Understanding caustic crossings in giant arcs: Characteristic scales, event rates, and constraints on compact dark matter

The recent discovery of fast transient events near critical curves of massive galaxy clusters, which are interpreted as highly magnified individual stars in giant arcs due to caustic crossing, opens up the possibility of using such microlensing events to constrain a range of dark matter models such as primordial black holes and scalar field dark matter. Based on a simple analytic model, we study lensing properties of a point mass lens embedded in a high magnification region, and we derive the dependence of the peak brightness, microlensing time scales, and event rates on the mass of the point mass lens, as well as the radius of a source star that is magnified. We find that the lens mass and source radius of the first event MACS J1149 Lensed Star 1 (LS1) are constrained, with the lens mass range of $0.1M_{\odot }\lesssim M\lesssim 4\times {10}^3{M}_{\odot }$ and the source radius range of $40R_{\odot }\lesssim R\lesssim 260R_{\odot }$. In the most plausible case with $M\approx 0.3M_{\odot }$ and $R\approx 180R_{\odot }$, the source star should have been magnified by a factor of $\approx 4300$ at the peak. The derived lens properties are fully consistent with the interpretation that MACS J1149 LS1 is a microlensing event produced by a star that contributes to the intracluster light. We argue that compact dark matter models with high fractional mass densities for the mass range ${10}^{-5}{M}_{\odot }\lesssim M\lesssim {10}^2{M}_{\odot }$ are inconsistent with the observation of MACS J1149 LS1 because such models predict too low magnifications. Our work demonstrates a potential use of caustic crossing events in giant arcs to constrain compact dark matter.

Figure 1

Critical curves (upper panels) and caustics (lower panels) of a point mass lens in the high magnification region. Left: Positive parity case with ${\mu }_{\mathrm{t}}^{-1}=0.001$ and ${\mu }_{\mathrm{r}}^{-1}=0.401$. Right: Negative parity case with ${\mu }_{\mathrm{t}}^{-1}=-0.001$ and ${\mu }_{\mathrm{r}}^{-1}=0.399$. These are computed with glafic [36]. Note the difference of scales between the $x$ and $y$ axes in the lower panels.

Figure 2

Constraints on the source radius ($R$) and lens mass ($M$) for MACS J1149 LS1. Shaded regions show excluded regions from various constraints. Specifically, we consider constraints from the peak magnitude [$m_{\mathrm{peak}}$, Eq. (51)]; the magnification during the caustic crossing [${\mu }_{\mathrm{obs}}$, Eq. (52)]; the source crossing time [$t_{\mathrm{src}}$, Eq. (57)]; the time scale between caustic crossings [$t_{\text{Ein}}$, Eq. (58)], and the unresolved shape during caustic crossing [unresolved, Eq. (59)]. The saturation condition given by Eq. (54) is always satisfied in the allowed region of this plot. Contours show the constant peak magnification [Eq. (48)] in this parameter space. From top to bottom, we show contours of ${\mu }_{\max }={10}^6$, $10^5$, $10^4$, and $10^3$.

Figure 3

Constraints in the $M-f_{\mathrm{p}}$ plane for MACS J1149 LS1, where $M$ is the lens mass and $f_{\mathrm{p}}$ is the mass fraction of the point mass component to the total mass. Shaded regions show excluded regions from the event rate [rate, Eq. (70)]. The small rectangular region shows the rough mass fraction and the mass range of ICL stars. Contours show the constant event rate in this plane. From inner to outer contours, we show contours for $dN/dt={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$.

Figure 4

Similar to Fig. 2, but an additional constraint on the lens mass range from the event rate (see Fig. 3) is included.

Figure 5

The expected distribution of the positions of caustic crossing events from the macro model critical curve ${\theta }_{\mathrm{h}}$, i.e., the normalized differential distribution of the event rate, $(d^{2}N/d{\theta }_{\mathrm{h}}dt)/(\int d{\theta }_{\mathrm{h}}{d}^{2}N/d{\theta }_{\mathrm{h}}dt)$. The distribution is essentially the integrand of Eq. (63). Parameters are tuned for those of MACS J1149 LS1 as considered in Sec. 4b. We consider two different masses of the point mass lens component, $M=0.0003M_{\odot }$ (solid) and $30M_{\odot }$ (dashed). Given the relation given in Eq. (31), the distribution of ${\theta }_{\mathrm{h}}$ can also be converted to that of the macro model magnification ${\mu }_{\mathrm{t}}$.

Figure 6

Dependence of constraints in the $R-M$ plane shown in Fig. 4 on model parameters. The solid line indicates an allowed region in the $R-M$ plane in our fiducial setup, as shown in Fig. 4. Dotted and short dashed lines show how the allowed region changes by changing the transverse velocity $v$ by factors of 2 and 0.5, respectively. Long dashed and dotted-dashed lines show how the allowed region changes by changing the macro model magnification ${\mu }_{\mathrm{h}}$ by factors of 2 and 0.5, respectively.

Figure 7

The dependences of the event rate [Eq. (63)] on various model parameters. We consider model parameters that are tuned for the MACS J1149 LS1 (see Sec. 4b), and fix the mass fraction of the point mass lens $f_{\mathrm{p}}=0.01$ and the lens mass $M=0.3M_{\odot }$. From the top panel to the bottom, we show the dependences of the event rate on the source redshift $z_s$, the surface brightness of the arc, and the limiting magnitude of the monitoring observation $m_{\lim }$. The vertical dotted lines show our fiducial values for MACS J1149 LS1. In the top panel, we also show the results for the limiting magnitudes brighter and fainter by 1 mag (i.e., $m_{\lim }=25$ and 27) by dashed lines.

Figure 8

Constraints on the mass ($M$) and abundance ($f_{\mathrm{p}}$) of compact dark matter. Shaded regions show excluded regions from the caustic crossing studied in this paper, microlensing observations of M31 with Subaru/Hyper Suprime-Cam (HSC) [11], EROS/MACHO microlensing [6, 9], ultrafaint dwarf galaxies (UFDs) [45], and Planck cosmic microwave background observations (Planck) [46]. We also show constraints from the caustic crossing with different assumptions on the transverse velocity (factors of 2 and 0.5 different from the fiducial value) by dotted lines. For UFDs and Planck, conservative limits are shown by solid lines, whereas more stringent limits are shown by dashed lines. For the Planck constraint, the stringent limit assumes the collisional ionization around PBHs, whereas the conservative limit assumes the photoionization due to the PBH radiation. For the UFDs constraint, different constraints reflect different assumptions on the dark matter densities and initial sizes of star clusters in UFDs.