It is known that geometrical optics no longer applies to the gravitational lensing if the wavelength of a propagating wave becomes comparable to or larger than the Schwarzshild radius of a lensing object. We investigate the propagation of gravitational waves in wave optics, particularly focusing on the difference between their arrival time and the arrival time of light. We argue that, contrary to the observation in the previous work, gravitational waves never arrive at an observer earlier than light when both gravitational waves and light are emitted from a same source simultaneously.
It is of fundamental importance to know the mass of gravitons. A simple method for constraining the graviton mass is to compare the arrival time of light and that of gravitational waves provided that both waves are simultaneously emitted from the same source. To date, from observations of gravitational waves by the LIGO, the upper bound on the graviton mass $m_g$ is given by $m_g\lesssim 5.0 \times 10^{-23}$eV. However, when we compare the arrival time of light and gravitational waves, lensing effects could be important for some cases. Moreover, in many cases, the wavelength of gravitational w…
The standard siren approach of gravitational wave cosmology appeals to the direct luminosity distance estimation through the waveform signals from inspiralling double compact binaries, especially those with electromagnetic counterparts providing redshifts. It is limited by the calibration uncertainties in strain amplitude and relies on the fine details of the waveform. The Einstein Telescope is expected to produce $10^4-10^5$ gravitational wave detections per year, $50-100$ of which will be lensed. Here we report a waveform-independent strategy to achieve precise cosmography by combining the a…
We propose a new method to test the cosmic distance duality relation using the strongly lensed gravitational waves. The simultaneous observations of image positions, the relative time delay between different images, the redshift measurements of lens and source, together with the mass modelling of the lens galaxy, provide the angular diameter distance to the gravitational wave source. On the other hand, from the observation of gravitational wave signals the luminosity distance to the source can be obtained. This is to our knowledge the first method to simultaneously measure both the angular dia…
It has been recently shown that quadruply lensed gravitational-wave (GW) events due to coalescing binaries can be localized to one or just a few galaxies, even in the absence of an electromagnetic counterpart. We discuss how this can be used to extract information on modified GW propagation, which is a crucial signature of modifications of gravity at cosmological scales. We show that, using quadruply lensed systems, it is possible to constrain the parameter $Ξ_0$ that characterizes modified GW propagation, without the need of imposing a prior on $H_0$. A LIGO/Virgo/Kagra network at target sens…
The rapid accumulation of observed Fast Radio Bursts (FRBs) originating from cosmological distances makes it likely that some will be strongly lensed by intervening matter along the line of sight. Detection of lensed FRB repeaters, which account for a noteworthy fraction of the total population, will allow not only an accurate measurement of the lensing time delay, but also follow-up high-resolution observations to pinpoint the location of the lensed images. Recent works proposed to use such strongly-lensed FRBs to derive constraints on the current expansion rate $ H_{0} $ as well as on cosmic…
The cosmic distance relation (DDR) associates the angular diameters distance ($D_A$) and luminosity distance ($D_L$) by a simple formula, i.e., $D_L=(1+z)^2D_A$. The strongly lensed gravitational waves (GWs) provide a unique way to measure $D_A$ and $D_L$ simultaneously to the GW source, hence can be used as probes to test DDR. In this paper, we prospect the use of strongly lensed GW events from the future Einstein Telescope to test DDR. We write the possible deviation of DDR as $(1+z)^2D_A/D_L=η(z)$, and consider two different parametrizations of $η(z)$, namely, $η_1(z)=1+η_0 z$ and $η_2(z)=1…
In this paper, we discuss the possibility of using strongly lensed gravitational waves (GWs) and their electromagnetic (EM) counterparts as powerful cosmic rulers. In the EM domain, it has been suggested that joint observations of the time delay ($Δτ$) between lensed quasar images and the velocity dispersion ($σ$) of the lensing galaxy (i.e., the combination $Δτ/σ^{2}$) are able to constrain the cosmological parameters more strongly than $Δτ$ or $σ^{2}$ separately. Here, for the first time, we propose that this $Δτ/σ^{2}$ method can be applied to the strongly lensed systems observed in both GW…
General relativity (GR) predicts concordant trajectories for photons and gravitational waves (GW). We propose a new multi-messenger avenue (GW-CMB-CMB) to prove this aspect of fundamental physics by cross-correlating the GW signal of astrophysical origin with the lensing field derived from the cosmic microwave background (CMB). This new window will allow robust measurement of the prediction from GR with high signal-to-noise and will be able to unveil the true nature of gravity using the GW sources detected by missions such as the Laser Interferometer Space Antenna (LISA), Einstein Telescope an…
Modified gravitational wave propagation is a smoking gun of modifications of gravity at cosmological scales, and can be the most promising observable for testing such theories. The observation of gravitational waves (GW) in recent years has allowed us to start probing this effect, and here we briefly review two promising ways of testing it. We will show that, already with the current network of detectors, it is possible to reach an interesting accuracy in the estimation of the $Ξ_0$ parameter (that characterizes modified gravitational wave propagation, with $Ξ_{0, {\rm GR}} = 1$) and with next…
We investigate the potential of high-energy astrophysical events, from which both massless and massive signals are detected, to probe fundamental physics. In particular, we consider how strong gravitational lensing can induce time delays in multi-messenger signals originating from the same source. Obvious messenger examples are massless photons and gravitational waves, and massive neutrinos, although more exotic applications can also be imagined, such as to massive gravitons or axions. The different propagation times of the massive and massless particles can, in principle, place bounds on the …
Forecasting Constraints on Cosmology and Modified Gravitational-wave Propagation by Combining Strongly Lensed Gravitational Waves and Galaxy Surveys
International Center for Theoretical Physics Asia-Pacific, University of Chinese Academy of Sciences（国际理论物理中心亚太分部，中国科学院大学）；Taiji Laboratory for Gravitational Wave Universe, University of Chinese Academy of Sciences（太极引力波宇宙实验室，中国科学院大学）
Due to the expansion of our Universe, the redshift of distant objects changes with time. Although the amplitude of this redshift drift is small, it will be measurable with a decade-long campaigns on the next generation of telescopes. Here we present an alternative view of the redshift drift which captures the expansion of the universe in single epoch observations of the multiple images of gravitationally lensed sources. Considering a sufficiently massive lens, with an associated time delay of order decades, simultaneous photons arriving at a detector would have been emitted decades earlier in …
Since their serendipitous discovery, Fast Radio Bursts (FRBs) have garnered a great deal of attention from both observers and theorists. A new class of radio telescopes with wide fields of view have enabled a rapid accumulation of FRB observations, confirming that FRBs originate from cosmological distances. The high occurrence rate of FRBs and the development of new instruments to observe them create opportunities for FRBs to be utilized for a host of astrophysical and cosmological studies. We focus on the rare, and as yet undetected, subset of FRBs that undergo repeated bursts and are strongl…
Wave-Optics Imprints of Dark Matter Subhalos on Strongly Lensed Gravitational Waves
GRAPPA Institute, University of Amsterdam（阿姆斯特丹大学 GRAPPA 研究所）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学 Kavli 宇宙物理与数学研究所（WPI））
We use simulated strongly lensed gravitational wave events from the Einstein Telescope to demonstrate how the luminosity and angular diameter distances, $d_L(z)$ and $d_A(z)$ respectively, can be combined to test in a model independent manner for deviations from the cosmic distance duality relation and the standard cosmological model. In particular, we use two machine learning approaches, the Genetic Algorithms and Gaussian Processes, to reconstruct the mock data and we show that both approaches are capable of correctly recovering the underlying fiducial model and can provide percent-level con…
Uncovering faint lensed gravitational-wave signals and reprioritizing their follow-up analysis using galaxy lensing forecasts with detected counterparts
Similarly to light, gravitational waves can be gravitationally lensed as they propagate near massive astrophysical objects such as galaxies, stars, or black holes. In recent years, forecasts have suggested a reasonable chance of strong gravitational-wave lensing detections with the LIGO-Virgo-KAGRA detector network at design sensitivity. As a consequence, methods to analyse lensed detections have seen rapid development. However, the impact of higher-order modes on the lensing analyses is still under investigation. In this work, we show that the presence of higher-order modes enables the identi…
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…
Similar to light, gravitational waves (GWs) can be lensed. Such lensing phenomena can magnify the waves, create multiple images observable as repeated events, and superpose several waveforms together, inducing potentially discernible patterns on the waves. In particular, when the lens is small, $\lesssim 10^5 M_\odot$, it can produce lensed images with time delays shorter than the typical gravitational-wave signal length that conspire together to form ``beating patterns''. We present a proof-of-principle study utilizing deep learning for identification of such a lensing signature. We bring the…
Gravitational lensing of gravitational waves: universal characteristics of strongly lensed memory waveforms
College of Physics, Henan University of Technology（河南工业大学物理学院）；Department of Applied Physics, College of Science, China Agricultural University（中国农业大学理学院应用物理系）；School of Physics and Technology, Wuhan University（武汉大学物理科学与技术学院）
We present a new idea of testing the validity of the Friedman-Lema\^ıtre-Robertson-Walker metric, through the multiple measurements of galactic-scale strong gravitational lensing systems with type Ia supernovae in the role of sources. Each individual lensing system will provide a model-independent measurement of the spatial curvature parameter referring only to geometrical optics independently of the matter content of the universe. This will create a valuable opportunity to test the FLRW metric directly. Our results show that with hundreds of strongly lensed SNe Ia observed by LSST, one would …
Like light, gravitational waves can be gravitationally lensed by massive objects along their travel path. Strong lensing produces several images from the same binary coalescence and is forecasted to have a promising rate in ground-based gravitational detectors. To search for this effect in the data, one would, in principle, have to analyze all the possible combinations of the individual detected events, whose number will be ever-increasing. To keep up with the rising computational cost, we propose a fast and precise methodology to analyze strongly-lensed gravitational wave events. The method w…
We prose a new measurement strategy of the estimation of the lens mass, as well as the actual amplification, of the lens through the modulation patten of lensed gravitational wave signals alone. This can be done by measuring the frequency and amplitude ratio of peaks and valleys of the modulation pattern of CBC signals, known as "beat", which is a time domain phenomenon of strong lens effect.
Gravitational waves, like light, can be gravitationally lensed by massive astrophysical objects such as galaxies and galaxy clusters. Strong gravitational-wave lensing, forecasted at a reasonable rate in ground-based gravitational-wave detectors such as Advanced LIGO, Advanced Virgo, and KAGRA, produces multiple images separated in time by minutes to months. These images appear as repeated events in the detectors: gravitational-wave pairs, triplets, or quadruplets with identical frequency evolution originating from the same sky location. To search for these images, we need to, in principle, an…
The expected event rate of lensed gravitational wave sources scales with the merger rate at redshift $z\geq 1$, where the optical depth for lensing is high. It is commonly assumed that the merger rate of the astrophysical compact objects is closely connected with the star formation rate, which peaks around redshift $z\sim 2$. However, a major source of uncertainty is the delay time between the formation and merger of compact objects. We explore the impact of delay time on the lensing event rate. We show that as the delay time increases, the peak of the merger rate of gravitational wave sources…
Alternative theories of gravity predict up to six distinct polarization modes for gravitational waves. Strong gravitational lensing of gravitational waves allows us to probe the polarization content of these signals by effectively increasing the number of observations from the same astrophysical source. The lensing time delays due to the multiple observed lensed images combined with the rotation of the Earth allows for effective non-collocated interferometers to be defined with respect to the source location and hence probe the alternative polarization amplitudes with more observations. To mea…
The diffraction patterns of lensed gravitational waves encode information about their propagation speeds. If gravitons have mass, the dispersion relation and speed of gravitational waves will be affected in a frequency-dependent manner, which would leave traces in the diffraction pattern if the waves are lensed. In this paper, we study how the alternative dispersion relation induced by massive gravitons affects gravitational waves lensed by point-mass lenses, such as intermediate-mass black holes. We find that the waveform morphology of lensed dispersive gravitational waves depends on the grav…
We discuss the prospects of gravitational lensing of gravitational waves (GWs) coming from core-collapse supernovae (CCSN). As the CCSN GW signal can only be detected from within our own Galaxy and the local group by current and upcoming ground-based GW detectors, we focus on microlensing. We introduce a new technique based on analysis of the power spectrum and association of peaks of the power spectrum with the peaks of the amplification factor to identify lensed signals. We validate our method by applying it on the CCSN-like mock signals lensed by a point mass lens. We find that the lensed a…
Strong gravitational lensing is a gravitational wave (GW) propagation effect that influences the inferred GW source parameters and the cosmological environment. Identifying strongly-lensed GW images is challenging as waveform amplitude magnification is degenerate with a shift in the source intrinsic mass and redshift. However, even in the geometric-optics limit, type II strongly-lensed images cannot be fully matched by type I (or unlensed) waveform templates, especially with large binary mass ratios and orbital inclination angles. We propose to use this mismatch to distinguish individual type …
Gravitational waves are theorized to be gravitationally lensed when they propagate near massive objects. Such lensing effects cause potentially detectable repeated gravitational wave patterns in ground- and space-based gravitational wave detectors. These effects are difficult to discriminate when the lens is small and the repeated patterns superpose. Traditionally, matched filtering techniques are used to identify gravitational-wave signals, but we instead aim to utilize machine learning techniques to achieve this. In this work, we implement supervised machine learning classifiers (support vec…
We study the prospects of detecting continuous gravitational waves (CGWs) from spinning neutron stars (NSs), gravitationally lensed by the galactic supermassive black hole. Assuming various astrophysically motivated spatial distributions of galactic NSs, we find that CGW signals from a few ($\sim 0-6$) neutron stars should be strongly lensed. Lensing will produce two copies of the signal (with time delays of seconds to minutes) that will interfere with each other. The relative motion of the NS with respect to the lensing optical axis will change the interference pattern, which will help us to …
The possible variation of the electromagnetic fine structure constant $α_e$ on cosmological scales arouses great interests in recent years. The strongly lensed gravitational waves and the electromagnetic counterparts could be used to test this variation. Under the assumption that the speed of photon could be modified, while the speed of GW is the same as GR predicated, and they both propagate in a flat Friedman-Robertson-Walker universe, we investigate the difference of time delays of the images and derive the upper bound of the variation of $α_e$. For a typical lensing system in the standard …
Testing Screened Modified Gravity with Strongly Lensed Gravitational Waves
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）；Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University（北京师范大学前沿天体物理研究所）
We propose a new model-independent measurement strategy for the propagation speed of gravitational waves (GWs) based on strongly lensed GWs and their electromagnetic (EM) counterparts. This can be done in two ways: by comparing arrival times of GWs and their EM counterparts and by comparing the time delays between images seen in GWs and their EM counterparts. The lensed GW-EM event is perhaps the best way to identify an EM counterpart. Conceptually, this method does not rely on any specific theory of massive gravitons or modified gravity. Its differential setting (i.e., measuring the differenc…
A small fraction of the gravitational-wave (GW) signals that will be detected by second and third generation detectors are expected to be strongly lensed by galaxies and clusters, producing multiple observable copies. While optimal Bayesian model selection methods are developed to identify lensed signals, processing tens of thousands (billions) of possible pairs of events detected with second (third) generation detectors is both computationally intensive and time consuming. To mitigate this problem, we propose to use machine learning to rapidly rule out a vast majority of candidate lensed pair…
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…
