Probing Dark Matter Substructure with Image Number Anomaly in Strong Lensing Systems

TL;DR

This work proposes image-number anomalies in strong gravitational lensing as a novel probe of small-scale dark matter substructure, specifically testing PBHs and FDM. It develops a two-component lens model and simulates PBH and FDM perturbations to quantify how extra images arise and depend on angular resolution. Using a null-detection analysis on 3,500 simulated lens systems, it derives 95% confidence limits on PBH abundance and FDM particle mass, demonstrating that higher-resolution observations significantly improve constraints. The study also provides methodologies for identifying anomalies in complex configurations and discusses practical mitigation of microlensing effects, highlighting the potential of next-generation surveys and extended-source data to advance our understanding of DM microphysics.

Abstract

Gravitational lensing observables, including anomalies in image positions, flux ratios, and time delays, serve as usual probes of dark matter (DM) substructure. When dark matter substructure possesses sufficient perturbations, it may lead to the formation of extra images in otherwise canonical doubly or quadruply imaged systems. With the advent of increasingly precise observational instruments, previously undetectable images may become measurable and image number anomalies therefore could be an increasingly viable method. In this paper, we utilize the gravitational lensing phenomenon of image number anomaly to derive constraints on dark matter substructure. We present the extra images induced by distinct forms of DM substructure, specifically primordial black holes (PBHs) and fuzzy dark matter (FDM) and show that higher angular resolution observations increase the probability of detecting additional lensed images. Based on a null detection of image number anomalies in a sample of 3500 lens systems generated from the \textit{Strong Lensing Halo model-based mock catalogs} (SL-Hammocks), we derive upper limits on the abundance of PBHs. At the 95\% confidence level, the PBH abundance is constrained to $\lesssim 0.125\%$, $0.08\%$, and $0.04\%$ for PBH masses in the range $\sim 10^{7}$--$10^{9}~M_{\odot}$, corresponding to angular resolutions of $0.1''$, $0.05''$, and $0.01''$, respectively. Similarly, we exclude particle masses below $0.4$, $0.6$, and $3.5 \times 10^{-22} \ \mathrm{eV}$ for FDM at the same confidence level for the respective resolutions. Furthermore, the abundance of PBHs $\lesssim 0.9\%$ could be constrained at an angular resolution of $0.5''$ for the Legacy Survey of Space and Time (LSST) Observations. Finally, we discuss methodologies for identifying image number anomalies in special cases and demonstrate feasibility using a fitting procedure.

Figures (10)

• Figure 1: Lensed image numbers and critical curves for a strong lensing system with parameters in Table \ref{['macro']}. The left panel shows original quad system, i.e., without dark matter substructure. The blue star denotes the fixed source position. The red points indicate the lensed images. The black solid curves correspond to the critical curves. The middle panel presents the image number anomaly induced by PBHs. The right panel shows the one arising from FDM-induced fluctuations corresponding to an ultra-light boson mass of $m_{\psi} = 10^{-22}\,{\mathrm{eV}}$. All spatial units are given in arcseconds.
• Figure 2: Effect of angular resolution on the number of observed lensed images. The left panel (angular resolution $0.1"$) and the right panel (angular resolution $0.01"$) show a clear increase in the number of resolved images with higher resolution, observing 4 and 5 images, respectively. Observed lensed images are marked by red dots, while the true source position is indicated by a blue cross. The black solid lines denote critical curves for the lens model.
• Figure 3: Source positions leading to image number anomalies related to FDM-induced perturbations. The caustics, as perturbed by the FDM, are illustrated by the green line. The red crosses denote source positions corresponding to 4 images, while the blue points indicate source positions yielding more than 4 images, i.e., image number anomalies, all within the identical FDM simulation.
• Figure 4: Redshift distribution of all strong gravitational lensing systems in the 100 samples. Systems classified as doubles and quads are denoted by green and red points, respectively. The median lens and source redshifts are indicated by black dashed lines. Marginal panels (upper and right) present the probability density functions of the lens and source redshift distributions, derived using Gaussian kernel density estimation.
• Figure 5: Probability density curve of the Einstein radii, obtained by applying a Gaussian kernel density estimator to the selected 100 strong lensing systems. The black dashed line indicates the median of the Einstein radii.