arXiv subjects

Liliya L. R. Williams

Publications and source records attributed to Liliya L. R. Williams.

At least 19 recordsLinked to original sources

arXiv2026

How much can shape degeneracies affect the Hubble constant measurement?

Cluster lens models are constrained almost entirely by observed lensed image positions, leaving a large space of mass perturbations with arbitrary amplitudes such that they reproduce every image exactly. Such degeneracies are known as shape degeneracies (ShaDes). We show that these ShaDes shift the observed time delays, by the difference of the perturbing potential between the two images, without depending on the number of source/lens planes. Bounding their amplitude by preserving the positivity of the convergence, image multiplicity, and parity, in MACS J0138.0$-$2155, they can contribute at least $\simeq7$ days to the delay between the detected images of SN-Encore (i.e., $17.5\%$ of the measured value of $39.8$ days) and $\simeq112$ days (i.e., $2.8$--$3.7\%$ of predicted value of $3,000-4000$ days) to the reappearance intervals predicted for SN-Encore and SN-Requiem. These error bars transfer one-to-one to $H_0$, and remain invisible in comparisons of independent models.

astro-ph.CO↗

arXiv2026

(Monty Python and the) HoliGRALE: A Hybrid GRALE Lens Inversion Methodology Diagnostically Evaluated with Synthetic and Real Data

We present HoliGRALE (Hybrid, only light included, GRAvitational LEnsing), a hybrid lens modeling extension of the freeform method GRALE. HoliGRALE incorporates physically motivated parametric components within the freeform genetic algorithm GRALE. Using three synthetic clusters from IllustrisTNG, each of varying complexity, we evaluate its performance according to its accuracy in reconstructing the density distribution, image positions, magnifications, and time delays. We find that HoliGRALE generally outperforms GRALE on all fronts, and shows greater accuracy in regions containing a high image number density. We estimate the error budget in projected density to be $<5\%$ for simple, few component clusters, and $\sim5-10\%$ for complex, multi-component clusters in the regions constrained by observed images. We then use HoliGRALE to model three lenses: RX J2129.7$+$0005, MACS J0138.0$-$2155, and Abell S1063. In each case, HoliGRALE is able to accurately reconstruct observed magnifications and time delays. In its predictions, HoliGRALE is found to be more precise than GRALE. We particularly highlight the predicted time delay for the next arriving image of SN Requiem to be $4346.61^{+490.27}_{-818.36}$ days ($\approx$ April 2026 -- October 2029), consistent with previous work, albeit with less precision. In Abell S1063, we report the reconstruction of a main dark matter halo core radius of $89.9\pm33.2$ kpc, in excellent agreement with past work and useful in testing theories of dark matter. Our results emphasize the expected error budget of HoliGRALE, which should be mentioned and scrutinized in future applications of HoliGRALE for other lens systems.

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arXiv2026

Shape Degeneracies: the likely culprit for the differences between lens mass models of galaxy clusters

Sky-projected mass distributions of galaxy clusters are widely used to constrain particle properties of dark matter, to magnify in flux and angular extent high redshift background galaxies, and to construct their luminosity function down to faint absolute magnitudes that are otherwise hard or impossible to reach. Obtaining accurate and precise mass maps of clusters is therefore of central importance for these studies. Yet, a visual inspection of any two mass maps of the same cluster shows that the shapes of their isodensity contours are often dissimilar in detail on <30kpc scales, and there is no simple relation connecting the two. These differences are due to systematic uncertainties in reconstructions, which are in turn due to lensing degeneracies. However, commonly known degeneracies, like the mass sheet degeneracy or the source position transformation cannot account for most of these differences, implying that more generalized shape degeneracies are responsible. In this paper we study shape degeneracies using an analytical approach. We conclude with three main takeaways: (i) Shape degeneracies are the likely cause of differences in the mass maps of cluster lenses recovered by different techniques. (ii) Shape degeneracies are significantly suppressed in regions of the lens plane that have tight groupings of lensed images. (iii) In clusters where image numbers are high, such as MACS J0416, their effect on time delays between images from the same source are generally small, of order 1%, but can reach ~10% for some multiply imaged background sources, affecting the determination of H_0.

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arXiv2026

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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arXiv2026

Lensed stars in galaxy-galaxy strong lensing -- a JWST prediction for the Cosmic Horseshoe

We explore for the first time the possibility of detecting lensed star transients in galaxy-galaxy strong lensing systems upon repeated, deep imaging using the {\it James-Webb Space Telescope} ({\it JWST}). Our calculation predicts that the extremely high recent star formation rate of $\sim 140\,M_{\odot}\textrm{yr}^{-1}$ over the last 50 Myr (not accounting for image multiplicity) in the ``Cosmic Horseshoe'' lensed system ($z = 2.381$) generates many young, bright stars, of which their large abundance is expected to lead to a detection rate of $\sim 60$ transients per pointing in {\it JWST} observations with a $5σ$ limiting magnitude of $\sim 29\,m_{AB}$. With the high expected detection rate and little room for uncertainty for the lens model compared with cluster lenses, our result suggests that the Cosmic Horseshoe could be an excellent tool to test the nature of dark matter based on the spatial distribution of transients, and can be used to constrain axion mass if dark matter is constituted of ultra-light axions. We also argue that the large distance modulus of $\sim46.5\,$mag at $z \approx 2.4$ can act as a filter to screen out less massive stars as transients and allow one to better constrain the high-mass end of the stellar initial mass function based on the transient detection rate. Follow-up {\it JWST} observations of the Cosmic Horseshoe would allow one to better probe the nature of dark matter and the star formation properties, such as the initial mass function at the cosmic noon, via lensed star transients.

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arXiv2026

Dents in the Mirror: A Novel Probe of Dark Matter Substructure in Galaxy Clusters from the Astrometric Asymmetry of Lensed Arcs

Astrometric perturbations of lensed arcs behind galaxy clusters have been recently suggested as promising probes of small-scale ($\lesssim10^9 M_{\odot}$) dark matter substructure. Populations of cold dark matter (CDM) subhalos, predicted in hierarchical structure formation theory, can break the symmetry of arcs near the critical curve, leading to positional shifts in the observed images. We present a novel statistical method to constrain the average subhalo mass fraction ($f_{\rm sub}$) in clusters that takes advantage of this induced positional asymmetry. Focusing on CDM, we extend a recent semi-analytic model of subhalo tidal evolution to accurately simulate realistic subhalos within a cluster-scale host. We simulate the asymmetry of lensed arcs from these subhalo populations using Approximate Bayesian Computation. Using mock data, we demonstrate that our method can reliably recover the simulated $f_{\rm sub}$ to within 68\% CI in 73\% of cases, regardless of the lens model, astrometric precision, and image morphology. We show that the constraining power of our method is optimized for larger samples of well observed arcs, ideal for recent JWST observations of cluster lenses. As a preliminary test, we apply our method to the MACSJ0416 Warhol arc and AS1063 System 1. For Warhol we constrain the upper limit on $\log f_{\rm sub} < -3.48^{+1.00}_{-0.91}$, while for AS1063 System 1 we constrain $\log f_{\rm sub} = -2.44^{+0.61}_{-0.86}$ (both at 68\% CI), consistent with CDM predictions. We elaborate on our method's limitations and its future potential to place stringent constraints on dark matter properties in cluster environments.

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arXiv2025

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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arXiv2025

A Fuzzy Situation Eased: Cold Dark Matter with Multipoles Can Explain The Double Radio Quad Lens HS 0810+2554

Originally observed in isophotal density contours of elliptical galaxies, higher order perturbations in the form of Fourier modes, or multipoles, are becoming increasingly recognized as necessary to account for angular mass complexity in strong lensing analyses. When smooth, elliptical CDM mass models fail, multipoles often emerge as solutions. With the discovery of two radio jets in the source quasar, the strong gravitational lens HS 0810+2554 can no longer be well fit by elliptical mass models, suggesting perturbations on small-scales. In this paper, we investigate the efficacy of multipoles $m=1$ (lopsidedness), $m=3$ (triangleness), and $m=4$ (boxiness and diskiness) in addressing the image positional anomalies of the two radio quads of HS 0810+2554. Due to the exact pairing and arrival sequence of the images being unknown, we consider all feasible image configurations. With 64 unique best-fit models, we achieve a fit of $χ=1.59$ ($χ^2=2.53$), with $m=1,3,4$ multipole strengths of 0.9%, 0.4%, and 0.6%, respectively, with images in the reverse time ordering. Elliptical+shear models from previous works find $χ\!\sim\!7\!-\!10$, for comparison. With the morphological (i.e., standard) arrival sequence, we achieve a fit of $χ=2.95$ with two images being assigned to opposite sources. Therefore, CDM mass models with mass complexity in the form of multipoles are able to adequately explain the positional anomalies in HS 0810+2554. Alternative dark matter theories, like fuzzy dark matter, need not be invoked.

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arXiv2025

Testing Light Unaffiliated Mass Clumps in MACS 0416 on galaxy and galaxy cluster scales using JWST

Light unaffiliated mass clumps (LUMCs), i.e. dark matter (DM) components without any stellar counterparts, have been reported in strong lensing mass reconstructions of MACS 0416, both on galaxy and galaxy cluster scales. On galaxy cluster scale, the most recent parametric study based on 303 multiple images features a LUMC, in the south of the cluster. On galaxy scale, the most recent GRALE non-parametric study based on 237 multiple images features two LUMCs, M1 and M2. Given the implications of these findings in the context of structure formation and evolution, we test these features parametrically, using LENSTOOL. First, we show that a mass model where each large scale DM component introduced in the modelling is associated with a stellar counterpart can reproduce the 303 multiple images, removing the need for any cluster scale LUMC in \lens. We then update the GRALE model using the 303 multiple images, finding that one of the two galaxy scale LUMC, M1, is no longer significant, while M2 remains. We test M2 by explicitely including it in our parametric model. We find that the inclusion of this LUMC does not improve the global RMS, but mildly improves locally the RMS for one multiple image located close to M2. Besides, the preferred mass for M2 corresponds to the lowest mass allowed by the adopted prior. M2 is therefore not strongly favoured by a parametric approach but it is not ruled out. We present a comparison of parametric and non-parametric models in the M2 area. Both approaches show very similar surface mass density at this location, with a 5-6 % difference between the mass maps. The difference is that GRALE favors a distinct mass substructure when LENSTOOL favors a more diffuse mass distribution. We have been able to propose a parametric mass model without including any LUMCs, providing further evidences for DM being associated with light in clusters.

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arXiv2025

Addressing the core-cusp and diversity problem of dwarf and disk galaxies using cold collisionless DARKexp theory

Dwarf galaxies are observed to have linearly rising rotation curves, which indicate constant density cores in their centers. These do not arise naturally from dark matter only simulations, giving rise to the ``core-cusp'' problem. Hydrodynamic simulations incorporating baryonic feedback can create cored profiles, but this solution requires some fine tuning. Additionally, there is a related ``diversity'' problem in which simulations cannot reproduce the large range in rotation curve shapes for galaxies of similar mass. Here we investigate, for the first time, whether a theoretical model based on statistical mechanics (DARKexp) can produce flat density cores with its single shape parameter. We find that these theoretical profiles are able to fit the observed rotation curves of galaxies with last measured velocities in the range ~ 20-200 km/s by producing fits to 96 SPARC catalog galaxies covering this range. We additionally show that the projected stellar density distributions of ultrafaint dwarfs that show cores are also well fit, and that wave, or fuzzy dark matter is unable to fit the full range of SPARC galaxies. Thus, DARKexp appears to be able to address the core-cusp problem and the diversity of rotation curves in these dark matter dominated galaxies with cold collisionless dark matter alone, i.e., without introducing baryonic feedback, or alternative nature of dark matter.

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arXiv2025

Flashlights: Prospects for constraining the Initial Mass Function around cosmic noon with caustic-crossing events

The Flashlights program with the Hubble Space Telescope imaged the six Hubble Frontier Fields galaxy clusters in two epochs and detected twenty transients. These are primarily expected to be caustic-crossing events (CCEs) where bright stars in distant lensed galaxies, typically at redshift $z\approx1$--3, get temporarily magnified close to cluster caustics. Since CCEs are generally biased toward more massive and luminous stars, they offer a unique route for probing the high end of the stellar mass function. We take advantage of the Flashlights event statistics to place preliminary constraints on the stellar initial mass function (IMF) around cosmic noon. The photometry (along with spectral information) of lensed arcs is used to infer their various stellar properties, and stellar synthesis models are used to evolve a recent stellar population in them. We estimate the microlens surface density near each arc and, together with existing lens models and simple formalism for CCEs, calculate the expected rate for a given IMF. We find that, on average, a Salpeter-like IMF ($α=2.35$) underpredicts the number of observed CCEs by a factor of ${\sim}0.7$, and a top-heavy IMF ($α=1.00$) overpredicts by a factor of ${\sim}1.7$, suggesting that the average IMF slope may lie somewhere in between. However, given the large uncertainties associated with estimating the stellar populations, these results are strongly model-dependent. Nevertheless, we introduce a useful framework for constraining the IMF using CCEs. Observations with JWST are already yielding many more CCEs and will soon enable more stringent constraints on the IMF at a range of redshifts.

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arXiv2025

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.

astro-ph.CO↗

arXiv2025

Are Models of Strong Gravitational Lensing by Clusters Converging or Diverging?

The increasingly large numbers of multiple images in cluster-scale gravitational lenses have allowed for tighter constraints on the mass distributions of these systems. Most lens models have progressed alongside this increase in image number. The general assumption is that these improvements would result in lens models converging to a common solution, suggesting that models are approaching the true mass distribution. To test whether or not this is occurring, we examine a sample of lens models of MACS J0416.1$-$2403 containing varying number of images as input. Splitting the sample into two bins (those including $<150$ and $>150$ images), we quantify the similarity of models in each bin using three comparison metrics, two of which are novel: Median Percent Difference, Frechet Distance, and Wasserstein Distance. In addition to quantifying similarity, the Frechet distance metric seems to also be an indicator of the mass sheet degeneracy. Each metric indicates that models with a greater number of input images are no more similar between one another than models with fewer input images. This suggests that lens models are neither converging nor diverging to a common solution for this system, regardless of method. With this result, we suggest that future models more carefully investigate lensing degeneracies and anomalous mass clumps (mass features significantly displaced from baryonic counterparts) to rigorously evaluate their model's validity. We also recommend further study into alternative, underutilized lens model priors (e.g. flux ratios) as an additional input constraint to image positions in hopes of breaking existing degeneracies.

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arXiv2025

Entropy Evolution Towards DARKexp of IllustrisTNG Dark Matter Halos

Dark matter (DM) halos simulated via N-body techniques are known to exhibit nearly universal profiles at equilibrium; however, their origin remains uncertain despite thorough investigation. This work aims to probe the origin of simulated DM halo structure by testing DARKexp, a first-principles approach that describes the maximum entropy state of collisionless, self-gravitating systems in equilibrium via their energy distributions, and proposes an entropy functional that is expected to increase during evolution. We fit the DARKexp energy distribution to a set of massive equilibrium halos from the cosmological simulation IllustrisTNG. For the first time, we calculate the entropy of these halos as a function of cosmological time by tracking halos that are in equilibrium at z = 0.0, to z = 3.0 and calculating their entropy at various epochs between. We find that DARKexp provides an excellent fit to the energy distributions of equilibrium DM halos and that such halos exhibit an overall increase in entropy during evolution. Our results indicate that DM halos evolve to become their maximum entropy state at equilibrium and that this state is described by DARKexp.

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arXiv2024

BUFFALO Wild Wings: A High Precision Free-Form Lens Model of MACSJ0416 with Constraints on Dark Matter from Substructure and Highly Magnified Arcs

We present new free-form and hybrid mass reconstructions of the galaxy cluster lens MACS J0416.1$-$2403 at $z=0.396$ using the lens inversion method GRALE. The reconstructions use 237 spectroscopically confirmed multiple images from Bergamini et. al. 2023 as the main input. Our primary model reconstructs images to a positional accuracy of 0.191", thus representing one of the most precise reconstructions of this lens to date. Our models find broad agreement with previous reconstructions, and identify two $\sim 10^{12} M_{\odot}$ light-unaffiliated substructures. We focus on two highly magnified arcs: Spock and Mothra. Our model features a unique critical curve structure around the Spock arc with 2 crossings. This structure enables sufficient magnification across this arc to potentially explain the large number of transients as microlensing events of supergiant stars. Additionally, we develop a model of the millilens substructure expected to be magnifying Mothra, which may be a binary pair of supergiants with $μ\sim 6000$. This model accounts for flexibility in the millilens position while preserving the observed flux and minimizing image position displacements along the Mothra arc. We constrain the millilens mass and core radius to $\lesssim 10^6 M_{\odot}$ and $\lesssim 17$ pc, respectively, which would render it one of the smallest and most compact substructures constrained by lensing. If the millilens is dominated by wave dark matter, the axion mass is constrained to be $\lesssim 3.0 \times 10^{-21}$ eV. Further monitoring of this lens with JWST will uncover more transients, permitting tighter constraints on the structure surrounding these two arcs.

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arXiv2024

Flashlights: Transients among Gravitationally-Lensed Star Clusters in the Dragon Arc. I. Stellar Microlensing vs Stellar Outbursts

We report the discovery of transients among star clusters in a distant galaxy that is gravitationally lensed by a foreground galaxy cluster, and explore whether these transients correspond to: (i) intrinsic variations associated with stellar outbursts; or (ii) extrinsic variations imposed through microlensing by intraclusters stars along, perhaps, with primordial black holes. From images at two epochs separated by nearly a year, we discovered ten such transients -- displaying brightness variations of $\sim$10\%--20\% -- among 55 persistent knots identified as young star clusters in the Dragon arc. Two of these transients are associated with a triply-lensed star cluster, permitting a test of intrinsic variability by checking whether their light variations are repeated among the different lensed counterparts with a suitable time delay given their different light arrival times at the observer. Despite considerable care in constructing a lens model for Abell 370 that is optimized at the Dragon arc, we found that the predicted lensing magnifications are not sufficiently accurate to provide a definitive test of intrinsic variability based only on two images -- although such a test will become feasible as more observations are made. On the other hand, we perform simulations demonstrating that the observed level of brightness variations, as well as the observed transient event rate, can be explained entirely by stellar microlensing: as stars in the background star cluster move across the sky relative to intracluster stars, changes in their individual brightnesses can result in an overall change in the brightness of their parent star cluster.

astro-ph.CO↗

arXiv2024

Elucidating Galaxy Population Properties Using a Model-Free Analysis of Quadruply Imaged Quasar Lenses From Large Surveys

The population of strong lensing galaxies is a sub-set of intermediate-redshift massive galaxies, whose population-level properties are not yet well understood. In the near future, thousands of multiply imaged systems are expected to be discovered by wide-field surveys like Rubin Observatory's Legacy Survey of Space and Time (LSST) and Euclid. With the soon-to-be robust population of quadruply lensed quasars, or quads, in mind, we introduce a novel technique to elucidate the empirical distribution of the galaxy population properties. Our re-imagining of the prevailing strong lensing analysis does not fit mass models to individual lenses, but instead starts with parametric models of many galaxy populations, which include generally ignored mass distribution complexities and exclude external shear for now. We construct many mock galaxy populations with different properties and obtain populations of quads from each of them. The mock `observed' population of quads is then compared to those from the mocks using a model-free analysis based on a 3D sub-space of directly observable quad image properties. The distance between two quad populations in the space of image properties is measured by a metric $η$, and the distance between their parent galaxy populations in the space of galaxy properties is measured by $ζ$. We find a well defined relation between $η$ and $ζ$. The discovered relation between the space of image properties and the space of galaxy properties allows for the observed galaxy population properties to be estimated from the properties of their quads, which will be conducted in a future paper.

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arXiv2024

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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