Using wavelets to capture deviations from smoothness in galaxy-scale strong lenses

Abstract

Modeling the mass distribution of galaxy-scale strong gravitational lenses is a task of increasing difficulty. The high-resolution and depth of imaging data now available render simple analytical forms ineffective at capturing lens structures spanning a large range in spatial scale, mass scale, and morphology. In this work, we address the problem with a novel multiscale method based on wavelets. We tested our method on simulated Hubble Space Telescope (HST) imaging data of strong lenses containing the following different types of mass substructures making them deviate from smooth models: (1) a localized small dark matter subhalo, (2) a Gaussian random field (GRF) that mimics a nonlocalized population of subhalos along the line of sight, and (3) galaxy-scale multipoles that break elliptical symmetry. We show that wavelets are able to recover all of these structures accurately. This is made technically possible by using gradient-informed optimization based on automatic differentiation over thousands of parameters, which also allow us to sample the posterior distributions of all model parameters simultaneously. By construction, our method merges the two main modeling paradigms-analytical and pixelated-with machine-learning optimization techniques into a single modular framework. It is also well-suited for the fast modeling of large samples of lenses. All methods presented here are publicly available in our new Herculens package .

Modeling lens potentials with continuous neural fields in galaxy-scale strong lenses

“…This is in direct analogy to the regularization strength in pixelated models, like the semi-linear inversion technique (e.g. Vernardos & Koopmans 2022) and when using wavelets (Galan et al 2021(Galan et al , 2022. In this work, we chose a value obtained by trial and error, which works well for analyzing the mock data and the given signal-to-noise explored here.…”

“…It is parameterized by a collection of weights and biases, which we denote by η Φ , and x ∈ R 2 is a point in the (continuous) lens plane. The network has been implemented as a module within the Herculens code (Galan et al 2022) using Flax (Heek et al 2020), a JAX-compatible library for neural network development. Since the model is fully differentiable, training the network (i.e.…”

“…For example, there can be ellipticity gradients (Van de Vyvere et al 2021), the presence of a bar (Hsueh et al 2018), or deviations from ellipticity that can be attributed to multipolar azimuthal perturbations (Van de Vyvere et al 2021). In the latter case, Galan et al (2022) consider the effects of an octupole; instead, here we consider the more common case of a quadrupole through an external shear perturbation, which simulates first-order effects of massive structures in the vicinity of the lens. This moment is widely used to describe lensing deflections at distances much larger than the Einstein radius, usually induced at the location of a given lens by mass concentrations (e.g.…”

“…However, the success of the technique can strongly depend on the choice of regularization used to reconstruct the pixelated perturbations (Vernardos & Koopmans 2022). Recently, Galan et al (2022) introduced a novel multiscale regularization technique based on wavelet transforms and sparsity priors to reconstruct different types perturbations on a pixelated grid.…”

“…Differentiable programming has also been used in Karchev et al (2022) to model lensed sources with Gaussian processes and variational inference, and Gu et al (2022) constructed a GPU-accelerated Bayesian framework for fast modeling and parameter inference applied to large samples of lenses. Another recent effort toward fully differentiable strong lens modeling is Herculens 1 (Galan et al 2022), which combines analytical and pixelated models together with deep learning optimization methods into a single modular framework. Herculens allows for fast gradient-informed optimization over thousands of parameters, in particular the reconstruction of pixelated source light distribution and lens potential perturbations regularized with wavelets.…”