We show that gravitational lensing can provide a direct method to probe the nature of dark energy at astrophysical scales. For lensing system as an isolated astrophysical object, we derive the dark energy contribution to gravitational potential as a repulsive power-law term, containing a generic equation of state parameter $w$. We find that it generates $w$-dependent and position-dependent modification to the conventional light orbital equation of $w=-1$. With post-Newtonian approximation, we compute its direct effect for an isolated lensing system at astrophysical scales and find that the dark energy force can deflect the path of incident light rays. We demonstrate that the dark-energy-induced deflection angle $Δα_{DE}^{}\propto M^{(1+\frac{1}{3w})}$ (with $1+\frac{1}{3w}>0$), which increases with the lensing mass $M$ and consistently approaches zero in the limit $M \to 0$. This effect is distinctive because dark energy tends to diffuse the rays and generates concave lensing effect. This is in contrast to the conventional convex lensing effect caused by both visible and dark matter. Measuring such concave lensing effect can directly probe the existence and nature of dark energy. We estimate this effect and show that the current gravitational lensing experiments are sensitive to the direct probe of dark energy at astrophysical scales. For the special case $w=-1$, our independent study favors the previous works that the cosmological constant can affect light bending, although our prediction qualitatively and quantitatively differ from the literature, including our consistent realization of $Δα_{DE}\to 0$ (under $M \to 0$) at the leading order.
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…
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…
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…
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.
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…
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.
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…
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.
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 …
The cubic Galileon model containing the lowest non-trivial order action of the full Galileon action can produce the stable late-time cosmic acceleration. This model can have a significant role in the growth of structures. The signatures of the cubic Galileon model in the structure formation can be probed by the weak lensing statistics. Weak lensing convergence statistics is one of the strongest probes to the structure formation and hence it can probe the dark energy or modified theories of gravity models. In this work, we investigate the detectability of the cubic Galileon model from the $ Λ$C…
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…
Intermediate redshifts between galaxy surveys and the cosmic microwave background (CMB) remain unexplored territory. Line intensity mapping (LIM) offers a way to probe the $z\gtrsim 1$ Universe, including the epoch of reionization and the dark ages. Via exact nulling of the lensing kernel, we show that LIM lensing, in combination with galaxy (resp., CMB) lensing, can uniquely probe the $z\gtrsim 1$ (resp., pre-reionization) Universe. However, LIM foregrounds are a key hurdle to this futuristic technique. While continuum foregrounds can be controlled by discarding modes perpendicular to the lin…
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.
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 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.
The gravitational lensing of Bullet Clusters and early-type galaxies pose serious challenges on the validity of MOND. Recently, Finslerian MOND, a generalization of MOND in the framework of Finsler gravity, has been proposed to explain the mass discrepancy problem of Bullet Cluster 1E 0657\ 558. In this paper, we check the validity of the Finslerian MOND in describing the strong gravitational lensing of early-type galaxies. The investigation on ten strong lenses of the CASTLES samples shows that there is no strong evidence for the existence of dark matter.
We develop a model of gravitational lensing in a non-uniform plasma. When a gravitating body is surrounded by a plasma, the lensing angle depends on the frequency of the electromagnetic wave, due to dispersion properties of plasma, in presence of a plasma inhomogeneity, and of a gravity. The second effect leads, even in a uniform plasma, to a difference of the gravitational photon deflection angle from the vacuum case, and to its dependence on the photon frequency. We take into account both effects, and derive the expression for the lensing angle in the case of a strongly nonuniform plasma in …
We compute time delays for gravitational lensing in a flat LambdaCDM Swiss cheese universe. We assume a primary and secondary pair of light rays are deflected by a single point mass condensation described by a Kottler metric (Schwarzschild with Lambda) embedded in an otherwise homogeneous cosmology. We find that the cosmological constant's effect on the difference in arrival times is non-linear and at most around 0.002% for a large cluster lens; however, we find differences from time delays predicted by conventional linear lensing theory that can reach ~4% for these large lenses. The differenc…
Continuous gravitational waves are analogous to monochromatic light and therefore could be used to detect wave effects like interference or diffraction. This would be possible with strongly lensed gravitational waves. This article reviews and summaries the theory of gravitational lensing in the context of gravitational waves in two different regimes: geometric optics and wave optics, for two widely used lens models such as point mass lens and Singular Isothermal Sphere (SIS). Observable effects due to wave nature of gravitational waves are discussed. As a consequence of interference GWs produc…
The Cosmic Microwave Background (CMB) radiation lensing is a promising tool to study the physics of early universe. In this work we probe the imprints of deviations from isotropy and scale invariance of primordial curvature perturbation power spectrum on CMB lensing potential and convergence. Specifically, we consider a scale-dependent hemispherical asymmetry in primordial power spectrum. We show that the CMB lensing potential and convergence and also the cross-correlation of the CMB lensing and late time galaxy convergence can probe the amplitude and the scale dependence of the dipole modulat…
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…
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 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…
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 …
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…
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 …
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…
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…
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…
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…
We introduce here a cross-correlation term between CMB lensing and galaxy intrinsic alignment, noted hereby as $ϕI$. This effect acts as a contaminant to the cross-correlation between CMB lensing and galaxy lensing. The latter cross-correlation has recently been detected for the first time, and measurements will greatly improve as the area of overlap between galaxy and CMB surveys increases and measurements of the CMB polarization become more significant. This will constitute a powerful probe for studying the structure and evolution of the universe. The magnitude of the $ϕI$ term is found to b…
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.
We propose a new method for Point Spread Function (PSF) correction in weak gravitational lensing shear analysis using an artificial image with the same ellipticity as the lensed image. This avoids the systematic error associated with the approximation in PSF correction used in previous approaches. We test the new method with simulated objects which have Gaussian or Cersic profiles smeared by a Gaussian PSF, and confirm that there is no systematic error.
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.
