Einstein Ring Research Articles

Shadow and strong gravitational lensing of new wormhole solutions supported by embedding Class-I condition

This study deals with the new class of embedded wormhole solutions in the background of general relativity. Two newly calculated wormhole solutions satisfy all the required properties. The embedded diagrams for both calculated wormhole solutions are provided. All the energy conditions are discussed through their validity regions for the different ranges of involved parameters. In maximum regions, all energy conditions are violated. We investigate the shadow and strong gravitational lensing by the wormhole throat for the two new wormhole models, namely Model-I and Model-II. The present paper considers the wormhole throat to act as a photon sphere. We first derive null geodesics using the Hamilton-Jacobi separation method to investigate the shadow and strong gravitational lensing caused by the wormhole throat. We then numerically obtain the radius of wormhole shadow, strong deflection angle, and various lensing observables by taking the example of supermassive black M87* and Sgr A* in the context of both Model-I and Model-II. Keeping all other parameters fixed, it is observed that the parameters ζ1 and ζ2 for Model-I; and χ1 and χ2 for Model-II have significant effects on the wormhole shadow and strong gravitational lensing phenomena. Our conclusion is that it is possible to detect relativistic images, such as Einstein rings, produced by wormholes with throat radii of rth=3M. The stability analysis via Tolman–Oppenheimer–Volkov equation is included for both wormhole solutions. Additionally, current technology enables us to test hypotheses related to astrophysical wormholes.

Holographic Einstein Rings of AdS Black Holes in Horndeski Theory

Abstract By utilizing the AdS/CFT correspondence and wave optics techniques, we conducted an extensive study of the imaging properties of holographic Einstein rings in the context of Anti-de Sitter (AdS) black holes (BHs) in Horndeski theory. Our results indicate that the optical characteristics of these holographic Einstein rings are significantly influenced by the observer's position, the physical parameters of the BH, the nature of the wave source, and the configuration of the optical system. Specifically, when the observer is positioned at the north pole of the AdS boundary, the holographic image prominently displays a ring structure aligning with the BH's photon sphere. We thoroughly analyzed how various physical parameters---including the observation position, event horizon radius, temperature, and the parameter $\gamma$ in Horndeski theory---affect the holographic Einstein rings. These parameters play a crucial role in determining the rings' radius and brightness, with variations potentially causing the ring structures to deform or even transform into bright spots. Furthermore, our comparative analysis between wave optics and geometric optics reveals a strong agreement in predicting the positions and brightness of both the photon ring and the Einstein ring. This research offers new insights into the spacetime geometry of BHs in Horndeski theory and proposes a promising framework for exploring the gravitational duals of strongly coupled systems.

Dark Matter Distinguished by Skewed Microlensing in the “Dragon Arc”

Abstract Many microlensed stars discovered by JWST closely follow the winding critical curve of A370 along the “Dragon Arc” with m AB > 26.5, which we show comprises asymptotic giant branch stars microlensed by the observed level of diffuse cluster stars, corresponding to ≃1% of the dark matter density. Most events appear along the inner edge of the critical curve, following an asymmetric band of width ≃4.5 kpc that is skewed by −0.7 ± 0.2 kpc. This asymmetry, we argue, follows from the parity difference in caustic structure inherent to microlensing that extends to higher magnification in the negative parity regions. This parity difference predicts a modest net shift of −0.04 kpc to the inside of the cluster critical curve within a narrower band of ≃1.4 kpc than observed. Adding cold-dark-matter-like subhalos of 106−8 M ⊙ doubles the width, but detections are predicted to favor the outside of the critical curve, where the subhalos generate local Einstein rings, and subhalos inside the critical curve depress the magnification, reducing microlensing. Instead, the density perturbations of “wave dark matter” as a Bose–Einstein condensate (ψDM) can generate a wide band of corrugated critical curves with a large negative asymmetry. We find that a de Broglie wavelength of ≃10 pc reproduces the observed width of 4.5 kpc, with a negative skewness ≃−0.6 kpc, like the data, corresponding to a boson mass of ≃10−22 eV, in agreement with dwarf galaxy dynamical estimates. Independently, we also find clear asymmetry in the Jupiter Arc, with 12 microlensed stars lying along the inside of the critical curve, like the Dragon Arc.

The deflection angle and quasi-periodic oscillations of an extended gravitational decoupled black hole solution

In weak field limits, we compute the deflection angle of a gravitational decoupling extended black hole (BH) solution. We obtained the Gaussian optical curvature by examining the null geodesic equations with the help of Gauss–Bonnet theorem (GBT). We also looked into the deflection angle of light by a black hole in weak field limits with the use of the Gibbons–Werner method. We verify the graphical behavior of the black hole after determining the deflection angle of light. Additionally, in the presence of the plasma medium, we also determine the deflection angle of the light and examine its graphical behavior. Furthermore, we compute the Einstein ring via gravitational decoupling extended black hole solution. We also compute the quasi-periodic oscillations and discuss their graphical behavior.

Multifractality signatures in lensed quasars

ABSTRACT Variations in scaling behaviour in the flux and emissions of gravitational lensed quasars can provide valuable information about the dynamics within the sources and their cosmological evolution with time. Here, we study the multifractal behaviour of the light curves (LCs) of 14 lensed quasars with multiple images in the r band, with redshift ranging from 0.657 to 2.730, in the search for potential differences in non-linearity between the signals of the quasar multiple images. Among these lensed systems, nine present two images, two present three images, and three present four images. To this end, we apply the wavelet transform-based multifractal analysis formalism called wavelet transform modulus maxima. We identify strong multifractal signatures in the LCs of the images of all analysed lensed quasar systems, independently of the number of images, with a significant difference between the degree of multifractality of all the images and combinations. We have also searched for a possible connection between the degree of multifractality and the characteristic parameters related to the quasar source and the lensing galaxy. These parameters include the Einstein ring radius and the accretion disc size and the characteristic time-scales related to microlensing variability. The analysis reveals some apparent trends, pointing to a decrease in the degree of multifractality with the increase of the quasar’s source size and time-scale. Using a larger sample and following a similar approach, this study confirms a previous finding for the quasar Q0957 + 561.

A simple model of a gravitational lens from geometric optics

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n(r)2=1+C2/r2, where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images.

Astrophysical implications of Weyl geometric black holes: Shadows and strong gravitational lensing

We consider the astrophysical properties of an exact black hole solution obtained in Weyl geometric gravity theory from the simplest conformally invariant action, constructed from the square of the Weyl scalar and the strength of the Weyl vector only. The action is linearized in the Weyl scalar by introducing an auxiliary scalar field. In static spherical symmetry, this theory admits an exact black hole solution, which generalizes the standard Schwarzschild solution through the presence of two new terms in the metric, having a linear and a quadratic dependence on the radial coordinate. To test the astrophysical validity properties of the solution, we perform a detailed analysis of its lensing properties in the strong field regimes. In particular, we consider the shadow of the Weyl geometric black hole, and we obtain a first set of constraints on the solution parameters by using the observational data from the shadows of the M87* and Sgr A* supermassive black holes. As a second possibility of observationally testing the Weyl geometric black hole and constraining its free parameters, we consider the strong lensing in this geometry. We investigate in detail the basic strong lensing observables, including the angular position of the images, the angular separation, the relative magnification, the radii of the Einstein ring, and the relativistic time delay. For all these quantities, a comparison with the predictions of the standard Schwarzschild black hole solution is performed by using realistic astrophysical data. The obtained results may lead to the possibility of testing Weyl geometry, Weyl geometric gravity, and their effects at galactic and extragalactic levels by using the observational properties of the black hole solutions of the theory.

Witt’s Hyperbola Is Both Predicted and Observed to Pass Close to the Lensing Galaxies in Quadruple Quasars

When a rectangular hyperbola is constructed from the image positions of a quadruply lensed quasar, as proposed by Witt, it passes very close to the lensing galaxy. The median measured perpendicular offset between the observed light center of the lens and Witt’s hyperbola is 0.″013 for a sample for 39 systems lensed by a relatively isolated galaxy. The family of lens models adopted by Witt predicts that the lens lies on the hyperbola, but its position is not used for its construction. The median offset corresponds to roughly 1% of the Einstein ring radius and suggests that the centers of the lensing potential are close to the light centers of the lens. By putting a restrictive prior on the perpendicular distance to Witt’s hyperbola (or on the distance between the galaxy and the potential), one reduces by one the dimensionality of a model space when fitting data. Taking the brightest pixel of a lensing galaxy as its center avoids a shortcoming of using the average light center for a constraint.

Optics equation provides approximation of gravitational lenses for students

Straighforward ray trajectory equation uses trigonometric and hyperbolic functions to model curves in space-time, including Einstein rings and fatal attraction near black holes.

Gravitational lensing of spherically symmetric black holes in dark matter halos

The gravitational lensing of supermassive black holes surrounded by dark matter halo has attracted a great number of interests in recent years. However, many studies employed simplified dark matter density models, which makes it very hard to give a precise prediction on the dark matter effects in real astrophysical galaxies. In this work, to more accurately describe the distribution of dark matter in real astrophysical galaxies, we study the gravitational lensing of black holes in astrophysical dark matter halo models (Beta, Burkert, Brownstein, and Moore). The deflection angle is obtained using a generalized Gibbons-Werner approach. The visual angular positions and the Einstein rings are also calculated by adopting the gravitational lens equation. Specifically, we choose the supermassive black holes in Milky Way Galaxy, Andromeda galaxy (M31), Virgo galaxy (M87), and ESO138-G014 galaxy as examples, including the corresponding fitted value of dark matter halos. The results suggest that the dark matter halo described by the Beta model has non-negligible influences on the gravitational deflection angle and gravitational lensing observations. However, the Burkert, Brownstein, and Moore models have relatively small influences on angular position of images and the Einstein ring.

Gravitational lensing and shadow by a Schwarzschild-like black hole in metric-affine bumblebee gravity

In this paper, we investigate the gravitational lensing effect and the shadow around a Schwarzschild-like black hole in metric-affine bumblebee gravity, which leads to the Lorentz symmetry breaking. We first present a generalized formalism for calculating higher-order corrections to light weak bending angle in a static, spherically symmetric and not asymptotically flat spacetime, and then applying this general formalism to the metric-affine bumblebee gravity. Moreover, we derive the light deflection angle and the size of the Einstein ring within the weak field in this scenario. In addition, we analyze the black hole shadow in this theory framework. By using observational data from the Einstein’s ring of the galaxy ESO325-G004 and the black hole shadow of the M87\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{M}87$$\\end{document} galaxy, we estimate the upper bounds of the Lorentz symmetry breaking coefficient ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}, respectively.

Investigating the weak gravitational lensing in Starobinsky–Bel–Robinson gravity black hole

This paper is motivated to study the weak gravitational lensing of the black hole in the context of Starobinsky–Bel–Robinsion gravity. We deduce the deflection angle of light in the weak field limits. For this purpose, we find the Gaussian curvature and apply the Gauss–Bonnet theorem. We also investigate the deflection angle at which light is deflected by a plasma medium. Furthermore, we analyze the graphical behavior of the deflection angle in the framework of SBR gravity. We also calculate the energy extracted from the black hole in SBR gravity, Einstein ring, and lens equation in the non-plasma and plasma framework. We conclude this study with a discussion of how light beams behave when they approach such a four-dimensional black hole by calculating the deflection angle.

Inflation and dark matter from the low entropy hypothesis and modeling mechanism of modified gravity

Abstract The hypothesis of low entropy in the initial state of the Universe usually explains the observed entropy increase is in only one time direction: the thermodynamic arrow of time. The Hamiltonian formalism is commonly used in the context of general relativity. The set of Lagrange multipliers are introduced in the formalism, and they are corresponding to the Hamiltonian constraints which are written in terms of ‘weak equality’—the equality is satisfied if the constraints hold. Follow the low-entropy hypothesis, we postulate a modeling mechanism—a weak equality (of modeling) that holds only on the subspace of the theory space of physical models defined by some modeling constraints. By applying the modeling mechanism, we obtain a specific model of modified gravity under specific modeling conditions. We derive a novel equation of modeling from the mechanism, that describes how different gravitational models emerge. The solution of the modeling equation naturally turns out to be the model of R 2-gravity (with additional terms) if ordinary matter is negligible. We also found that this mechanism leads to two models: large-field inflation and wave-like dark matter (DM). Interestingly, the wave-like DM model is supported by the most recent observations of Einstein rings.

An effective model for the quantum Schwarzschild black hole: Weak deflection angle, quasinormal modes and bounding of greybody factor

In this paper, we thoroughly explore two crucial aspects of a quantum Schwarzschild black solution within four-dimensional space–time: (i) the weak deflection angle, (ii) the rigorous greybody factor and, (iii) the Dirac quasinormal modes. Our investigation involves employing the Gauss–Bonnet theorem to precisely compute the deflection angle and establishing its correlation with the Einstein ring. Additionally, we derive the rigorous bounds for greybody factors through the utilization of general bounds for reflection and transmission coefficients in the context of Schrodinger-like one-dimensional potential scattering. We also compute the corresponding Dirac quasinormal modes using the WKB approximation. We reduce the Dirac equation to a Schrodinger-like differential equation and solve it with appropriate boundary conditions to obtain the quasinormal frequencies. To visually underscore the quantum effect, we present figures that illustrate the impact of varying the parameter r0, or more specifically, in terms of the parameter α. This comprehensive examination enhances our understanding of the quantum characteristics inherent in the Schwarzschild black solution, shedding light on both the deflection angle and greybody factors in a four-dimensional space–time framework.

Black hole in a generalized Chaplygin–Jacobi dark fluid: Shadow and light deflection angle

We investigate a generalized Chaplygin-like gas with an anisotropic equation of state, characterizing a dark fluid within which a static spherically symmetric black hole is assumed. By solving the Einstein equations for this black hole spacetime, we explicitly derive the metric function. The spacetime is parametrized by two critical parameters, B and α, which measure the deviation from the Schwarzschild black hole and the extent of the dark fluid’s anisotropy, respectively. We explore the behavior of light rays in the vicinity of the black hole by calculating its shadow and comparing our results with the Event Horizon Telescope observations. This comparison constrains the parameters to 0≤B≲0.03 and 0<α≲0.1. Additionally, we calculate the deflection angles to determine the extent to which light is bent by the black hole. These calculations are further utilized to formulate possible Einstein rings, estimating the angular radius of the rings to be approximately 37.6μas. Throughout this work, we present analytical solutions wherever feasible, and employ reliable approximations where necessary to provide comprehensive insights into the spacetime characteristics and their observable effects.

Holographic Einstein rings of a black hole with a global monopole

The global monopole solutions which give rise to quite unusual physical phenomena have captured considerable attention. In this paper, we study the Einstein ring of the spherically symmetric AdS black hole solution with a global monopole based on the AdS/CFT correspondence. With the help of the given response function of the QFT on the boundary, we construct the holographic images of the black hole in the bulk. Our results exhibit the absolute amplitude of total response function |⟨O⟩|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\langle \\mathcal {O}\\rangle |$$\\end{document} increases with the decrease of the monopole parameter b and the temperature T. And the frequency ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} of the Gaussian source will also increase the absolute amplitude. These parameters also affects the Einstein ring. With the change of the observation position, this ring will change from the concentric stripe to a luminosity-deformed ring, or light points. And the monopole parameter has an effect on the brightness and the position of Einstein ring. All these results imply that the holographic images can be used as an effective tool to distinguish different types of black holes for fixed wave source and optical system. These theoretical proposal opens a door to gravitational phenomena on strongly correlated material. Therefore, the holographic Einstein ring is of great theoretical and experimental significance for our research. In theory, it provides us a new way to find a necessary condition for the existence of the dual black hole. And experimentally speaking, if we can realize suitable targets for observing AdS black holes successfully, we would be able to observe Einstein rings by tabletop experiments when we apply localized sources on such materials and measure their responses.

Holographic Einstein ring of a charged Rastall AdS black hole with bulk electromagnetic field* *Supported by the National Natural Science Foundation of China (11675140, 11705005, 12375043), the Innovation and Development Joint Foundation of Chongqing Natural Science Foundation (CSTB2022NSCQ-LZX0021), and the Basic Research Project of Science and Technology Committee of Chongqing (CSTB2023NSCQ-MSX0324)

We study the Einstein images of a charged Rastall AdS black hole (BH) within the fabric of AdS/CFT correspondence. Considering the holographic setup, we analyze the amplitude of the total response function for various values of model parameters. With an increase in parameter λ and temperature T, the amplitude of the response function decreases, while it increases with an increase in electric charge e and chemical potential μ. The influence of frequency ω also plays an important role in the bulk field, as it is found that decreasing ω leads to an increase in the periods of the waves, which means that the amplitude of the response function also depends on the wave source. The relation between T and the inverse of the horizon for various values of parameter λ is interpreted under fixed values of other involved parameters. These, in turn, affect the behavior of the response function and the Einstein ring, which may be used to differentiate the present study from previous ones. We construct the holographic images of the BH in bulk via a special optical system. The results show that the Einstein ring always appears with concentric stripes at the position of the north pole, and this ring transforms into a luminosity-deformed ring or bright light spot when the distant observer lies away from the north pole. Finally, we discuss the influence of the associated parameters on the Einstein ring radius, which is consistent with wave optics.

Detailed study of a rare hyperluminous rotating disk in an Einstein ring 10 billion years ago

Hyperluminous infrared galaxies (HyLIRGs) are the rarest and most extreme starbursts and found only in the distant Universe (z ≳ 1). They have intrinsic infrared (IR) luminosities LIR ≥ 1013 L⊙ and are commonly found to be major mergers. Recently, the Planck All-Sky Survey to Analyze Gravitationally-lensed Extreme Starbursts project (PASSAGES) searched ~104 deg2 of the sky and found ~20 HyLIRGs. We describe a detailed study of PJ0116-24, the brightest (μLIR ≈ 2.6 × 1014 L⊙, magnified with μ ≈ 17) Einstein-ring HyLIRG in the southern sky, at z = 2.125, with observations from the near-IR integral-field spectrograph VLT/ERIS and the submillimetre interferometer ALMA. We detected Hα, Hβ, [N ii] and [S ii] lines and obtained an extreme Balmer decrement (Hα/Hβ ≈ 8.73 ± 1.14). We modelled the molecular-gas and ionized-gas kinematics with CO(3–2) and Hα data at ~100–300 pc and (sub)kiloparsec delensed scales, respectively, finding consistent regular rotation. We found PJ0116-24 to be highly rotationally supported (vrot/σ0, mol. gas ≈ 9.4) with a richer gaseous substructure than other known HyLIRGs. Our results imply that PJ0116-24 is an intrinsically massive (Mbaryon ≈ 1011.3 M⊙) and rare starbursty disk (star-formation rate, SFR = 1,490 M⊙ yr−1) probably undergoing secular evolution. This indicates that the maximal SFR (≳1,000 M⊙ yr−1) predicted by simulations could occur during a galaxy’s secular evolution, away from major mergers.

Four microlensing giant planets detected through signals produced by minor-image perturbations

Aims. We investigated the nature of the anomalies appearing in four microlensing events KMT-2020-BLG-0757, KMT-2022-BLG-0732, KMT-2022-BLG-1787, and KMT-2022-BLG-1852. The light curves of these events commonly exhibit initial bumps followed by subsequent troughs that extend across a substantial portion of the light curves. Methods. We performed thorough modeling of the anomalies to elucidate their characteristics. Despite their prolonged durations, which differ from the usual brief anomalies observed in typical planetary events, our analysis revealed that each anomaly in these events originated from a planetary companion located within the Einstein ring of the primary star. It was found that the initial bump arouse when the source star crossed one of the planetary caustics, while the subsequent trough feature occurred as the source traversed the region of minor image perturbations lying between the pair of planetary caustics. Results. The estimated masses of the host and planet, their mass ratios, and the distance to the discovered planetary systems are (Mhost/M⊙, Mplanet/MJ, q/10−3, DL/kpc) = (0.58−0.30+0.33, 10.71−5.61+6.17, 17.61 ± 2.25, 6.67−1.30+0.93) for KMT-2020-BLG-0757, (0.53−0.31+0.31, 1.12−0.65+0.65, 2.01 ± 0.07, 6.66−1.84+1.19) for KMT-2022-BLG-0732, (0.42−0.23+0.32, 6.64−3.64+4.98, 15.07 ± 0.86, 7.55−1.30+0.89) for KMT-2022-BLG-1787, and (0.32−0.19+0.34, 4.98−2.94+5.42, 8.74 ± 0.49, 6.27−1.15+0.90) for KMT-2022-BLG-1852. These parameters indicate that all the planets are giants with masses exceeding the mass of Jupiter in our solar system and the hosts are low-mass stars with masses substantially less massive than the Sun.

The COSMOS-Web ring: In-depth characterization of an Einstein ring lensing system at z ∼ 2

Aims. We provide an in-depth analysis of the COSMOS-Web ring, an Einstein ring at z ≈ 2 that we serendipitously discovered during the data reduction of the COSMOS-Web survey and that could be the most distant lens discovered to date. Methods. We extracted the visible and near-infrared photometry of the source and the lens from more than 25 bands. We combined these observations with far-infrared detections to study the dusty nature of the source and we derived the photometric redshifts and physical properties of both the lens and the source with three different spectral energy distribution (SED) fitting codes. Using JWST/NIRCam images, we also produced two lens models to (i) recover the total mass of the lens, (ii) derive the magnification of the system, (iii) reconstruct the morphology of the lensed source, and (iv) measure the slope of the total mass density profile of the lens. Results. We find the lens to be a very massive elliptical galaxy at z = 2.02 ± 0.02 with a total mass within the Einstein radius of Mtot(&lt;θEin = (3.66 ± 0.36) × 1011 M⊙ and a total stellar mass of M⋆ = 1.37−0.11+0.14 × 1011 M⊙. We also estimate it to be compact and quiescent with a specific star formation rate below 10−13 yr. Compared to stellar-to-halo mass relations from the literature, we find that the total mass of the lens within the Einstein radius is consistent with the presence of a dark matter (DM) halo of total mass Mh = 1.09−0.57+1.46 × 1013 M⊙. In addition, the background source is a M⋆ = (1.26 ± 0.17) × 1010 M⊙ star-forming galaxy (SFR ≈ (78 ± 15) M⊙ yr) at z = 5.48 ± 0.06. The morphology reconstructed in the source plane shows two clear components with different colors. Dust attenuation values from SED fitting and nearby detections in the far infrared also suggest that the background source could be at least partially dust-obscured. Conclusions. We find the lens at z ≈ 2. Its total, stellar, and DM halo masses are consistent within the Einstein ring, so we do not need any unexpected changes in our description of the lens such as changing its initial mass function or including a non-negligible gas contribution. The most likely solution for the lensed source is at z ≈ 5.5. Its reconstructed morphology is complex and highly wavelength dependent, possibly because it is a merger or a main sequence galaxy with a heterogeneous dust distribution.