Evidence for lensing of gravitational waves from LIGO-Virgo data

Recently, the LIGO-Virgo Collaboration (LVC) has concluded there is no evidence for lensed gravitational waves (GWs) in the first half of the O3 run [LIGO Scientific Collaboration and Virgo Collaboration, Search for lensing signatures in the gravitational-wave observations from the first half of LIGO-Virgo’s third observing run], claiming “We find the observation of lensed events to be unlikely, with the fractional rate at $\mu >2$ being $3.3\times {10}^{-4}$.” While we agree that the chance of an individual GW event being lensed at $\mu >2$ is smaller than $<{10}^{-3}$, the number of observed events depends on the product of this small probability times the rate of mergers at high redshift. Observational constraints from the stochastic GW background indicate that the rate of conventional mass binary black hole (BBH) mergers ($8<M/M_{\odot }<15$) in the redshift range $1<z<2$ could be as high as O(${10}^7$) events per year, more than sufficient to compensate for the intrinsically low probability of lensing. To reach the LVC trigger threshold, these events require high magnification but would still produce up to 10–30 LVC observable events per year. Thus, all the LVC observed ordinary stellar mass BBH mergers from this epoch must be strongly lensed with as many as ${10}^6{\mathrm{yr}}^{-1}$ events lying below the current detection threshold. By adopting a low BBH coalescence rate at high redshift, LVC implicitly assume that lensed events are insignificant and, thus, incorrectly underestimate the distances of most BBH events and correspondingly overestimate masses by factors of 2–5. Furthermore, the LVC adopted priors on time delay for ideal circularly symmetric lenses are in tension with the known distribution of observed time delays of lensed quasars that require elliptical potentials with a broad spread of time delays. Pairs of events like GW190421_213856GW190910_112807 and GW190424_180648GW190910_112807, which are directly assigned a probability of zero by LVC, should instead be considered as candidate lensed BBH pairs, since their separation in time is typical of lensed quasars. Replacing the LVC model prior for the time delay distribution with the empirical quasar-based distribution reverses the LVC conclusions and says that a significant fraction of BBH pairs identified by LVC are viable multiply lensed events, including quadruple systems.

Figure 1

Observed distribution of time delays of lensed QSOs (orange dashed line) from Refs. [12, 13], compared with the prediction from one NFW lens with ellipticity $e=0.3$, mass ${10}^{13}{M}_{\odot }$, located at $z_l=0.4$ and for a background source at $z_s=2$ (dark blue dotted line). Only pairs of lensed images where the minimum magnification in one of the images is larger than 5 are considered. The red solid line shows the distribution of time intervals for the LVC lensed candidate pairs in Ref. [1]. This distribution peaks at time delays similar to the case of the observed lensed QSOs. The two blue dots in the bottom right part of the plot show the time delay of the two pairs of events which were directly excluded by LVC on the (erroneous) basis that they are not consistent with being lensed owing to their large time delay.

Figure 2

Scaling of the LIGO-Virgo prior ratio on time delays, ${\mathcal{R}}^{\mathrm{gal}}$, with the time separation between GW events. The circles show the values tabulated in Table 3 in Ref. [1]. The two diamond-shape yellow symbols at $\mathrm{\Delta }T>100$ days have no values in Table 3 of Ref. [1], and for presentation purposes we assign them a value of ${\log }_{10}{\mathcal{R}}^{\mathrm{gal}}=-2.75$. The dotted lines and dashed line show the scaling ${\mathcal{R}}^{\mathrm{gal}}\propto \mathrm{\Delta }{T}^{\alpha }$ for three values of $\alpha =-1,-3,-0.5$, with $\alpha =-1$ reproducing the LIGO-Virgo prior for time delays sorter than $\approx 1$ month, while $\alpha =-3$ better describes the prior for time delays longer than $\approx 1$ month. Overall, the LIGO-Virgo prior can be described with an exponent $\alpha =-2$. This should be compared with the value $\alpha =-0.5$, that reproduces well the observed time delay distribution for lensed QSOs as shown in Fig. 1.

Figure 3

Histogram of inferred black hole masses published by the LIGO/Virgo Collaboration vs expectations from the BDS model. The blue bars show the histogram of published masses. The solid dark blue curve shows the expected number of not-lensed events from the BDS model in one year of LIGO/Virgo observations at O3 sensitivity. In this case the inferred mass is unbiased, and the distribution of masses follows Eq. (3). The solid red line shows the corresponding number of events that are being lensed from the same model and for the same sensitivity and observation period. In this case, the inferred mass is biased due to lensing being ignored (the inferred redshift is smaller than the true one, and this bias is compensated by a larger inferred mass). The dotted, dashed, and dot-dashed lines correspond to the number of events from gen11, gen12, and gen22, respectively. For clarity, these curves are not shown for the not-lensed case.