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 waves (GWs) are subject to gravitational lensing in the same way as electromagnetic radiation. However, to date, no unequivocal observation of a lensed GW transient has been reported. Independently, GW observatories continue to search for the stochastic GW signal which is produced by many transient events at high redshift. We exploit a surprising connection between the lensing of individual transients and limits to the background radiation produced by the unresolved population of binary back hole mergers: we show that it constrains the fraction of individually resolvable lensed b…
Strong-lensing effects in high-redshift massive black-hole binary population inference
University of Chinese Academy of Sciences（中国科学院大学）；Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）
We show how LIGO is expected to detect coalescing binary black holes at $z>1$, that are lensed by the intervening galaxy population. Gravitational magnification, $μ$, strengthens gravitational wave signals by $\sqrtμ$, without altering their frequencies, which if unrecognised leads to an underestimate of the event redshift and hence an overestimate of the binary mass. High magnifications can be reached for coalescing binaries because the region of intense gravitational wave emission during coalescence is so small ($\sim$100km), permitting very close projections between lensing caustics and gra…
Wave effects in lensing form a rich phenomenon at the intersection of classical caustic singularities and quantum interference, yet are notoriously difficult to model. A large number of recently observed pulsars and fast radio bursts in radio astronomy and the prospected increase in sensitivity of gravitational wave detectors suggest that wave effects will likely be observed in the near future. The interference fringes are sensitive to physical parameters which cannot be inferred from geometric optics. In particular, for multi-plane lensing, the pattern depends on the redshifts of the lens pla…
Strong lensing cosmography using binary-black-hole mergers: Prospects for the near future
International Centre for Theoretical Science, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）；Department of Physics, The Chinese University of Hong Kong（香港中文大学物理系）；Department of Physics, University of California at Santa Barbara（加州大学圣塔芭芭拉分校物理系）
Gravitational wave (GW) transients from binary neutron star (BNS) coalescences can, in principle, be subject to gravitational lensing thereby increasing the amplitude and signal-to-noise ratio. We estimate the rate of lensed BNS events resolvable by LIGO and Virgo and find that it is constrained by the current non-detection of a stochastic GW background. Following closely the formalism we developed (10.1103/PhysRevLett.125.141102) in the context of binary black hole lensing, we show that at current sensitivities the fraction of BNS coalescences with lensing magnifications $μ> 1.02$ is less tha…
We search for signatures of gravitational lensing in the binary black hole events detected by Advanced LIGO and Virgo during their first two observational runs. In particular, we look for three effects: 1) evidence of lensing magnification in the individual signals due to galaxy lenses, 2) evidence of multiple images due to strong lensing by galaxies, 3) evidence of wave optics effects due to point-mass lenses. We find no compelling evidence of any of these signatures in the observed gravitational wave signals. However, as the sensitivities of gravitational wave detectors improve in the future…
We present a method to include lensing selection effects due to the finite horizon of a given detector when studying lensing of gravitational wave (GW) sources. When selection effects are included, the mean of the magnification distribution is shifted from one to higher values for sufficiently high-redshift sources. This introduces an irreducible (multiplicative) bias on the luminosity distance reconstruction, in addition to the typical source of uncertainty in the distance determination. We apply this method to study lensing of GWs emitted by massive black hole binary mergers at high redshift…
We consider binary black holes (BBHs) in a hierarchical triple system where a more compact, less-massive binary is emitting detectable gravitational waves (GWs), and the tertiary is a supermassive BH at the center of a nuclear star cluster. As previous works have shown, the orbital motion of the outer binary can generate a detectable relativistic Doppler boost of the GWs emitted by the orbiting inner binary. We show here that for outer-binary orbits with a period of order one year, there can be a non-negligible probability for repeated gravitational lensing of the GWs emitted by the inner bina…
We discuss image formation in gravitational lensing systems using wave optics. Applying the Fresnel-Kirchhoff diffraction formula to waves scattered by a gravitational potential of a lens object, we demonstrate how images of source objects are obtained directly from wave functions without using a lens equation for gravitational lensing.
Gravitational waves (GWs) may be magnified or de-magnified due to lensing. This phenomenon will bias the distance estimation based on the matched filtering technique. Via the multi-sphere ray-tracing technique, we study the GW magnification effect and selection effect with particular attention to the stellar-mass binary black holes (BBHs). We find that, for the observed luminosity distance $\lesssim 3~\mathrm{Gpc}$, which is the aLIGO/Virgo observational horizon limit, the average magnification keeps as unity, namely unbiased estimation, with the relative distance uncertainty $σ(\hat{d})/\hat{…
We investigate binary lenses with $1/r^n$ potentials in the asymmetric case with two lenses with different indexes $n$ and $m$. These kinds of potentials have been widely used in several contexts, ranging from galaxies with halos described by different power laws to lensing by wormholes or exotic matter. In this paper, we present a complete atlas of critical curves and caustics for mixed binaries, starting from the equal-strength case, and then exploring unequal-strength systems. We also calculate the transitions between all different topology regimes. Finally we find some useful analytic appr…
Advanced LIGO and Advanced Virgo could observe the first lensed gravitational wave sources in the coming years, while the future Einstein Telescope could observe hundreds of lensed events. It is, therefore, crucial to develop methodologies to distinguish between lensed from unlensed gravitational-wave observations. A lensed signal not identified as such will lead to biases during the interpretation of the source. In particular, sources will appear to have intrinsically higher masses. No robust method currently exists to distinguish between the magnification bias caused by lensing and intrinsic…
We investigate the detectability of gravitational waves (GWs) lensed by a system that consists of binary black holes as lenses using time-domain numerical simulations. The gravitational lensing potential of this system is no longer static but evolves with time. When GWs from the source pass through the binary lens, their amplitudes can be modulated, which is similar to the phenomenon of amplitude modulation (AM) in radio communication. We find that even the frequency of the binary lens itself is too low to be detected by the LISA detection band, the sidebands in the spectrum of the lensed GWs …
The recent detection of gravitational waves indicates that stellar-mass black hole binaries are likely to be a key population of sources for forthcoming observations. With future upgrades, ground-based detectors could detect merging black hole binaries out to cosmological distances. Gravitational wave bursts from high redshifts ($z \gtrsim 1$) can be strongly magnified by gravitational lensing due to intervening galaxies along the line of sight. In the absence of electromagnetic counterparts, the mergers' intrinsic mass scale and redshift are degenerate with the unknown magnification factor $μ…
We consider the possible effects of gravitational lensing by globular clusters on gravitational waves from asymmetric neutron stars in our galaxy. In the lensing of gravitational waves, the long wavelength, compared with the usual case of optical lensing, can lead to the geometrical optics approximation being invalid, in which case a wave optical solution is necessary. In general, wave optical solutions can only be obtained numerically. We describe a computational method that is particularly well suited to numerical wave optics. This method enables us to compare the properties of several lens …
Self-lensing of gravitational waves from binary extreme-mass-ratio inspirals: Extended caustics and paired flashes
Peking University（北京大学）；Beijing Normal University（北京师范大学）；Tsung-Dao Lee Institute, Shanghai Jiao-Tong University（上海交大李政道研究所）；School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院）
We analyzed the influence of static gravitational field on the vacuum and proposed the concept of inhomogeneous vacuum. According to the corresponding Fermat's principle in the general relativity, we derived a graded refractive index of vacuum in a static gravitational field. We found that the light deflection in a gravitational field can be calculated correctly with the use of this refractive index and therefore the gravitational lensing can be treated conveniently with the optical method. For illustration, we simulated the imaging of gravitational lensing, figured out the time delay between …
The measurement of the Shapiro time delay in binary pulsar systems with highly inclined orbit can be affected both by the motion of the pulsar's companion because of the finite time it takes a photon to cross the binary, and by the gravitational light bending if the orbit is sufficiently edge-on relative to the line of sight. Here we calculate the effect of retardation due to the companion's motion on various time delays in pulsar binaries, including the Shaipro delay, the geometric lensing delay, and the lens-induced delays associated with the pulsar rotation. Our results can be applied to sy…
Supermassive black hole (BH) binaries are thought to produce self-lensing flares (SLF) when the two BHs are aligned with the line-of-sight. If the binary orbit is observed nearly edge-on, we find a distinct feature in the light curve imprinted by the BH shadow from the lensed BH. We study this feature by ray-tracing in a binary model and predict that 1\% of the current binary candidates could show this feature. Our BH tomography method proposed here could make it possible to extract BH shadows that are spatially unresolvable by high-resolution VLBI.
Although most gravitational wave events are claimed to be mergers of unusually massive, $25-65M_\odot$, black holes, it is now clear that 20\% of all reported events comprise modest mass black holes, $5-15M_\odot$, like the stellar black holes in the Milky Way. We show that such stellar mass black hole binaries (BBH) if magnified by lensing galaxies can be detected at high redshift, 1$< $z$ <$5, with chirp masses increased by $1+z$, accounting for the majority of apparently high mass BBH events. This simple lensing explanation is manifested by the evident bimodality of BBH chirp masses now vis…
We present explicit expressions for the optical scalars and the deection angle in terms of the energymomentum tensor components of matter distributions. Our work generalizes standard references in the literature where normally stringent assumptions are made on the sources.
Binary black holes have been in limelight off late due to the detection of gravitational waves from coalescing compact binaries in the events GW150914 and GW151226. In this paper we study gravitational lensing by the binary black holes modeled as equal mass Majumdar-Papapetrou dihole metric and show that this system displays features that are quite unprecedented and absent in any other lensing configuration investigated so far. We restrict our attention to the light rays which move on the plane midway between the two identical black holes, which allows us to employ techniques developed for the…
The current gravitational-wave localization methods rely mainly on sources with electromagnetic counterparts. Unfortunately, a binary black hole does not emit light. Due to this, it is generally not possible to localize these objects precisely. However, strongly lensed gravitational waves, which are forecasted in this decade, could allow us to localize the binary by locating its lensed host galaxy. Identifying the correct host galaxy is challenging because there are hundreds to thousands of other lensed galaxies within the sky area spanned by the gravitational-wave observation. However, we can…
In a previous article concerning image distortion in non-perturbative gravitational lensing theory we described how to introduce shape and distortion parameters for small sources. We also showed how they could be expressed in terms of the scalar products of the geodesic deviation vectors of the source's pencil of rays in the past lightcone of an observer. In the present work we give an alternative approach to the description of the shape and distortion parameters and their evolution along the null geodesic from the source to the observer, but now in terms of the optical scalars (the convergenc…
Based on the rate of gravitational-wave (GW) detections by Advanced LIGO and Virgo, we expect these detectors to observe hundreds of binary black hole mergers as they achieve their design sensitivities (within a few years). A small fraction of them can undergo strong gravitational lensing by intervening galaxies, resulting in multiple images of the same signal. To a very good approximation, the lensing magnifies/de-magnifies these GW signals without affecting their frequency profiles. We develop a Bayesian inference technique to identify pairs of strongly lensed images among hundreds of binary…
The laws of geometric optics and their corrections are derived for scalar, electromagnetic, and gravitational waves propagating in generic curved spacetimes. Local peeling-type results are obtained, where different components of high-frequency fields are shown to scale with different powers of their frequencies. Additionally, finite-frequency corrections are identified for a number of conservation laws and observables. Among these observables are a field's energy and momentum densities, as well as several candidates for its corrected "propagation directions."
A new geometric method to determine the deflection of light in the equatorial plane of the Kerr solution is presented, whose optical geometry is a surface with a Finsler metric of Randers type. Applying the Gauss-Bonnet theorem to a suitable osculating Riemannian manifold, adapted from a construction by Nazım, it is shown explicitly how the two leading terms of the asymptotic deflection angle of gravitational lensing can be found in this way.
The self-lensing of a massive black hole binary (MBHB), which occurs when the two BHs are aligned close to the line of sight, is expected to produce periodic, short-duration flares. Here we study the shapes of self-lensing flares (SLFs) via general-relativistic ray tracing in a superimposed binary BH metric, in which the emission is generated by geometrically thin accretion flows around each component. The suite of models covers eccentric binary orbits, black hole spins, unequal mass binaries, and different emission model geometries. We explore the above parameter space, and report how the lig…
We present a method of calculating the strong-field gravitational lensing caused by many analytic and numerical spacetimes. We use this procedure to calculate the distortion caused by isolated black holes and by numerically evolved black hole binaries. We produce both demonstrative images illustrating details of the spatial distortion and realistic images of collections of stars taking both lensing amplification and redshift into account. On large scales the lensing from inspiraling binaries resembles that of single black holes, but on small scales the resulting images show complex and in some…
We propose the use of modulated spectra of astronomical sources due to gravitational lensing to probe Ellis wormholes. The modulation factor due to gravitational lensing by the Ellis wormhole is calculated. Within the geometrical optics approximation, the normal point mass lens and the Ellis wormhole are indistinguishable unless we know the source's unlensed luminosity. This degeneracy is resolved with the significant wave effect in the low frequency domain if we take the deviation from the geometrical optics into account. We can roughly estimate the upper bound for the number density of Ellis…
The gravitational waves (GWs) has been a topic of interest for its versatile capabilities of probing several aspects of cosmology and early Universe. Gravitational lensing enhances further the extent of this sort of waves and upgrade our understanding to a next level. Besides several similarities with optical waves, GWs are capable of passing through optically opaque celestial objects like stars, exoplanets unlike light waves and manifest a different kind of lensing effect. In this work we have explored the lensing action of compact objects on gravitational waves using numerical means. After m…
A geometrical-optics analogy for gravitational lensing using axicon-type lenses
Institut für Astronomie und Astrophysik, Eberhard Karls Universität Tübingen（蒂宾根大学天文与天体物理研究所）；Department of Optics, Faculty of Physical Sciences, Universidad Complutense de Madrid（马德里康普顿斯大学物理科学学院光学系）
Typical applications of gravitational lensing use the properties of electromagnetic or gravitational waves to infer the geometry through which those waves propagate. Nevertheless, the optical fields themselves - as opposed to their interactions with material bodies - encode very little of that geometry: It is shown here that any given configuration is compatible with a very large variety of spacetime metrics. For scalar fields in geometric optics, or observables which are not sensitive to the detailed polarization content of electromagnetic or gravitational waves, seven of the ten metric compo…
We study how the angular resolution of LISA for merging massive black-hole binaries would be improved if we observe multiple gravitational wave ``images'' due to strong gravitational lensing. The correlation between fitting parameters is reduced by the additional information of the second image which significantly reduces the error box on the sky. This improvement would be very helpful for identifying the host galaxy of a binary. The angular resolution expected with multiple detectors is also discussed.
In this geometrical approach to gravitational lensing theory, we apply the Gauss-Bonnet theorem to the optical metric of a lens, modelled as a static, spherically symmetric, perfect non-relativistic fluid, in the weak deflection limit. We find that the focusing of the light rays emerges here as a topological effect, and we introduce a new method to calculate the deflection angle from the Gaussian curvature of the optical metric. As examples, the Schwarzschild lens, the Plummer sphere and the singular isothermal sphere are discussed within this framework.
