PHYSICAL REVIEW D 97, 023518 (2018)
Understanding caustic crossings in giant arcs: Characteristic scales, event rates, and constraints on compact dark matter
Masamune Oguri,1,2,3 Jose M. Diego,4 Nick Kaiser,5 Patrick L. Kelly,6 and Tom Broadhurst7,8 1Research Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan 2Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 3Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo, Chiba 277-8582, Japan 4IFCA, Instituto de Física de Cantabria (UC-CSIC), Avenida de Los Castros s/n, 39005 Santander, Spain 5Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822-1839, USA 6School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455, USA 7Department of Theoretical Physics, University of the Basque Country, Bilbao 48080, Spain 8IKERBASQUE, Basque Foundation for Science, Alameda Urquijo, 36-5 48008 Bilbao, Spain
(Received 5 October 2017; published 19 January 2018)
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 0.1 M⊙ ≲ M ≲ 4 × 3 10 M⊙ and the source radius range of 40R⊙ ≲ R ≲ 260R⊙. In the most plausible case with M ≈ 0.3 M⊙ and R ≈ 180R⊙, the source star should have been magnified by a factor of ≈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 −5 2 high fractional mass densities for the mass range 10 M⊙ ≲ M ≲ 10 M⊙ 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.
DOI: 10.1103/PhysRevD.97.023518
I. INTRODUCTION behind the cluster MACS J0416.1-2403 in a strongly lensed galaxy at z ¼ 1.0054. While the Spock events Recently, Kelly et al. [1] reported the discovery of can be explained by the outburst of a luminous blue MACS J1149 Lensed Star 1 (LS1, also known as variable star or a recurrent nova, one possible interpretation “Icarus”), a faint transient near the critical curve of the is that these two events are also caustic crossing events. massive cluster MACS J1149.6+2223. The transient is These caustic crossing events in giant arcs behind massive interpreted as a luminous star in the host galaxy of z ¼ 1 49 clusters remarkably differ from traditional microlensing supernova Refsdal [2] at . , which is magnified observations. Microlensing observations in our Galaxy or by a compact object very close to the critical curve of the in nearby galaxies (e.g., [4–11]) are usually produced by foreground lens. The light curve is consistent with caustic isolated stars (or compact objects), whereas caustic crossing crossing of the background star, and from the comparison events in giant arcs are produced by stars embedded in high with ray-tracing simulations it was suggested that the star magnification regions due to the cluster potential. As shown was probably magnified by a factor of several thousands at in previous works (e.g., [12–16]), microlensing properties the peak brightness. There was also an additional transient are significantly modified by the presence of such conver- (“Iapyx”) detected at roughly the same distance from the gence and shear field due to the cluster potential. critical curve of the cluster but on the opposite side, which Perhaps quasar microlensing (e.g., [17–24]) more resem- can be the counterimage of LS1. Furthermore, Rodney bles caustic crossing events in giant arcs in the sense that it et al. [3] reported two peculiar fast transients (“Spock”) is also caused by stars embedded in high magnification
2470-0010=2018=97(2)=023518(16) 023518-1 © 2018 American Physical Society OGURI, DIEGO, KAISER, KELLY, and BROADHURST PHYS. REV. D 97, 023518 (2018) regions due to the lensing galaxy. However there are several This paper is organized as follows. We present an notable differences between microlensing in giant arcs and analytic lens model which sets the theoretical background quasar microlensing. For example, in the former case the in Sec. II. We then discuss what kinds of constraints we can source is a star, whereas in the latter case the source is a place from observed caustic crossing events in Sec. III.We quasar. While the internal structure of a quasar is compli- apply our results to MACS J1149 LS1 to derive constraints cated with largely different sizes for different wavelengths, on lens and source properties of this particular microlensing the surface brightness distribution of a star is uniform event in Sec. IV. We discuss event rates and derive expected (neglecting limb darkening) and its radius can be predicted event rates for MACS J1149 LS1 as well as for general from observations of nearby stars. Also, the radius of a star cases in Sec. V. In Sec. VI, we discuss constraints on is typically much smaller than the size of the light-emitting compact dark matter in the presence of ICL. Finally we region of a quasar. The smaller radius translates into a summarize our results in Sec. VII. Throughout the paper higher maximum magnification that scales as the inverse of we adopt a cosmological model with the matter density Ω ¼ 0 3 Ω ¼ 0 7 the square root of the radius (see Sec. II). In addition, while m . , cosmological constant Λ . , and the −1 −1 typical magnifications of the brightest images of lensed Hubble constant H0 ¼ 70 km s Mpc . quasars are ≈10–20, MACS J1149 LS1 is observed very close to the critical curve of the macro lens model where the II. GENERAL THEORY magnification due to the macro lens model is estimated to Here we summarize basic properties of gravitational ≳300 be . Such high magnification of the macro mass lensing by a point mass lens embedded in a high magni- model is possible because the giant arc directly crosses the fication region. The high magnification region concentrates critical curve and many stars in the giant arcs are located around the caustics that are produced by a macro lens very near the critical curve. model of, e.g., a massive cluster of galaxies, and we Caustic crossing events in giant arcs are new phenomena consider a perturbation by a compact object to a highly that may probe a different parameter space from previously magnified background object (star) near the caustics. The known microlensing events. This possibility of observing lensing properties of such a compound system have been highly magnified images of individual stars in giant arcs was studied in depth in the literature [12–16] and more recently first discussed in [25], although only the smooth cluster by [26,27] in the context of interpreting MACS J1149 LS1 potential was considered in that work. Motivated by the [1]. We present some key results which are necessary for discovery of MACS J1149 LS1, the authors of [26,27] the discussions in the following sections. revisited this problem. They argued that even a small fraction of compact objects in the lens disrupts the critical curve into a A. Lens equation network of microcaustics and drastically modifies the micro- We consider a point mass lens in a constant convergence lensing properties near the critical curve. Even if dark matter (κ¯) and shear (γ¯) field which comes from a macro lens consists entirely of a smooth component, such compact model. Then we have objects are expected to exist, like for instance the stars that are responsible for the so-called intracluster light (ICL). Given μ−1 ¼ 1 − κ¯ − γ¯; ð Þ t 1 the drastic change of lensing properties near the critical curve, it has been argued that caustic crossing events in giant μ−1 ¼ 1 − κ¯ þ γ¯: ð Þ r 2 arcs may serve as a powerful probe of a range of dark matter scenarios such as primordial black holes (PBHs) [28–32] and The total magnification by the macro lens model is – μ¯ ¼ μ μ scalar field dark matter [33 35]. t r. We consider a region near the tangential critical μ−1 ≈ 0 In this paper, we adopt a simple analytic model that curve where t gives rise to the high magnification. consists of a point mass lens and a constant convergence A point mass lens with mass M, in the absence of the and shear component, and we study basic microlensing macro mass model, has the Einstein radius of properties of this lens model. The result is used to derive 4GM D 1=2 characteristic scales of caustic crossing events in giant arcs θ ¼ ls : ð Þ E 2 3 and their dependences on the lens and source properties. c DosDol The result is used to interpret MACS J1149 LS1 and place constraints on the lens mass as well as the property of the The lens equation for the point mass lens embedded in the source star. We also discuss the event rates of caustic macro lens model is crossing. Such analytic studies of caustic crossing events θ θ2 θ should complement the ray-tracing simulations presented β ¼ 1 − E 1 ; ð Þ 1 μ θ2 4 in [1,26] for which repeating simulations with many r different model parameters may be computationally expen- θ θ2 θ sive. Our approach more resembles that in [27], which β ¼ 2 − E 2 : ð Þ 2 μ θ2 5 appeared recently. t
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Note that (β1, β2) is the position of the source and (θ1, θ2)is the position of the image. The origin of the coordinates is taken at the position of the point mass lens. For simplicity we assume that the shear direction is aligned with the x axis. The inverse magnification matrix is ∂β⃗ μ−1 þ θ2 cosð2ϕÞ=θ2 −θ2 sinð2ϕÞ=θ2 ¼ r E E ; ⃗ −θ2 ð2ϕÞ=θ2 μ−1 − θ2 ð2ϕÞ=θ2 ∂θ E sin t E cos ð6Þ where ϕ is the polar angle of (θ1, θ2). The magnification is
θ2 θ4 μ−1 ¼ðμ μ Þ−1 − ðμ−1 − μ−1Þ ð2ϕÞ E − E : ð Þ t r r t cos θ2 θ4 7
B. Critical curve and caustic The critical curve can be derived from μ−1 ¼ 0. Specifically, θ 2 ð2ϕÞ FIG. 1. Critical curves (upper panels) and caustics (lower ¼ cos ðμ − μ Þ panels) of a point mass lens in the high magnification region. θ 2 t r μ−1 ¼ 0 001 μ−1 ¼ 0 401 E Left: Positive parity case with t . and r . . " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# μ−1 ¼ −0 001 μ−1 ¼ 0 399 4μ μ Right: Negative parity case with t . and r . . × 1 1 þ t r These are computed with GLAFIC [36]. Note the difference of ðμ − μ Þ2 2ð2ϕÞ x y t r cos scales between the and axes in the lower panels. " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# μ ð2ϕÞ 4μ ð μ Þ ≈ t cos 1 1 þ r sgn t : ð Þ pffiffiffiffi 2 jμ j 2ð2ϕÞ 8 μ t cos β ≈ t θ ; ð Þ caus;1 μ E 11 r A similar equation is also given in [26]. The caustic is 1 obtained by converting the critical curve in the image plane β ≈ pffiffiffiffi θ : ð Þ caus;2 μ E 12 to the source plane via the lens equation. Figure 1 shows the r critical curves and caustics for positive and negative parities, which have already been given in the literature We also want to know the typical width of the caustic. As (e.g., [12]). shown in Fig. 1, it becomes very long and thin. Given that μ The solutions of Eq. (8) depend on the sign of t (or the size ofpffiffiffiffi the critical curves at the region of interest is θ ≈ Oð μ θ Þ parity), which is different on each side of the critical curve. crit t E , from Eq. (5) we can infer the typical μ pffiffiffiffi r maintains the same sign on both sides of the critical width of the caustic to be (assuming μ ≫ 1) μ t curve, and without loss of generality, we assume that r is always positive (i.e., tangential critical curves). θ β ≈ pEffiffiffiffi ; ð Þ w μ 13 t 1. Positive parity case: μt > 0 From Eq. (8) we can easily estimate the size of the which implies that the total area enclosed by the caustic is critical curve along the θ1 and θ2 axes. Along θ1 [or not significantly enhanced compared with the case of an horizontal axis, i.e., cosð2ϕÞ¼1] we have isolated point mass lens. Figure 1 indicates that the area pffiffiffiffi enclosed by the caustic at around β1 ≈ 0 has the same order. θ ≈ μ θ ; ð Þ crit;1 t E 9
2. Negative parity case: μt < 0 and along θ2 [or vertical axis, i.e., cosð2ϕÞ¼−1]wehave θ pffiffiffiffi In this case, the critical curves do not form along the 1 θ ≈ μ θ : ð Þ crit;2 r E 10 axis because the right-hand side of Eq. (8) is always negative. On the other hand, there are two solutions of θ θ θ Using the lens equation, the corresponding size of the Eq. (8) along the 2 axis, which we denote crit;þ and crit;−. caustic along the β1 and β2 axes is estimated as They are
023518-3 OGURI, DIEGO, KAISER, KELLY, and BROADHURST PHYS. REV. D 97, 023518 (2018) pffiffiffiffiffiffiffi θ ≈ jμ jθ ; ð Þ C. Light curves crit;þ t E 14 pffiffiffiffi In order to predict light curves we need to assume θ ≈ μ θ : ð Þ crit;− r E 15 velocities of the lens and source with respect to the observer. The transverse velocity in the lens plane vl is The corresponding size of the caustic along the β2 axis is converted to the angular unit as
2 vl 1 β ;þ ≈ pffiffiffiffiffiffiffi θ ð16Þ ul ¼ ; ð21Þ crit jμ j E 1 þ z D t l ol
1 where Dol is the angular diameter distance from the β ;− ≈ pffiffiffiffi θ : ð17Þ crit μ E observer to the lens and the factor 1 þ zl accounts for r the dilution of the unit time; i.e., the angular velocity above We also want to estimate the size of the critical curve and indicates the change of the position on the sky per unit caustic along the θ1 direction. First, the maximum θ1 of the observed-frame time. We divide the lens velocity into two critical curves is estimated as follows. In the region of components: one is a bulk velocity of the whole lens system ðθ=θ Þ2 ≈ μ ð2ϕÞ (e.g., a peculiar velocity of a galaxy cluster for the case of interest, we can approximate E t cos . At the θ θ θ2 ¼ θ2 2 ϕ MACS J1149 LS1), and the other is a relative motion of a maximum along 1, crit;max, 1 cos must be sta- ϕ ¼ 1=2 ð2ϕÞ¼−1=2 point mass lens within the whole lens system. We denote tionary, leading to cos and cos . v v Therefore, these velocity components as m and p, respectively. We consider these two components separately as they have rffiffiffiffiffiffiffi jμ j different dependences on the macro lens model when they θ ≈ t θ : ð Þ are converted to velocities in the source plane (see below). crit;max 8 E 18 We also consider the transverse velocity in the source plane vs. Again, it can be converted to the angular unit as The corresponding size of the caustic along the β1 direction is v 1 u ¼ s : ð Þ rffiffiffiffiffiffiffi s 1 þ z D 22 1 jμ j s os β ≈ t θ : ð Þ caus;max μ 8 E 19 r We derive the relative velocity of the source and lens in the source plane by converting the transverse velocity in the We can use the same argument as above to show that the lens plane to the source plane as (see [14]) typical width of the caustics is given by Eq. (13). Thus, the area enclosed by the caustics is again largely unchanged μ−1 0 r compared with the case of an isolated point mass lens, u ¼ um þ up þ us: ð23Þ 0 μ−1 which can also be seen in Fig. 1. By comparing Eqs. (11) t and (19), it is found that the length of the caustic is longer in This indicates that the relative velocity can be anisotropic. thepffiffiffi positive parity than in the negative parity by a factor of 2 2. However, Fig. 1 indicates that in the negative parity The magnification tensor comes in because distances there are twice as many caustic crossings for each lens, between the point mass lenses in the image plane translate which compensates for the shorter length of the caustic into smaller distances in the source plane due to the cluster potential. The bulk velocity of a cluster is typically jvmj ∼ when we estimate the rate of caustic crossings (see Sec. V). −1 β ¼ β 500 km s (e.g., [37]). Although the relative motion of The source plane region surrounded by crit;þ demagnifies the star. As an example, we consider multiple the point mass lens, which is of the order of velocity jv j ∼ 1000 −1 images for a source placed at the origin ðβ1; β2Þ¼ð0; 0Þ. dispersions of massive clusters, p km s ,is larger than the bulk velocity, it is suppressed by the From the lens equation, we findp thatffiffiffiffi there are two multiple ðθ ; θ Þ¼ð μ θ ; 0Þ images located at 1 2 r E . The magnifica- magnification factors as shown in Eq. (23). The contri- tion of the individual images is computed as bution from the source motion is expected to be smaller given the larger redshift and distance to the source. μ2 Therefore, a simple approximation which we adopt in μ ¼ − r ; ð Þ 2 20 the following discussions is which is much smaller than the macro magnification u ≈ um: ð24Þ μ ¼ μ μ t r, indicating that any sources near the origin are significantly less magnified compared with the case with- On the other hand, Fig. 1 indicates that thep causticsffiffiffiffi x μ ≫ 1 out the point mass lens. are elongated along the axis by a factor of t .
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Thus, caustic crossing typically occurs by a source If the source size is too big compared with the Einstein moving along the y axis with the velocity ∼jumj. radius, we do not observe any microlensing magnifications. When a source crosses the caustics vertically along the y From Eq. (26), we can argue that the source size needs to axis, crossing occurs two times in total for the positive satisfy the following condition to have sensitivity to parity case, and four times for the negative parity case. As microlensing: discussed above, for the negative parity case, a source is demagnified near the center. pffiffiffiffi μ βR ≲ θ : ð29Þ The behavior of the total magnification near the caustic is t E important for estimating the possible maximum magnifi- cation. The total magnification near the caustics is known Suppose that we observe a specific microlensing event in to behave as μ ∝ Δβ−1=2, where Δβ is the distance between which we can estimate the lower limit of the relative μ the source and the caustic in the source plane. From the magnification factor during the caustic crossing event, obs, analytic examination and numerical calculations with which is derived from the difference of the magnitudes GLAFIC, we find that the magnifications near the caustics measured at the beginning of the event and when the lensed are crudely approximated as source is brightest. This gives the lower limit because the true magnification factor during the caustic crossing event θ 1=2 is larger given the limited sampling and detection limit of μðΔβÞ ≈ μ μ pffiffiffiffiE ; ð Þ t r μ Δβ 25 monitoring observations. Thus μ must be smaller than t obs the relative maximum magnification due to caustic cross- μ < μ =ðμ μ Þ which agrees with numerical results within a factor of ≲2. ing, i.e., obs max t r . Using this condition, we can replace the condition in Eq. (29) with This approximation is reasonable givenpffiffiffiffi that the width of the “ ” μ caustic is shrunk by a factor of t [Eq. (13)]. pffiffiffiffi θ Equation (25) suggests that the magnification becomes μ β ≲ E : ð Þ larger as the source approaches the caustic, which is a t R μ2 30 obs universal property of lensing near the caustic (see, e.g., [38,39]). However, the magnification saturates when the This means that, while the peak brightness is higher for the distance to the caustic becomes comparable to the size of larger star simply because of its large intrinsic brightness, the source in the source plane, βR (e.g., [25]). From this the microlensing magnification is more prominent for the condition, we can estimate the maximum magnification as smaller star. θ 1=2 μ ≈ μ μ pffiffiffiffiE : ð Þ max t r μ β 26 t R B. Macro model magnification The analysis presented in Sec. II assumed a uniform III. EXPECTED PROPERTIES OF THE LENS macro model magnification for simplicity. In practice, μ μ AND SOURCE however, the macro model magnification t and r depend on the image position with respect to the critical curve of A. Dependence on the source star μ the macro model. It has been known that t quickly We discuss how the peak magnification scales with the increases as the image approaches the critical curve (see, radius R and luminosity L of a background star. Assuming e.g., [38,39]). More specifically, given that we denote the θ a blackbody with temperature T, they are related as distance between the image and the critical curve as h,we μ ∝ θ−1 generally expect t h . We parametrize the dependence L R 2 T 4 θ ¼ : ð Þ on h as L R T 27 ⊙ ⊙ ⊙ θ −1 μ ðθ Þ¼μ h ; ð Þ In practice the spectral energy distribution (SED) of a star t h h 31 does not strictly follow the blackbody, but this relation still arcsec holds approximately. Using this relation, we find that the μ observed maximum flux of the star scales as where h is a constant factor that depends on the macro lens model, as well as the location on the critical curve. On the f ∝ μ L ∝ R−1=2L ∝ R3=2T4: ð Þ μ max max 28 other hand, r is approximately constant near the critical curve. Therefore, for a given temperature (which can be inferred There is also a well-known asymptotic behavior between β from the observation of the SED of the star), the maximum h (the angular distance to the caustic in the source plane) θ magnified flux of a larger star is larger than that of a and h (the angular distance to the critical curve in the β ∝ θ2 smaller star. image plane), h h. In particular, we parametrize it as
023518-5 OGURI, DIEGO, KAISER, KELLY, and BROADHURST PHYS. REV. D 97, 023518 (2018) θ 2 θ h the distance to the macro model critical line, h, becomes β ¼ β0 : ð32Þ h arcsec comparable to the Einstein radius of the point mass lens, the critical curves by the point mass lens merge with those from μ ¼ μ ðβ =β Þ−1=2 the macro lens model, and our basic assumption breaks From these equations we have t h h 0 for the macro model magnification of one of the merging pairs of down. Therefore, to have enough magnifications by the images. point mass lens, we need the following condition: The maximum magnification [Eq. (26)] of caustic cross- pffiffiffiffi μ μ θ ≲ θ : ð Þ ing is larger for larger t, which suggests that stars that are t E h 36 closer to the critical curve can have higher magnifications. However, the authors of [26] (see also [27]) argued that Using Eq. (31), this condition is rewritten as even a small fraction of point mass lenses significantly μ changes the asymptotic behavior of the macro model θ ≲ h : ð37Þ E μ3=2 magnification toward the critical curve. This is because t μ thep Einsteinffiffiffiffi radius of the point masspffiffiffiffi lens depends on t as ∝ μ μ The similar condition in the source plane is t, and hence for very large t the Einstein radii for different point mass lenses overlap, even when the number θ density of point mass lenses is small. As shown by the ray- pEffiffiffiffi ≲ β ; ð Þ μ h 38 tracing simulations in [26], beyond this “saturation” point t the source breaks into many microimages, and as a result it loses its sensitivity to the source position with respect to the which results in macro model caustic. Therefore, the macro model magni- 2 β0μ fication in fact does not diverge as predicted by Eq. (31), θ ≲ h : ð39Þ E μ3=2 but saturates at a finite value. t To estimate where the saturation happens, the authors of [26] considered the optical depth τ defined by In practice, Eqs. (37) and (39) give quite similar conditions, so in what follows we consider only Eq. (37). From this Σ pffiffiffiffi 2 condition, we have another condition for the maximum τ ¼ πð μ θ DolÞ ; ð33Þ M t E magnification of the macro mass model as where Σ is the surface mass density of the point mass μ 2=3 μ ≈ h ; ð Þ component. Here we implicitly assumed that the point t;max θ 40 masses have the same mass M, although we note that this E approximation is reasonably good when compared with the −1=3 which indicates μ ; ∝ M . The true maximum macro realistic ray-tracing simulations. The authors of [26] argued t max magnification is given by the smaller of μ ; given in that the saturation happens when τ ≈ 1. From the definition t max Eqs. (35) and (40). of the Einstein radius [Eq. (3)], it is found that
τ ∝ μ Σ; ð Þ t 34 C. Light curve time scales There are important time scales that characterize caustic which indicates that the maximum macro model magnifi- crossing events. One is the so-called source crossing time cation where the saturation happens is inversely propor- defined by tional to the surface mass density of the point mass component Σ, and it does not depend on the mass M. 2β t ¼ R ; ð Þ This means that, in order to achieve high peak magnifica- src u 41 tions [Eq. (26)], lower Σ is preferred. Specifically, we can compute the maximum macro model magnification by where βR is the angular size of the source in the source setting τ ¼ 1 in Eq. (33) as plane, and u is the source plane velocity as defined in Eq. (23). This source crossing time determines the time M μ ≈ : ð Þ scale of the light curve near the peak. Another important t;max πΣðθ D Þ2 35 E ol time scale is given by the time to cross between caustics. β Using the width of the caustics, w [Eq. (13)], the time to Σ μ Again, for a fixed surface density , t;max does not depend cross between caustics is expressed as on mass M. β We can consider another condition for the saturation t ¼ w ; ð42Þ from the Einstein radius (see [27]). Even for τ ≪ 1, when Ein u
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−1 which gives the typical time scale between multiple caustic lensing cluster, jvj ≈ jvmj ∼ 500 km s . We can convert it t crossing events. The former time scale src does not depend to the angular velocity on the sky as t on the lens property, whereas the latter time scale Ein scales ∝ M1=2 v with the lens mass as . u ≈ 5.2 × 10−8 arcsec yr−1: ð46Þ 500 km s−1 D. Apparent size of microlensed image While we fix the v ¼ 500 km s−1 in our main analysis, we If the Einstein radius is sufficiently large, multiple also check how the uncertainty on the velocity propagates images can be resolved. In this case, from the separations into various constraints that we obtain in this paper. of the multiple images we can directly infer the mass scale For the macro mass model using the best-fitting model of of the lens. In usual cases, however, the Einstein radius of the GLAFIC mass model [36,40],wehaveμ ≈ 13 in the point mass lens is sufficiently small compared with the h Eq. (31) and β0 ≈ 0.045 in Eq. (32). MACS J1149 LS1 angular resolution of observations. As a result, the observed θ ≈ 0 13 was discovered at h . arcsec, at which the model microlensed image is pointlike, from which we can set the μ ≈ 100 constraint on the lens mass as predicts the magnification of t . On the other hand, μ ≈ 3 near MACS J1149 LS1. Therefore, the total macro pffiffiffiffi r μ θ ≲ σ ; ð Þ mass model magnification at the position of MACS J1149 t E θ;obs 43 μ μ ≈ 300 LS1 is t r . σ We note that there is uncertainty associated with the where θ;obs is the angular resolution of observations. For the case of Hubble Space Telescope imaging observations, macro mass model. For example, the authors of [26] noted that the GLAFIC and WSLAP+ [41] mass models of MACS we typically have σθ; ≈ 0.05 arcsec. obs J1149.6+2223 predict roughly a factor of 2 different macro model magnifications near MACS J1149 LS1 (see also [42] IV. CONSTRAINING THE SOURCE AND LENS for a test of the accuracy of strong lens mass modeling). PROPERTIES OF MACS J1149 LS1 This difference in the macro mass model affects our We now tune the parameters to the observations of quantitative results. Again, while we fix mass model MACS J1149 LS1 [1] and see what kind of constraints we parameter values to those mentioned above in our analysis, can place on properties of both the lens object and the we also examine the dependence of our results on macro source star. mass model uncertainties by checking the dependence of μ our results on h that differs considerably between GLAFIC A. Parameters and WSLAP+ mass models. The ICL plays a crucial role in the interpretation of the The lens redshift of MACS J1149 LS1 is zl ¼ 0.544 and z ¼ 1 49 caustic crossing event. At the position of MACS J1149 the source redshift is s . . Both the redshifts are Σ ≈ LS1, the surface density of ICL is estimated as ICL spectroscopic redshifts and therefore are sufficiently accu- −2 11–19 M⊙pc depending on assumed stellar initial mass rate. For a cosmology with matter density ΩM ¼ 0.3, functions [1,43]. Given the critical surface density cosmological constant ΩΛ ¼ 0.7, and the dimensionless Σ ≈ 2 4 103 M −2 Hubble constant h ¼ 0.7, we have 6.4 kpc arcsec−1 at the crit . × ⊙pc , the convergence from the ICL κ ≈ 0 0046–0 0079 8 5 −1 reads ICL . . . This should be compared with lens and . kpc arcsec at the source. The distance κ ≈ 0 83 modulus to the source is 45.2. With these distances, the the total surface density . predicted by the best- Einstein radius of the (isolated) point mass lens is fitting GLAFIC mass model [36,40]. In what follows, we may use the mass fraction of the point mass lens component M 1=2 defined by θ ≈ 1 8 10−6 : ð Þ E . × arcsec 44 M⊙ Σ f ¼ ; ð Þ p 2000 M −2 47 The angular size of a star as a function of solar radius in the ⊙pc source plane is instead of the surface mass density Σ. −12 R βR ≈ 2.7 × 10 arcsec: ð45Þ B. Constraints on MACS J1149 LS1 R⊙ Based on discussions given in Sec. III, we constrain the We note that uncertainties of our analysis originating from properties of the lens that is responsible for the observed cosmological parameter uncertainties are much smaller caustic crossing event, as well as the properties of the compared with other uncertainties that we will discuss source star. below. First, from Eqs. (26), (44), and (45), at the position of μ ≈ 100 μ ≈ 3 As discussed in Sec. II C, we can assume that the MACS J1149 LS1 with t and r , the maximum velocity is dominated by that of the bulk motion of the magnification becomes
023518-7 OGURI, DIEGO, KAISER, KELLY, and BROADHURST PHYS. REV. D 97, 023518 (2018) M 1=4 R −1=2 R v −1 μ ≈ 7 8 104 : ð Þ t ≈ 0 038 : ð Þ max . × 48 src . −1 days 55 M⊙ R⊙ R⊙ 500 km s
Assuming the temperature to be T ¼ 12 000 K (see [1]), Similarly, the time scale between caustic crossings is the absolute V-band magnitude of the star before the magnification is computed using Eq. (27) as M 1=2 v −1 t ≈ 3 5 : ð Þ Ein . −1 yr 56 M⊙ 500 km s R M ≈ 2 2 − 5 ; ð Þ star . log 49 R⊙ In the case of MACS J1149 LS1, the source crossing time, which is the time scale where the light curve is affected by M ≈ 4 8 V T ¼ 5777 where we used L⊙ . in the -band, ⊙ K, the finiteness of the source star radius very near the peak, and included the bolometric correction. Therefore, taking appears to be smaller than ∼10 days [1]. The condition t ≲ 10 into account the cross-filter K-correction derived in [1], the src days becomes minimum apparent magnitude of the star magnified by R v −1 microlensing at the peak is ≲ 260: ð Þ −1 57 R⊙ 500 km s R m ≈ 46.4 − 5 log − 2.5 log μ ; peak R max For this source size, the apparent magnitude of the star ⊙ R M without microlensing by the point mass lens is ≲28, which ≈ 34 1 − 3 75 − 0 625 : ð Þ t . . log R . log M 50 appears to be consistent with the observation [1]. Also Ein ⊙ ⊙ ∼1 t ≳ 1 seems to be at least larger than yr, so Ein yr gives rise to The observation indicates that the peak magnitude is m ¼ 26 brighter than [1], i.e., M v −2 ≳ 0 082: ð Þ −1 . 58 R M M⊙ 500 km s 3 75 þ 0 625 ≳ 8 1: ð Þ . log R . log M . 51 ⊙ ⊙ Since MACS J1149 LS1 was unresolved during the caustic crossing event, we use Eq. (43) to set the constraint During the caustic crossing event, the source star is μ ≈ 3 on the mass of the lens as magnified by at least a factor of 3 or so. Setting obs in Eq. (30), we obtain a constraint on the source size as 6 M ≲ 7.6 × 10 M⊙: ð59Þ R M −1=2 ≲ 7600: ð Þ R M 52 By using this argument we may also exclude the possibility ⊙ ⊙ that the caustic crossing event was produced by massive dark matter substructures. We now consider the condition that the saturation does We now put together all these constraints and derive not happen, because the quantitative constraints derived allowed ranges of the lens mass M and the source size R. above assumed no saturation at the position of MACS The result is shown in Fig. 2, where we fixed the bulk J1149 LS1. By setting μ ¼ 100 in Eq. (35), we obtain −1 t velocity of the lens to v ¼ 500 km s . We find that there are large ranges of the lens mass and the source size that can Σ ≲ 24 M −2 ð Þ ⊙pc . 53 explain MACS J1149 LS1. However, it is expected that the lens and source pop- Again, we caution that this is the result assuming that ulations are not distributed uniformly in this parameter M point mass lenses have the same mass . As shown above, space. As discussed in the next section, the size distribution Σ ≈ the ICL component has the surface density of ICL of the source star is expected to be significantly bottom −2 11–19 M⊙pc (which corresponds to the mass fraction heavy; i.e., stars with smaller radii are more abundant than f ¼ Σ =Σ ≈ 0 0055–0 0095 of ICL ICL tot . . ) and therefore satis- those with larger radii. The same argument also holds for fies this condition. On the other hand, from the other the lens mass, if we assume standard stars and stellar condition for nonsaturation [Eq. (40)], we have remnants as the lens population, but with the minimum mass of ≈0.3 M⊙ below which the stellar initial mass 7 M ≲ 5.1 × 10 M⊙: ð54Þ function is truncated. Therefore, in the allowed parameter space, the most likely set of parameters is R ≈ 180R⊙ and For MACS J1149 LS1, the source crossing time M ≈ 0.3 M⊙. In this case, the star is magnified by a factor [Eq. (41)]is of ≈4300 at the peak. The result is fully consistent with the
023518-8 UNDERSTANDING CAUSTIC CROSSINGS IN GIANT … PHYS. REV. D 97, 023518 (2018) 6 5 109 −2 L L n ðL