Gravitational Lensing by a Massive Object in a Dark Matter Halo. I. Critical Curves and Caustics

We study the gravitational lensing properties of a massive object in a dark matter halo, concentrating on the critical curves and caustics of the combined lens. We model the system in the simplest approximation by a point mass embedded in a spherical Navarro–Frenk–White density profile. The low number of parameters of such a model permits a systematic exploration of its parameter space. We present galleries of critical curves and caustics for different masses and positions of the point in the halo. We demonstrate the existence of a critical mass, above which the gravitational influence of the centrally positioned point is strong enough to eliminate the radial critical curve and caustic of the halo. In the point-mass parameter space we identify the boundaries at which critical-curve transitions and corresponding caustic metamorphoses occur. The number of transitions as a function of the position of the point is surprisingly high, ranging from three for higher masses to as many as eight for lower masses. On the caustics we identify the occurrence of six different types of caustic metamorphoses. We illustrate the peculiar properties of the single radial critical curve and caustic appearing in an additional unusual nonlocal metamorphosis for a critical mass positioned at the halo center. Although we construct the model primarily to study the lensing influence of individual galaxies in a galaxy cluster, it can also be used to study the lensing by dwarf satellite galaxies in the halo of a host galaxy, as well as (super)massive black holes at a general position in a galactic halo.

Phase transitions for spatially extended pinning

We consider a directed polymer of length N interacting with a linear interface. The monomers carry i.i.d. random charges $$(\omega _i)_{i=1}^N$$ ( ω i ) i = 1 N taking values in $${\mathbb {R}}$$ R with mean zero and variance one. Each monomer i contributes an energy $$(\beta \omega _i-h)\varphi (S_i)$$ ( β ω i - h ) φ ( S i ) to the interaction Hamiltonian, where $$S_i \in {\mathbb {Z}}$$ S i ∈ Z is the height of monomer i with respect to the interface, $$\varphi :\,{\mathbb {Z}}\rightarrow [0,\infty )$$ φ : Z → [ 0 , ∞ ) is the interaction potential, $$\beta \in [0,\infty )$$ β ∈ [ 0 , ∞ ) is the inverse temperature, and $$h \in {\mathbb {R}}$$ h ∈ R is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $${\mathbb {Z}}$$ Z . We study both the quenched and the annealed free energy per monomer in the limit as $$N\rightarrow \infty $$ N → ∞ . We show that each exhibits a phase transition along a critical curve in the $$(\beta ,h)$$ ( β , h ) -plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as $$\beta ,h \downarrow 0$$ β , h ↓ 0 in three different regimes for the tail exponent of $$\varphi $$ φ .

Prediction of rupture risk in cerebral aneurysms by comparing clinical cases with fluid–structure interaction analyses

Cerebral aneurysms should be treated on the basis of accurate rupture risk prediction. Nowadays, the rupture risk in aneurysms has been estimated using hemodynamic parameters. In this paper, we suggest a new way to predict the rupture risks in cerebral aneurysms by using fluid–structure interaction (FSI) analysis for better decision-making regarding treatment. A patient-specific model was constructed using digital subtraction angiography of 51 cerebral aneurysms. For each model, a thin-walled area (TWA) was first predicted using computational fluid dynamics (CFD), and then the highest equivalent strain in the TWA was calculated with FSI by varying wall thicknesses and mechanical properties. A critical curve was made from 16 FSI results for each patient-specific model to estimate the rupture risk. On average, the equivalent strains of the ruptured aneurysms were higher than those of the unruptured aneurysms. Furthermore, the patterns of critical curves between unruptured and ruptured aneurysms were clearly distinguishable. From the rupture risk evaluation based on the cut-off value, 24 of the 27 unruptured aneurysms and 15 of the 24 ruptured aneurysms were matched with actual-clinical setting cases. The critical curve proposed in the present study could be an effective tool for the prediction of the rupture risk of aneurysm.

Direct collapse to supermassive black hole seeds: the critical conditions for suppression of H2 cooling

ABSTRACT Observations of high-redshift quasars imply the presence of supermassive black holes (SMBHs) already at $z$ ∼ 7.5. An appealing and promising pathway to their formation is the direct collapse scenario of a primordial gas in atomic-cooling haloes at $z$ ∼ 10–20, when the $\rm H_{2}$ formation is inhibited by a strong background radiation field, whose intensity exceeds a critical value, Jcrit. To estimate Jcrit, typically, studies have assumed idealized spectra, with a fixed ratio of $\rm H_{2}$ photodissociation rate $k_{\rm H_2}$ to the $\rm H^-$ photodetachment rate $k_{\rm H^-}$. This assumption, however, could be too narrow in scope as the nature of the background radiation field is not known precisely. In this work we argue that the critical condition for suppressing the H2 cooling in the collapsing gas could be described in a more general way by a combination of $k_{\rm H_2}$ and $k_{\rm H^-}$ parameters, without any additional assumptions about the shape of the underlying radiation spectrum. By performing a series of cosmological zoom-in simulations covering a wide range of relevant $k_{\rm H_2}$ and $k_{\rm H^-}$ parameters, we examine the gas flow by following evolution of basic parameters of the accretion flow. We test under what conditions the gas evolution is dominated by $\rm H_{2}$ and/or atomic cooling. We confirm the existence of a critical curve in the $k_{\rm H_2}{\!-\!}k_{\rm H^-}$ plane and provide an analytical fit to it. This curve depends on the conditions in the direct collapse, and reveals domains where the atomic cooling dominates over the molecular cooling. Furthermore, we have considered the effect of $\rm H_{2}$ self-shielding on the critical curve, by adopting three methods for the effective column density approximation in $\rm H_{2}$. We find that the estimate of the characteristic length scale for shielding can be improved by using λJeans25, which is 0.25 times that of the local Jeans length, which is consistent with previous one-zone modelling.

Asymmetric surface brightness structure of caustic crossing arc in SDSS J1226+2152: a case for dark matter substructure

ABSTRACT We study the highly magnified arc SGAS J122651.3+215220 caused by a star-forming galaxy at zs = 2.93 crossing the lensing caustic cast by the galaxy cluster SDSS J1226+2152 (zl = 0.43), using Hubble Space Telescope observations. We report in the arc several asymmetric surface brightness features whose angular separations are a fraction of an arcsecond from the lensing critical curve and appear to be highly but unequally magnified image pairs of underlying compact sources, with one brightest pair having clear asymmetry consistently across four filters. One explanation of unequal magnification is microlensing by intracluster stars, which induces independent flux variations in the images of individual or groups of source stars in the lensed galaxy. For a second possibility, intracluster dark matter subhaloes invisible to telescopes effectively perturb lensing magnifications near the critical curve and give rise to persistently unequal image pairs. Our modelling suggests, at least for the most prominent identified image pair, that the microlensing hypothesis is in tension with the absence of notable asymmetry variation over a six-year baseline, while subhaloes of ∼106–$10^8\, \mathrm{ M}_\odot$ anticipated from structure formation with cold dark matter typically produce stationary and sizable asymmetries. We judge that observations at additional times and more precise lens models are necessary to stringently constrain temporal variability and robustly distinguish between the two explanations. The arc under this study is a scheduled target of a Director’s Discretionary Early Release Science program of the James Webb Space Telescope, which will provide deep images and a high-resolution view with integral field spectroscopy.

Magnifications of paired micro-images emerging from a micro-lensing critical curve

Studies of the inner regions of micro-lensed active galactic nucleus during caustic crossing events have often relied upon the approximation that the magnification near a fold caustic is inversely proportional to the square root of the source-caustic distance. We examine here the behaviour of the individual micro-images (one a micro-minimum of the light traveltime, the other a micro-saddle) that emerge as a point source crosses a micro-fold caustic. We provide a variety of statistics on both the behaviour of the two newly created micro-images and some parameters that appear in higher order approximations for the magnification. We compare the predictions of these higher order approximations to the actual image magnifications of our simulations.