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.

Figure 1

An example of fluctuated critical curves by CDM subhalos for each model summarized in Table 1. The red dashed vertical line shows the original macrocritical curve described in Sec. 2. The gray points show critical curves calculated in glafic, while the blue points show the fluctuated critical curves obtained by filtering, as explained in the text. We also show the value of $⟨\delta {\theta }_x^2{⟩}^{1/2}$ for each panel in units of arcsec. For the mean value of each model, see Table 1. Top left: model (i). Top center: model (ii). Top right: model (iii). Middle left: model (v). Middle center: model (vii). Middle right: model (viii). Bottom left: model (ix). Bottom center: model (x). Bottom right: model (xii). The acronym “CC” stands for “critical curve” in the figure.

Figure 2

Comparison between analytically calculated values of $⟨\delta {\theta }_x^2{⟩}^{1/2}$ from Eq. (16) and the ones numerically estimated using glafic. Here we set $M_{\mathrm{hh}}={10}^{15}{M}_{\odot }/h$.

Figure 3

Contour of $⟨\delta {\theta }_x^2{⟩}^{1/2}$ as a function of $M_{\max }$ and $M_{\min }/M_{\max }$ for the Mothra-like lens system. The value of each contour is shown in units of arcsec. The red cross shows the model that we calculate in detail to interpret the Mothra’s anomalous position.

Figure 4

The correlation between $\delta {\theta }_x$ and the magnification ratio between $c$ and $c^{\prime }$, multiple image pair near Mothra. From the configuration of multiple images, the magnification ratio is predicted to be about 1.22, represented by the horizontal red dotted line. The vertical red line shows $\delta {\theta }_x$ to make the image at the Mothra’s offset an unresolved pair of images.

Figure 5

The relation between the critical curve perturbation and the FDM mass in the case of Mothra. The horizontal dotted line shows the fluctuation needed to explain the observed offset of Mothra.