Analytic approach to astrometric perturbations of critical curves by substructures

Astrometric perturbations of critical curves in strong lens systems are thought to be one of the most promising probes of substructures down to small-mass scales. While a smooth mass distribution creates a symmetric geometry of critical curves with radii of curvature about the Einstein radius, substructures introduce small-scale distortions on critical curves, which can break the symmetry of gravitational lensing events near critical curves, such as highly magnified individual stars. We derive a general formula that connects the fluctuation of critical curves with the fluctuation of the surface density caused by substructures, which is useful when constraining models of substructures from observed astrometric perturbations of critical curves. We numerically check that the formula is valid and accurate as long as substructures are not dominated by a small number of massive structures. As a demonstration of the formula, we also explore the possibility that an anomalous position of an extremely magnified star, recently reported as ``Mothra,'' can be explained by fluctuations in the critical curve due to substructures. We find that cold dark matter subhalos with masses ranging from $5 \times 10^7 M_{\odot}/h$ to $10^9 M_{\odot}/h$ can well explain the anomalous position of Mothra, while in the fuzzy dark matter model, the very small mass of $\sim 10^{-24}~\mathrm{eV}$ is needed to explain it.