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

TL;DR

This work advances subhalo detection in strong gravitational lensing by developing a fully free-form, Matérn-regularised perturbation framework built on PyAutoLens. By linearising the impact of potential perturbations around a macro lens and solving for pixelised corrections in conjunction with a pixelised source, the method jointly recovers both extended and highly localised mass structures while optimising regularisation via Bayesian evidence. Applied to mock data and the SLACS Jackpot lens (SLACS0946+1006), it robustly detects a highly concentrated subhalo and characterises its mass distribution in a model-independent way, highlighting potential tensions with standard cold dark matter and underscoring the importance of unbiased initialization and adaptive regularisation. The approach is automated and scalable, offering a promising path for analysing the large samples expected from upcoming surveys like Euclid, CSST, and Roman for constraining dark matter on sub-galactic scales.

Abstract

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.

Figures (14)

• Figure 1: Mock datasets used in this work (see Table \ref{['table:mock_setting']} for specific parameter values). The first row depicts the intrinsic source image, the lensing potential of the main lens, and the resulting lensed image. The second row shows the lensing potential maps of various perturbative signals: a $10^{10}$ M$_{\odot}$ NFW subhalo, an external shear field, a $m=4$ multipole component, and a Gaussian random field. The third row presents the corresponding image perturbations induced by these lensing potential perturbations. Unless otherwise noted, lensing potentials are shown in units of $\text{arcsec}^2$, and lensing images are in units of $\mathrm{e}^- \ \mathrm{s}^{-1} \ \mathrm{pixel}^{-1}$.
• Figure 2: Lens-light subtracted image of SLACS0946+1006 in the HST F814W band, using the multiple Gaussian expansion model presented in He2024_mge.
• Figure 3: Reconstruction of the lensing potential perturbation induced by a $10^{10} M_{\odot}$ NFW subhalo using the Matérn regularisation scheme, assuming that the true source and the image residuals induced by the perturber are perfectly known. Top-left: Image residual induced by the $10^{10} M_{\odot}$ NFW subhalo. Top-middle: Image residual reconstructed using the potential correction model. Top-right: Difference between the top-left and top-middle panels, normalised by the noise map. Bottom-left: Lensing potential map of the input perturber. Bottom-middle: Lensing potential map of the model perturber, reconstructed using the potential correction model. Bottom-right: Difference between the bottom-left and bottom-middle panels. The 'plus' symbol in the top-left and top-middle panels marks the position of the input subhalo.
• Figure 4: Reconstruction of the convergence perturbation induced by a $10^{10}\,M_{\odot}$ NFW subhalo using different regularisation schemes for the perturbative lensing potential. Top-left: Input convergence map of the $10^{10}\,M_{\odot}$ NFW subhalo. Top-right: Reconstructed convergence map using the potential correction method with Matérn regularisation. The black circle (radius $0.5\arcsec$) indicates the region where a parametric NFW model is fit to the reconstructed lensing potential to assess consistency with the input perturbation. Bottom-left: Same as top-right, but with Exponential regularisation. Bottom-right: Same as top-right, but with Gaussian regularisation. Exponential and Gaussian regularisations tend to produce under- or over-smoothed solutions, whereas the Matérn kernel yields a good reconstruction.
• Figure 5: Comparison of reconstructed lensing potential perturbations induced by an NFW subhalo (left column), external shear (middle-left column), $m_4$ multipole (middle-right column), and a Gaussian random field (GRF) (right column) with the ground truth. The first row presents the input lensing potential maps of perturbers. The second row shows the perturbative lensing potential maps derived from the potential correction model. The third row presents a quantitative analysis: for the NFW subhalo, external shear, and $m_4$ multipole, parametric models are fitted to the lensing potential map in the second row, with point estimates reported in Table \ref{['tab:diagnose_inversion']}; for the GRF, the power spectra of the input (blue lines) and reconstructed (red lines) lensing potential maps are compared. Solid lines indicate power spectra calculated within the unmasked modelling region, while dashed lines represent power-law model fits.