Polar Coordinates Improve Detection of Dark Subhalos in Strong Lenses
Strong gravitational lensing lets us probe dark matter on small scales through perturbations that subhalos introduce into arc images. The study explores whether switching images to polar coordinates boosts the ability of convolutional neural networks to estimate subhalo mass. An architecture is proposed that predicts mass along with an uncertainty estimate. Using simulated data mimicking Hubble Space Telescope observations, models were pitted against each other on Cartesian versus polar representations under varied initializations, noise levels, and subhalo concentrations (c=60, c=30). Polar representations consistently score a higher detection fraction of subhalos across all tested masses; for masses 10^9–10^9.5 M⊙, that fraction climbs by roughly 15%. The edge holds even under tough conditions: at low signal-to-noise ratios and for low-concentration subhalos, where perturbations are barely visible. Transforming to polar coordinates turns out to be a computationally cheap way to sharpen neural network detection of subhalos.
Context
Within the standard model that describes the evolution of the Universe from the Big Bang and includes dark energy, the nature of dark matter remains one of the biggest unsolved mysteries. On galactic scales, dark matter forms hierarchical structures, including galactic halos and populations of subhalos whose mass is sensitive to the specific dark matter model. Detecting these subhalos, especially completely dark ones, is only possible through their gravitational influence—for example, via perturbations in the arcs of strong gravitational lensing. Early pioneers of dark matter research such as Fritz Zwicky and Vera Rubin demonstrated the need for hidden mass, and observations with Hubble, named after Edwin Hubble, made it possible to study gravitational lenses in distant galaxies.
Methods
The study used simulated observations of the Hubble Space Telescope (WFC3 IR camera, F160W filter) at different signal-to-noise ratios (50 orbits and 1 orbit). The images contained a single subhalo modeled with a truncated NFW profile, set against a smooth lensing potential with multipole distortions. The data were transformed from Cartesian to polar coordinates: for each grid point (64×64 in radius and angle), pixel values were bilinearly interpolated. The neural network architecture is based on ConvNeXt-Tiny with ImageNet pretraining and a special regression head that predicts the mean and log variance of the mass. The loss function is the negative log-likelihood, incorporating aleatoric uncertainty. Four variants were compared: Cartesian/polar inputs, with random initialization and with pretraining.
Results
In all tested regimes (high and low signal-to-noise ratio, subhalo concentrations c=60 and c=30), networks trained on polar data achieved a better detection fraction. For subhalos with masses from 10^9 to 10^{9.5} M⊙, the improvement was about 15% compared to the Cartesian representation. Pretraining also significantly improved the results, and the relative advantage of the polar transform increased with decreasing subhalo mass and when moving to less concentrated profiles (c=30). In the low signal-to-noise case, the gain narrowed to 1–5% but persisted for sources with faint magnitudes. The networks successfully separated confident from unconfident predictions using an uncertainty threshold, allowing unreliable estimates to be filtered out.
Implications
The study shows that a simple polar transform is an effective way to improve feature extraction from strong lensing images without modifying the neural network architecture. This is especially important for future large surveys (for example, with Hubble and its successors), where rapid processing of large data volumes is required. Improved detectability of low-mass subhalos will help place tighter constraints on the subhalo mass function, which in turn will narrow down the viable dark matter models and shed light on the "core-cusp problem".
Future development
Further development of the method will involve testing on realistic simulations with multiple subhalos, as well as on real observational data, where precise centering of the polar grid will be required. In addition, it is promising to explore combining the polar transform with other architectures, such as U-Net for segmentation or transformers, and to compare it with group equivariant convolutions that inherently account for rotational symmetry.
Impact
The results will impact observational cosmology and particle astrophysics, providing a more reliable tool for indirect detection of dark matter through gravitational lensing. This also advances the methodology of applying deep learning in astronomy, where accounting for data symmetries can significantly boost performance.
Next steps
The next steps include adapting the method to real telescope data, including images from Hubble, and testing the robustness of the polar transform advantage in the presence of multiple subhalos or complex lens configurations.
Key open problems
This work is directly connected to the fundamental problem of the nature of dark matter and the small-scale issues of the standard cosmological model, in particular to determining the subhalo mass function, which critically depends on the properties of dark matter particles. Improved detection of subhalos via strong lensing brings us closer to distinguishing between cold, warm, and self-interacting dark matter, one of the key challenges in modern physics.
🎯 If you turn an Einstein ring into a straight line using polar coordinates, the neural network begins to 'see' it as an ordinary texture, reminiscent of the trick with cylindrical anamorphosis, where a distorted image makes sense when viewed from the right angle.
🎬 In Liu Cixin's novel 'The Dark Forest', gravitational lenses are mentioned as a tool for interstellar communication, and the idea of 'unfolding' an image around the center of mass echoes the concept of transforming space to extract hidden information—almost like a lie detector for dark matter.
Key numbers
- increase in detection fraction: ~15% for subhalos with masses 10^9–10^{9.5} M⊙
- subhalo concentrations: c=60 and c=30
- HST orbits for SNR: 50 orbits (high) and 1 orbit (low)
- pixel scale: 0.08 arcseconds/pixel
- image size: 64×64 pixels