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Ashish K. Meena

Publications and source records attributed to Ashish K. Meena.

At least 19 recordsLinked to original sources

Resolving 3 Exotic Hyperbolic-Umbilic Lensing Configurations in the "Cosmic Mantis": An Exploration of RXJ0437.1+0043 with JWST

We present an updated strong lensing model for the galaxy cluster RXJ0437.1+0043 (z=0.285) using new JWST NIRCam imaging in four pass-bands. The cluster field notably displays at least three separate gravitationally lensed galaxies in hyperbolic-umbilic (H-U) configurations, which in the JWST imaging show exquisitely resolved stellar clumps at scales as small as 0.04''. We identify 13 new systems of multiply-imaged lensed galaxies and determine the redshifts for two sources based on archival spectroscopic observations. We also identify over 50 individually resolved substructure clumps in the multiple images of eight galaxies and add them into the lens model, for a total of 275 positional lensing constraints between all 26 lensed sources used in the model. We investigate the impact of different modeling choices on our results. In the final lens model, we reproduce the observed multiple images with a precision of 0.23'' , and achieve 0.13'' specifically for the H-U systems. We estimate a total enclosed mass in the cluster's strong lensing regime of M(<110 kpc) ~ 8.2 x 10^13 Msun, reconstruct the source images of the most highly magnified galaxies, and measure time delays in the most magnified, morphologically detailed systems. The isotropic lensing effect from the H-U configurations enables future detailed studies of the properties of their source galaxies, and the number and spatial density of additional constraints improve the model's ability to detect dark matter substructures within the cluster core.

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SN 2022riv in RX J2129: Discovery, Spectroscopic Classification, and Microlensing of a Strongly Lensed Type Ia Supernova from JWST and HST Observations

The multiply imaged SN 2022riv was discovered through a search of galaxy cluster fields as part of a Hubble Space Telescope (HST) SNAP program to find highly magnified stars. The supernova (SN) was detected in the image corresponding to the longest time delay of a galaxy at redshift $z=1.522$ strongly lensed by the foreground galaxy cluster RX J2129.7+0005. Follow up James Webb Space Telescope (JWST) NIRSpec G140M and PRISM spectroscopy yields a Type Ia SN classification. Using the SALT3-NIR light-curve fitter, we obtain a cosmology-independent measurement of the magnification of $5.35\pm1.01$ for the last-to-arrive image of the SN, with multiple SALT SN spectral time-series models yielding consistent constraints. The last-to-arrive image of SN 2022riv we detect appeared adjacent to the brightest cluster galaxy (BCG) at a location with an exceptionally high stellar mass density ($\sim 1-2$ dex higher than that of SN Refsdal), where microlensing is expected to introduce a 20-50% modulation of the magnification. Analyzing six independent lens models of the cluster, we find that four predict the magnification with much greater precision ($p < 0.05$) than would be expected by random chance, given the large effect anticipated from microlensing. Five models yield magnifications of roughly $4-7$ (within $1σ$) prior to accounting for microlensing, whereas HoliGRALE favors a significantly higher value of $15.39 \pm 0.85$. After incorporating nominal microlensing, the HoliGRALE prediction is within $1σ$ tension with our measurement. A companion paper (Dalrymple et al.) will present constraints on the relative time delay of the image that arrived earlier.

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VENUS: Strong-lensing model of MACS J1931.8-2635 -- revealing the farthest multiply imaged supernova

We present a parametric strong-lensing model for the galaxy cluster MACS J1931.8-2635 ($z_l = 0.35$), accompanying the detection of the spectroscopically confirmed SN Eos at $z = 5.13$ (Coulter et al. 2026). We identify 10 new multiple-image systems in recent VENUS JWST/NIRCam imaging, so that the model is constrained with a total of 19 robust multiple-image systems -- nine of which also have a spectroscopic redshift. For the point-like source corresponding to SN Eos, our model predicts a total of five images, with the observed radial image pair having a similar magnification of $μ\simeq 25 - 30$ and a small time delay of $< 5$ days, in agreement with their simultaneous observation. According to the model, the other three predicted images arrived earlier, with time delays of $3.6 \pm 0.7$, $3.4 \pm 0.7$ and $53.9 \pm 10.8$ years prior to the two observed images, and with magnifications of $14.5 \pm 2.9$, $11.9 \pm 2.4$ and $2.2 \pm 0.4$, respectively. The absence of detections at the predicted positions, where the host galaxy's images are also visible, confirms the transient nature of the source. SN Eos and its host galaxy are studied in separate articles, and we here focus on the lens model. The final model reaches a very good $r.m.s.$ distance between model and observations of $0.44''$. We present the lens-modeling results, including newly identified systems such as a triply imaged, grand-design spiral galaxy candidate at $z \simeq 3.65_{-0.09}^{+0.04}$, and discuss the potential of using high-redshift lensed SNe for cosmography.

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Predisposition of galaxy clusters to producing exotic hyperbolic umbilic lensing configurations

Strong gravitational lensing is a powerful tool for investigating the universe's large-scale structure and understanding the properties of dark matter and dark energy. The magnification and distortion of distant background sources by cluster lenses have enabled detailed studies of both lens and source populations, making these systems promising probes for precision cosmology. While classical strong-lenses are well understood, much remains to be explored for hyperbolic-umbilic (HU) exotic lenses, which produce unique telescopic effects and uncommon images with potentially very high magnifications. Identifying and quantifying these objects, along with characterising their geometric configurations, could have broad implications for studies of galaxy clusters and lensed galaxy populations. Using parametric cluster mass models, we mapped regions in the source plane where HU exotic images can form and integrate these areas over redshift to define an exotic comoving volume (V_z<10). We validated this approach on confirmed exotic systems (RXJ0437.1+0043 and Abell 1703), then applied it to a sample of 74 cluster models. We show HU-region contours for the most promising clusters, assess both systematic and stochastic uncertainties on exotic area and volume estimates, and confirm that our error remains sufficiently small to support robust conclusions. Next, we explore correlations between six cluster parameters and (V_z<10), finding that pairs of parameters, especially ellipticity with Einstein radius or cuspiness, best distinguish high-(V_z<10) systems. Finally, we estimate that each cluster contributes ~0.125 galaxies to its exotic volume on average (as a conservative lower bound), meaning that observing 19 clusters yields a 90% chance of detecting at least one HU system in a random sample.

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Other red dots: A possible GLIMPSE of normal AGB stars at Cosmic Noon through extreme lensing

We report the discovery of four extremely faint ($m_{\mathrm{F444W}}\gtrsim29$) red point sources in recent ultra-deep JWST/NIRCam images of the strong lensing galaxy cluster Abell S1063. All four sources sit in lensed arcs, on the symmetry points very close to the critical curves for their host-galaxies' redshifts ($z\sim1-4$). Remarkably, these point sources appear in most arcs that are sufficiently faint close to the critical curve's position ($<21\,\mathrm{nJy}\,\mathrm{arcsec}^{-2}$ in F115W). This suggests that -- unlike previous caustic-crossing events or lensed stars -- thanks to the unprecedented depth of the GLIMPSE observations paired with the extreme lensing magnification (up to $μ\sim10^4$) we might be resolving the lower-mass ($M\sim1-11\,\mathrm{M}_{\odot}$) red stellar population. Concretely, we detect three likely extremely magnified asymptotic giant branch (AGB) stars ($T_{\mathrm{eff}}\sim3200-3750$ K), and one yellow super-giant star ($T_{\mathrm{eff}}\sim6750$ K) -- possibly a yellow hyper-giant or a Cepheid. In addition to offering the first glimpse at low-mass extremely magnified stars, these detections open a possible window into stellar populations, evolution, and chemical enrichment at high redshifts, and could pave the way for using lensed stars such as these as standard candles to populate the distance ladder at cosmological redshifts.

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A spectroscopically confirmed, strongly lensed, metal-poor Type II supernova at z = 5.13

Observing supernovae (SNe) in the early Universe (z > 3) provides a window into how both galaxies and individual stars have evolved over cosmic time, yet a detailed study of high-redshift stars and SNe has remained difficult due to their extreme distances and cosmological redshifting. To overcome the former, searches for gravitationally lensed sources allow for the discovery of magnified SNe that appear as multiple images - further providing the opportunity for efficient follow-up. Here we present the discovery of "SN Eos": a strongly lensed, multiply-imaged, SN II at a spectroscopic redshift of z = 5.133 +/- 0.001. SN Eos exploded in a Lyman-α emitting galaxy when the Universe was only ~1 billion years old, shortly after it reionized and became transparent to ultraviolet radiation. A year prior to our discovery in JWST data, archival HST imaging of SN Eos reveals rest-frame far ultraviolet (~1,300Å) emission, indicative of shock breakout or interaction with circumstellar material in the first few (rest-frame) days after explosion. The JWST spectroscopy of SN Eos, now the farthest spectroscopically confirmed SN ever discovered, shows that SN Eos's progenitor star likely formed in a metal-poor environment (<= 0.1 Z_{\odot}), providing the first direct evidence of massive star formation in the metal-poor, early Universe. SN Eos would not have been detectable without the extreme lensing magnification of the system, highlighting the potential of such discoveries to eventually place constraints on the faint end of the cosmic star-formation rate density in the very early Universe.

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Caught in Swallowtails: Discovery of Two Swallowtail Image Formations in MS 0451.6-0305

We report the discovery of two swallowtail image formations at $z=2.91$ and $z=6.70$ behind the galaxy cluster MS 0451.6-0305 in JWST-NIRCam imaging. We find that in both of the above lensed systems, the complex image morphology cannot be reproduced by simple fold/cusp caustics, and detailed lens modeling reveals higher-order swallowtail caustic configurations. In the $z=2.91$ lens system, a small part of the source galaxy (which itself is part of a galaxy group) containing atleast two compact knots sits inside the swallowtail caustic, producing a quadruply imaged arc. At two of the image positions of these knots, we infer point source magnifications of $\gtrsim 300$, implying lensing-corrected effective radii of $\lesssim 0.8-1.5$ pc. The $z=6.70$ system exhibits even more complex image formation. We therefore only use the most confidently identified counter-images of knots in this system as constraints in our lens modeling. The resulting model predicts magnifications $\sim20-200$ and lensing-corrected effective radii of $\lesssim 0.8-18.5$ pc for various knots. Together, these two systems represent the first example of observations of multiple swallowtail image formations in a single galaxy cluster and demonstrate the ability of swallowtail caustics to magnify individual substructures at sub-parsec scales, from intermediate redshifts to the first billion years of the Universe.

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Transient star B/R ratio and star formation in $z\gtrsim 1$ lensed galaxies

The extreme magnification from galaxy clusters and microlenses therein allows the detection of individual, luminous stars in lensed galaxies as transient events, and hence provides a valuable window into the high mass stellar population in $z>1$ galaxies. As these bright stars can only be formed at specific ages, the relative abundance of transient events at blue (B) and red (R) optical wavelengths ($B/R$ ratio) can provide insights into the recent star formation history of galaxies that are not well constrained by their spectral energy distributions (SEDs). Here, we forward model the transient detection rates in an idealized mock scenario to find that the $B/R$ ratio of strongly lensed $z>1$ galaxies decreases quickly with increasing age. This ratio has moderate sensitivity to metallicity and comparatively low sensitivity to dust attenuation, with no significant dependency on the stellar initial mass function. Fitting model stellar populations to either the SED or $B/R$ ratio alone of ``Warhol'' arc ($z = 0.94$), we find that neither a simple single starburst nor a more complex star formation can simultaneously reproduce both constraints. We then demonstrate that a best-fit model constrained by both the B/R ratio and SED requires a star-formation rate that has varied quite dramatically over the past $\sim$50 Myr, for which the total stellar mass formed over this time is a factor of 10 (with $2-3σ$ significance) different from the best-fit models to the SED alone. Our work shows that the transient $B/R$ ratio can be used as an additional powerful constraint on the recent star formation history of higher-redshift galaxies in future works that are strongly lensed by galaxy clusters.

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SLICE: SPT-CL J0546-5345 -- A prominent strong-lensing cluster at $z=1.07$

Massive galaxy clusters act as prominent strong-lenses. Due to a combination of observational biases, cluster evolution and lensing efficiency, most of the known cluster lenses lie typically at $z_{l}\sim0.2-0.7$, with only a few prominent examples at higher redshifts. Here we report a first strong-lensing analysis of the massive galaxy cluster SPT-CL J0546-5345 at a redshift $z_l=1.07$. This cluster was first detected through the Sunyaev-Zel'dovich effect, with a high estimated mass for its redshift of $M_{200,c} = (7.95 \pm 0.92) \times 10^{14}\,M_{\odot}$. Using recent JWST/NIRCam and archival HST imaging, we identify at least 10 secure and 6 candidate sets of multiply imaged background galaxies, which we use to constrain the mass distribution in the cluster. We derive effective Einstein radii of $θ_{E}= 18.1 \pm 1.8 ''$ for a source at $z_{s}=3$, and $θ_{E}= 27.9 \pm 2.8 ''$ for a source at $z_{s}=9$. The total projected mass within a $200$ kpc radius around the strong-lensing region is $M(<200\,\mathrm{kpc}) = (1.9 \pm 0.3) \times 10^{14}\,M_{\odot}$. While our results rely on photometric redshifts warranting spectroscopic follow-up, this central mass resembles that of the Hubble Frontier Fields clusters - although SPT-CL J0546-5345 is observed when the Universe was $\sim 3-4$ Gyr younger. Amongst the multiply-imaged sources, we identify a hyperbolic-umbilic-like configuration, and, thanks to its point-like morphology, a possible Active Galactic Nucleus (AGN). If confirmed spectroscopically, it will add to just a handful of other quasars and AGN known to be multiply lensed by galaxy clusters.

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Constraining the z $\sim$ 1 Initial Mass Function with {\it HST} and {\it JWST} Lensed Stars in MACS J0416.1-2403

Our understanding of galaxy properties and evolution is contingent on knowing the initial mass function (IMF), and yet to date, the IMF is constrained only to local galaxies. Individual stars are now becoming routinely detected at cosmological distances, where luminous stars such as supergiants in background galaxies strongly lensed by galaxy clusters are temporarily further magnified by huge factors (up to $10^{4}$) by intracluster stars, thus being detected as transients. The detection rate of these events depends on the abundance of luminous stars in the background galaxy and is thus sensitive to the IMF and the star-formation history (SFH), especially for the blue supergiants detected as transients in the rest-frame ultraviolet/optical filters. As a proof of concept, we use simple SFH and IMF models constrained by spectral energy distributions (SEDs) to see how well we can predict the {\it HST} and {\it JWST} transient detection rate in a lensed arc dubbed ``Spock'' ($z = 1.0054$). We find that demanding a simultaneous fit of the SED and the transient detection rate places constraints on the IMF, independent of the assumed simple SFH model. We conclude our likelihood analysis indicates that the data definitively prefers the ``Spock'' galaxy to have a Salpeter IMF ($α= 2.35$) rather than a Top-heavy IMF ($α= 1$) -- which is thought to be the case in the early universe -- with no clear excess of supergiants above the standard IMF.

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Microlensing at Cosmological Distances: Event Rate Predictions in the Warhol Arc of MACS 0416

Highly magnified stars ($μ$ $>$ 100) are now outinely identified as transient events at cosmological distances thanks to microlensing by intra-cluster stars near the critical curves of galaxy clusters. Using the {\it James Webb} Space Telescope (JWST) in combination with the {\it Hubble} Space Telescope (HST), we outline here an analytical framework that is applied to the Warhol arc (at $z=0.94$) in the MACS 0416 galaxy cluster (at $z=0.396)$ where over a dozen microlensed stars have been detected to date. This method is general and can be applied to other lensed arcs. Within this lensed galaxy we fit the spatially resolved SED spanned by eight JWST-NIRCam filters combined with three ACS filters, for accurate lensed star predictions in 2D. With this tool we can generate 2D maps of microlensed stars for well resolved arcs in general, including dependence on wavelength and limiting apparent magnitude, for comparison with with planned cadenced campaigns for JWST and Hubble, for constraining directly the IMF and the level of dark matter substructure.

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Identification of $>$40 gravitationally magnified stars in a galaxy at redshift of 0.725

Strong gravitational magnifications enable to detect faint background sources, resolve their internal structures, and even identify individual stars in distant galaxies. Highly magnified individual stars allow various applications, including studies of stellar populations in distant galaxies and constraining dark matter structures in the lensing plane. However, these applications have been hampered by the small number of individual stars observed, as typically one or a few stars are identified from each distant galaxy. Here, we report the discovery of more than 40 microlensed stars in a single galaxy behind Abell 370 at redshift of 0.725 when the Universe was half of its current age (dubbed ``the Dragon arc''), using James Webb Space Telescope (JWST) observations with the time-domain technique. These events are found near the expected lensing critical curves, suggesting that these are magnified stars that appear as transients from intracluster stellar microlenses. Through multi-wavelength photometry, we constrain stellar types and find that many of them are consistent with red giants/supergiants magnified by factors of hundreds. This finding reveals an unprecedented high occurrence of microlensing events in the Dragon arc, and proves that {\it JWST}'s time-domain observations open up the possibility of conducting statistical studies of high-redshift stars.

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Reverberation mapping of high-mass and high-redshift quasars using gravitational time delays

Mass estimates of black holes (BHs) in the centers of Active Galactic Nuclei (AGN) often rely on the radius-luminosity relation. However, this relation, usually probed by reverberation mapping (RM), is poorly constrained in the high-luminosity and high-redshift ends due to the very long expected RM lag times. Multiply imaged AGN may offer a unique opportunity to explore the radius-luminosity relation at these ends. In addition to comprising several magnified images enabling a more efficient light-curve sampling, the time delay between multiple images of strongly lensed quasars can also aid in making such RM measurements feasible on reasonable timescales: If the strong-lensing time delay is, for example, of the order of the expected RM time lag, changes in the emission lines in the leading image can be observed around the same time as the changes in the continuum in the trailing image. In this work we probe the typical time-delay distribution in galaxy-cluster lenses and estimate the number of both high-mass ($\sim10^9-10^{10}$ M$_{\odot}$), and high-redshift ($z\gtrsim4-12$) quasars that are expected to be strongly lensed by clusters. We find that up to several dozen thousand M$_{BH}\sim10^{6}$-$10^{8}$ M$_{\odot}$ broad-line AGN at $z>4$ should be multiply imaged by galaxy clusters and detectable with JWST, hundreds with Euclid and several thousands with the Roman Space Telescope, across the whole sky. These could supply an important calibration for the BH mass scaling in the early Universe.

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SN H0pe: The First Measurement of $H_0$ from a Multiply-Imaged Type Ia Supernova, Discovered by JWST

The first James Webb Space Telescope ({\it JWST}) Near InfraRed Camera (NIRCam) imaging in the field of the galaxy cluster PLCK G165.7+67.0 ($z=0.35$) uncovered a Type Ia supernova (SN Ia) at $z=1.78$, called ``SN H0pe." Three different images of this one SN were detected as a result of strong gravitational lensing, each one traversing a different path in spacetime, thereby inducing a relative delay in the arrival of each image. Follow-up {\it JWST} observations of all three SN images enabled photometric and rare spectroscopic measurements of the two relative time delays. Following strict blinding protocols which oversaw a live unblinding and regulated post-unblinding changes, these two measured time delays were compared to the predictions of seven independently constructed cluster lens models to measure a value for the Hubble constant, $H_0=71.8^{+9.2}_{-8.1}$ km s$^{-1}$ Mpc$^{-1}$. The range of admissible $H_0$ values predicted across the lens models limits further precision, reflecting the well-known degeneracies between lens model constraints and time delays. It has long been theorized that a way forward is to leverage a standard candle, but this has not been realized until now. For the first time, the lens models are evaluated by their agreement with the SN absolute magnifications, breaking degeneracies and producing our best estimate, $H_0=75.7^{+8.1}_{-5.5}$ km s$^{-1}$ Mpc$^{-1}$. This is the first precision measurement of $H_0$ from a multiply-imaged SN Ia and only the second from any multiply-imaged SN.

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A complex node of the cosmic web associated with the massive galaxy cluster MACS J0600.1-2008

MACS J0600.1-2008 (MACS0600) is an X-ray luminous, massive galaxy cluster at $z_{\mathrm{d}}=0.43$, studied previously by the REionization LensIng Cluster Survey (RELICS) and ALMA Lensing Cluster Survey (ALCS) projects which revealed a complex, bimodal mass distribution and an intriguing high-redshift object behind it. Here, we report on the results of a combined analysis of the extended strong lensing (SL), X-ray, Sunyaev-Zeldovich (SZ), and galaxy luminosity-density properties of this system. Using new JWST and ground-based Gemini-N and Keck data, we obtain 13 new spectroscopic redshifts of multiply imaged galaxies and identify 12 new photometric multiple-image systems and candidates, including two multiply imaged $z\sim7$ objects. Taking advantage of the larger areal coverage, our analysis reveals an additional bimodal, massive SL structure which we measure spectroscopically to lie adjacent to the cluster and whose existence was implied by previous SL-modeling analyses. While based in part on photometric systems identified in ground-based imaging requiring further verification, our extended SL model suggests that the cluster may have the second-largest critical area and effective Einstein radius observed to date, $A_{\mathrm{crit}}\simeq2.16 \mathrm{arcmin}^2$ and $θ_{\mathrm{E}}=49.7''\pm5.0''$ for a source at $z_{\mathrm{s}}=2$, enclosing a total mass of $M(<θ_{\mathrm{E}})=(4.7\pm0.7)\times10^{14} \mathrm{M}_{\odot}$. These results are also supported by the galaxy luminosity distribution, the SZ and X-ray data. Yet another, probably related massive cluster structure, discovered in X-rays $5'$ (1.7 Mpc) further north, suggests that MACS0600 is part of an even larger filamentary structure. This discovery adds to several recent detections of massive structures around SL galaxy clusters and establishes MACS0600 as a prime target for future high-redshift surveys with JWST.

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A high-resolution view of the source-plane magnification near cluster caustics in wave dark matter models

We present the highest resolution images to date of caustics formed by wave dark matter ($ψ$DM) fluctuations near the critical curves of cluster gravitational lenses. We describe the basic magnification features of $ψ$DM in the source plane at high macromodel magnification and discuss specific differences between the $ψ$DM and standard cold dark matter (CDM) models. The unique generation of demagnified counterimages formed outside the Einstein radius for $ψ$DM is highlighted. Substructure in CDM cannot generate such demagnified images of positive parity, thus providing a definitive way to distinguish $ψ$DM from CDM. Highly magnified background sources with sizes $r\approx 1pc$, or approximately a factor of ten smaller than the expected de Broglie wavelength of $ψ$DM, offer the best possibility of discriminating between $ψ$DM and CDM. These include objects such as very compact stellar clusters at high redshift that JWST is finding in abundance.

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Dark Matter distinguished by skewed microlensing in the "Dragon Arc"

Microlensed stars recently discovered by JWST & HST follow closely the winding critical curve of A370 along all sections of the ``Dragon Arc" traversed by the critical curve. These transients are fainter than $m_{AB}>26.5$, corresponding to the Asymptotic Giant Branch (AGB) and microlensed by diffuse cluster stars observed with $\simeq 18M_\odot/pc^2$, or about $\simeq 1$\% of the projected dark matter density. Most microlensed stars appear along the inner edge of the critical curve, following an asymmetric band of width $\simeq 4$kpc that is skewed by $-0.7\pm0.2$kpc. Some skewness is expected as the most magnified images should form along the inner edge of the critical curve with negative parity, but the predicted shift is small $\simeq -0.04$kpc and the band of predicted detections is narrow, $\simeq 1.4$kpc. Adding CDM-like dark halos of $10^{6-8}M_\odot$ broadens the band as desired but favours detections along the outer edge of the critical curve, in the wrong direction, where sub-halos generate local Einstein rings. Instead, the interference inherent to ``Wave Dark Matter" as a Bose-Einstein condensate ($ψ$DM) forms a symmetric band of critical curves that favours negative parity detections. A de Broglie wavelength of $\simeq 10$pc matches well the observed $4$kpc band of microlenses and predicts negative skewness $\simeq -0.6$kpc, similar to the data. The implied corresponding boson mass is $\simeq 10^{-22}$eV, in good agreement with estimates from dwarf galaxy cores when scaled by momentum. Further JWST imaging may reveal the pattern of critical curves by simply ``joining the dots" between microlensed stars, allowing wave corrugations of $ψ$DM to be distinguished from CDM sub-halos

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Imaging dark matter at the smallest scales with $z\approx1$ lensed stars

Observations of caustic-crossing galaxies at redshift $0.7 2$ and the number density of events is greater around substructureand the number density of events is greater around substructures, and (ii) negative imaging regime where $β<2$. We study the particular case of seven microlensing events found by HST in the Dragon arc (at z=0.725). We find that a population of supergiant stars with a steep LF with $β=2.55$ fits the distribution of these events. We identify a small region of high density of microlensing events, and interpret it as evidence of a possible invisible substructure, for which we derive a mass of $\sim 1.3 \times 10^8\,\Msun$ (within its Einstein radius).

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