Cosmological numerical simulations of galaxy formation have led to the cuspy density profile of a pure cold dark matter halo toward the center, which is in sharp contradiction with the observations of the rotation curves of cold dark matter-dominated dwarf and low surface brightness disk galaxies, with the latter tending to favor mass profiles with a flat central core. Many efforts have been devoted to resolve this cusp-core problem in recent years, among them, baryon-cold dark matter interactions are considered to be the main physical mechanisms erasing the cold dark matter (CDM) cusp into a flat core in the centers of all CDM halos. Clearly, baryon-cold dark matter interactions are not customized only for CDM-dominated disk galaxies, but for all types, including giant ellipticals. We first fit the most recent high resolution observations of rotation curves with the Burkert profile, then use the constrained core size-halo mass relation to calculate the lensing frequency, and compare the predicted results with strong lensing observations. Unfortunately, it turns out that the core size constrained from rotation curves of disk galaxies cannot be extrapolated to giant ellipticals. We conclude that, in the standard cosmological paradigm, baryon-cold dark matter interactions are not universal mechanisms for galaxy formation, and therefore, they cannot be true solutions to the cusp-core problem.
We study approximate solutions of the gravitational lens equation and corresponding lens magnification factor near the critical point. This consideration is based on the Taylor expansion of the lens potential in powers of coordinates and an introduction of a proximity parameter characterising the closeness of a point source to the caustic. Second-order corrections to known approximate solutions and magnification are found in case of a general fold point. The first-order corrections near a general cusp are found as well. Key words: gravitational lensing, methods: analytical
Lensing flux-ratio anomalies have been frequently observed and taken as evidence for the presence of abundant dark matter substructures in lensing galaxies, as predicted by the cold dark matter (CDM) model of cosmogony. In previous work, we examined the cusp-caustic relations of the multiple images of background quasars lensed by galaxy-scale dark matter haloes, using a suite of high-resolution N-body simulations (the Aquarius simulations). In this work, we extend our previous calculations to incorporate both the baryonic and diffuse dark components in lensing haloes. We include in each lensin…
We have been able to connect the statistics of the observed double image gravitational lenses to the general properties of the internal structure of dark matter haloes. Our analytical theory for the GNFW lenses with parametrized cusp slope (alpha) gives us a relation connecting the cusp slope of the lensing profile to the observed magnification ratio of the produced images and location of the optical axis. The relation does not depend on cosmology, total lens mass, concentration or redshifts of the the lens and the lensed object. Simple geometry of axially symmetric lensing and aforementioned …
We extend our analyses of the dark matter (DM) distribution in relaxed clusters to the case of Abell 383, a luminous X-ray cluster at z=0.189 with a dominant central galaxy and numerous strongly-lensed features. Following our earlier papers, we combine strong and weak lensing constraints secured with Hubble Space Telescope and Subaru imaging with the radial profile of the stellar velocity dispersion of the central galaxy, essential for separating the baryonic mass distribution in the cluster core. Hydrostatic mass estimates from Chandra X-ray observations further constrain the solution. These …
The time delay measured between the images of gravitationally lensed quasars probes a combination of the angular diameter distance to the source-lens system and the mass density profile of the lens. Observational campaigns to measure such systems have reported a determination of the Hubble parameter H0 that shows significant tension with independent determination based on the cosmic microwave background (CMB) and large scale structure (LSS). We show that lens mass models that exhibit a cored component, coexisting with a cusp, probe a degenerate direction in the lens model parameter space, bein…
The cross-correlation of gravitational wave strain with upcoming galaxy surveys probe theories of gravity in a new way. This method enables testing the theory of gravity by combining the effects from both gravitational lensing of gravitational waves and the propagation of gravitational waves in spacetime. We find that within 10 years, the combination of the Advanced-LIGO and VIRGO detector networks with planned galaxy surveys should detect weak gravitational lensing of gravitational waves in the low redshift Universe ($z<0.5$). With the next generation gravitational wave experiments such as Vo…
Weak gravitational lensing of background galaxies is a unique, direct probe of the distribution of matter in clusters of galaxies. We review several important aspects of cluster weak gravitational lensing together with recent advances in weak lensing techniques for measuring cluster lensing profiles and constraining cluster structure parameters.
We find the leading-order effect of gravitational back-reaction on cosmic strings for points near kinks and cusps. Near a kink, the effect diverges as the inverse cube root of the distance to the kink, and acts in a direction transverse to the worldsheet. Over time the kink is rounded off, but only regions fairly close to the kink are significantly affected. Near cusps, the effect diverges inverse linearly with the distance to the cusp, and acts against the direction of the cusp motion. This results in a fractional loss of string energy that diverges logarithmically with the distance of closes…
We compute the contribution of kinks on cosmic string loops to stochastic background of gravitational waves (SBGW).We find that kinks contribute at the same order as cusps to the SBGW.We discuss the accessibility of the total background due to kinks as well as cusps to current and planned gravitational wave detectors, as well as to the big bang nucleosynthesis (BBN), the cosmic microwave background (CMB), and pulsar timing constraints. As in the case of cusps, we find that current data from interferometric gravitational wave detectors, such as LIGO, are sensitive to areas of parameter space of…
This Resource Letter provides a guide to a selection of the literature on gravitational lensing and its applications. Journal articles, books, popular articles, and websites are cited for the following topics: foundations of gravitational lensing, foundations of cosmology, history of gravitational lensing, strong lensing, weak lensing, and microlensing.
Gravitational lensing has been identified as a powerful tool to address fundamental problems in astrophysics at different scales, ranging from exoplanet identification to dark energy and dark matter characterization in cosmology. Image simulations have played a fundamental role in the realization of the full potential of gravitational lensing by providing a means to address needs such as systematic error characterization, pipeline testing, calibration analyses, code validation, and model development. We present a general overview of the generation and applications of image simulations in stron…
Advanced LIGO and Advanced Virgo could observe the first lensed gravitational waves in the coming years, while the future Einstein Telescope could observe hundreds of lensed events. Ground-based gravitational-wave detectors can resolve arrival time differences of the order of the inverse of the observed frequencies. As LIGO/Virgo frequency band spans from a few $\rm Hz$ to a few $ \rm kHz$, the typical time resolution of current interferometers is of the order of milliseconds. When microlenses are embedded in galaxies or galaxy clusters, lensing can become more prominent and result in observab…
The detailed observation of the distribution of redshifts and chirp masses of binary black hole mergers is expected to provide a clue to their origin. In this paper, we develop a hybrid model of the probability distribution function of gravitational lensing magnification taking account of both strong and weak gravitational lensing, and use it to study the effect of gravitational lensing magnification on the distribution of gravitational waves from distant binary black hole mergers detected in ongoing and future gravitational wave observations. We find that the effect of gravitational lensing m…
We discuss the gravitational lensing of gravitational wave signals from coalescing binaries. We delineate the regime where wave effects are significant from the regime where geometric limit can be used. Further, we focus on the effect of micro-lensing and the combined effect of strong lensing and micro-lensing. We find that micro-lensing combined with strong lensing can introduce time varying phase shift in the signal and hence can lead to detectable differences in the signal observed for different images produced by strong lensing. This, coupled with the coarse localization of signal source i…
According to the theory of general relativity, masses deflect light in a way similar to convex glass lenses. This gravitational lensing effect is astigmatic, giving rise to image distortions. These distortions allow to quantify cosmic structures statistically on a broad range of scales, and to map the spatial distribution of dark and visible matter. We summarise the theory of weak gravitational lensing and review applications to galaxies, galaxy clusters and larger-scale structures in the Universe.
Strong gravitational lensing is an important tool to probe the universe. In the theoretical analysis of gravitational lensing, it is assumed that continuous density profile can correctly describe the lens galaxy. But in fact this assumption has never been rigorously tested. In this paper, we discuss this issue, and point out that if we use discrete density profile to model the lens galaxy, then the position of the images does not change, but the magnification will be increased. Strongly lensed gravitational waves could test this conclusion in the future.
Gravitational lensing has developed into one of the most powerful tools for the analysis of the dark universe. This review summarises the theory of gravitational lensing, its main current applications and representative results achieved so far. It has two parts. In the first, starting from the equation of geodesic deviation, the equations of thin and extended gravitational lensing are derived. In the second, gravitational lensing by stars and planets, galaxies, galaxy clusters and large-scale structures is discussed and summarised.
Gravitational lensing by large scale structure introduces non-Gaussianity into the Cosmic Microwave Background and imprints a new observable, which can be used as a cosmological probe. We apply a four-point estimator to the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year coadded temperature maps alone to reconstruct the gravitational lensing signal. The Gaussian bias is simulated and subtracted, and the higher order bias is investigated. We measure a gravitational lensing signal with a statistical amplitude of $\mathcal {C}$ = $1.27\pm 0.98$ using all the correlations of the W- and V-band D…
The cosmic infrared background (CIB) is gravitationally lensed. A quadratic-estimator technique that is inherited from lensing analyses of the cosmic microwave background (CMB) can be applied to detect the CIB lensing effects. However, the CIB fluctuations are intrinsically strongly non-Gaussian, making CIB lensing reconstruction highly biased. We perform numerical simulations to estimate the intrinsic non-Gaussianity and establish a cross-correlation approach to precisely extract the CIB lensing signal from raw data. We apply this technique to CIB data from the Planck satellite and cross-corr…
We gain insight into the effects of gravitational lensing on the estimated distribution of merging binaries observed through gravitational waves. We quantify the efficiency of magnification for gravitational wave events in the geometric optics limit, and we compare it to the electromagnetic case by making minimal assumptions about the distribution of intrinsic properties for the source population. We show that lensing effects leave a recognizable signature on the observed rates, and that they can be prominent only in the presence of an extremely steep mass function (or redshift evolution) and …
The difficult task of observing Dark Matter subhaloes is of paramount importance since it would constrain Dark Matter particle properties (cold or warm relic) and confirm once again the longstanding $Λ$CDM model. In the near future the new generation of ground and space surveys will observe thousands of strong gravitational lensing systems providing a unique probe of Dark Matter substructures. Here, we describe a new strong lensing analysis pipeline that combines deep Convolutional Neural Networks with physical models and exploits traditional sampling techniques such as Hamiltonian Monte Carlo…
Dark matter halos represent the highest density peaks in the matter distribution. Conversely, cosmic voids are under-dense patches of the universe. Probing the mass distribution of the universe requires various approaches, including weak gravitational lensing that subtly modifies the shape of distant sources, and Doppler lensing that changes the apparent size and magnitude of objects due to peculiar velocities. In this work, we adopt both gravitational and Doppler lensing effects to study the underlying matter distribution in and around cosmic voids/halos. We use the relativistic $N$-body code…
We study the relative contribution of cusps and pseudocusps, on cosmic (super)strings, to the emitted bursts of gravitational waves. The gravitational wave emission in the vicinity of highly relativistic points on the string follows, for a high enough frequency, a logarithmic decrease. The slope has been analytically found to be $^{-4}/_3$ for points reaching exactly the speed of light in the limit $c=1$. We investigate the variations of this high frequency behaviour with respect to the velocity of the points considered, for strings formed through a numerical simulation, and we then compute nu…
We present the first detailed computations of wave optics effects in the gravitational lensing of binary systems. The field is conceptually rich, combining the caustic singularities produced in classical gravitational lensing with quantum (wave) interference effects. New techniques have enabled us to overcome previous barriers to computation. Recent developments in radio astronomy present observational opportunities which, while still futuristic, appear promising.
Gravitational wave sources are a promising cosmological standard candle because their intrinsic luminosities are determined by fundamental physics (and are insensitive to dust extinction). They are, however, affected by weak lensing magnification due to the gravitational lensing from structures along the line of sight. This lensing is a source of uncertainty in the distance determination, even in the limit of perfect standard candle measurements. It is commonly believed that the uncertainty in the distance to an ensemble of gravitational wave sources is limited by the standard deviation of the…
The weak gravitational lensing formalism can be extended to the strong lensing regime by integrating a nonlinear version of the geodesic deviation equation. The resulting "roulette" expansion generalises the notion of convergence, shear and flexion to arbitrary order. The independent coefficients of this expansion are screen space gradients of the optical tidal tensor which approximates to the usual lensing potential in the weak field limit. From lensed images, knowledge of the roulette coefficients can in principle be inverted to reconstruct the mass distribution of a lens. In this paper, we …
Ordinary baryonic particles (such as protons and neutrons) account for only one-sixth of the total matter in the Universe. The remainder is a mysterious "dark matter" component, which does not interact via the electromagnetic force and thus neither emits nor reflects light. However, evidence is mounting for its gravitational influence. The past few years have seen particular progress in observations of weak gravitational lensing, the slight deflection of light from distant galaxies due to the curvature of space around foreground mass. Recent surveys from the Hubble Space Telescope have provide…
Gravitational lensing refers to the deflection of light by the gravity of celestial bodies, often predominantly composed of dark matter. Seen through a gravitational lens, the images of distant galaxies appear distorted. In this paper we discuss simulation of the image distortion by gravitational lensing. The objective is to enhance our understanding of how gravitational lensing works through a simple tool to visualise hypotheses. The simulator can also generate synthetic data for the purpose of machine learning, which will hopefully allow us to invert the distortion function, something which …
We present a framework for studying gravitational lensing in spherically symmetric spacetimes using 1+1+2 covariant methods. A general formula for the deflection angle is derived and we show how this can be used to recover the standard result for the Schwarzschild spacetime.
Gravitational lensing of fast radio bursts (FRBs) offers an exciting avenue for several cosmological applications. However, it is not yet clear how many such events future surveys will detect nor how to optimally find them. We use the known properties of FRBs to forecast detection rates of gravitational lensing on delay timescales from microseconds to years, corresponding to lens masses spanning fifteen orders of magnitude. We highlight the role of the FRB redshift distribution on our ability to observe gravitational lensing. We consider cosmological lensing of FRBs by stars in foreground gala…
Gravitational and plasma lensing share the same mathematical formalism in the limit of geometrical optics. Both phenomena can be effectively described by a projected, two-dimensional deflection potential whose gradient causes an instantaneous light deflection in a single, thin lens plane. We highlight the differences in the time-delay and lensing equations that occur because plasma lensing is caused by a potential directly proportional to the deflecting electron number density and gravitational lensing is caused by a potential related to the deflecting mass density by a Poisson equation. Since…
Shear fields due to weak gravitational lensing have characteristic coherent patterns. We describe the topological defects in shear fields in terms of the curvature of the surface described by the lensing potential. A simple interpretation of the characteristic defects is given in terms of the the umbilical points of the potential surface produced by ellipsoidal halos. We show simulated lensing shear maps and point out the typical defect configurations. Finally, we show how statistical properties such as the abundance of defects can be expressed in terms of the correlation function of the lensi…
We investigate the effect of the redshift drift in strong gravitational lensing. The redshift drift produces a time variation of $i)$ the apparent position of a lensed source and $ii)$ the time delay among incoming signals from different images. We dub these effects as angular drift and time delay drift, respectively, and analyze their relevance in cosmology.
Current sky surveys have been conducted very accurately in order to understand our universe. One of the phenomena survey maps provide is gravitational effect. Albert Einstein (1936) first discussed the possibilities of gravitational lensing effects based on his theory of general relativity, which might give us some information about cosmology. We often categorize the effects by micro, weak, and strong lensing types. This paper focuses on the strong gravitational lensing, especially the double-source plane lens systems. It was recently recognized that the double lensing events can provide us wi…
We discuss how small-scale density perturbations on the Fresnel scale affect amplitudes and phases of gravitational waves that are magnified by gravitational lensing in geometric optics. We derive equations that connect the small-scale density perturbations with the amplitude and phase fluctuations to show that such perturbative wave optics effects are significantly boosted in the presence of macro model magnifications such that amplitude and phase fluctuations can easily be observed for highly magnified gravitational waves. We discuss expected signals due to microlensing by stars, dark matter…
We present the results of computational gravitational backreaction on simple models of cosmic string loops. These results give us insight into the general behavior of cusps and kinks on loops, in addition to other features of evolution. Kinks are rounded off via an asymmetric and divergent correction to the string direction. The result is that cusps emerge in the place of kinks but the resulting smooth string section has a small amount of energy. Existing cusps persist, but quickly lose strength as backreaction removes energy from the string surrounding the cusp. Both kinks and cusps have thei…
We present a pedagogical review of the weak gravitational lensing measurement process and its connection to major scientific questions such as dark matter and dark energy. Then we describe common ways of parametrizing systematic errors and understanding how they affect weak lensing measurements. Finally, we discuss several instrumental systematics and how they fit into this context, and conclude with some future perspective on how progress can be made in understanding the impact of instrumental systematics on weak lensing measurements.
For a general class of analytic f(R)-gravity theories, we discuss the weak field limit in view of gravitational lensing. Though an additional Yukawa term in the gravitational potential modifies dynamics with respect to the standard Newtonian limit of General Relativity, the motion of massless particles results unaffected thanks to suitable cancellations in the post-Newtonian limit. Thus, all the lensing observables are equal to the ones known from General Relativity. Since f(R)-gravity is claimed, among other things, to be a possible solution to overcome for the need of dark matter in virializ…
Weak-gravitational-lensing distortions to the intensity pattern of 21-cm radiation from the dark ages can be decomposed geometrically into curl and curl-free components. Lensing by primordial gravitational waves induces a curl component, while the contribution from lensing by density fluctuations is strongly suppressed. Angular fluctuations in the 21-cm background extend to very small angular scales, and measurements at different frequencies probe different shells in redshift space. There is thus a huge trove of information with which to reconstruct the curl component of the lensing field, all…
Based on a binary tree model for the self intersection of cosmic string loops containing high harmonics we estimate the number of self-intersections of the parent and daughter loops and the associated cusp production to determine the most likely number of cusp events per period on the resultant non-self intersecting loops, and provide an updated calculation for the gravitational wave signal that arrives on Earth from cusps on such loops. This is done for different numbers of cusps supported from the cosmic strings of the network, and for different harmonic distributions on the loops. We plot o…
Utilizing gravitational-wave (GW) lensing opens a new way to understand the small-scale structure of the universe. We show that, in spite of its coarse angular resolution and short duration of observation, LIGO can detect the GW lensing induced by compact structures, in particular by compact dark matter (DM) or primordial black holes of $10 - 10^5 \, M_\odot$, which remain interesting DM candidates. The lensing is detected through GW frequency chirping, creating the natural and rapid change of lensing patterns: \emph{frequency-dependent amplification and modulation} of GW waveforms. As a highe…
The equation in the title describes the number of bright images of a point source under lensing by an elliptic object with isothermal density. We prove that this equation has at most 6 solutions. Any number of solutions from 1 to 6 can actually occur.
