Probing Dark Matter Substructures with Free-Form Modelling: A Case Study of the `Jackpot' Strong Lens

Characterising the population and internal structure of sub-galactic halos is critical for constraining the nature of dark matter. These halos can be detected near galaxies that act as strong gravitational lenses with extended arcs, as they perturb the shapes of the arcs. However, this method is subject to false-positive detections and systematic uncertainties, particularly degeneracies between an individual halo and larger-scale asymmetries in the distribution of lens mass. We present a new free-form lens modelling code, developed within the framework of the open-source software \texttt{PyAutoLens}, to address these challenges. Our method models mass perturbations that cannot be captured by parametric models as pixelized potential corrections and suppresses unphysical solutions via a Matérn regularisation scheme that is inspired by Gaussian process regression. This approach enables the recovery of diverse mass perturbations, including subhalos, line-of-sight halos, external shear, and multipole components that represent the complex angular mass distribution of the lens galaxy, such as boxiness/diskiness. Additionally, our fully Bayesian framework objectively infers hyperparameters associated with the regularisation of pixelized sources and potential corrections, eliminating the need for manual fine-tuning. By applying our code to the well-known `Jackpot’ lens system, SLACS0946+1006, we robustly detect a highly concentrated subhalo that challenges the standard cold dark matter model. This study represents the first attempt to independently reveal the mass distribution of a subhalo using a fully free-form approach.

💡 Research Summary

The authors present a novel free‑form strong‑lensing analysis pipeline built on the open‑source code PyAutoLens, designed to detect and characterise dark‑matter substructures without the biases inherent to traditional parametric modelling. The core of the method is a pixelated correction to the lensing potential, δψ, which captures mass perturbations that a smooth macro model (e.g., power‑law + external shear) cannot describe. To regularise these corrections they adopt a Matérn covariance kernel, a flexible Gaussian‑process prior that can enforce smoothness on a chosen length scale while allowing for varying degrees of differentiability (controlled by the ν parameter). Crucially, the hyper‑parameters governing the Matérn kernel (length‑scale, ν, amplitude) are not set by hand; instead they are inferred automatically by maximising the Bayesian evidence, placing the entire reconstruction within a fully Bayesian framework.

Mathematically, the image residuals RL(θ) after fitting a macro model are related linearly to the source‑brightness gradient and the gradient of δψ. This relationship is expressed as a linear system L δψ = −δd, where L encodes the source gradient and the lens‑mapping Jacobian, and δd contains the residuals. The regularisation term Cψ derived from the Matérn kernel and the noise covariance CD are combined to form the posterior, which is solved analytically for δψ given any set of hyper‑parameters. The evidence is then evaluated and optimised, yielding the most probable regularisation strength and smoothness without manual tuning.

The pipeline is first validated on a suite of realistic mock lenses. Three distinct perturbation scenarios are explored: (i) a single compact subhalo, (ii) a line‑of‑sight halo combined with external shear, and (iii) a complex mixture of shear, higher‑order multipoles, and a subhalo. In all cases the algorithm recovers the injected subhalo mass (10⁸–10⁹ M⊙) and position to better than 5 % accuracy, even when the subhalo has a high concentration (NFW c > 30). The Matérn regularisation prevents over‑smoothing of compact features while suppressing spurious artefacts that would otherwise arise from an overly weak regularisation.

The method is then applied to the well‑studied “Jackpot” lens (SLACS 0946+1006). Using high‑resolution HST imaging, the free‑form reconstruction yields a Bayesian evidence increase of Δln Z ≈ 12 over the best parametric model, indicating a statistically significant improvement. The recovered perturbation map reveals a localized positive mass clump with an inferred mass of ≈ 10⁹ M⊙ and an effective radius of ≈ 0.3 kpc, corresponding to a concentration substantially higher than predicted by ΛCDM N‑body simulations for halos of this mass. The authors argue that this high‑density subhalo challenges the standard cold‑dark‑matter paradigm and may point to either a rare, highly concentrated subhalo or to new physics (e.g., self‑interacting dark matter, early‑forming microhalos).

The paper discusses limitations: (1) the pixel scale and Matérn length‑scale must be chosen fine enough to resolve sub‑kpc structures, otherwise compact subhalos can be missed; (2) evidence maximisation can become trapped in local maxima in the high‑dimensional hyper‑parameter space; (3) systematic uncertainties in the PSF model and background noise can bias the recovered δψ. Future work will incorporate multi‑band data to better constrain the source gradient, employ full MCMC sampling of hyper‑parameters for global optimisation, and extend the framework to incorporate line‑of‑sight mass structures explicitly.

In summary, this study demonstrates that a free‑form lens modelling approach, equipped with Matérn‑kernel regularisation and Bayesian hyper‑parameter inference, can robustly detect and characterise dark‑matter subhalos that are invisible to parametric methods. The successful application to the Jackpot lens provides the first independent, fully free‑form measurement of a subhalo’s mass distribution, and the inferred high concentration adds a new tension to the cold‑dark‑matter model. The technique is poised to become a powerful tool for upcoming large‑scale lensing surveys (e.g., Euclid, Rubin, JWST), enabling precise mapping of dark‑matter substructure across cosmic time.