Lens Modeling of STRIDES Strongly Lensed Quasars using Neural Posterior Estimation

TL;DR

This work applies neural posterior estimation (NPE) for modeling galaxy-scale strongly lensed quasars from the Strong Lensing Insights into the Dark Energy Survey (STRIDES) sample and finds the population mean of the power-law elliptical mass distribution slope, $\gamma_{\text{lens}}$, to be the first population-level constraint for these systems.

Abstract

Strongly lensed quasars can be used to constrain cosmological parameters through time-delay cosmography. Models of the lens masses are a necessary component of this analysis. To enable time-delay cosmography from a sample of $\mathcal{O}(10^3)$ lenses, which will soon become available from surveys like the Rubin Observatory's Legacy Survey of Space and Time (LSST) and the Euclid Wide Survey, we require fast and standardizable modeling techniques. To address this need, we apply neural posterior estimation (NPE) for modeling galaxy-scale strongly lensed quasars from the Strong Lensing Insights into the Dark Energy Survey (STRIDES) sample. NPE brings two advantages: speed and the ability to implicitly marginalize over nuisance parameters. We extend this method by employing sequential NPE to increase precision of mass model posteriors. We then fold individual lens models into a hierarchical Bayesian inference to recover the population distribution of lens mass parameters, accounting for out-of-distribution shift. After verifying our method using simulated analogs of the STRIDES lens sample, we apply our method to 14 Hubble Space Telescope single-filter observations. We find the population mean of the power-law elliptical mass distribution slope, $γ_{\text{lens}}$, to be $\mathcal{M}_{γ_{\text{lens}}}=2.13 \pm 0.06$. Our result represents the first population-level constraint for these systems. This population-level inference from fully automated modeling is an important stepping stone towards cosmological inference with large samples of strongly lensed quasars.

Figures (16)

• Figure 1: Diagram of the neural posterior estimation (NPE) technique. The first round of training is shown in the top row and described in Section \ref{['section:NPE']}. 5e5 lenses are sampled from the interim prior, $\nu_{\text{int}}$. Then, a neural network is trained on those examples. At test time, the network takes in an image of a lens, and outputs the parameters describing an approximate lens model posterior $q_\phi(\xi_{k}|d_{k},\nu_{\text{int}})$. The second round of training is sequential, shown in the bottom row and described in Section \ref{['section:SNPE']}. In this step, 5e4 sequential training examples are generated for each lens in the test set. These new lenses are sampled from a proposal distribution $\tilde{p}(\xi_{k})$ that is informed by the NPE posterior $q_\phi(\xi_{k}|d_{k},\nu_{\text{int}})$. Then, a copy of the neural network is trained for each lens on the sequential training examples. At test time, each lens is passed through its copy of the neural network to produce parameters describing an approximate lens model posterior $q_{\phi_k}(\xi_{k}|d_{k},\nu_{\text{int}})$
• Figure 2: We demonstrate the statistical framework of our hierarchical inference. To generate an image of a strong lens, $d_{\text{k}}$, we need to specify the lens mass parameters, $\xi_{\text{k}}$ and a broader set of nuisance parameters $n_{\text{k}}$. In this work, nuisance parameters include source light, lens light, microlensing effects, PSF kernel, and other instrumental effects. In step 1, NPE, we infer lens model posteriors, $p(\xi_{k}|$$d_{k}$,$\nu_{\text{int}})$, for each lens k. The NPE method allows us to implicitly marginalize over $\eta_{\text{k}}$ at this step. The population-level distribution at this stage is fixed at the training prior, $\nu_{\text{int}}$. In step 2, HBI, we infer a population model, $p(\nu | \{d\})$, over the lens model parameters $\xi$. The population distribution over nuisance parameters, $\nu_{\text{int},\eta}$, remains fixed.
• Figure 3: Distribution of the shifted test set (solid purple) compared to the training distribution $\nu_{\text{int}}$ (dashed grey). The shifted test set is designed to test our ability to recover from distribution shift between the training set and the test set. The Gaussian distribution of $\theta_E$ is shifted from $\mathcal{N}$($\mu$=0.8, $\sigma$=0.15) to $\mathcal{N}$($\mu$=0.7, $\sigma$=0.08). The Gaussian distribution of $\gamma_{\text{lens}}$ is shifted from $\mathcal{N}$($\mu$=2.0, $\sigma$=0.2) to $\mathcal{N}$($\mu$=2.05, $\sigma$=0.1).
• Figure 4: Comparison of real HST data to the simulated doppelganger test set in the F814W band. Images are 80$\times$80 pixels with 0.04$\hbox{$^{\prime\prime}$}$ resolution. Images are oriented with East to the left, and North to the top. All images are plotted with log-scaled color. Note when using the doppelganger simulation procedure described in Appendix \ref{['appendix:doppels']}, we were unable to recreate DES J0530$-$3730.
• Figure 5: Calibration curves for the verification test sets. In perfectly calibrated posteriors (dashed line), a given x% of the probability volume contains the truth x% of the time. Calibration of NPE posteriors is shown in purple. Calibration of SNPE posteriors is shown in green. The shaded region encompasses 1$\sigma$ uncertainty. Doppelganger posteriors are more overconfident than shifted set posteriors for both modeling methods, which is discussed in Section \ref{['section:shifted_vs_doppels']}.