Arrival Time Differences Between Gravitational Waves and Electromagnetic Signals Due to Gravitational Lensing

Abstract

In this study, we demonstrate that general relativity predicts arrival time differences between gravitational wave (GW) and electromagnetic (EM) signals caused by the wave effects in gravitational lensing. The GW signals can arrive earlier than the EM signals in some cases if the GW/EM signals have passed through a lens, even if both signals were emitted simultaneously by a source. GW wavelengths are much larger than EM wavelengths; therefore, the propagation of the GWs does not follow the laws of geometrical optics, including the Shapiro time delay, if the lens mass is less than approximately 10 5 M ⊙ (f /Hz) −1 , where f is the GW frequency. The arrival time difference can reach ∼ 0.1 s (f /Hz) −1 if the signals have passed by a lens of mass ∼ 8000M ⊙ (f /Hz) −1 with the impact parameter smaller than the Einstein radius; therefore, it is more prominent for lower GW frequencies. For example, when a distant super massive black hole binary (SMBHB) in a galactic center is lensed by an intervening galaxy, the time lag becomes of the order of 10 days. Future pulsar timing arrays including SKA (the Square Kilometre Array) and X-ray detectors may detect several time lags by measuring the orbital phase differences between the GW/EM signals in the SMBHBs. Gravitational lensing imprints a characteristic modulation on a chirp waveform; therefore, we can deduce whether a measured arrival time lag arises from intrinsic source properties or gravitational lensing. Determination of arrival time differences would be extremely useful in multimessenger observations and tests of general relativity.

What if LIGO’s gravitational wave detections are strongly lensed by massive galaxy clusters?

“…The GW detections have stimulated a flurry of articles on strong lensing of GWs, discussing the effect of lens magnification on the detectability of GWs (Dai et al 2017), forecast event rates including the effects of strong lensing by galaxies (Ng et al 2017), relative arrival times of GW and electromagnetic (EM) signals (Takahashi 2017), prospects for measuring the speed of GWs (Collett & Bacon 2017;Fan et al 2017), and the impact of strong lensing on cosmography (Baker & Trodden 2017;Liao et al 2017). In this article, we investigate the probability that one or more of the GW sources detected to date was strongly lensed by a massive galaxy cluster.…”

The Legacy of Einstein’s Eclipse, Gravitational Lensing

“…For instance, Ryuichi Takahashi reported that the transition to wave optics occurs, if the lens mass is less than approximately 10 5 M⊙(f/Hz) −1 , where f is the GW frequency. The GW is predicted to be early by typically ~1 ms (f/100 Hz) −1 for ground-based detectors, ~2 min (f/mHz) −1 for space-based interferometers, and ~4 months (f/10−8 Hz) −1 for pulsar timing arrays [183]; similar estimates are given in [184].…”

High accuracy on H0 constraints from gravitational wave lensing events

“…In that case, with, e.g., 10 observations, we could have a much better uncertainty on H 0 , comparable with present values from cosmic distance ladder. In this case, though, we need to consider that the probability of detection is quite low [28] and therefore it will be hard to collect an high number of observations.…”

“…The boundary between the two approximations is defined, among others, in [28]: geometrical optics approximation breaks when…”

“…It was found that a significant ATD is expected when considering a super massive binary black hole (SMBBH) emitting both EM radiation and GWs, with a very small frequency (f ∼ 10 −8 Hz), lensed by a galaxy with a mass M lens ∼ 10 11 M . Although the probability to observe such event(s) is of the order of 10 −3 − 10 −4 [28], it has been shown that they can modify with a clear signature the waveform of a detected GW. Thus, it would be interesting to investigate which kind of additional information, on the cosmological side, one could retain from at least one event.…”