_{-3.2}$ km s$^{-1}{\rm Mpc}^{-1}$, assuming that the TDCOSMO and SLACS galaxies are drawn from the same parent population. The blind H0LiCOW, TDCOSMO-only and TDCOSMO+SLACS analyses are in mutual statistical agreement. The TDCOSMO+SLACS analysis prefers marginally shallower mass profiles than H0LiCOW or TDCOSMO-only. While our new analysis does not statistically invalidate the mass profile assumptions by H0LiCOW, and thus their $H_0$ measurement relying on those, it demonstrates the importance of understanding the mass density profile of elliptical galaxies. The uncertainties on $H_0$ derived in this paper can be reduced by physical or observational priors on the form of the mass profile, or by additional data, chiefly spatially resolved kinematics of lens galaxies.

Figures (26)

• Figure 1: Illustration of a composite profile consisting of a stellar component (Hernquist profile, dotted lines) and a dark matter component (NFW + cored component (Eqn. \ref{['eqn:core_profile_projected']}), dashed lines) which transform according to an approximate MST (joint as solid lines). The stellar component gets rescaled by the MST while the cored component transforms the dark matter component. Left: profile components in three dimensions. Right: profile components in projection. The transforms presented here cannot be distinguished by imaging data alone and require i.e., stellar kinematics constraints. https://github.com/TDCOSMO/hierarchy_analysis_2020_public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/MST_impact/MST_composite_cored.ipynb
• Figure 2: Illustration of the constraining power of imaging data on a cored mass component (Eqn. \ref{['eqn:approx_mst_core']}). Shown are the parameter inference of the power-law profile mock quadruply lensed quasar of Figure \ref{['fig:mock_lens']} when including a marginalization of an additional cored power law profile (Eqn. \ref{['eqn:core_profile_projected']}). Orange lines indicate the input truth of the model without a cored component. $\lambda_{\rm c}$ is the scaled core model parameter (Eqn. \ref{['eqn:approx_mst_core']}) resembling the pure MST for large core radii ($\lambda_{\rm c} \approx \lambda_{\rm int}$). https://github.com/TDCOSMO/hierarchy_analysis_2020_public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/MST_impact/MST_pl_cored.ipynb
• Figure 3: Constraints on an approximate internal MST transform with a cored component, $\lambda_{\rm c}$, of an NFW profile as a function of core radius. In gray are the 1-$\sigma$ exclusion limits that imaging data can provide. In orange is the region where the total mass of the core within a three-dimensional radius exceeds the mass of the NFW profile in the same sphere. In blue is the region where the transformed profile results in negative convergence at the core radius. The white region is effectively allowed by the imaging data and simple plausibility considerations and where we can use the mathematical MST as an approximation ($\lambda_{\rm c} \approx \lambda_{\rm int}$). The halo mass, concentration and the redshift configuration is displayed in the lower left box. https://github.com/TDCOSMO/hierarchy_analysis_2020_public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/MST_impact/MST_pl_cored.ipynb
• Figure 4: Comparison of the actual predicted kinematics from the modeling of the physical three-dimensional mass distribution $\kappa_{\lambda_{\rm int}}$ (Eqn. \ref{['eqn:approx_mst_core']}) for varying core sizes (solid) and the analytic relation of a perfect MST (Eqn. \ref{['eqn:kinematics_mst']}, dashed) for the mock lens presented in Figure \ref{['fig:mock_lens']}. Lower panel shows the fractional differences between the exact prediction and a perfect MST calculation. The MST prediction matches to <1% in the considered range. Minor numerical noise is present at the subpercent level. https://github.com/TDCOSMO/hierarchy_analysis_2020_public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/MST_impact/MST_pl_cored.ipynb
• Figure 5: Mock data from the TDLMC Rung3 inference with the parameters and prior specified in Table \ref{['table:param_summary_tdlmc']}. Orange contours indicate the inference with a uniform prior in $a_{\rm ani}$ while the purple contours indicate the inference with a uniform priors in $\log(a_{\rm ani})$. The thin vertical line indicates the ground truth $H_0$ value in the challenge. https://github.com/TDCOSMO/hierarchy_analysis_2020_public/blob/6c293af582c398a5c9de60a51cb0c44432a3c598/TDLMC/TDLMC_rung3_inference.ipynb