Where do gravitational waves like GW170817 come from? Using our network of detectors, we cannot pinpoint a source, but we can make a good estimate—the amplitude of the signal tells us about the distance; the time delay between the signal arriving at different detectors, and relative amplitudes of the signal in different detectors tells us about the sky position (see the excellent video by Leo Singer below).
In this paper we look at full three-dimensional localization of gravitational-wave sources; we important a (rather cunning) technique from computer vision to construct a probability distribution for the source’s location, and then explore how well we could localise a set of simulated binary neutron stars. Knowing the source location enables lots of cool science. First, it aids direct follow-up observations with non-gravitational-wave observatories, searching for electromagnetic or neutrino counterparts. It’s especially helpful if you can cross-reference with galaxy catalogues, to find the most probable source locations (this technique was used to find the kilonova associated with GW170817). Even without finding a counterpart, knowing the most probable host galaxy helps us figure out how the source formed (have lots of stars been born recently, or are all the stars old?), and allows us measure the expansion of the Universe. Having a reliable technique to reconstruct source locations is useful!
This was a fun paper to write [bonus note]. I’m sure it will be valuable, both for showing how to perform this type of reconstruction of a multi-dimensional probability density, and for its implications for source localization and follow-up of gravitational-wave signals. I go into details of both below, first discussing our statistical model (this is a bit technical), then looking at our results for a set of binary neutron stars (which have implications for hunting for counterparts) .
Dirichlet process Gaussian mixture model
When we analyse gravitational-wave data to infer the source properties (location, masses, etc.), we map out parameter space with a set of samples: a list of points in the parameter space, with there being more around more probable locations and fewer in less probable locations. These samples encode everything about the probability distribution for the different parameters, we just need to extract it…
For our application, we want a nice smooth probability density. How do we convert a bunch of discrete samples to a smooth distribution? The simplest thing is to bin the samples. However, picking the right bin size is difficult, and becomes much harder in higher dimensions. Another popular option is to use kernel density estimation. This is better at ensuring smooth results, but you now have to worry about the size of your kernels.
Our approach is in essence to use a kernel density estimate, but to learn the size and position of the kernels (as well as the number) from the data as an extra layer of inference. The “Gaussian mixture model” part of the name refers to the kernels—we use several different Gaussians. The “Dirichlet process” part refers to how we assign their properties (their means and standard deviations). What I really like about this technique, as opposed to the usual rule-of-thumb approaches used for kernel density estimation, is that it is well justified from a theoretical point of view.
I hadn’t come across a Dirchlet process before. Section 2 of the paper is a walkthrough of how I built up an understanding of this mathematical object, and it contains lots of helpful references if you’d like to dig deeper.
In our application, you can think of the Dirichlet process as being a probability distribution for probability distributions. We want a probability distribution describing the source location. Given our samples, we infer what this looks like. We could put all the probability into one big Gaussian, or we could put it into lots of little Gaussians. The Gaussians could be wide or narrow or a mix. The Dirichlet distribution allows us to assign probabilities to each configuration of Gaussians; for example, if our samples are all in the northern hemisphere, we probably want Gaussians centred around there, rather than in the southern hemisphere.
With the resulting probability distribution for the source location, we can quickly evaluate it at a single point. This means we can rapidly produce a list of most probable source galaxies—extremely handy if you need to know where to point a telescope before a kilonova fades away (or someone else finds it).
To verify our technique works, and develop an intuition for three-dimensional localizations, we used studied a set of simulated binary neutron star signals created for the First 2 Years trilogy of papers. This data set is well studied now, it illustrates performance it what we anticipated to be the first two observing runs of the advanced detectors, which turned out to be not too far from the truth. We have previously looked at three-dimensional localizations for these signals using a super rapid approximation.
The plots below show how well we could localise the sources of our binary neutron star sources. Specifically, the plots show the size of the volume which has a 90% probability of containing the source verses the signal-to-noise ratio (the loudness) of the signal. Typically, volumes are –, which is about – Olympic swimming pools. Such a volume would contain something like – galaxies.
Localization volume as a function of signal-to-noise ratio. The top panel shows results for two-detector observations: the LIGO-Hanford and LIGO-Livingston (HL) network similar to in the first observing run, and the LIGO and Virgo (HLV) network similar to the second observing run. The bottom panel shows all observations for the HLV network including those with all three detectors which are colour coded by the fraction of the total signal-to-noise ratio from Virgo. In both panels, there are fiducial lines scaling inversely with the sixth power of the signal-to-noise ratio. Adapted from Fig. 4 of Del Pozzo et al. (2018).
Looking at the results in detail, we can learn a number of things
The localization volume is roughly inversely proportional to the sixth power of the signal-to-noise ratio [bonus note]. Loud signals are localized much better than quieter ones!
The localization dramatically improves when we have three-detector observations. The extra detector improves the sky localization, which reduces the localization volume.
To get the benefit of the extra detector, the source needs to be close enough that all the detectors could get a decent amount of the signal-to-noise ratio. In our case, Virgo is the least sensitive, and we see the the best localizations are when it has a fair share of the signal-to-noise ratio.
Considering the cases where we only have two detectors, localization volumes get bigger at a given signal-to-noise ration as the detectors get more sensitive. This is because we can detect sources at greater distances.
Putting all these bits together, I think in the future, when we have lots of detections, it would make most sense to prioritise following up the loudest signals. These are the best localised, and will also be the brightest since they are the closest, meaning there’s the greatest potential for actually finding a counterpart. As the sensitivity of the detectors improves, its only going to get more difficult to find a counterpart to a typical gravitational-wave signal, as sources will be further away and less well localized. However, having more sensitive detectors also means that we are more likely to have a really loud signal, which should be really well localized.
Left: Localization (yellow) with a network of two low-sensitivity detectors. The sky location is uncertain, but we know the source must be nearby. Right: Localization (green) with a network of three high-sensitivity detectors. We have good constraints on the source location, but it could now be at a much greater range of distances. Not to scale.
Using our localization volumes as a guide, you would only need to search one galaxy to find the true source in about 7% of cases with a three-detector network similar to at the end of our second observing run. Similarly, only ten would need to be searched in 23% of cases. It might be possible to get even better performance by considering which galaxies are most probable because they are the biggest or the most likely to produce merging binary neutron stars. This is definitely a good approach to follow.
Galaxies within the 90% credible volume of an example simulated source, colour coded by probability. The galaxies are from the GLADE Catalog; incompleteness in the plane of the Milky Way causes the missing wedge of galaxies. The true source location is marked by a cross [bonus note]. Part of Figure 5 of Del Pozzo et al. (2018).
We started writing this paper back before the first observing run of Advanced LIGO. We had a pretty complete draft on Friday 11 September 2015. We just needed to gather together a few extra numbers and polish up the figures and we’d be done! At 10:50 am on Monday 14 September 2015, we made our first detection of gravitational waves. The paper was put on hold. The pace of discoveries over the coming years meant we never quite found enough time to get it together—I’ve rewritten the introduction a dozen times. It’s extremely satisfying to have it done. This is a shame, as it meant that this study came out much later than our other three-dimensional localization study. The delay has the advantage of justifying one of my favourite acknowledgement sections.
We find that the localization volume is inversely proportional to the sixth power of the signal-to-noise ration . This is what you would expect. The localization volume depends upon the angular uncertainty on the sky , the distance to the source , and the distance uncertainty ,
The signal-to-noise ratio itself is inversely proportional to the distance to the source (sources further way are quieter. Therefore, putting everything together gives
We all know that treasure is marked by a cross. In the case of a binary neutron star merger, dense material ejected from the neutron stars will decay to heavy elements like gold and platinum, so there is definitely a lot of treasure at the source location.
On 4 January 2017, Advanced LIGO made a new detection of gravitational waves. The signal, which we call GW170104 [bonus note], came from the coalescence of two black holes, which inspiralled together (making that characteristic chirp) and then merged to form a single black hole.
On 4 January 2017, I was just getting up off the sofa when my phone buzzed. My new year’s resolution was to go for a walk every day, and I wanted to make use of the little available sunlight. However, my phone informed me that PyCBC (one or our search algorithms for signals from coalescing binaries) had identified an interesting event. I sat back down. I was on the rota to analyse interesting signals to infer their properties, and I was pretty sure that people would be eager to see results. They were. I didn’t leave the sofa for the rest of the day, bringing my new year’s resolution to a premature end.
Since 4 January, my time has been dominated by working on GW170104 (you might have noticed a lack of blog posts). Below I’ll share some of my war stories from life on the front line of gravitational-wave astronomy, and then go through some of the science we’ve learnt. (Feel free to skip straight to the science, recounting the story was more therapy for me).
Time–frequency plots for GW170104 as measured by Hanford (top) and Livingston (bottom). The signal is clearly visible as the upward sweeping chirp. The loudest frequency is something between E3 and G♯3 on a piano, and it tails off somewhere between D♯4/E♭4 and F♯4/G♭4. Part of Fig. 1 of the GW170104 Discovery Paper.
In the second observing run, the Parameter Estimation group have divided up responsibility for analysing signals into two week shifts. For each rota shift, there is an expert and a rookie. I had assumed that the first slot of 2017 would be a quiet time. The detectors were offline over the holidays, due back online on 4 January, but the instrumentalists would probably find some extra tinkering they’d want to do, so it’d probably slip a day, and then the weather would be bad, so we’d probably not collect much data anyway… I was wrong. Very wrong. The detectors came back online on time, and there was a beautifully clean detection on day one.
My partner for the rota was Aaron Zimmerman. 4 January was his first day running parameter estimation on live signals. I think I would’ve run and hidden underneath my duvet in his case (I almost did anyway, and I lived through the madness of our first detection GW150914), but he rose to the occasion. We had first results after just a few hours, and managed to send out a preliminary sky localization to our astronomer partners on 6 January. I think this was especially impressive as there were some difficulties with the initial calibration of the data. This isn’t a problem for the detection pipelines, but does impact the parameters which we infer, particularly the sky location. The Calibration group worked quickly, and produced two updates to the calibration. We therefore had three different sets of results (one per calibration) by 6 January [bonus note]!
Producing the final results for the paper was slightly more relaxed. Aaron and I conscripted volunteers to help run all the various permutations of the analysis we wanted to double-check our results [bonus note].
Recovered gravitational waveforms from analysis of GW170104. The broader orange band shows our estimate for the waveform without assuming a particular source (wavelet). The narrow blue bands show results if we assume it is a binary black hole (BBH) as predicted by general relativity. The two match nicely, showing no evidence for any extra features not included in the binary black hole models. Figure 4 of the GW170104 Discovery Paper.
I started working on GW170104 through my parameter estimation duties, and continued with paper writing.
Ahead of the second observing run, we decided to assemble a team to rapidly write up any interesting binary detections, and I was recruited for this (I think partially because I’m not too bad at writing and partially because I was in the office next to John Veitch, one of the chairs of the Compact Binary Coalescence group,so he can come and check that I wasn’t just goofing off eating doughnuts). We soon decided that we should write a paper about GW170104, and you can decide whether or not we succeeded in doing this rapidly…
Being on the paper writing team has given me huge respect for the teams who led the GW150914 and GW151226 papers. It is undoubtedly one of the most difficult things I’ve ever done. It is extremely hard to absorb negative remarks about your work continuously for months [bonus note]—of course people don’t normally send comments about things that they like, but that doesn’t cheer you up when you’re staring at an inbox full of problems that need fixing. Getting a collaboration of 1000 people to agree on a paper is like herding cats while being a small duckling.
On of the first challenges for the paper writing team was deciding what was interesting about GW170104. It was another binary black hole coalescence—aren’t people getting bored of them by now? The signal was quieter than GW150914, so it wasn’t as remarkable. However, its properties were broadly similar. It was suggested that perhaps we should title the paper “GW170104: The most boring gravitational-wave detection”.
One potentially interesting aspect was that GW170104 probably comes from greater distance than GW150914 or GW151226 (but perhaps not LVT151012) [bonus note]. This might make it a good candidate for testing for dispersion of gravitational waves.
Dispersion occurs when different frequencies of gravitational waves travel at different speeds. A similar thing happens for light when travelling through some materials, which leads to prisms splitting light into a spectrum (and hence the creation of Pink Floyd album covers). Gravitational waves don’t suffered dispersion in general relativity, but do in some modified theories of gravity.
It should be easier to spot dispersion in signals which have travelled a greater distance, as the different frequencies have had more time to separate out. Hence, GW170104 looks pretty exciting. However, being further away also makes the signal quieter, and so there is more uncertainty in measurements and it is more difficult to tell if there is any dispersion. Dispersion is also easier to spot if you have a larger spread of frequencies, as then there can be more spreading between the highest and lowest frequencies. When you throw distance, loudness and frequency range into the mix, GW170104 doesn’t always come out on top, depending upon the particular model for dispersion: sometimes GW150914’s loudness wins, other times GW151226’s broader frequency range wins. GW170104 isn’t too special here either.
Even though GW170104 didn’t look too exciting, we started work on a paper, thinking that we would just have a short letter describing our observations. The Compact Binary Coalescence group decided that we only wanted a single paper, and we wouldn’t bother with companion papers as we did for GW150914. As we started work, and dug further into our results, we realised that actually there was rather a lot that we could say.
I guess the moral of the story is that even though you might be overshadowed by the achievements of your siblings, it doesn’t mean that you’re not awesome. There might not be one outstanding feature of GW170104, but there are lots of little things that make it interesting. We are still at the beginning of understanding the properties of binary black holes, and each new detection adds a little more to our picture.
I think GW170104 is rather neat, and I hope you do too.
As we delved into the details of our results, we realised there was actually a lot of things that we could say about GW170104, especially when considered with our previous observations. We ended up having to move some of the technical details and results to Supplemental Material. With hindsight, perhaps it would have been better to have a companion paper or two. However, I rather like how packed with science this paper is.
The paper, which Physical Review Letters have kindly accommodated, despite its length, might not be as polished a classic as the GW150914 Discovery Paper, but I think they are trying to do different things. I rarely ever refer to the GW150914 Discovery Paper for results (more commonly I use it for references), whereas I think I’ll open up the GW170104 Discovery Paper frequently to look up numbers.
Although perhaps not right away, I’d quite like some time off first. The weather’s much better now, perfect for walking…
Advanced LIGO’s first observing run was hugely successful. Running from 12 September 2015 until 19 January 2016, there were two clear gravitational-wave detections, GW1501914 and GW151226, as well as a less certain candidate signal LVT151012. All three (assuming that they are astrophysical signals) correspond to the coalescence of binary black holes.
The second observing run started 30 November 2016. Following the first observing run’s detections, we expected more binary black hole detections. On 4 January, after we had collected almost 6 days’ worth of coincident data from the two LIGO instruments [bonus note], there was a detection.
The signal was first spotted by an online analysis. Our offline analysis of the data (using refined calibration and extra information about data quality) showed that the signal, GW170104, is highly significant. For both GstLAL and PyCBC, search algorithms which use templates to search for binary signals, the false alarm rate is estimated to be about 1 per 70,000 years.
The signal is also found in unmodelled (burst) searches, which look for generic, short gravitational wave signals. Since these are looking for more general signals than just binary coalescences, the significance associated with GW170104 isn’t as great, and coherent WaveBurst estimates a false alarm rate of 1 per 20,000 years. This is still pretty good! Reconstructions of the waveform from unmodelled analyses also match the form expected for binary black hole signals.
The search false alarm rates are the rate at which you’d expect something this signal-like (or more signal-like) due to random chance, if you data only contained noise and no signals. Using our knowledge of the search pipelines, and folding in some assumptions about the properties of binary black holes, we can calculate a probability that GW170104 is a real astrophysical signal. This comes out to be greater than .
As for the previous gravitational wave detections, GW170104 comes from a binary black hole coalescence. The initial black holes were and (where is the mass of our Sun), and the final black hole was . The quoted values are the median values and the error bars denote the central 90% probable range. The plot below shows the probability distribution for the masses; GW170104 neatly nestles in amongst the other events.
Estimated masses for the two black holes in the binary . The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours for all events. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 2 of the GW170104 Discovery Paper.
GW150914 was the first time that we had observed stellar-mass black holes with masses greater than around . GW170104 has similar masses, showing that our first detection was not a fluke, but there really is a population of black holes with masses stretching up into this range.
Black holes have two important properties: mass and spin. We have good measurements on the masses of the two initial black holes, but not the spins. The sensitivity of the form of the gravitational wave to spins can be described by two effective spin parameters, which are mass-weighted combinations of the individual spins.
The effective inspiral spin parameter qualifies the impact of the spins on the rate of inspiral, and where the binary plunges together to merge. It ranges from +1, meaning both black holes are spinning as fast as possible and rotate in the same direction as the orbital motion, to −1, both black holes spinning as fast as possible but in the opposite direction to the way that the binary is orbiting. A value of 0 for could mean that the black holes are not spinning, that their rotation axes are in the orbital plane (instead of aligned with the orbital angular momentum), or that one black hole is aligned with the orbital motion and the other is antialigned, so that their effects cancel out.
The effective precession spin parameter qualifies the amount of precession, the way that the orbital plane and black hole spins wobble when they are not aligned. It is 0 for no precession, and 1 for maximal precession.
We can place some constraints on , but can say nothing about . The inferred value of the effective inspiral spin parameter is . Therefore, we disfavour large spins aligned with the orbital angular momentum, but are consistent with small aligned spins, misaligned spins, or spins antialigned with the angular momentum. The value is similar to that for GW150914, which also had a near-zero, but slightly negative of .
Estimated effective inspiral spin parameter and effective precession spin parameter. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models, as well as the prior probability distribution. The dotted lines mark the edge of our 90% probability intervals. We learn basically nothing about precession. Part of Figure 3 of the GW170104 Discovery Paper.
Converting the information about , the lack of information about , and our measurement of the ratio of the two black hole masses, into probability distributions for the component spins gives the plots below [bonus note]. We disfavour (but don’t exclude) spins aligned with the orbital angular momentum, but can’t say much else.
Estimated orientation and magnitude of the two component spins. The distribution for the more massive black hole is on the left, and for the smaller black hole on the right. The probability is binned into areas which have uniform prior probabilities, so if we had learnt nothing, the plot would be uniform. Part of Figure 3 of the GW170104 Discovery Paper.
One of the comments we had on a draft of the paper was that we weren’t making any definite statements about the spins—we would have if we could, but we can’t for GW170104, at least for the spins of the two inspiralling black holes. We can be more definite about the spin of the final black hole. If two similar mass black holes spiral together, the angular momentum from the orbit is enough to give a spin of around . The spins of the component black holes are less significant, and can make it a bit higher of lower. We infer a final spin of ; there is a tail of lower spin values on account of the possibility that the two component black holes could be roughly antialigned with the orbital angular momentum.
Estimated mass and spin for the final black hole. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 6 of the GW170104 Supplemental Material (Figure 11 of the arXiv version).
If you’re interested in parameter describing GW170104, make sure to check out the big table in the Supplemental Material. I am a fan of tables [bonus note].
Adding the first 11 days of coincident data from the second observing run (including the detection of GW170104) to the results from the first observing run, we find merger rates consistent with those from the first observing run.
To calculate the merger rates, we need to assume a distribution of black hole masses, and we use two simple models. One uses a power law distribution for the primary (larger) black hole and a uniform distribution for the mass ratio; the other uses a distribution uniform in the logarithm of the masses (both primary and secondary). The true distribution should lie somewhere between the two. The power law rate density has been updated from to , and the uniform in log rate density goes from to . The median values stay about the same, but the additional data have shrunk the uncertainties a little.
The discoveries from the first observing run showed that binary black holes exist and merge. The question is now how exactly they form? There are several suggested channels, and it could be there is actually a mixture of different formation mechanisms in action. It will probably require a large number of detections before we can make confident statements about the the probable formation mechanisms; GW170104 is another step towards that goal.
There are two main predicted channels of binary formation:
Isolated binary evolution, where a binary star system lives its life together with both stars collapsing to black holes at the end. To get the black holes close enough to merge, it is usually assumed that the stars go through a common envelope phase, where one star puffs up so that the gravity of its companion can steal enough material that they lie in a shared envelope. The drag from orbiting inside this then shrinks the orbit.
Dynamical evolution where black holes form in dense clusters and a binary is created by dynamical interactions between black holes (or stars) which get close enough to each other.
It’s a little artificial to separate the two, as there’s not really such a thing as an isolated binary: most stars form in clusters, even if they’re not particularly large. There are a variety of different modifications to the two main channels, such as having a third companion which drives the inner binary to merge, embedding the binary is a dense disc (as found in galactic centres), or dynamically assembling primordial black holes (formed by density perturbations in the early universe) instead of black holes formed through stellar collapse.
All the channels can predict black holes around the masses of GW170104 (which is not surprising given that they are similar to the masses of GW150914).
The updated rates are broadly consistent with most channels too. The tightening of the uncertainty of the rates means that the lower bound is now a little higher. This means some of the channels are now in tension with the inferred rates. Some of the more exotic channels—requiring a third companion (Silsbee & Tremain 2017; Antonini, Toonen & Hamers 2017) or embedded in a dense disc (Bartos et al. 2016; Stone, Metzger & Haiman 2016; Antonini & Rasio 2016)—can’t explain the full rate, but I don’t think it was ever expected that they could, they are bonus formation mechanisms. However, some of the dynamical models are also now looking like they could predict a rate that is a bit low (Rodriguez et al. 2016; Mapelli 2016; Askar et al. 2017; Park et al. 2017). Assuming that this result holds, I think this may mean that some of the model parameters need tweaking (there are more optimistic predictions for the merger rates from clusters which are still perfectly consistent), that this channel doesn’t contribute all the merging binaries, or both.
The spins might help us understand formation mechanisms. Traditionally, it has been assumed that isolated binary evolution gives spins aligned with the orbital angular momentum. The progenitor stars were probably more or less aligned with the orbital angular momentum, and tides, mass transfer and drag from the common envelope would serve to realign spins if they became misaligned. Rodriguez et al. (2016) gives a great discussion of this. Dynamically formed binaries have no correlation between spin directions, and so we would expect an isotropic distribution of spins. Hence it sounds quite simple: misaligned spins indicates dynamical formation (although we can’t tell if the black holes are primordial or stellar), and aligned spins indicates isolated binary evolution. The difficulty is the traditional assumption for isolated binary evolution potentially ignores a number of effects which could be important. When a star collapses down to a black hole, there may be a supernova explosion. There is an explosion of matter and neutrinos and these can give the black hole a kick. The kick could change the orbital plane, and so misalign the spin. Even if the kick is not that big, if it is off-centre, it could torque the black hole, causing it to rotate and so misalign the spin that way. There is some evidence that this can happen with neutron stars, as one of the pulsars in the double pulsar system shows signs of this. There could also be some instability that changes the angular momentum during the collapse of the star, possibly with different layers rotating in different ways [bonus note]. The spin of the black hole would then depend on how many layers get swallowed. This is an area of research that needs to be investigated further, and I hope the prospect of gravitational wave measurements spurs this on.
For GW170104, we know the spins are not large and aligned with the orbital angular momentum. This might argue against one variation of isolated binary evolution, chemically homogeneous evolution, where the progenitor stars are tidally locked (and so rotate aligned with the orbital angular momentum and each other). Since the stars are rapidly spinning and aligned, you would expect the final black holes to be too, if the stars completely collapse down as is usually assumed. If the stars don’t completely collapse down though, it might still be possible that GW170104 fits with this model. Aside from this, GW170104 is consistent with all the other channels.
Estimated effective inspiral spin parameter for all events. To indicate how much (or little) we’ve learnt, the prior probability distribution for GW170104 is shown (the other priors are similar).All of the events have at 90% probability. Figure 5 of the GW170104 Supplemental Material (Figure 10 of the arXiv version). This is one of my favourite plots [bonus note].
If we start looking at the population of events, we do start to notice something about the spins. All of the inferred values of are close to zero. Only GW151226 is inconsistent with zero. These values could be explained if spins are typically misaligned (with the orbital angular momentum or each other) or if the spins are typically small (or both). We know that black holes spins can be large from observations of X-ray binaries, so it would be odd if they are small for binary black holes. Therefore, we have a tentative hint that spins are misaligned. We can’t say why the spins are misaligned, but it is intriguing. With more observations, we’ll be able to confirm if it is the case that spins are typically misaligned, and be able to start pinning down the distribution of spin magnitudes and orientations (as well as the mass distribution). It will probably take a while to be able to say anything definite though, as we’ll probably need about 100 detections.
Tests of general relativity
As well as giving us an insight into the properties of black holes, gravitational waves are the perfect tools for testing general relativity. If there are any corrections to general relativity, you’d expect them to be most noticeable under the most extreme conditions, where gravity is strong and spacetime is rapidly changing, exactly as in a binary black hole coalescence.
We added extra terms to to the waveform and constrained their potential magnitudes. The results are pretty much identical to at the end of the first observing run (consistent with zero and hence general relativity). GW170104 doesn’t add much extra information, as GW150914 typically gives the best constraints on terms that modify the post-inspiral part of the waveform (as it is louder), while GW151226 gives the best constraint on the terms which modify the inspiral (as it has the longest inspiral).
We also chopped the waveform at a frequency around that of the innermost stable orbit of the remnant black hole, which is about where the transition from inspiral to merger and ringdown occurs, to check if the low frequency and high frequency portions of the waveform give consistent estimates for the final mass and spin. They do.
We have also done something slightly new, and tested for dispersion of gravitational waves. We did something similar for GW150914 by putting a limit on the mass of the graviton. Giving the graviton mass is one way of adding dispersion, but we consider other possible forms too. In all cases, results are consistent with there being no dispersion. While we haven’t discovered anything new, we can update our gravitational wave constraint on the graviton mass of less than .
The search for counterparts
We don’t discuss observations made by our astronomer partners in the paper (they are not our results). A number (28 at the time of submission) of observations were made, and I expect that there will be a series of papers detailing these coming soon. So far papers have appeared from:
AGILE—hard X-ray and gamma-ray follow-up. They didn’t find any gamma-ray signals, but did identify a weak potential X-ray signal occurring about 0.46 s before GW170104. It’s a little odd to have a signal this long before the merger. The team calculate a probability for such a coincident to happen by chance, and find quite a small probability, so it might be interesting to follow this up more (see the INTEGRAL results below), but it’s probably just a coincidence (especially considering how many people did follow-up the event).
ANTARES—a search for high-energy muon neutrinos. No counterparts are identified in a ±500 s window around GW170104, or over a ±3 month period.
AstroSat-CZTI and GROWTH—a collaboration of observations across a range of wavelengths. They don’t find any hard X-ray counterparts. They do follow up on a bright optical transient ATLASaeu, suggested as a counterpart to GW170104, and conclude that this is a likely counterpart of long, soft gamma-ray burst GRB 170105A.
ATLAS and Pan-STARRS—optical follow-up. They identified a bright optical transient 23 hours after GW170104, ATLAS17aeu. This could be a counterpart to GRB 170105A. It seems unlikely that there is any mechanism that could allow for a day’s delay between the gravitational wave emission and an electromagnetic signal. However, the team calculate a small probability (few percent) of finding such a coincidence in sky position and time, so perhaps it is worth pondering. I wouldn’t put any money on it without a distance estimate for the source: assuming it’s a normal afterglow to a gamma-ray burst, you’d expect it to be further away than GW170104’s source.
Borexino—a search for low-energy neutrinos. This paper also discusses GW150914 and GW151226. In all cases, the observed rate of neutrinos is consistent with the expected background.
Fermi (GBM and LAT)—gamma-ray follow-up. They covered an impressive fraction of the sky localization, but didn’t find anything.
INTEGRAL—gamma-ray and hard X-ray observations. No significant emission is found, which makes the event reported by AGILE unlikely to be a counterpart to GW170104, although they cannot completely rule it out.
The intermediate Palomar Transient Factory—an optical survey. While searching, they discovered iPTF17cw, a broad-line type Ic supernova which is unrelated to GW170104 but interesting as it an unusual find.
Mini-GWAC—a optical survey (the precursor to GWAC). This paper covers the whole of their O2 follow-up including GW170608.
NOvA—a search for neutrinos and cosmic rays over a wide range of energies. This paper covers all the events from O1 and O2, plus triggers from O3.
If you’re looking for the most up-to-date results regarding GW170104, check out the O2 Catalogue Paper.
Gravitational wave signals (at least the short ones, which are all that we have so far), are named by their detection date. GW170104 was discovered 2017 January 4. This isn’t too catchy, but is at least better than the ID number in our database of triggers (G268556) which is used in corresponding with our astronomer partners before we work out if the “GW” title is justified.
Previous detections have attracted nicknames, but none has stuck for GW170104. Archisman Ghosh suggested the Perihelion Event, as it was detected a few hours before the Earth reached its annual point closest to the Sun. I like this name, its rather poetic.
More recently, Alex Nitz realised that we should have called GW170104 the Enterprise-D Event, as the USS Enterprise’s registry number was NCC-1701. For those who like Star Trek: the Next Generation, I hope you have fun discussing whether GW170104 is the third or fourth (counting LVT151012) detection: “There are four detections!“
The 6 January sky map
I would like to thank the wi-fi of Chiltern Railways for their role in producing the preliminary sky map. I had arranged to visit London for the weekend (because my rota slot was likely to be quiet… ), and was frantically working on the way down to check results so they could be sent out. I’d also like to thank John Veitch for putting together the final map while I was stuck on the Underground.
Binary black hole waveforms
The parameter estimation analysis works by matching a template waveform to the data to see how well it matches. The results are therefore sensitive to your waveform model, and whether they include all the relevant bits of physics.
In the first observing run, we always used two different families of waveforms, to see what impact potential errors in the waveforms could have. The results we presented in discovery papers used two quick-to-calculate waveforms. These include the effects of the black holes’ spins in different ways
SEOBNRv2 has spins either aligned or antialigned with the orbital angular momentum. Therefore, there is no precession (wobbling of orientation, like that of a spinning top) of the system.
IMRPhenomPv2 includes an approximate description of precession, packaging up the most important information about precession into a single parameter .
For GW150914, we also performed a follow-up analysis using a much more expensive waveform SEOBNRv3 which more fully includes the effect of both spins on precession. These results weren’t ready at the time of the announcement, because the waveform is laborious to run.
For GW170104, there were discussions that using a spin-aligned waveform was old hat, and that we should really only use the two precessing models. Hence, we started on the endeavour of producing SEOBNRv3 results. Fortunately, the code has been sped up a little, although it is still not quick to run. I am extremely grateful to Scott Coughlin (one of the folks behind Gravity Spy), Andrea Taracchini and Stas Babak for taking charge of producing results in time for the paper, in what was a Herculean effort.
I spent a few sleepless nights, trying to calculate if the analysis was converging quickly enough to make our target submission deadline, but it did work out in the end. Still, don’t necessarily expect we’ll do this for a all future detections.
Since the waveforms have rather scary technical names, in the paper we refer to IMRPhenomPv2 as the effective precession model and SEOBNRv3 as the full precession model.
Distance measurements for gravitational wave sources have significant uncertainties. The distance is difficult to measure as it determined from the signal amplitude, but this is also influences by the binary’s inclination. A signal could either be close and edge on or far and face on-face off.
Estimated luminosity distance and binary inclination angle . The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 4 of the GW170104 Supplemental Material (Figure 9 of the arXiv version).
The uncertainty on the distance rather awkwardly means that we can’t definitely say that GW170104 came from a further source than GW150914 or GW151226, but it’s a reasonable bet. The 90% credible intervals on the distances are 250–570 Mpc for GW150194, 250–660 Mpc for GW151226, 490–1330 Mpc for GW170104 and 500–1500 Mpc for LVT151012.
Translating from a luminosity distance to a travel time (gravitational waves do travel at the speed of light, our tests of dispersion are consistent wit that!), the GW170104 black holes merged somewhere between 1.3 and 3.0 billion years ago. This is around the time that multicellular life first evolved on Earth, and means that black holes have been colliding longer than life on Earth has been reproducing sexually.
A first draft of the paper (version 2; version 1 was a copy-and-paste of the Boxing Day Discovery Paper) was circulated to the Compact Binary Coalescence and Burst groups for comments on 4 March. This was still a rough version, and we wanted to check that we had a good outline of the paper. The main feedback was that we should include more about the astrophysical side of things. I think the final paper has a better balance, possibly erring on the side of going into too much detail on some of the more subtle points (but I think that’s better than glossing over them).
A first proper draft (version 3) was released to the entire Collaboration on 12 March in the middle of our Collaboration meeting in Pasadena. We gave an oral presentation the next day (I doubt many people had read the paper by then). Collaboration papers are usually allowed two weeks for people to comment, and we followed the same procedure here. That was not a fun time, as there was a constant trickle of comments. I remember waking up each morning and trying to guess how many emails would be in my inbox–I normally low-balled this.
I wasn’t too happy with version 3, it was still rather rough. The members of the Paper Writing Team had been furiously working on our individual tasks, but hadn’t had time to look at the whole. I was much happier with the next draft (version 4). It took some work to get this together, following up on all the comments and trying to address concerns was a challenge. It was especially difficult as we got a series of private comments, and trying to find a consensus probably made us look like the bad guys on all sides. We released version 4 on 14 April for a week of comments.
The next step was approval by the LIGO and Virgo executive bodies on 24 April. We prepared version 5 for this. By this point, I had lost track of which sentences I had written, which I had merely typed, and which were from other people completely. There were a few minor changes, mostly adding technical caveats to keep everyone happy (although they do rather complicate the flow of the text).
The paper was circulated to the Collaboration for a final week of comments on 26 April. Most comments now were about typos and presentation. However, some people will continue to make the same comment every time, regardless of how many times you explain why you are doing something different. The end was in sight!
The paper was submitted to Physical Review Letters on 9 May. I was hoping that the referees would take a while, but the reports were waiting in my inbox on Monday morning.
The referee reports weren’t too bad. Referee A had some general comments, Referee B had some good and detailed comments on the astrophysics, and Referee C gave the paper a thorough reading and had some good suggestions for clarifying the text. By this point, I have been staring at the paper so long that some outside perspective was welcome. I was hoping that we’d have a more thorough review of the testing general relativity results, but we had Bob Wald as one of our Collaboration Paper reviewers (the analysis, results and paper are all reviewed internally), so I think we had already been held to a high standard, and there wasn’t much left to say.
We put together responses to the reports. There were surprisingly few comments from the Collaboration at this point. I guess that everyone was getting tired. The paper was resubmitted and accepted on 20 May.
One of the suggestions of Referee A was to include some plots showing the results of the searches. People weren’t too keen on showing these initially, but after much badgering they were convinced, and it was decided to put these plots in the Supplemental Material which wouldn’t delay the paper as long as we got the material submitted by 26 May. This seemed like plenty of time, but it turned out to be rather frantic at the end (although not due to the new plots). The video below is an accurate representation of us trying to submit the final version.
I have an email which contains the line “Many Bothans died to bring us this information” from 1 hour and 18 minutes before the final deadline.
After this, things were looking pretty good. We had returned the proofs of the main paper (I had a fun evening double checking the author list. Yes, all of them). We were now on version 11 of the paper.
Of course, there’s always one last thing. On 31 May, the evening before publication, Salvo Vitale spotted a typo. Nothing serious, but annoying. The team at Physical Review Letters were fantastic, and took care of it immediately!
There’ll still be one more typo, there always is…
Looking back, it is clear that the principal bottle-neck in publishing the results is getting the Collaboration to converge on the paper. I’m not sure how we can overcome this… Actually, I have some ideas, but none that wouldn’t involve some form of doomsday device.
The sensitivities of the LIGO Hanford and Livinston detectors are around the same as they were in the first observing run. After the success of the first observing run, the second observing run is the difficult follow up album. Livingston has got a little better, while Hanford is a little worse. This is because the Livingston team concentrate on improving low frequency sensitivity whereas the Hanford team focused on improving high frequency sensitivity. The Hanford team increased the laser power, but this introduces some new complications. The instruments are extremely complicated machines, and improving sensitivity is hard work.
The current plan is to have a long commissioning break after the end of this run. The low frequency tweaks from Livingston will be transferred to Hanford, and both sites will work on bringing down other sources of noise.
While the sensitivity hasn’t improved as much as we might have hoped, the calibration of the detectors has! In the first observing run, the calibration uncertainty for the first set of published results was about 10% in amplitude and 10 degrees in phase. Now, uncertainty is better than 5% in amplitude and 3 degrees in phase, and people are discussing getting this down further.
As the binary inspirals, the orientation of the spins will evolve as they precess about. We always quote measurements of the spins at a point in the inspiral corresponding to a gravitational wave frequency of 20 Hz. This is most convenient for our analysis, but you can calculate the spins at other points. However, the resulting probability distributions are pretty similar at other frequencies. This is because the probability distributions are primarily determined by the combination of three things: (i) our prior assumption of a uniform distribution of spin orientations, (ii) our measurement of the effective inspiral spin, and (iii) our measurement of the mass ratio. A uniform distribution stays uniform as spins evolve, so this is unaffected, the effective inspiral spin is approximately conserved during inspiral, so this doesn’t change much, and the mass ratio is constant. The overall picture is therefore qualitatively similar at different moments during the inspiral.
I love footnotes. It was challenging for me to resist having any in the paper.
It is possible that internal gravity waves (that is oscillations of the material making up the star, where the restoring force is gravity, not gravitational waves, which are ripples in spacetime), can transport angular momentum from the core of a star to its outer envelope, meaning that the two could rotate in different directions (Rogers, Lin & Lau 2012). I don’t think anyone has studied this yet for the progenitors of binary black holes, but it would be really cool if gravity waves set the properties of gravitational wave sources.
I really don’t want to proof read the paper which explains this though.
For our plots, we use a consistent colour coding for our events. GW150914 is blue; LVT151012 is green; GW151226 is red–orange, and GW170104 is purple. The colour scheme is designed to be colour blind friendly (although adopting different line styles would perhaps be more distinguishable), and is implemented in Python in the Seaborn package as colorblind. Katerina Chatziioannou, who made most of the plots showing parameter estimation results is not a fan of the colour combinations, but put a lot of patient effort into polishing up the plots anyway.
Binary black hole mergers are the ultimate laboratory for testing gravity. The gravitational fields are strong, and things are moving at close to the speed of light. these extreme conditions are exactly where we expect our theories could breakdown, which is why we were so exciting by detecting gravitational waves from black hole coalescences. To accompany the first detection of gravitational waves, we performed several tests of Einstein’s theory of general relativity (it passed). This paper outlines the details of one of the tests, one that can be extended to include future detections to put Einstein’s theory to the toughest scrutiny.
One of the difficulties of testing general relativity is what do you compare it to? There are many alternative theories of gravity, but only a few of these have been studied thoroughly enough to give an concrete idea of what a binary black hole merger should look like. Even if general relativity comes out on top when compared to one alternative model, it doesn’t mean that another (perhaps one we’ve not thought of yet) can be ruled out. We need ways of looking for something odd, something which hints that general relativity is wrong, but doesn’t rely on any particular alternative theory of gravity.
The test suggested here is a consistency test. We split the gravitational-wave signal into two pieces, a low frequency part and a high frequency part, and then try to measure the properties of the source from the two parts. If general relativity is correct, we should get answers that agree; if it’s not, and there’s some deviation in the exact shape of the signal at different frequencies, we can get different answers. One way of thinking about this test is imagining that we have two experiments, one where we measure lower frequency gravitational waves and one where we measure higher frequencies, and we are checking to see if their results agree.
To split the waveform, we use a frequency around that of the last stable circular orbit: about the point that the black holes stop orbiting about each other and plunge together and merge [bonus note]. For GW150914, we used 132 Hz, which is about the same as the C an octave below middle C (a little before time zero in the simulation below). This cut roughly splits the waveform into the low frequency inspiral (where the two black hole are orbiting each other), and the higher frequency merger (where the two black holes become one) and ringdown (where the final black hole settles down).
We are fairly confident that we understand what goes on during the inspiral. This is similar physics to where we’ve been testing gravity before, for example by studying the orbits of the planets in the Solar System. The merger and ringdown are more uncertain, as we’ve never before probed these strong and rapidly changing gravitational fields. It therefore seems like a good idea to check the two independently [bonus note].
We use our parameter estimation codes on the two pieces to infer the properties of the source, and we compare the values for the mass and spin of the final black hole. We could use other sets of parameters, but this pair compactly sum up the properties of the final black hole and are easy to explain. We look at the difference between the estimated values for the mass and spin, and , if general relativity is a good match to the observations, then we expect everything to match up, and and to be consistent with zero. They won’t be exactly zero because we have noise in the detector, but hopefully zero will be within the uncertainty region [bonus note]. An illustration of the test is shown below, including one of the tests we did to show that it does spot when general relativity is not correct.
Results from the consistency test. The top panels show the outlines of the 50% and 90% credible levels for the low frequency (inspiral) part of the waveform, the high frequency (merger–ringdown) part, and the entire (inspiral–merger–ringdown, IMR) waveform. The bottom panel shows the fractional difference between the high and low frequency results. If general relativity is correct, we expect the distribution to be consistent with , indicated by the cross (+). The left panels show a general relativity simulation, and the right panel shows a waveform from a modified theory of gravity. Figure 1 of Ghosh et al. (2016).
A convenient feature of using and to test agreement with relativity, is that you can combine results from multiple observations. By averaging over lots of signals, you can reduce the uncertainty from noise. This allows you to pin down whether or not things really are consistent, and spot smaller deviations (we could get precision of a few percent after about 100 suitable detections). I look forward to seeing how this test performs in the future!
I became involved in this work as a reviewer. The LIGO Scientific Collaboration is a bit of a stickler when it comes to checking its science. We had to check that the test was coded up correctly, that the results made sense, and that calculations done and written up for GW150914 were all correct. Since most of the team are based in India [bonus note], this involved some early morning telecons, but it all went smoothly.
One of our checks was that the test wasn’t sensitive to exact frequency used to split the signal. If you change the frequency cut, the results from the two sections do change. If you lower the frequency, then there’s less of the low frequency signal and the measurement uncertainties from this piece get bigger. Conversely, there’ll be more signal in the high frequency part and so we’ll make a more precise measurement of the parameters from this piece. However, the overall results where you combine the two pieces stay about the same. You get best results when there’s a roughly equal balance between the two pieces, but you don’t have to worry about getting the cut exactly on the innermost stable orbit.
In order for the test to work, we need the two pieces of the waveform to both be loud enough to allow us to measure parameters using them. This type of signals are referred to as golden. Earlier work on tests of general relativity using golden binaries has been done by Hughes & Menou (2015), and Nakano, Tanaka & Nakamura (2015). GW150914 was a golden binary, but GW151226 and LVT151012 were not, which is why we didn’t repeat this test for them.
For The Event, we ran this test, and the results are consistent with general relativity being correct. The plots below show the estimates for the final mass and spin (here denoted rather than ), and the fractional difference between the two measurements. The points is at the 28% credible level. This means that if general relativity is correct, we’d expect a deviation at this level to occur around-about 72% of the time due to noise fluctuations. It wouldn’t take a particular rare realisation of noise to cause the assume true value of to be found at this probability level, so we’re not too suspicious that something is amiss with general relativity.
Results from the consistency test for The Event. The top panels final mass and spin measurements from the low frequency (inspiral) part of the waveform, the high frequency (post-inspiral) part, and the entire (IMR) waveform. The bottom panel shows the fractional difference between the high and low frequency results. If general relativity is correct, we expect the distribution to be consistent with , indicated by the cross. Figure 3 of the Testing General Relativity Paper.
Abhirup Ghosh and Archisman Ghosh were two of the leads of this study. They are both A. Ghosh at the same institution, which caused some confusion when compiling the LIGO Scientific Collaboration author list. I think at one point one of them (they can argue which) was removed as someone thought there was a mistaken duplication. To avoid confusion, they now have their full names used. This is a rare distinction on the Discovery Paper (I’ve spotted just two others). The academic tradition of using first initials plus second name is poorly adapted to names which don’t fit the typical western template, so we should be more flexible.