The importance of Prior-itizing

The quest for lensed gravitational waves

We are all subject to personal biases, even when we try not to be biased. In Science, a popular tool to select among a variety of models is inspired on Bayesian theory. According to this theory, the most likely model is the one that offers the best description to the current data (i.e fits the data), and given by its likelihood, but also is consistent with previous knowledge on some of the parameters that are being fitted. Previous information is accounted for by a term known as the prior. Good priors result in fairs results. Biased priors often lead to the wrong conclusion.

In a recent paper, the LIGO-Virgo collaboration (LVC) studies the possibility that gravitational waves (GW) are being strongly lensed. If lensing is taking place, a fundamental prediction from gravitational lensing theory is that multiple occurrences of the same GW should take place, with a time separation between them of typically days to months. LVC presents a series of candidates which show high consistency with the lensing hypothesis, or in the Bayesian terminology, have a good likelihood of being pairs of images of the same GW. All these pairs are later discarded as possible lensed images based on the prior term, that heavily penalizes pairs of GWs separated by more than a few weeks. They finally concluded that lensing of GWs is unlikely, based on their final score.

Distribution of time delays of observed lensed quasars (orange) and gravitational waves (red)

This is however a conclusion that is obtained after adopting potentially bad priors. The LVC does not provide sufficient information regarding their prior, but in a recent study we show how the prior used by LVC is in tension with observations of time delays from real gravitationally lensed pairs of images. In fact, the distribution of time delays between the pairs of gravitational waves which LVC found to be good candidates to be strongly lensed, is consistent with the known distribution of time delays from quasars and from analytical models.

LVC finds that approximately half the events published in the O3 catalog can form pairs of lensed GW events (that is, there is another GW which shows high consistency in terms of GW parameters and sky localization, as predicted by lensing). As shown in the image accompanying this post, the separation in time between the two GWs forming the pair (solid red curve) is consistent with the known distribution of time delays (dashed orange curve). In the LVC analysis, pairs of GWs with time separations of approx 4 months are directly assigned a probability of zero, contradicting real observations where such time delays are possible. LVC concludes that none of their candidates to be pairs of gravitationally lensed GWs favours the lensing hypothesis, but this conclusion is biased by the adoption of a prior that intrinsically negates the possibility that realistic time delays are possible.

The use of a better (unbiased) prior does not necessarily mean that the opposite conclusion is true (ie. that lensing is favored), but it begs the question of whether the conclusions from LVC could have been reversed.

If you want to find more about this discussion you can check our latest work, or our earlier work where we present a model that predicts that gravitational lensing of GWs has already been observed.

Is LIGO really seeing Mass Gap events?

The recent publication by LIGO of two events (GW190412 and GW190814) with high mass ratios, and with one of the masses close to the mass gap (that is, a mass between 3 and 5 solar masses which are difficult to explain with standard models) has created an intense debate on the nature of these objects. If confirmed, the implications of these observations are important since they can give us information about the equation of state of neutron stars (where one can study exotic forms of matter, such as axions or hyperons), or reveal a new type of black hole, including a leading dark matter candidate, primordial black holes.

However, a simpler alternative could be that these events are strongly lensed. Gravitational lensing can amplify the signal of observed gravitational waves, allowing their observation from much farther distances (and hence much larger volumes). In earlier work, we showed that if the rate of mergers (that produce gravitational waves) at redshift z>1 is sufficently high, observation of these distant events by LIGO is not only possible, but unavoidable. On fact, for rates larger than a few times 10^4 mergers per year and Gpc^3, lensed events will dominate over not-lensed events, in a similar fashion as lensed gravitational lensed IR galaxies dominated over not-lensed IR galaxies in the bright end of Herschel observations.

The left plot shows the prediction from our lensing model (colored circles) compared with the observations (squares and diamonds with error bars). All events concentrate around two locus regions. BBH and NSBH. Note how observations match perfectly the prediction.


Lensed gravitational waves get stretched due to cosmic expansion as they travel from their originating source to the detector. The farther the source is, the lager the stretch. The stretch is proportional to (1+z), where z is the redshift if the source. Higher z translates into gravitational waves that, when observed, appear as having a longer wavelength. If the gravitational wave is being magnified by strong lensing, and this magnification goes unnoticed (there is no way a priori to know if a gravitational wave is being magnified), the longer wavelength will be missinterpreted as being due to a larger mass of the two compact objects that ar causing the gravitational wave. For instance, if a neutron star with a mass of 1.3 solar masses is merging with a black hole of 12 solar masses (these are the typical masses found in our Galaxy for these objects) at redhift z=1, the observed gravitational wave will appear identical as the one from a much closer merger (z =0) with a neutron star of 2.6 solar masses and a blackhole with 24 solar masses. This example is not arbitrary since it was chosen to match the observed masses of the latest published gravitational wave event , GW190814, interpreted as being a local event (z=0), but which based on our interpretation could be also a lensed event at z=1 or z>1.

The lensing model interpretauon makes a series of interesting predictions, but among these, it is interesting to pay attention to the predicted mass ratio for binary black holes and neutron star black hole mergers. As shown in the figure illustrating this blog, lensing predicts these type of events will appear in the M1-M2 plane in two well defined regions. Interestingly, all observed data points so far agree remarkably well with this prediction. The two possible mass gap events are marked with a big yellow circle and follow well the predicted locus for lensed NSBH events. If confirmed, our lensing model would offer a simple solution to the mass gap problem, and would imply a much higher rate of events at z>1 that previously thought. 

You can see our full work below. A Fun Fact about this paper is that it was put “On Hold” by arxiv moderators after being submitted to arxiv on June 19 (Friday). After requesting an explanation from arxiv, none was given. At the time of our original submission, we where unaware of the upcoming publication, by the LIGO team, of the GW190814 “Mass Gap” event. The following week, in June 24 we saw in arxiv the LIGO paper with the values of M1 and M2 for GW190814. The same day, the “Hold” on our paper was lifted and we where able to just add the new data point to our figure (see point marked GW190814 in the figure above), before it appeared on arxiv the following day (June 25). I do understand (and support) the need for moderation in arxiv, but this process is far from transparent. The lack of communication and explanation of why papers are being put on hold, inhebitably leads to one suspect foul play, which is something that should be avoided at all cost, specially in portals such as arxiv, that makes research freely available. 


Link to paper

Click to access 2006.13219.pdf