# Dirichlet Process Gaussian-mixture model: An application to localizing coalescing binary neutron stars with gravitational-wave observations

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).

### Gravitational-wave localization

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 $10^4$$10^5~\mathrm{Mpc}^3$, which is about $10^{68}$$10^{69}$ Olympic swimming pools. Such a volume would contain something like $100$$1000$ 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

1. 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!
2. The localization dramatically improves when we have three-detector observations. The extra detector improves the sky localization, which reduces the localization volume.
3. 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.
4. 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).

arXiv: 1801.08009 [astro-ph.IM]
Journal: Monthly Notices of the Royal Astronomical Society; 479(1):601–614; 2018
Code: 3d_volume
Buzzword bingo: Interdisciplinary (we worked with computer scientist Tom Haines); machine learning (the inference involving our Dirichlet process Gaussian mixture model); multimessenger astronomy (as our results are useful for following up gravitational-wave signals in the search for counterparts)

### Bonus notes

#### Writing

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.

#### Sixth power

We find that the localization volume $\Delta V$ is inversely proportional to the sixth power of the signal-to-noise ration $\varrho$. This is what you would expect. The localization volume depends upon the angular uncertainty on the sky $\Delta \Omega$, the distance to the source $D$, and the distance uncertainty $\Delta D$,

$\Delta V \sim D^2 \Delta \Omega \Delta D$.

Typically, the uncertainty on a parameter (like the masses) scales inversely with the signal-to-noise ratio. This is the case for the logarithm of the distance, which means

$\displaystyle \frac{\Delta D}{D} \propto \varrho^{-1}$.

The uncertainty in the sky location (being two dimensional) scales inversely with the square of the signal-to-noise ration,

$\Delta \Omega \propto \varrho^{-2}$.

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

$\Delta V \propto \varrho^{-6}$.

#### Treasure

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.

# Prospects for observing and localizing gravitational-wave transients with Advanced LIGO, Advanced Virgo and KAGRA

This paper, known as the Observing Scenarios Document with the Collaboration, outlines the observing plans of the ground-based detectors over the coming decade. If you want to search for electromagnetic or neutrino signals from our gravitational-wave sources, this is the paper for you. It is a living review—a document that is continuously updated.

This is the second published version, the big changes since the last version are

1. We have now detected gravitational waves
2. We have observed our first gravitational wave with a mulitmessenger counterpart [bonus note]
3. We now include KAGRA, along with LIGO and Virgo

As you might imagine, these are quite significant updates! The first showed that we can do gravitational-wave astronomy. The second showed that we can do exactly the science this paper is about. The third makes this the first joint publication of the LIGO Scientific, Virgo and KAGRA Collaborations—hopefully the first of many to come.

I lead both this and the previous version. In my blog on the previous version, I explained how I got involved, and the long road that a collaboration must follow to get published. In this post, I’ll give an overview of the key details from the new version together with some behind-the-scenes background (working as part of a large scientific collaboration allows you to do amazing science, but it can also be exhausting). If you’d like a digest of this paper’s science, check out the LIGO science summary.

### Commissioning and observing phases

The first section of the paper outlines the progression of detector sensitivities. The instruments are incredibly sensitive—we’ve never made machines to make these types of measurements before, so it takes a lot of work to get them to run smoothly. We can’t just switch them on and have them work at design sensitivity [bonus note].

Target evolution of the Advanced LIGO and Advanced Virgo detectors with time. The lower the sensitivity curve, the further away we can detect sources. The distances quoted are binary neutron star (BNS) ranges, the average distance we could detect a binary neutron star system. The BNS-optimized curve is a proposal to tweak the detectors for finding BNSs. Figure 1 of the Observing Scenarios Document.

The plots above show the planned progression of the different detectors. We had to get these agreed before we could write the later parts of the paper because the sensitivity of the detectors determines how many sources we will see and how well we will be able to localize them. I had anticipated that KAGRA would be the most challenging here, as we had not previously put together this sequence of curves. However, this was not the case, instead it was Virgo which was tricky. They had a problem with the silica fibres which suspended their mirrors (they snapped, which is definitely not what you want). The silica fibres were replaced with steel ones, but it wasn’t immediately clear what sensitivity they’d achieve and when. The final word was they’d observe in August 2017 and that their projections were unchanged. I was sceptical, but they did pull it out of the bag! We had our first clear three-detector observation of a gravitational wave 14 August 2017. Bravo Virgo!

Plausible time line of observing runs with Advanced LIGO (Hanford and Livingston), advanced Virgo and KAGRA. It is too early to give a timeline for LIGO India. The numbers above the bars give binary neutron star ranges (italic for achieved, roman for target); the colours match those in the plot above. Currently our third observing run (O3) looks like it will start in early 2019; KAGRA might join with an early sensitivity run at the end of it. Figure 2 of the Observing Scenarios Document.

### Searches for gravitational-wave transients

The second section explain our data analysis techniques: how we find signals in the data, how we work out probable source locations, and how we communicate these results with the broader astronomical community—from the start of our third observing run (O3), information will be shared publicly!

The information in this section hasn’t changed much [bonus note]. There is a nice collection of references on the follow-up of different events, including GW170817 (I’d recommend my blog for more on the electromagnetic story). The main update I wanted to include was information on the detection of our first gravitational waves. It turned out to be more difficult than I imagined to come up with a plot which showed results from the five different search algorithms (two which used templates, and three which did not) which found GW150914, and harder still to make a plot which everyone liked. This plot become somewhat infamous for the amount of discussion it generated. I think we ended up with something which was a good compromise and clearly shows our detections sticking out above the background of noise.

Offline transient search results from our first observing run (O1). The plot shows the number of events found verses false alarm rate: if there were no gravitational waves we would expect the points to follow the dashed line. The left panel shows the results of the templated search for compact binary coalescences (binary black holes, binary neutron stars and neutron star–black hole binaries), the right panel shows the unmodelled burst search. GW150914, GW151226 and LVT151012 are found by the templated search; GW150914 is also seen in the burst search. Arrows indicate bounds on the significance. Figure 3 of the Observing Scenarios Document.

### Observing scenarios

The third section brings everything together and looks at what the prospects are for (gravitational-wave) multimessenger astronomy during each observing run. It’s really all about the big table.

Summary of different observing scenarios with the advanced detectors. We assume a 70–75% duty factor for each instrument (including Virgo for the second scenario’s sky localization, even though it only joined our second observing run for the final month). Table 3 from the Observing Scenarios Document.

I think there are three really awesome take-aways from this

1. Actual binary neutron stars detected = 1. We did it!
2. Using the rates inferred using our observations so far (including GW170817), once we have the full five detector network of LIGO-Hanford, LIGO-Livingston, Virgo, KAGRA and LIGO-India, we could be detected 11–180 binary neutron stars a year. That something like between one a month to one every other day! I’m kind of scared…
3. With the five detector network the sky localization is really good. The median localization is about 9–12 square degrees, about the area the LSST could cover in a single pointing! This really shows the benefit of adding more detector to the network. The improvement comes not because a source is much better localized with five detectors than four, but because when you have five detectors you almost always have at least three detectors(the number needed to get a good triangulation) online at any moment, so you get a nice localization for pretty much everything.

In summary, the prospects for observing and localizing gravitational-wave transients are pretty great. If you are an astronomer, make the most of the quiet before O3 begins next year.

arXiv: 1304.0670 [gr-qc]
Journal: Living Reviews In Relativity21:3(57); 2018
Science summary: A Bright today and brighter tomorrow: Prospects for gravitational-wave astronomy With Advanced LIGO, Advanced Virgo, and KAGRA
Prospects for the next update:
After two updates, I’ve stepped down from preparing the next one. Wooh!

### Bonus notes

#### GW170817 announcement

The announcement of our first multimessenger detection came between us submitting this update and us getting referee reports. We wanted an updated version of this paper, with the current details of our observing plans, to be available for our astronomer partners to be able to cite when writing their papers on GW170817.

Predictably, when the referee reports came back, we were told we really should include reference to GW170817. This type of discovery is exactly what this paper is about! There was avalanche of results surrounding GW170817, so I had to read through a lot of papers. The reference list swelled from 8 to 13 pages, but this effort was handy for my blog writing. After including all these new results, it really felt like this was version 2.5 of the Observing Scenarios, rather than version 2.

#### Design sensitivity

We use the term design sensitivity to indicate the performance the current detectors were designed to achieve. They are the targets we aim to achieve with Advanced LIGO, Advance Virgo and KAGRA. One thing I’ve had to try to train myself not to say is that design sensitivity is the final sensitivity of our detectors. Teams are currently working on plans for how we can upgrade our detectors beyond design sensitivity. Reaching design sensitivity will not be the end of our journey.

#### Binary black holes vs binary neutron stars

Our first gravitational-wave detections were from binary black holes. Therefore, when we were starting on this update there was a push to switch from focusing on binary neutron stars to binary black holes. I resisted on this, partially because I’m lazy, but mostly because I still thought that binary neutron stars were our best bet for multimessenger astronomy. This worked out nicely.

# GW170817—The papers

After three months (and one binary black hole detection announcement), I finally have time to write about the suite of LIGO–Virgo papers put together to accompany GW170817.

### The papers

There are currently 9 papers in the GW170817 family. Further papers, for example looking at parameter estimation in detail, are in progress. Papers are listed below in order of arXiv posting. My favourite is the GW170817 Discovery Paper. Many of the highlights, especially from the Discovery and Multimessenger Astronomy Papers, are described in my GW170817 announcement post.

Keeping up with all the accompanying observational results is a task not even Sisyphus would envy. I’m sure that the details of these will be debated for a long time to come. I’ve included references to a few below (mostly as [citation notes]), but these are not guaranteed to be complete (I’ll continue to expand these in the future).

#### 0. The GW170817 Discovery Paper

Title: GW170817: Observation of gravitational waves from a binary neutron star inspiral
arXiv:
1710.05832 [gr-qc]
Journal:
Physical Review Letters; 119(16):161101(18); 2017
LIGO science summary:
GW170817: Observation of gravitational waves from a binary neutron star inspiral

This is the paper announcing the gravitational-wave detection. It gives an overview of the properties of the signal, initial estimates of the parameters of the source (see the GW170817 Properties Paper for updates) and the binary neutron star merger rate, as well as an overview of results from the other companion papers.

I was disappointed that “the era of gravitational-wave multi-messenger astronomy has opened with a bang” didn’t make the conclusion of the final draft.

More details: The GW170817 Discovery Paper summary

#### −1. The Multimessenger Astronomy Paper

Title: Multi-messenger observations of a binary neutron star merger
arXiv:
1710.05833 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 848(2):L12(59); 2017
LIGO science summary:
The dawn of multi-messenger astrophysics: observations of a binary neutron star merger

I’ve numbered this paper as −1 as it gives an overview of all the observations—gravitational wave, electromagnetic and neutrino—accompanying GW170817. I feel a little sorry for the neutrino observers, as they’re the only ones not to make a detection. Drawing together the gravitational wave and electromagnetic observations, we can confirm that binary neutron star mergers are the progenitors of (at least some) short gamma-ray bursts and kilonovae.

Do not print this paper, the author list stretches across 23 pages.

More details: The Multimessenger Astronomy Paper summary

#### 1. The GW170817 Gamma-ray Burst Paper

Title: Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A
arXiv:
1710.05834 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 848(2):L13(27); 2017
LIGO science summary:
Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A

Here we bring together the LIGO–Virgo observations of GW170817 and the Fermi and INTEGRAL observations of GRB 170817A. From the spatial and temporal coincidence of the gravitational waves and gamma rays, we establish that the two are associated with each other. There is a 1.7 s time delay between the merger time estimated from gravitational waves and the arrival of the gamma-rays. From this, we make some inferences about the structure of the jet which is the source of the gamma rays. We can also use this to constrain deviations from general relativity, which is cool. Finally, we estimate that there be 0.3–1.7 joint gamma ray–gravitational wave detections per year once our gravitational-wave detectors reach design sensitivity!

More details: The GW170817 Gamma-ray Burst Paper summary

#### 2. The GW170817 Hubble Constant Paper

Title: A gravitational-wave standard siren measurement of the Hubble constant [bonus note]
arXiv:
1710.05835 [astro-ph.CO]
Journal:
Nature; 551(7678):85–88; 2017 [bonus note]
LIGO science summary:
Measuring the expansion of the Universe with gravitational waves

The Hubble constant quantifies the current rate of expansion of the Universe. If you know how far away an object is, and how fast it is moving away (due to the expansion of the Universe, not because it’s on a bus or something, that is important), you can estimate the Hubble constant. Gravitational waves give us an estimate of the distance to the source of GW170817. The observations of the optical transient AT 2017gfo allow us to identify the galaxy NGC 4993 as the host of GW170817’s source. We know the redshift of the galaxy (which indicates how fast its moving). Therefore, putting the two together we can infer the Hubble constant in a completely new way.

More details: The GW170817 Hubble Constant Paper summary

#### 3. The GW170817 Kilonova Paper

Title: Estimating the contribution of dynamical ejecta in the kilonova associated with GW170817
arXiv:
1710.05836 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 850(2):L39(13); 2017
LIGO science summary:
Predicting the aftermath of the neutron star collision that produced GW170817

During the coalescence of two neutron stars, lots of neutron-rich matter gets ejected. This undergoes rapid radioactive decay, which powers a kilonova, an optical transient. The observed signal depends upon the material ejected. Here, we try to use our gravitational-wave measurements to predict the properties of the ejecta ahead of the flurry of observational papers.

More details: The GW170817 Kilonova Paper summary

#### 4. The GW170817 Stochastic Paper

Title: GW170817: Implications for the stochastic gravitational-wave background from compact binary coalescences
arXiv:
1710.05837 [gr-qc]
Journal: Physical Review Letters; 120(9):091101(12); 2018
LIGO science summary: The background symphony of gravitational waves from neutron star and black hole mergers

We can detect signals if they are loud enough, but there will be many quieter ones that we cannot pick out from the noise. These add together to form an overlapping background of signals, a background rumbling in our detectors. We use the inferred rate of binary neutron star mergers to estimate their background. This is smaller than the background from binary black hole mergers (black holes are more massive, so they’re intrinsically louder), but they all add up. It’ll still be a few years before we could detect a background signal.

More details: The GW170817 Stochastic Paper summary

#### 5. The GW170817 Progenitor Paper

Title: On the progenitor of binary neutron star merger GW170817
arXiv:
1710.05838 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 850(2):L40(18); 2017
LIGO science summary:
Making GW170817: neutron stars, supernovae and trick shots (I’d especially recommend reading this one)

We know that GW170817 came from the coalescence of two neutron stars, but where did these neutron stars come from? Here, we combine the parameters inferred from our gravitational-wave measurements, the observed position of AT 2017gfo in NGC 4993 and models for the host galaxy, to estimate properties like the kick imparted to neutron stars during the supernova explosion and how long it took the binary to merge.

More details: The GW170817 Progenitor Paper summary

#### 6. The GW170817 Neutrino Paper

Title: Search for high-energy neutrinos from binary neutron star merger GW170817 with ANTARES, IceCube, and the Pierre Auger Observatory
arXiv:
1710.05839 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 850(2):L35(18); 2017

This is the search for neutrinos from the source of GW170817. Lots of neutrinos are emitted during the collision, but not enough to be detectable on Earth. Indeed, we don’t find any neutrinos, but we combine results from three experiments to set upper limits.

More details: The GW170817 Neutrino Paper summary

#### 7. The GW170817 Post-merger Paper

Title: Search for post-merger gravitational waves from the remnant of the binary neutron star merger GW170817
arXiv:
1710.09320 [astro-ph.HE]
Journal:
Astrophysical Journal Letters; 851(1):L16(13); 2017
LIGO science summary:
Searching for the neutron star or black hole resulting from GW170817

After the two neutron stars merged, what was left? A larger neutron star or a black hole? Potentially we could detect gravitational waves from a wibbling neutron star, as it sloshes around following the collision. We don’t. It would have to be a lot closer for this to be plausible. However, this paper outlines how to search for such signals; the GW170817 Properties Paper contains a more detailed look at any potential post-merger signal.

More details: The GW170817 Post-merger Paper summary

#### 8. The GW170817 Properties Paper

Title: Properties of the binary neutron star merger GW170817
arXiv:
1805.11579 [gr-qc]

In the GW170817 Discovery Paper we presented initial estimates for the properties of GW170817’s source. These were the best we could do on the tight deadline for the announcement (it was a pretty good job in my opinion). Now we have had a bit more time we can present a new, improved analysis. This uses recalibrated data and a wider selection of waveform models. We also fold in our knowledge of the source location, thanks to the observation of AT 2017gfo by our astronomer partners, for our best results. if you want to know the details of GW170817’s source, this is the paper for you!

If you’re looking for the most up-to-date results regarding GW170817, check out the O2 Catalogue Paper.

More details: The GW170817 Properties Paper summary

#### 9. The GW170817 Equation-of-state Paper

Title: GW170817: Measurements of neutron star radii and equation of state
arXiv:
1805.11581 [gr-qc]

Neutron stars are made of weird stuff: nuclear density material which we cannot replicate here on Earth. Neutron star matter is often described in terms of an equation of state, a relationship that explains how the material changes at different pressures or densities. A stiffer equation of state means that the material is harder to squash, and a softer equation of state is easier to squish. This means that for a given mass, a stiffer equation of state will predict a larger, fluffier neutron star, while a softer equation of state will predict a more compact, denser neutron star. In this paper, we assume that GW170817’s source is a binary neutron star system, where both neutron stars have the same equation of state, and see what we can infer about neutron star stuff™.

More details: The GW170817 Equation-of-state Paper summary

### The GW170817 Discovery Paper

Synopsis: GW170817 Discovery Paper
Read this if: You want all the details of our first gravitational-wave observation of a binary neutron star coalescence
Favourite part: Look how well we measure the chirp mass!

GW170817 was a remarkable gravitational-wave discovery. It is the loudest signal observed to date, and the source with the lowest mass components. I’ve written about some of the highlights of the discovery in my previous GW170817 discovery post.

Binary neutron stars are one of the principal targets for LIGO and Virgo. The first observational evidence for the existence of gravitational waves came from observations of binary pulsars—a binary neutron star system where (at least one) one of the components is a pulsar. Therefore (unlike binary black holes), we knew that these sources existed before we turned on our detectors. What was less certain was how often they merge. In our first advanced-detector observing run (O1), we didn’t find any, allowing us to estimate an upper limit on the merger rate of $12600~\mathrm{Gpc^{-1}\,yr^{-1}}$. Now, we know much more about merging binary neutron stars.

GW170817, as a loud and long signal, is a highly significant detection. You can see it in the data by eye. Therefore, it should have been a easy detection. As is often the case with real experiments, it wasn’t quite that simple. Data transfer from Virgo had stopped over night, and there was a glitch (a non-stationary and non-Gaussian noise feature) in the Livingston detector, which meant that this data weren’t automatically analysed. Nevertheless, GstLAL flagged something interesting in the Hanford data, and there was a mad flurry to get the other data in place so that we could analyse the signal in all three detectors. I remember being sceptical in these first few minutes until I saw the plot of Livingston data which blew me away: the chirp was clearly visible despite the glitch!

Time–frequency plots for GW170104 as measured by Hanford, Livingston and Virgo. The Livinston data have had the glitch removed. The signal is clearly visible in the two LIGO detectors as the upward sweeping chirp; it is not visible in Virgo because of its lower sensitivity and the source’s position in the sky. Figure 1 of the GW170817 Discovery Paper.

Using data from both of our LIGO detectors (as discussed for GW170814, our offline algorithms searching for coalescing binaries only use these two detectors during O2), GW170817 is an absolutely gold-plated detection. GstLAL estimates a false alarm rate (the rate at which you’d expect something at least this signal-like to appear in the detectors due to a random noise fluctuation) of less than one in 1,100,000 years, while PyCBC estimates the false alarm rate to be less than one in 80,000 years.

Parameter estimation (inferring the source properties) used data from all three detectors. We present a (remarkably thorough given the available time) initial analysis in this paper (more detailed results are given in the GW170817 Properties Paper, and the most up-to-date results are in O2 Catalogue Paper). This signal is challenging to analyse because of the glitch and because binary neutron stars are made of stuff™, which can leave an imprint on the waveform. We’ll be looking at the effects of these complications in more detail in the future. Our initial results are

• The source is localized to a region of about $28~\mathrm{deg^2}$ at a distance of $40^{+8}_{-14}~\mathrm{Mpc}$ (we typically quote results at the 90% credible level). This is the closest gravitational-wave source yet.
• The chirp mass is measured to be $1.188_{-0.002}^{+0.004} M_\odot$, much lower than for our binary black hole detections.
• The spins are not well constrained, the uncertainty from this means that we don’t get precise measurements of the individual component masses. We quote results with two choices of spin prior: the astrophysically motivated limit of 0.05, and the more agnostic and conservative upper bound of 0.89. I’ll stick to using the low-spin prior results be default.
• Using the low-spin prior, the component masses are $m_1 = 1.36$$1.60 M_\odot$ and $m_2 = 1.17$$1.36 M_\odot$. We have the convention that $m_1 \geq m_2$, which is why the masses look unequal; there’s a lot of support for them being nearly equal. These masses match what you’d expect for neutron stars.

As mentioned above, neutron stars are made of stuff™, and the properties of this leave an imprint on the waveform. If neutron stars are big and fluffy, they will get tidally distorted. Raising tides sucks energy and angular momentum out of the orbit, making the inspiral quicker. If neutron stars are small and dense, tides are smaller and the inspiral looks like that for tow black holes. For this initial analysis, we used waveforms which includes some tidal effects, so we get some preliminary information on the tides. We cannot exclude zero tidal deformation, meaning we cannot rule out from gravitational waves alone that the source contains at least one black hole (although this would be surprising, given the masses). However, we can place a weak upper limit on the combined dimensionless tidal deformability of $\tilde{\Lambda} \leq 900$. This isn’t too informative, in terms of working out what neutron stars are made from, but we’ll come back to this in the GW170817 Properties Paper and the GW170817 Equation-of-state Paper.

Given the source masses, and all the electromagnetic observations, we’re pretty sure this is a binary neutron star system—there’s nothing to suggest otherwise.

Having observed one (and one one) binary neutron star coalescence in O1 and O2, we can now put better constraints on the merger rate. As a first estimate, we assume that component masses are uniformly distributed between $1 M_\odot$ and $2 M_\odot$, and that spins are below 0.4 (in between the limits used for parameter estimation). Given this, we infer that the merger rate is $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$, safely within our previous upper limit [citation note].

There’s a lot more we can learn from GW170817, especially as we don’t just have gravitational waves as a source of information, and this is explained in the companion papers.

### The Multimessenger Paper

Synopsis: Multimessenger Paper
Read this if: Don’t. Use it too look up which other papers to read.
Favourite part: The figures! It was a truly amazing observational effort to follow-up GW170817

The remarkable thing about this paper is that it exists. Bringing together such a diverse (and competitive) group was a huge effort. Alberto Vecchio was one of the editors, and each evening when leaving the office, he was convinced that the paper would have fallen apart by morning. However, it hung together—the story was too compelling. This paper explains how gravitational waves, short gamma-ray bursts, kilonovae all come from a single source [citation note]. This is the greatest collaborative effort in the history of astronomy.

The paper outlines the discoveries and all of the initial set of observations. If you want to understand the observations themselves, this is not the paper to read. However, using it, you can track down the papers that you do want. A huge amount of care went in to trying to describe how discoveries were made: for example, Fermi observed GRB 170817A independently of the gravitational-wave alert, and we found GW170817 without relying on the GRB alert, however, the communication between teams meant that we took everything much seriously and pushed out alerts as quickly as possible. For more on the history of observations, I’d suggest scrolling through the GCN archive.

The paper starts with an overview of the gravitational-wave observations from the inspiral, then the prompt detection of GRB 170817A, before describing how the gravitational-wave localization enabled discovery of the optical transient AT 2017gfo. This source, in nearby galaxy NGC 4993, was then the subject of follow-up across the electromagnetic spectrum. We have huge amount of photometric and spectroscopy of the source, showing general agreement with models for a kilonova. X-ray and radio afterglows were observed 9 days and 16 days after the merger, respectively [citation note]. No neutrinos were found, which isn’t surprising.

### The GW170817 Gamma-ray Burst Paper

Synopsis: GW170817 Gamma-ray Burst Paper
Read this if: You’re interested in the jets from where short gamma-ray bursts originate or in tests of general relativity
Favourite part: How much science come come from a simple time delay measurement

This joint LIGO–Virgo–FermiINTEGRAL paper combines our observations of GW170817 and GRB 170817A. The result is one of the most contentful of the companion papers.

Detection of GW170817 and GRB 170817A. The top three panels show the gamma-ray lightcurves (first: GBM detectors 1, 2, and 5 for 10–50 keV; second: GBM data for 50–300 keV ; third: the SPI-ACS data starting approximately at 100 keV and with a high energy limit of least 80 MeV), the red line indicates the background.The bottom shows the a time–frequency representation of coherently combined gravitational-wave data from LIGO-Hanford and LIGO-Livingston. Figure 2 of the GW170817 Gamma-ray Burst Paper.

The first item on the to-do list for joint gravitational-wave–gamma-ray science, is to establish that we are really looking at the same source.

From the GW170817 Discovery Paper, we know that its source is consistent with being a binary neutron star system. Hence, there is matter around which can launch create the gamma-rays. The Fermi-GBM and INTEGRAL observations of GRB170817A indicate that it falls into the short class, as hypothesised as the result of a binary neutron star coalescence. Therefore, it looks like we could have the right ingredients.

Now, given that it is possible that the gravitational waves and gamma rays have the same source, we can calculate the probability of the two occurring by chance. The probability of temporal coincidence is $5.0 \times 10^{-6}$, adding in spatial coincidence too, and the probability becomes $5.0 \times 10^{-8}$. It’s safe to conclude that the two are associated: merging binary neutron stars are the source of at least some short gamma-ray bursts!

#### Testing gravity

There is a $\sim1.74\pm0.05~\mathrm{s}$ delay time between the inferred merger time and the gamma-ray burst. Given that signal has travelled for about 85 million years (taking the 5% lower limit on the inferred distance), this is a really small difference: gravity and light must travel at almost exactly the same speed. To derive exact limit you need to make some assumptions about when the gamma-rays were created. We’d expect some delay as it takes time for the jet to be created, and then for the gamma-rays to blast their way out of the surrounding material. We conservatively (and arbitrarily) take a window of the delay being 0 to 10 seconds, this gives

$\displaystyle -3 \times 10^{-15} \leq \frac{v_\mathrm{GW} - v_\mathrm{EM}}{v_\mathrm{EM}} \leq 7 \times 10^{-16}$.

That’s pretty small!

General relativity predicts that gravity and light should travel at the same speed, so I wasn’t too surprised by this result. I was surprised, however, that this result seems to have caused a flurry of activity in effectively ruling out several modified theories of gravity. I guess there’s not much point in explaining what these are now, but they are mostly theories which add in extra fields, which allow you to tweak how gravity works so you can explain some of the effects attributed to dark energy or dark matter. I’d recommend Figure 2 of Ezquiaga & Zumalacárregui (2017) for a summary of which theories pass the test and which are in trouble; Kase & Tsujikawa (2018) give a good review.

Table showing viable (left) and non-viable (right) scalar–tensor theories after discovery of GW170817/GRB 170817A. The theories are grouped as Horndeski theories and (the more general) beyond Horndeski theories. General relativity is a tensor theory, so these models add in an extra scalar component. Figure 2 of Ezquiaga & Zumalacárregui (2017).

We don’t discuss the theoretical implications of the relative speeds of gravity and light in this paper, but we do use the time delay to place bounds for particular on potential deviations from general relativity.

1. We look at a particular type of Lorentz invariance violation. This is similar to what we did for GW170104, where we looked at the dispersion of gravitational waves, but here it is for the case of $\alpha = 2$, which we couldn’t test.
2. We look at the Shapiro delay, which is the time difference travelling in a curved spacetime relative to a flat one. That light and gravity are effected the same way is a test of the weak equivalence principle—that everything falls the same way. The effects of the curvature can be quantified with the parameter $\gamma$, which describes the amount of curvature per unit mass. In general relativity $\gamma_\mathrm{GW} = \gamma_\mathrm{EM} = 1$. Considering the gravitational potential of the Milky Way, we find that $-2.6 \times 10^{-7} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM} \leq 1.2 \times 10 ^{-6}$ [citation note].

As you’d expect given the small time delay, these bounds are pretty tight! If you’re working on a modified theory of gravity, you have some extra checks to do now.

#### Gamma-ray bursts and jets

From our gravitational-wave and gamma-ray observations, we can also make some deductions about the engine which created the burst. The complication here, is that we’re not exactly sure what generates the gamma rays, and so deductions are model dependent. Section 5 of the paper uses the time delay between the merger and the burst, together with how quickly the burst rises and fades, to place constraints on the size of the emitting region in different models. The papers goes through the derivation in a step-by-step way, so I’ll not summarise that here: if you’re interested, check it out.

Isotropic energies (left) and luminosities (right) for all gamma-ray bursts with measured distances. These isotropic quantities assume equal emission in all directions, which gives an upper bound on the true value if we are observing on-axis. The short and long gamma-ray bursts are separated by the standard $T_{90} = 2~\mathrm{s}$ duration. The green line shows an approximate detection threshold for Fermi-GBM. Figure 4 from the GW170817 Gamma-ray Burst Paper; you may have noticed that the first version of this paper contained two copies of the energy plot by mistake.

GRB 170817A was unusually dim [citation note]. The plot above compares it to other gamma-ray bursts. It is definitely in the tail. Since it appears so dim, we think that we are not looking at a standard gamma-ray burst. The most obvious explanation is that we are not looking directly down the jet: we don’t expect to see many off-axis bursts, since they are dimmer. We expect that a gamma-ray burst would originate from a jet of material launched along the direction of the total angular momentum. From the gravitational waves alone, we can estimate that the misalignment angle between the orbital angular momentum axis and the line of sight is $\leq 55~\mathrm{deg}$ (adding in the identification of the host galaxy, this becomes $\leq 28~\mathrm{deg}$ using the Planck value for the Hubble constant and $36~\mathrm{deg}$ with the SH0ES value), so this is consistent with viewing the burst off-axis (updated numbers are given in the GW170817 Properties Paper). There are multiple models for such gamma-ray emission, as illustrated below. We could have a uniform top-hat jet (the simplest model) which we are viewing from slightly to the side, we could have a structured jet, which is concentrated on-axis but we are seeing from off-axis, or we could have a cocoon of material pushed out of the way by the main jet, which we are viewing emission from. Other electromagnetic observations will tell us more about the inclination and the structure of the jet [citation note].

Cartoon showing three possible viewing geometries and jet profiles which could explain the observed properties of GRB 170817A. Figure 5 of the GW170817 Gamma-ray Burst Paper.

Now that we know gamma-ray bursts can be this dim, if we observe faint bursts (with unknown distances), we have to consider the possibility that they are dim-and-close in addition to the usual bright-and-far-away.

The paper closes by considering how many more joint gravitational-wave–gamma-ray detections of binary neutron star coalescences we should expect in the future. In our next observing run, we could expect 0.1–1.4 joint detections per year, and when LIGO and Virgo get to design sensitivity, this could be 0.3–1.7 detections per year.

### The GW170817 Hubble Constant Paper

Synopsis: GW170817 Hubble Constant Paper
Read this if: You have an interest in cosmology
Favourite part: In the future, we may be able to settle the argument between the cosmic microwave background and supernova measurements

The Universe is expanding. In the nearby Universe, this can be described using the Hubble relation

$v_H = H_0 D$,

where $v_H$ is the expansion velocity, $H_0$ is the Hubble constant and $D$ is the distance to the source. GW170817 is sufficiently nearby for this relationship to hold. We know the distance from the gravitational-wave measurement, and we can estimate the velocity from the redshift of the host galaxy. Therefore, it should be simple to combine the two to find the Hubble constant. Of course, there are a few complications…

This work is built upon the identification of the optical counterpart AT 2017gfo. This allows us to identify the galaxy NGC 4993 as the host of GW170817’s source: we calculate that there’s a $4 \times 10^{-5}$ probability that AT 2017gfo would be as close to NGC 4993 on the sky by chance. Without a counterpart, it would still be possible to infer the Hubble constant statistically by cross-referencing the inferred gravitational-wave source location with the ensemble of compatible galaxies in a catalogue (you assign a probability to the source being associated with each galaxy, instead of saying it’s definitely in this one). The identification of NGC 4993 makes things much simpler.

As a first ingredient, we need the distance from gravitational waves. For this, a slightly different analysis was done than in the GW170817 Discovery Paper. We fix the sky location of the source to match that of AT 2017gfo, and we use (binary black hole) waveforms which don’t include any tidal effects. The sky position needs to be fixed, because for this analysis we are assuming that we definitely know where the source is. The tidal effects were not included (but precessing spins were) because we needed results quickly: the details of spins and tides shouldn’t make much difference to the distance. From this analysis, we find the distance is $41^{+6}_{-13}~\mathrm{Mpc}$ if we follow our usual convention of quoting the median at symmetric 90% credible interval; however, this paper primarily quotes the most probable value and minimal (not-necessarily symmmetric) 68.3% credible interval, following this convention, we write the distance as $44^{+3}_{-7}~\mathrm{Mpc}$.

While NGC 4993 being close by makes the relationship for calculating the Hubble constant simple, it adds a complication for calculating the velocity. The motion of the galaxy is not only due to the expansion of the Universe, but because of how it is moving within the gravitational potentials of nearby groups and clusters. This is referred to as peculiar motion. Adding this in increases our uncertainty on the velocity. Combining results from the literature, our final estimate for the velocity is $v_H= 3017 \pm 166~\mathrm{km\,s^{-1}}$.

We put together the velocity and the distance in a Bayesian analysis. This is a little more complicated than simply dividing the numbers (although that gives you a similar result). You have to be careful about writing things down, otherwise you might implicitly assume a prior that you didn’t intend (my most useful contribution to this paper is probably a whiteboard conversation with Will Farr where we tracked down a difference in prior assumptions approaching the problem two different ways). This is all explained in the Methods, it’s not easy to read, but makes sense when you work through. The result is $H_0 = 70^{+12}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (quoted as maximum a posteriori value and 68% interval, or $74^{+33}_{-12}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ in the usual median-and-90%-interval convention). An updated set of results is given in the GW170817 Properties Paper: $H_0 = 70^{+19}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (68% interval using the low-spin prior). This is nicely (and diplomatically) consistent with existing results.

The distance has considerable uncertainty because there is a degeneracy between the distance and the orbital inclination (the angle of the normal to the orbital plane relative to the line of sight). If you could figure out the inclination from another observation, then you could tighten constraints on the Hubble constant, or if you’re willing to adopt one of the existing values of the Hubble constant, you can pin down the inclination. Data (updated data) to help you try this yourself are available [citation note].

Two-dimensional posterior probability distribution for the Hubble constant and orbital inclination inferred from GW170817. The contours mark 68% and 95% levels. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SH0ES). Figure 2 of the GW170817 Hubble Constant Paper.

In the future we’ll be able to combine multiple events to produce a more precise gravitational-wave estimate of the Hubble constant. Chen, Fishbach & Holz (2017) is a recent study of how measurements should improve with more events: we should get to 4% precision after around 100 detections.

### The GW170817 Kilonova Paper

Synopsis: GW170817 Kilonova Paper
Read this if: You want to check our predictions for ejecta against observations
Favourite part: We might be able to create all of the heavy r-process elements—including the gold used to make Nobel Prizes—from merging neutron stars

When two neutron stars collide, lots of material gets ejected outwards. This neutron-rich material undergoes nuclear decay—now no longer being squeezed by the strong gravity inside the neutron star, it is unstable, and decays from the strange neutron star stuff™ to become more familiar elements (elements heavier than iron including gold and platinum). As these r-process elements are created, the nuclear reactions power a kilonova, the optical (infrared–ultraviolet) transient accompanying the merger. The properties of the kilonova depends upon how much material is ejected.

In this paper, we try to estimate how much material made up the dynamical ejecta from the GW170817 collision. Dynamical ejecta is material which escapes as the two neutron stars smash into each other (either from tidal tails or material squeezed out from the collision shock). There are other sources of ejected material, such as winds from the accretion disk which forms around the remnant (whether black hole or neutron star) following the collision, so this is only part of the picture; however, we can estimate the mass of the dynamical ejecta from our gravitational-wave measurements using simulations of neutron star mergers. These estimates can then be compared with electromagnetic observations of the kilonova [citation note].

The amount of dynamical ejecta depends upon the masses of the neutron stars, how rapidly they are rotating, and the properties of the neutron star material (described by the equation of state). Here, we use the masses inferred from our gravitational-wave measurements and feed these into fitting formulae calibrated against simulations for different equations of state. These don’t include spin, and they have quite large uncertainties (we include a 72% relative uncertainty when producing our results), so these are not precision estimates. Neutron star physics is a little messy.

We find that the dynamical ejecta is $10^{-3}$$10^{-2} M_\odot$ (assuming the low-spin mass results). These estimates can be feed into models for kilonovae to produce lightcurves, which we do. There is plenty of this type of modelling in the literature as observers try to understand their observations, so this is nothing special in terms of understanding this event. However, it could be useful in the future (once we have hoverboards), as we might be able to use gravitational-wave data to predict how bright a kilonova will be at different times, and so help astronomers decide upon their observing strategy.

Finally, we can consider how much r-process elements we can create from the dynamical ejecta. Again, we don’t consider winds, which may also contribute to the total budget of r-process elements from binary neutron stars. Our estimate for r-process elements needs several ingredients: (i) the mass of the dynamical ejecta, (ii) the fraction of the dynamical ejecta converted to r-process elements, (iii) the merger rate of binary neutron stars, and (iv) the convolution of the star formation rate and the time delay between binary formation and merger (which we take to be $\propto t^{-1}$). Together (i) and (ii) give the mass of r-process elements per binary neutron star (assuming that GW170817 is typical); (iii) and (iv) give total density of mergers throughout the history of the Universe, and combining everything together you get the total mass of r-process elements accumulated over time. Using the estimated binary neutron star merger rate of $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$, we can explain the Galactic abundance of r-process elements if more than about 10% of the dynamical ejecta is converted.

Present day binary neutron star merger rate density versus dynamical ejecta mass. The grey region shows the inferred 90% range for the rate, the blue shows the approximate range of ejecta masses, and the red band shows the band where the Galactic elemental abundance can be reproduced if at least 50% of the dynamical mass gets converted. Part of Figure 5 of the GW170817 Kilonova Paper.

### The GW170817 Stochastic Paper

Synopsis: GW170817 Stochastic Paper
Read this if: You’re impatient for finding a background of gravitational waves
Favourite part: The background symphony

For every loud gravitational-wave signal, there are many more quieter ones. We can’t pick these out of the detector noise individually, but they are still there, in our data. They add together to form a stochastic background, which we might be able to detect by correlating the data across our detector network.

Following the detection of GW150914, we considered the background due to binary black holes. This is quite loud, and might be detectable in a few years. Here, we add in binary neutron stars. This doesn’t change the picture too much, but gives a more accurate picture.

Binary black holes have higher masses than binary neutron stars. This means that their gravitational-wave signals are louder, and shorter (they chirp quicker and chirp up to a lower frequency). Being louder, binary black holes dominate the overall background. Being shorter, they have a different character: binary black holes form a popcorn background of short chirps which rarely overlap, but binary neutron stars are long enough to overlap, forming a more continuous hum.

The dimensionless energy density at a gravitational-wave frequency of 25 Hz from binary black holes is $1.1_{-0.7}^{+1.2} \times 10^{-9}$, and from binary neutron stars it is $0.7_{-0.6}^{+1.5} \times 10^{-9}$. There are on average $0.06_{-0.04}^{+0.06}$ binary black hole signals in detectors at a given time, and $15_{-12}^{+31}$ binary neutron star signals.

Simulated time series illustrating the difference between binary black hole (green) and binary neutron star (red) signals. Each chirp increases in amplitude until the point at which the binary merges. Binary black hole signals are short, loud chirps, while the longer, quieter binary neutron star signals form an overlapping background. Figure 2 from the GW170817 Stochastic Paper.

To calculate the background, we need the rate of merger. We now have an estimate for binary neutron stars, and we take the most recent estimate from the GW170104 Discovery Paper for binary black holes. We use the rates assuming the power law mass distribution for this, but the result isn’t too sensitive to this: we care about the number of signals in the detector, and the rates are derived from this, so they agree when working backwards. We evolve the merger rate density across cosmic history by factoring in the star formation rate and delay time between formation and merger. A similar thing was done in the GW170817 Kilonova Paper, here we used a slightly different star formation rate, but results are basically the same with either. The addition of binary neutron stars increases the stochastic background from compact binaries by about 60%.

Detection in our next observing run, at a moderate significance, is possible, but I think unlikely. It will be a few years until detection is plausible, but the addition of binary neutron stars will bring this closer. When we do detect the background, it will give us another insight into the merger rate of binaries.

### The GW170817 Progenitor Paper

Synopsis: GW170817 Progenitor Paper
Read this if: You want to know about neutron star formation and supernovae
Favourite part: The Spirography figures

The identification of NGC 4993 as the host galaxy of GW170817’s binary neutron star system allows us to make some deductions about how it formed. In this paper, we simulate a large number of binaries, tracing the later stages of their evolution, to see which ones end up similar to GW170817. By doing so, we learn something about the supernova explosion which formed the second of the two neutron stars.

The neutron stars started life as a pair of regular stars [bonus note]. These burned through their hydrogen fuel, and once this is exhausted, they explode as a supernova. The core of the star collapses down to become a neutron star, and the outer layers are blasted off. The more massive star evolves faster, and goes supernova first. We’ll consider the effects of the second supernova, and the kick it gives to the binary: the orbit changes both because of the rocket effect of material being blasted off, and because one of the components loses mass.

From the combination of the gravitational-wave and electromagnetic observations of GW170817, we know the masses of the neutron star, the type of galaxy it is found in, and the position of the binary within the galaxy at the time of merger (we don’t know the exact position, just its projection as viewed from Earth, but that’s something).

Orbital trajectories of simulated binaries which led to GW170817-like merger. The coloured lines show the 2D projection of the orbits in our model galaxy. The white lines mark the initial (projected) circular orbit of the binary pre-supernova, and the red arrows indicate the projected direction of the supernova kick. The background shading indicates the stellar density. Figure 4 of the GW170817 Progenitor Paper; animated equivalents can be found in the Science Summary.

We start be simulating lots of binaries just before the second supernova explodes. These are scattered at different distances from the centre of the galaxy, have different orbital separations, and have different masses of the pre-supernova star. We then add the effects of the supernova, adding in a kick. We fix then neutron star masses to match those we inferred from the gravitational wave measurements. If the supernova kick is too big, the binary flies apart and will never merge (boo). If the binary remains bound, we follow its evolution as it moves through the galaxy. The structure of the galaxy is simulated as a simple spherical model, a Hernquist profile for the stellar component and a Navarro–Frenk–White profile for the dark matter halo [citation note], which are pretty standard. The binary shrinks as gravitational waves are emitted, and eventually merge. If the merger happens at a position which matches our observations (yay), we know that the initial conditions could explain GW170817.

Inferred progenitor properties: (second) supernova kick velocity, pre-supernova progenitor mass, pre-supernova binary separation and galactic radius at time of the supernova. The top row shows how the properties vary for different delay times between supernova and merger. The middle row compares all the binaries which survive the second supernova compared with the GW170817-like ones. The bottom row shows parameters for GW170817-like binaries with different galactic offsets than the $1.8~\mathrm{kpc}$ to $2.2~\mathrm{kpc}$ range used for GW1708017. The middle and bottom rows assume a delay time of at least $1~\mathrm{Gyr}$. Figure 5 of the GW170817 Progenitor Paper; to see correlations between parameters, check out Figure 8 of the GW170817 Progenitor Paper.

The plot above shows the constraints on the progenitor’s properties. The inferred second supernova kick is $V_\mathrm{kick} \simeq 300_{-200}^{+250}~\mathrm{km\,s^{-1}}$, similar to what has been observed for neutron stars in the Milky Way; the per-supernova stellar mass is $M_\mathrm{He} \simeq 3.0_{-1.5}^{+3.5} M_\odot$ (we assume that the star is just a helium core, with the outer hydrogen layers having been stripped off, hence the subscript); the pre-supernova orbital separation was $R_\odot \simeq 3.5_{-1.5}^{+5.0} R_\odot$, and the offset from the centre of the galaxy at the time of the supernova was $2.0_{-1.5}^{+4.0}~\mathrm{kpc}$. The main strongest constraints come from keeping the binary bound after the supernova; results are largely independent of the delay time once this gets above $1~\mathrm{Gyr}$ [citation note].

As we collect more binary neutron star detections, we’ll be able to deduce more about how they form. If you’re interested more in the how to build a binary neutron star system, the introduction to this paper is well referenced; Tauris et al. (2017) is a detailed (pre-GW170817) review.

### The GW170817 Neutrino Paper

Synopsis: GW170817 Neutrino Paper
Read this if: You want a change from gravitational wave–electromagnetic multimessenger astronomy
Favourite part: There’s still something to look forward to with future detections—GW170817 hasn’t stolen all the firsts. Also this paper is not Abbot et al.

This is a joint search by ANTARES, IceCube and the Pierre Auger Observatory for neutrinos coincident with GW170817. Knowing both the location and the time of the binary neutron star merger makes it easy to search for counterparts. No matching neutrinos were detected.

Neutrino candidates at the time of GW170817. The map is is in equatorial coordinates. The gravitational-wave localization is indicated by the red contour, and the galaxy NGC 4993 is indicated by the black cross. Up-going and down-going regions for each detector are indicated, as detectors are more sensitive to up-going neutrinos, as the Cherenkov detectors are subject to a background from cosmic rays hitting the atmosphere. Figure 1 from the GW170817 Neutrino Paper.

Using the non-detections, we can place upper limits on the neutrino flux. These are summarised in the plots below. Optimistic models for prompt emission from an on axis gamma-ray burst would lead to a detectable flux, but otherwise theoretical predictions indicate that a non-detection is expected. From electromagnetic observations, it doesn’t seem like we are on-axis, so the story all fits together.

90% confidence upper limits on neutrino spectral fluence $F$ per flavour (electron, muon and tau) as a function of energy $E$ in $\pm 500~\mathrm{s}$ window (top) about the GW170817 trigger time, and a $14~\mathrm{day}$ window following GW170817 (bottom). IceCube is also sensitive to MeV neutrinos (none were detected). Fluences are the per-flavour sum of neutrino and antineutrino fluence, assuming equal fluence in all flavours. These are compared to theoretical predictions from Kimura et al. (2017) and Fang & Metzger (2017), scaled to a distance of 40 Mpc. The angles labelling the models are viewing angles in excess of the jet opening angle. Figure 2 from the GW170817 Neutrino paper.

Super-Kamiokande have done their own search for neutrinos, form $3.5~\mathrm{MeV}$ to around $100~\mathrm{PeV}$ (Abe et al. 2018). They found nothing in either the $\pm 500~\mathrm{s}$ window around the event or the $14~\mathrm{day}$ window following it. Similarly BUST looked for muon neutrinos and antineutrinos and found nothing in the $\pm 500~\mathrm{s}$ window around the event, and no excess in the $14~\mathrm{day}$ window following it (Petkov et al. 2019). NOvA looked for neutrinos and cosmic rays $1000~\mathrm{s}$ around the event and found nothing (Acero et al. 2020).

The only post-detection neutrino modelling paper I’ve seen is Biehl, Heinze, &Winter (2017). They model prompt emission from the same source as the gamma-ray burst and find that neutrino fluxes would be $10^{-4}$ of current sensitivity.

### The GW170817 Post-merger Paper

Synopsis: GW170817 Post-merger Paper
Read this if: You are an optimist
Favourite part: We really do check everywhere for signals

Following the inspiral of two black holes, we know what happens next: the black holes merge to form a bigger black hole, which quickly settles down to its final stable state. We have a complete model of the gravitational waves from the inspiral–merger–ringdown life of coalescing binary black holes. Binary neutron stars are more complicated.

The inspiral of two binary neutron stars is similar to that for black holes. As they get closer together, we might see some imprint of tidal distortions not present for black holes, but the main details are the same. It is the chirp of the inspiral which we detect. As the neutron stars merge, however, we don’t have a clear picture of what goes on. Material gets shredded and ejected from the neutron stars; the neutron stars smash together; it’s all rather messy. We don’t have a good understanding of what should happen when our neutron stars merge, the details depend upon the properties of the stuff™ neutron stars are made of—if we could measure the gravitational-wave signal from this phase, we would learn a lot.

There are four plausible outcomes of a binary neutron star merger:

1. If the total mass is below the maximum mass for a (non-rotating) neutron star ($M < M^\mathrm{Static}$), we end up with a bigger, but still stable neutron star. Given our inferences from the inspiral (see the plot from the GW170817 Gamma-ray Burst Paper below), this is unlikely.
2. If the total mass is above the limit for a stable, non-rotating neutron star, but can still be supported by uniform rotation ($M^\mathrm{Static} < M < M^\mathrm{Uniform}$), we have a supramassive neutron star. The rotation will slow down due to the emission of electromagnetic and gravitational radiation, and eventually the neutron star will collapse to a black hole. The time until collapse could take something like $10$$5 \times 10^4~\mathrm{s}$; it is unclear if this is long enough for supramassive neutron stars to have a mid-life crisis.
3. If the total mass is above the limit for support from uniform rotation, but can still be supported through differential rotation and thermal gradients($M^\mathrm{Uniform} < M < M^\mathrm{Differential}$), then we have a hypermassive neutron star. The hypermassive neutron star cools quickly through neutrino emission, and its rotation slows through magnetic braking, meaning that it promptly collapses to a black hole in $\lesssim 1~\mathrm{s}$.
4. If the total mass is big enough($M^\mathrm{Differential} < M$), the merging neutron stars collapse down to a black hole.

In the case of the collapse to a black hole, we get a ringdown as in the case of a binary black hole merger. The frequency is around $6~\mathrm{kHz}$, too high for us to currently measure. However, if there is a neutron star, there may be slightly lower frequency gravitational waves from the neutron star matter wibbling about. We’re not exactly sure of the form of these signals, so we perform an unmodelled search for them (knowing the position of GW170817’s source helps for this).

Comparison of inferred component masses with critical mass boundaries for different equations of state. The left panel shows the maximum mass of a non-rotating neutron star compared to the initial baryonic mass (ignoring material ejected during merger and gravitational binding energy); the middle panel shows the maximum mass for a uniformly rotating neutron star; the right panel shows the maximum mass of a non-rotating neutron star compared of the gravitational mass of the heavier component neutron star. Figure 3 of the GW170817 Gamma-ray Burst Paper.

Several different search algorithms were used to hunt for a post-merger signal:

1. coherent WaveBurst (cWB) was used to look for short duration ($< 1~\mathrm{s}$) bursts. This searched a $2~\mathrm{s}$ window including the merger time and covering the $1.7~\mathrm{s}$ delay to the gamma-ray burst detection, and frequencies of $1024$$4096~\mathrm{Hz}$. Only LIGO data were used, as Virgo data suffered from large noise fluctuations above $2.5~\mathrm{kHz}$.
2. cWB was used to look for intermediate duration ($< 500~\mathrm{s}$) bursts. This searched a $1000~\mathrm{s}$ window from the merger time, and frequencies $24$$2048~\mathrm{Hz}$. This used LIGO and Virgo data.
3. The Stochastic Transient Analysis Multi-detector Pipeline (STAMP) was also used to look for intermediate duration signals. This searched the merger time until the end of O2 (in $500~\mathrm{s}$ chunks), and frequencies $24$$4000~\mathrm{Hz}$. This used only LIGO data. There are two variations of STAMP: Zebragard and Lonetrack, and both are used here.

Although GEO is similar to LIGO and Virgo and the searched high-frequencies, its data were not used as we have not yet studied its noise properties in enough detail. Since the LIGO detectors are the most sensitive, their data is most important for the search.

No plausible candidates were found, so we set some upper limits on what could have been detected. From these, it is not surprising that nothing was found, as we would need pretty much all of the mass of the remnant to somehow be converted into gravitational waves to see something. Results are shown in the plot below. An updated analysis which puts upper limits on the post-merger signal is given in the GW170817 Properties Paper.

Noise amplitude spectral density $\sqrt{S_n}$ for the four detectors, and search upper limits $h_\mathrm{rss}$ as a function of frequency. The noise amplitude spectral densities compare the sensitivities of the detectors. The search upper limits are root-sum-squared strain amplitudes at 50% detection efficiency. The colour code of the upper-limit markers indicates the search algorithm and the shape indicates the waveform injected to set the limits (the frequency is the average for this waveform). The bar mode waveform come from the rapid rotation of the supramassive neutron star leading to it becoming distorted (stretched) in a non-axisymmetric way (Lasky, Sarin & Sammut 2017); the magnetar waveform assumes that the (rapidly rotating) supramassive neutron star’s magnetic field generates significant ellipticity (Corsi & Mészáros 2009); the short-duration merger waveforms are from a selection of numerical simulations (Bauswein et al. 2013; Takami et al. 2015; Kawamura et al. 2016; Ciolfi et al. 2017). The open squares are merger waveforms scaled to the distance and orientation inferred from the inspiral of GW170817. The dashed black lines show strain amplitudes for a narrow-band signal with fixed energy content: the top line is the maximum possible value for GW170817. Figure 1 of the GW170817 Post-merger Paper.

We can’t tell the fate of GW170817’s neutron stars from gravitational waves alone [citation note]. As high-frequency sensitivity is improved in the future, we might be able to see something from a really close by binary neutron star merger.

### The GW170817 Properties Paper

Synopsis: GW170817 Properties Paper
Read this if: You want the best results for GW170817’s source, our best measurement of the Hubble constant, or limits on the post-merger signal
Favourite part: Look how tiny the uncertainties are!

As time progresses, we often refine our analyses of gravitational-wave data. This can be because we’ve had time to recalibrate data from our detectors, because better analysis techniques have been developed, or just because we’ve had time to allow more computationally intensive analyses to finish. This paper is our first attempt at improving our inferences about GW170817. The results use an improved calibration of Virgo data, and analyses more of the signal (down to a low frequency of 23 Hz, instead of 30 Hz, which gives use about an extra 1500 cycles), uses improved models of the waveforms, and includes a new analysis looking at the post-merger signal. The results update those given in the GW170817 Discovery Paper, the GW170817 Hubble Constant Paper and the GW170817 Post-merger Paper.

#### Inspiral

Our initial analysis was based upon quick to calculate post-Newtonian waveform known as TaylorF2. We thought this should be a conservative choice: any results with more complicated waveforms should give tighter results. This worked out. We try several different waveform models, each based upon the point particle waveforms we use for analysing binary black hole signals with extra bits to model the tidal deformation of neutron stars. The results are broadly consistent, so I’ll concentrate on discussing our preferred results calculated using IMRPhenomPNRT waveform (which uses IMRPhenomPv2 as a base and adds on numerical-relativity calibrated tides). As in the GW170817 Discovery Paper, we perform the analysis with two priors on the binary spins, one with spins up to 0.89 (which should safely encompass all possibilities for neutron stars), and one with spins of up to 0.05 (which matches observations of binary neutron stars in our Galaxy).

The first analysis we did was to check the location of the source. Reassuringly, we are still perfectly consistent with the location of AT 2017gfo (phew!). The localization is much improved, the 90% sky area is down to just $16~\mathrm{deg^2}$! Go Virgo!

Having established that it still makes sense that AT 2017gfo pin-points the source location, we use this as the position in subsequent analyses. We always use the sky position of the counterpart and the redshift of the host galaxy (Levan et al. 2017), but we don’t typically use the distance. This is because we want to be able to measure the Hubble constant, which relies on using the distance inferred from gravitational waves.

We use the distance from Cantiello et al. (2018) [citation note] for one calculation: an estimation of the inclination angle. The inclination is degenerate with the distance (both affect the amplitude of the signal), so having constraints on one lets us measure the other with improved precision. Without the distance information, we find that the angle between the binary’s total angular momentum and the line of sight is $152^{+21}_{-27}~\mathrm{deg}$ for the high-spin prior and $146^{+25}_{-27}~\mathrm{deg}$ with the low-spin prior. The difference between the two results is because of the spin angular momentum slightly shifts the direction of the total angular momentum. Incorporating the distance information, for the high-spin prior the angle is $153^{+15}_{-11}~\mathrm{deg}$ (so the misalignment angle is $27^{+11}_{-15}~\mathrm{deg}$), and for the low-spin prior it is $151^{+15}_{-11}~\mathrm{deg}$ (misalignment $29^{+11}_{-15}~\mathrm{deg}$) [citation note].

Estimated orientation and magnitude of the two component spins. The left pair is for the high-spin prior and so magnitudes extend to 0.89, and the right pair are for the low-spin prior and extend to 0.05. In each, the distribution for the more massive component is on the left, and for the smaller component on the right. The probability is binned into areas which have uniform prior probabilities. The low-spin prior truncates the posterior distribution, but this is less of an issue for the high-spin prior. Results are shown at a point in the inspiral corresponding to a gravitational-wave frequency of $100~\mathrm{Hz}$. Parts of Figure 8 and 9 of the GW170817 Properties Paper.

Main results include:

• The luminosity distance is $38.7_{-14.3}^{+7.4}~\mathrm{Mpc}$ with the low-spin prior and $40.8_{-12.3}^{+5.6}~\mathrm{Mpc}$ with the high-spin prior. The difference is for the same reason as the difference in inclination measurements. The results are consistent with the distance to NGC 4993 [citation note].
• The chirp mass redshifted to the detector-frame is measured to be $1.1975^{+0.0001}_{-0.0001} M_\odot$ with the low-spin prior and $1.1976^{+0.0001}_{-0.0001} M_\odot$ with the high-spin. This corresponds to a physical chirp mass of $1.186_{-0.001}^{+0.001} M_\odot$.
• The spins are not well constrained. We get the best measurement along the direction of the orbital angular momentum. For the low-spin prior, this is enough to disfavour the spins being antialigned, but that’s about it. For the high-spin prior, we rule out large spins aligned or antialigned, and very large spins in the plane. The aligned components of the spin are best described by the effective inspiral spin parameter $\chi_\mathrm{eff}$, for the low-spin prior it is $0.00^{+0.02}_{-0.01}$ and for the high-spin prior it is $0.02^{+0.08}_{-0.02}$.
• Using the low-spin prior, the component masses are $m_1 = 1.36$$1.60 M_\odot$ and $m_2 = 1.16$$1.36 M_\odot$, and for the high-spin prior they are $m_1 = 1.36$$1.89 M_\odot$ and $m_2 = 1.00$$1.36 M_\odot$.

These are largely consistent with our previous results. There are small shifts, but the biggest change is that the errors are a little smaller.

Estimated masses for the two neutron stars in the binary using the high-spin (left) and low-spin (right) priors. The two-dimensional plot follows a line of constant chirp mass which is too narrow to resolve on this scale. Results are shown for four different waveform models. TaylorF2 (used in the initial analysis), IMRPhenomDNRT and SEOBNRT have aligned spins, while IMRPhenomPNRT includes spin precession. IMRPhenomPNRT is used for the main results.Figure 5 of the GW170817 Properties Paper.

For the Hubble constant, we find $H_0 = 70^{+19}_{-8}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ with the low-spin prior and $H_0 = 70^{+13}_{-7}~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ with the high-spin prior. Here, we quote maximum a posterior value and narrowest 68% intervals as opposed to the usual median and symmetric 90% credible interval. You might think its odd that the uncertainty is smaller when using the wider high-spin prior, but this is just another consequence of the difference in the inclination measurements. The values are largely in agreement with our initial values.

The best measured tidal parameter is the combined dimensionless tidal deformability $\tilde{\Lambda}$. With the high-spin prior, we can only set an upper bound of $\tilde{\Lambda} < 630$ . With the low-spin prior, we find that we are still consistent with zero deformation, but the distribution peaks away from zero. We have $\tilde{\Lambda} = 300^{+500}_{-190}$ using the usual median and symmetric 90% credible interval, and $\tilde{\Lambda} = 300^{+420}_{-230}$ if we take the narrowest 90% interval. This looks like we have detected matter effects, but since we’ve had to use the low-spin prior, which is only appropriate for neutron stars, this would be a circular argument. More details on what we can learn about tidal deformations and what neutron stars are made of, under the assumption that we do have neutron stars, are given in the GW170817 Equation-of-state Paper.

#### Post-merger

Previously, in the GW170817 Post-merger Paper, we searched for a post-merger signal. We didn’t find anything. Now, we try to infer the shape of the signal, assuming it is there (with a peak within $250~\mathrm{ms}$ of the coalescence time). We still don’t find anything, but now we set much tighter upper limits on what signal there could be there.

For this analysis, we use data from the two LIGO detectors, and from GEO 600! We don’t use Virgo data, as it is not well behaved at these high frequencies. We use BayesWave to try to constrain the signal.

Noise amplitude spectral density for the detectors used, prior and posterior strain upper limits, and selected numerical simulations as a function of frequency. The signal upper limits are Bayesian 90% credible bounds for the signal in Hanford, but is derived from a coherent analysis of all three indicated detectors. Figure 13 of the GW170817 Properties Paper.

While the upper limits are much better, they are still about 12–215 times larger than expectations from simulations. Therefore, we’d need to improve our detector sensitivity by about a factor of 3.5–15 to detect a similar signal. Fingers crossed!

### The GW170817 Equation-of-state Paper

Synopsis: GW170817 Equation-of-state Paper
Read this if: You want to know what neutron stars are made of
Favourite part: The beautiful butterfly plots

Usually in our work, we like to remain open minded and not make too many assumptions. In our analysis of GW170817, as presented in the GW170817 Properties Paper, we have remained agnostic about the components of the binary, seeing what the data tell us. However, from the electromagnetic observations, there is solid evidence that the source is a binary neutron star system. In this paper, we take it as granted that the source is made of two neutron stars, and that these neutron stars are made of similar stuff™ [citation note], to see what we can learn about the properties of neutron stars.

When a two neutron stars get close together, they become distorted by each other’s gravity. Tides are raised, kind of like how the Moon creates tides on Earth. Creating tides takes energy out of the orbit, causing the inspiral to proceed faster. This is something we can measure from the gravitational wave signal. Tides are larger when the neutron stars are bigger. The size of neutron stars and how easy they are the stretch and squash depends upon their equation of state. We can use the measurements of the neutron star masses and amount of tidal deformation to infer their size and their equation of state.

The signal is analysed as in the GW170817 Properties Paper (IMRPhenomPNRT waveform, low-spin prior, position set to match AT 2017gfo). However, we also add in some information about the composition of neutron stars.

Calculating the behaviour of this incredibly dense material is difficult, but there are some relations (called universal relations) between the tidal deformability of neutron stars and their radii which are insensitive to the details of the equation of state. One relates symmetric and antisymmetric combinations of the tidal deformations of the two neutron stars as a function of the mass ratio, allows us to calculate consistent tidal deformations. Another relates the tidal deformation to the compactness (mass divided by radius) allows us to convert tidal deformations to radii. The analysis includes the uncertainty in these relations.

In addition to this, we also use a parametric model of the equation of state to model the tidal deformations. By sampling directly in terms of the equation of state, it is easy to impose constraints on the allowed values. For example, we impose that the speed of sound inside the neutron star is less than the speed of light, that the equation of state can support neutron stars of that mass, that it is possible to explain the most massive confirmed neutron star (we use a lower limit for this mass of $1.97 M_\odot$), as well as it being thermodynamically stable. Accommodating the most massive neutron star turns out to be an important piece of information.

The plot below shows the inferred tidal deformation parameters for the two neutron stars. The two techniques, using the equation-of-state insensitive relations and using the parametrised equation-of-state model without included the constraint of matching the $1.97 M_\odot$ neutron star, give similar results. For a $1.4 M_\odot$ neutron star, these results indicate that the tidal deformation parameter would be $\Lambda_{1.4} = 190^{+390}_{-120}$. We favour softer equations of state over stiffer ones [citation note]. I think this means that neutron stars are more huggable.

Probability distributions for the tidal parameters of the two neutron stars. The tidal deformation of the more massive neutron star $\Lambda_1$ must be greater than that for the smaller neutron star $\Lambda_2$. The green shading and (50% and 90%) contours are calculated using the equation-of-state insensitive relations. The blue contours are for the parametrised equation-of-state model. The orange contours are from the GW170817 Properties Paper, where we don’t assume a common equation of state. The black lines are predictions from a selection of different equations of state Figure 1 of the GW170817 Equation-of-state Paper.

We can translate our results into estimates on the size of the neutron stars. The plots below show the inferred radii. The results for the parametrised equation-of-state model now includes the constraint of accommodating a $1.97 M_\odot$ neutron star, which is the main reason for the difference in the plots. Using the equation-of-state insensitive relations we find that the radius of the heavier ($m_1 = 1.36$$1.62M_\odot$) neutron star is $R_1 = 10.8^{+2.0}_{-1.7}~\mathrm{km}$ and the radius of the lighter ($m_2 = 1.15$$1.36M_\odot$) neutron star is $R_2 = 10.7^{+2.1}_{-1.5}~\mathrm{km}$. With the parametrised equation-of-state model, the radii are $R_1 = 11.9^{+1.4}_{-1.4}~\mathrm{km}$ ($m_1 = 1.36$$1.58M_\odot$) and $R_2 = 11.9^{+1.4}_{-1.4}~\mathrm{km}$ ($m_2 = 1.18$$1.36M_\odot$).

Posterior probability distributions for neutron star masses and radii (blue for the more massive neutron star, orange for the lighter). The left plot uses the equation-of-state insensitive relations, and the right uses the parametrised equation-of-state model. In the one-dimensional plots, the dashed lines indicate the priors. The lines in the top left indicate the size of a Schwarzschild Black hole and the Buchadahl limit for the collapse of a neutron star. Figure 3 of the GW170817 Equation-of-state Paper.

When I was an undergraduate, I remember learning that neutron stars were about $15~\mathrm{km}$ in radius. We now know that’s not the case.

If you want to investigate further, you can download the posterior samples from these analyses.

### Bonus notes

#### Standard sirens

In astronomy, we often use standard candles, objects like type IA supernovae of known luminosity, to infer distances. If you know how bright something should be, and how bright you measure it to be, you know how far away it is. By analogy, we can infer how far away a gravitational-wave source is by how loud it is. It is thus not a candle, but a siren. Sean Carrol explains more about this term on his blog.

#### Nature

I know… Nature published the original Schutz paper on measuring the Hubble constant using gravitational waves; therefore, there’s a nice symmetry in publishing the first real result doing this in Nature too.

#### Globular clusters

Instead of a binary neutron star system forming from a binary of two stars born together, it is possible for two neutron stars to come close together in a dense stellar environment like a globular cluster. A significant fraction of binary black holes could be formed this way. Binary neutron stars, being less massive, are not as commonly formed this way. We wouldn’t expect GW170817 to have formed this way. In the GW170817 Progenitor Paper, we argue that the probability of GW170817’s source coming from a globular cluster is small—for predicted rates, see Bae, Kim & Lee (2014).

Levan et al. (2017) check for a stellar cluster at the site of AT 2017gfo, and find nothing. The smallest 30% of the Milky Way’s globular clusters would evade this limit, but these account for just 5% of the stellar mass in globular clusters, and a tiny fraction of dynamical interactions. Fong et al. (2019) perform some detailed observations looking for a globular cluster, and also find nothing. This excludes a cluster down to $1.3\ times 10^4 M_\odot$, which is basically all (99.996%) of them. Therefore, it’s unlikely that a cluster is the source of this binary.

### Citation notes

#### Merger rates

From our gravitational-wave data, we estimate the current binary neutron star merger rate density is $1540_{-1220}^{+3200}~\mathrm{Gpc^{-3}\,yr^{-1}}$. Several electromagnetic observers performed their own rate estimates from the frequency of detection (or lack thereof) of electromagnetic transients.

Kasliwal et al. (2017) consider transients seen by the Palomar Transient Factory, and estimate a rate density of approximately $320~\mathrm{Gpc^{-3}\,yr^{-1}}$ (3-sigma upper limit of $800~\mathrm{Gpc^{-3}\,yr^{-1}}$), towards the bottom end of our range, but their rate increases if not all mergers are as bright as AT 2017gfo.

Siebert et al. (2017) works out the rate of AT 2017gfo-like transients in the Swope Supernova Survey. They obtain an upper limit of $16000~\mathrm{Gpc^{-3}\,yr^{-1}}$. They use to estimate the probability that AT 2017gfo and GW170817 are just a chance coincidence and are actually unrelated. The probability is $9 \times 10^{-6}$ at 90% confidence.

Smartt et al. (2017) estimate the kilonova rate from the ATLAS survey, they calculate a 95% upper limit of $30000~\mathrm{Gpc^{-3}\,yr^{-1}}$, safely above our range.

Yang et al. (2017) calculates upper limits from the DLT40 Supernova survey. Depending upon the reddening assumed, this is between $93000^{+16000}_{-18000}~\mathrm{Gpc^{-3}\,yr^{-1}}$ and $109000^{+28000}_{-18000}~\mathrm{Gpc^{-3}\,yr^{-1}}$. Their figure 3 shows that this is well above expected rates.

Zhang et al. (2017) is interested in the rate of gamma-ray bursts. If you know the rate of short gamma-ray bursts and of binary neutron star mergers, you can learn something about the beaming angle of the jet. The smaller the jet, the less likely we are to observe a gamma-ray burst. In order to do this, they do their own back-of-the-envelope for the gravitational-wave rate. They get $1100_{-910}^{+2500}~\mathrm{Gpc^{-3}\,yr^{-1}}$. That’s not too bad, but do stick with our result.

If you’re interested in the future prospects for kilonova detection, I’d recommend Scolnic et al. (2017). Check out their Table 2 for detection rates (assuming a rate of $1000~\mathrm{Gpc^{-3}\,yr^{-1}}$): LSST and WFIRST will see lots, about 7 and 8 per year respectively.

Using later observational constraints on the jet structure, Gupta & Bartos (2018) use the short gamma-ray burst rate to estimate a binary neutron star merger rate of $500~\mathrm{Gpc^{-3}\,yr^{-1}}$. They project that around 30% of gravitational-wave detections will be accompanied by gamma-ray bursts, once LIGO and Virgo reach design sensitivity.

Della Valle et al. (2018) calculate an observable kilonova rate of $352_{-281}^{+810}~\mathrm{Gpc^{-3}\,yr^{-1}}$. To match up to our binary neutron star merger rate, we either need only a fraction of binary neutron star mergers to produce kilonova or for them to only be observable for viewing angles of less than $40^\circ$. Their table 2 contains a nice compilation of rates for short gamma-ray bursts.

#### The electromagnetic story

Some notes on an incomplete overview of papers describing the electromagnetic discovery. For observational data, I’d recommend looking at the Open Kilonova Project.

Independently of our gravitational-wave detection, a short gamma-ray burst GRB 170817A was observed by Fermi-GBM (Goldstein et al. 2017). Fermi-LAT did not see anything, as it was offline for crossing through the South Atlantic Anomaly. At the time of the merger, INTEGRAL was following up the location of GW170814, fortunately this meant it could still observe the location of GW170817, and following the alert they found GRB 170817A in their data (Savchenko et al. 2017).

Following up on our gravitational-wave localization, an optical transient AT 2017gfo was discovered. The discovery was made by the One-Meter Two-Hemisphere (1M2H) collaboration using the Swope telescope at the Las Campanas Observatory in Chile; they designated the transient as SSS17a (Coulter et al. 2017). That same evening, several other teams also found the transient within an hour of each other:

• The Distance Less Than 40 Mpc (DLT40) search found the transient using the PROMPT 0.4-m telescope at the Cerro Tololo Inter-American Observatory in Chile; they designated the transient DLT17ck (Valenti et al. 2017).
• The VINROUGE collaboration (I think, they don’t actually identify themselves in their own papers) found the transient using VISTA at the European Southern Observatory in Chile (Tanvir et al. 2017). Their paper also describes follow-up observations with the Very Large Telescope, the Hubble Space Telescope, the Nordic Optical Telescope and the Danish 1.54-m Telescope, and has one of my favourite introduction sections of the discovery papers.
• The MASTER collaboration followed up with their network of global telescopes, and it was their telescope at the San Juan National University Observatory in Argentina which found the transient (Lipunov et al. 2017); they, rather catchily denote the transient as OTJ130948.10-232253.3.
• The Dark Energy Survey and the Dark Energy Camera GW–EM (DES and DECam) Collaboration found the transient with the DECam on the Blanco 4-m telescope, which is also at the Cerro Tololo Inter-American Observatory in Chile (Soares-Santos et al. 2017).
• The Las Cumbres Observatory Collaboration used their global network of telescopes, with, unsurprisingly, their 1-m telescope at the Cerro Tololo Inter-American Observatory in Chile first imaging the transient (Arcavi et al. 2017). Their observing strategy is described in a companion paper (Arcavi et al. 2017), which also describes follow-up of GW170814.

From these, you can see that South America was the place to be for this event: it was night at just the right time.

There was a huge amount of follow-up across the infrared–optical–ultraviolet range of AT 2017gfo. Villar et al. (2017) attempts to bring these together in a consistent way. Their Figure 1 is beautiful.

Assembled lightcurves from ultraviolet, optical and infrared observations of AT 2017gfo. The data points are the homogenized data, and the lines are fitted kilonova models. The blue light initially dominates but rapidly fades, while the red light undergoes a slower decay. Figure 1 of Villar et al. (2017).

Hinderer et al. (2018) use numerical relativity simulations to compare theory and observations for gravitational-wave constraints on the tidal deformation and the kilonova lightcurve. They find that observations could be consistent with a neutron star–black hole binary and well as a binary neutron star. Coughline & Dietrich (2019) come to a similar conclusion. I think it’s unlikely that there would be a black hole this low mass, but it’s interesting that there are some simulations which can fit the observations.

AT 2017gfo was also the target of observations across the electromagnetic spectrum. An X-ray afterglow was observed 9 days post merger, and 16 days post merger, just as we thought the excitement was over, a radio afterglow was found:

The afterglow will continue to brighten for a while, so we can expect a series of updates:

• Pooley, Kumar & Wheeler (2017) observed with Chandra 108 and 111 days post merger. Ruan et al. (2017) observed with Chandra 109 days post merger. The large gap in the X-ray observations from the initial observations is because the Sun got in the way.
• Mooley et al. (2017) update the GROWTH radio results up to 107 days post merger (the largest span whilst still pre-empting new X-ray observations), observing with the Very Large Array, Australia Telescope Compact Array and Giant Meterewave Radio Telescope.

Excitingly, the afterglow has also now been spotted in the optical:

• Lyman et al. (2018) observed with Hubble 110 (rest-frame) days post-merger (which is when the Sun was out of the way for Hubble). At this point the kilonova should have faded away, but they found something, and this is quite blue. The conclusion is that it’s the afterglow, and it will peak in about a year.
• Margutti et al. (2018) brings together Chandra X-ray observations, Very Large Array radio observations and Hubble optical observations. The Hubble observations are 137 days post merger, and the Chandra observations are 153 days and 163 days post-merger. They find that they all agree (including the tentative radio signal at 10 days post-merger). They argue that the emission disfavours on-axis jets and spherical fireballs.

Evolution of radio, optical and X-ray spectral energy density of the counterpart to GW170817. The radio and X-ray are always dominated by the afterglow, as indicated by them following the same power law. At early times, the optical is dominated by the kilonova, but as this fades, the afterglow starts to dominate. Figure 1 of Margutti et al. (2018).

• D’Avanzo et al. (2018) observed in X-ray 135 days post-merger with XMM-Newton. They find that the flux is faded compared to the previous trend. They suggest that we’re just at the turn-over, so this is consistent with the most recent Hubble observations.
• Resmi et al. (2018) observed at low radio frequencies with the Giant Meterwave Radio Telescope. They saw the signal at $1390~\mathrm{MHz}$ after 67 days post-merger, but this evolves little over the duration of their observations (to day 152 post-merger), also suggesting a turn-over.
• Dobie et al. (2018) observed in radio 125–200 days post-merger with the Very Large Array and Australia Telescope Compact Array, and they find that the afterglow is starting to fade, with a peak at 149 ± 2 days post-merger.
• Nynka et al. (2018) made X-ray observations at 260 days post-merger. They conclude the afterglow is definitely fading, and that this is not because of passing of the synchrotron cooling frequency.
• Mooley et al. (2018) observed in radio to 298 days. They find the turn-over around 170 days. They argue that results support a narrow, successful jet.
• Troja et al. (2018) observed in radio and X-ray to 359 days. The fading is now obvious, and starting to reveal something about the jet structure. Their best fits seem to favour a structured relativistic jet or a wide-angled cocoon.
• Lamb et al. (2018) observed in optical to 358 days. They infer a peak around 140–160 days. Their observations are well fit either by a Gaussian structured jet or a two-component jet (with the second component being the cocoon), although the two-component model doesn’t fit early X-ray observations well. They conclude there must have been a successful jet of some form.

Radio, optical and X-ray observations to 358 days after merger. The coloured lines show fitted Gaussian jet models. Figure 3 of Lamb et al. (2018).

• Fong et al. (2019) observe in optical to 584 days post-merger, combined with observation in radio to 585 days post-merger and in X-ray 583 days post-merger. These observations favour a structured jet over a quasi-spherical outflow. Hajela et al. (2019) extend the radio and X-ray observations even further, out to 743 days post-merger.

Left: Optical afterglow observed until 584 days post-merger together with predictions for a structured jet and a quasi-spherical outflow (Wu & MacFadyen 2018). Right: Radio, optical and X-ray observations to 535 days, 534 days and 533 days post-merger-respectively. Triangles denote upper limits. Figures 2 and 3 of Fong et al. (2019).

• Troja et al. (2020) observed with Chandra between 935 and 942 days post-merger, and see a nice decline, consistent with a spreading jet. They also looked in radio, but didn’t find anything.
• Makhathini et al. (2020) compile a uniform set of radio, optical and X-ray afterglow observations. Their data set covers 0.5 to 940 days post-merger. It really is a lovely data set!

Optical, radio and X-ray light-curves, scaled by a best-fit spectral index so that the different observations lie on top of each other, for GW170817’s afterglow. The top panel shows the individual observations, labelled by observatory and observing band. The bottom panel shows a moving average. Figure 1 of Makhathini et al. (2020).

• Balasubramanian et al. (2021) continue to obtain radio and X-ray observations until 1270 days  post-merger. The radio is as expected for a structured jet, but there may be some brighting in the X-ray?
• Hajela et al. (2021) do find that there is a brightening in the X-ray after around 900 days. However, there is nothing in the radio. This could suggest some form of kilonova afterglow (which may argue against a prompt collapse to a black hole), or it could be from accretion onto the remnant. Either would be an interesting observation.
• Troja et al. (2021) reanalyse the X-ray data, checking the calibration. They do not find a rise, but do find an excess at late times that is difficult to explain with just the jet afterglow, suggesting that there is some extra emission like a kilonova afterglow.
• Balasubramanian et al. (2022) perform 3 GHz Very Large Array until 29 March 2022. They no longer detect the radio emission, but instead place an upper limit. This suggests no rebrightening.

X-ray (top) and radio (bottom) observations from Chandra and the Very Large Array, respectively. The X-ray observations show an excess after around 900 days, but their is not sign in radio. The red and orange lines show estimated synchrotron emission for different power laws. The grey curve shows synchrotron emission from the dynamical ejecta of a kilonova from a numerical relativity simulation of a neutron star merger. Figure 2 of Hajela at al. (2021).

The story of the most ambitious cross-over of astronomical observations might now be coming to an end?

#### Shapiro delay

Using the time delay between GW170817 and GRB 170817A, a few other teams also did their own estimation of the Shapiro delay before they knew what was in our GW170817 Gamma-ray Burst Paper.

• Wang et al. (2017) consider the Milky Way potential and large scale structure to estimate $-4 \times 10^{-9} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM}$.
• Boran et al. (2017) consider all the galaxies in the GLADE catalogue which are within a radius of $400~\mathrm{kpc}$ of the line of sight, and derive $|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}| \leq 3.9 \times 10 ^{-9}$.
• Wei et al. (2017) estimate $|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}| \leq 5.9 \times 10 ^{-8}$ using the Milky Way’s potential and $|\gamma_\mathrm{GW} - \gamma_\mathrm{EM}| \leq 9.2 \times 10 ^{-11}$ using the Virgo cluster’s potential.

Our estimate of $-2.6 \times 10^{-7} \leq \gamma_\mathrm{GW} - \gamma_\mathrm{EM} \leq 1.2 \times 10 ^{-6}$ is the most conservative.

#### Comparison to other gamma-ray bursts

Are the electromagnetic counterparts to GW170817 similar to what has been observed before?

Yue et al. (2017) compare GRB 170817A with other gamma-ray bursts. It is low luminosity, but it may not be alone. There could be other bursts like it (perhaps GRB 070923, GRB 080121 and GRB 090417A), if indeed they are from nearby sources. They suggest that GRB 130603B may be the on-axis equivalent of GRB 170817A [citation note]; however, the non-detection of kilonovae for several bursts indicates that there needs to be some variation in their properties too. This agree with the results of Gompertz et al. (2017), who compares the GW170817 observations with other kilonovae: it is fainter than the other candidate kilonovae (GRB 050709, GRB 060614, GRB 130603B and tentatively GRB 160821B), but equally brighter than upper limits from other bursts. There must be a diversity in kilonovae observations. Fong et al. (2017) look at the diversity of afterglows (across X-ray to radio), and again find GW170817’s counterpart to be faint. This is probably because we are off-axis. The most comprehensive study is von Kienlin et al. (2019) who search ten years of Fermi archives and find 13 GRB 170817A-like short gamma-ray bursts: GRB 081209A, GRB 100328A, GRB 101224A, GRB 110717A; GRB 111024C, GRB 120302B, GRB 120915A, GRB 130502A, GRB 140511A, GRB 150101B, GRB 170111B, GRB 170817A and GRB 180511A. There is a range behaviours in these, with the shorter GRBs showing fast variability. Future observations will help unravel how much variation there is from viewing different angles, and how much intrinsic variation there is from the source—perhaps some short gamma-ray bursts come from neutron star–black hole binaries?

#### Inclination, jets and ejecta

Pretty much every observational paper has a go at estimating the properties of the ejecta, the viewing angle or something about the structure of the jet. I may try to pull these together later, but I’ve not had time yet as it is a very long list! Most of the inclination measurements assumed a uniform top-hat jet, which we now know is not a good model.

In my non-expert opinion, the later results seem more interesting. With very-long baseline interferometry radio observations to 230 days post-merger, Mooley et al. (2018) claim that while the early radio emission was powered by the wide cocoon of a structured jet, the later emission is dominated by a narrow, energetic jet. There was a successful jet, so we would have seen something like a regular short gamma-ray burst on axis. They estimate that the jet opening angle is $< 5~\mathrm{deg}$, and that we are viewing it at an angle of $20 \pm 5~\mathrm{deg}$. With X-ray and radio observations to 359 days, Troja et al. (2018) estimate (folding in gravitational-wave constraints too) that the viewing angle is $22 \pm 6~\mathrm{deg}$, and the width of a Gaussian structured jet would be $3.4 \pm 1.1~\mathrm{deg}$.

#### Hubble constant and misalignment

Guidorzi et al. (2017) try to tighten the measurement of the Hubble constant by using radio and X-ray observations. Their modelling assumes a uniform jet, which doesn’t look like a currently favoured option [citation note], so there is some model-based uncertainty to be included here. Additionally, the jet is unlikely to be perfectly aligned with the orbital angular momentum, which may add a couple of degrees more uncertainty.

Mandel (2018) works the other way and uses the recent Dark Energy Survey Hubble constant estimate to bound the misalignment angle to less than $28~\mathrm{deg}$, which (unsurprisingly) agrees pretty well with the result we obtained using the Planck value. Finstad et al. (2018) uses the luminosity distance from Cantiello et al. (2018) [citation note] as a (Gaussian) prior for an analysis of the gravitational-wave signal, and get a misalignment $32^{+10}_{-13}\pm 2~\mathrm{deg}$ (where the errors are statistical uncertainty and an estimate of systematic error from calibration of the strain).

Hotokezaka et al. (2018) use the inclination results from Mooley et al. (2018) [citation note] (together with the updated posterior samples from the GW170817 Properties Paper) to infer a value of $h = 0.689^{+0.047}_{-0.046}$ (quoting median and 68% symmetric credible interval). Using different jet models changes their value for the Hubble constant a little; the choice of spin prior does not (since we get basically all of the inclination information from their radio observations). The results is still consistent with Planck and SH0ES, but is closer to the Planck value.

Posterior probability distribution for the Hubble constant inferred from GW170817 using only gravitational waves (GWs), and folding in models for the power-law jet (PLJ) model and very-long baseline interferometry (VLBI) radio observations. The lines symmetric mark 68% intervals. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SH0ES). Figure 2 of Hotokezaka et al. (2018)

Dhawan et al. (2019) use broadband photometry of the kilonova to estimate the observation angle as $32.5^{+11.7}_{-9.7}~\mathrm{deg}$. Combining this with results from the Hubble Constant Paper they find $h = 0.724^{+0.079}_{-0.073}$.

#### NGC 4993 properties

In the GW170817 Progenitor Paper we used component properties for NGC 4993 from Lim et al. (2017): a stellar mass of $(10^{10.454}/h^2) M_\odot$ and a dark matter halo mass of $(10^{12.2}/h) M_\odot$, where we use the Planck value of $h = 0.679$ (but conclusions are similar using the SH0ES value for this).

Blanchard et al. (2017) estimate a stellar mass of about $\log(M_\ast/M_\odot) = 10.65^{+0.03}_{-0.03}$. They also look at the star formation history, 90% were formed by $6.8^{+2.2}_{-0.8}~\mathrm{Gyr}$ ago, and the median mass-weighted stellar age is $13.2^{+0.5}_{-0.9}~\mathrm{Gyr}$. From this they infer a merger delay time of $6.8$$13.6~\mathrm{Gyr}$. From this, and assuming that the system was born close to its current location, they estimate that the supernova kick $V_\mathrm{kick} \leq 200~\mathrm{km\,s^{-1}}$, towards the lower end of our estimate. They use $h = 0.677$.

Im et al. (2017) find a mean stellar mass of $0.3$$1.2 \times 10^{11} M_\odot$ and the mean stellar age is greater than about $3~\mathrm{Gyr}$. They also give a luminosity distance estimate of $38.4 \pm 8.9~\mathrm{Mpc}$, which overlaps with our gravitational-wave estimate. I’m not sure what value of $h$ they are using.

Levan et al. (2017) suggest a stellar mass of around $1.4 \times 10^{11} M_\odot$. They find that 60% of stars by mass are older than $5~\mathrm{Gyr}$ and that less than 1% are less than $0.5~\mathrm{Gyr}$ old. Their Figure 5 has some information on likely supernova kicks, they conclude it was probably small, but don’t quantify this. They use $h = 0.696$.

Pan et al. (2017) find $\log(M_\ast/M_\odot) = 10.49^{+0.08}_{-0.20}$. They calculate a mass-weighted mean stellar age of $10.97~\mathrm{Gyr}$ and a likely minimum age for GW170817’s source system of $2.8~\mathrm{Gyr}$. They use $h = 0.7$.

Troja et al. (2017) find a stellar mass of $\log(M_\ast/M_\odot) \sim 10.88$, and suggest an old stellar population of age $> 2~\mathrm{Gyr}$.

Ebrová & Bílek (2018) assume a distance of $41.0~\mathrm{kpc}$ and find a halo mass of $1.939 \times 10^{12} M_\odot$. They suggest that NGC 4993 swallowed a smaller late-type galaxy somewhere between $0.2~\mathrm{Gyr}$ and $1~\mathrm{Gyr}$ ago, most probably around $0.4~\mathrm{Gyr}$ ago.

The consensus seems to be that the stellar population is old (and not much else). Fortunately, the conclusions of the GW170817 Progenitor Paper are pretty robust for delay times longer than $1~\mathrm{Gyr}$ as seems likely.

A couple of other papers look at the distance of the galaxy:

• Hjoth et al. (2017) combine a redshift measurement from MUSE, and a fundamental plane estimate based upon Hubble observations, to obtain an distance of $41.0 \pm 3.1~\mathrm{Mpc}$.
• Cantiello et al. (2018) use Hubble observations to estimate the distance using surface brightness fluctuations. They obtain a distance of $40.7 \pm 1.4 \pm 1.9~\mathrm{Mpc}$. This implies a value for the Hubble constant of $h = 0.719 \pm 0.071$.

The values are consistent with our gravitational-wave estimates.

#### The remnant’s fate

We cannot be certain what happened to the merger remnant from gravitational-wave observations alone. However, electromagnetic observations do give some hints here.

Evans et al. (2017) argue that their non-detection of X-rays when observing with Swift and NuSTAR indicates that there is no neutron star remnant at this point, meaning we must have collapsed to form a black hole by 0.6 days post-merger. This isn’t too restricting in terms of the different ways the remnant could collapse, but does exclude a stable neutron star remnant. MAXI also didn’t detect any X-rays 4.6 hours after the merger (Sugita et al. 2018).

Pooley, Kumar & Wheeler (2017) consider X-ray observations of the afterglow. They calculate that if the remnant was a hypermassive neutron star with a large magnetic field, the early (10 day post-merger) luminosity would be much higher (and we could expect to see magnetar outbursts). Therefore, they think it is more likely that the remnant is a black hole. However, Piro et al. (2018) suggest that if the spin-down of the neutron star remnant is dominated by losses due to gravitational wave emission, rather than electromagnetic emission, then the scenario is still viable. They argue that a tentatively identified X-ray flare seen 155 days post-merger, could be evidence of dissipation of the neutron star’s toroidal magnetic field.

Kasen et al. (2017) use the observed red component of the kilonova to argue that the remnant must have collapsed to a black hole in $< 10~\mathrm{ms}$. A neutron star would irradiate the ejecta with neutrinos, lower the neutron fraction and making the ejecta bluer. Since it is red, the neutrino flux must have been shut off, and the neutron star must have collapsed. We are in case b in their figure below.

Cartoon of the different components of matter ejected from neutron star mergers. Red colours show heavy r-process elements and blue colours light r-process elements. There is a tidal tail of material forming a torus in the orbital plane, roughly spherical winds from the accretion disk, and material squeezed into the polar reasons during the collision. In case a, we have a long-lived neutron star, and its neutrino irradiation leads to blue ejecta. In case b the neutron star collapses, cutting off the neutrino flux. In case c, there is a neutron star–black hole merger, and we don’t have the polar material from the collision. Figure 1 of Kasen et al. (2017); also see Figure 1 of Margalit & Metzger (2017).

Ai et al. (2018) find that there are some corners of parameter space for certain equations of state where a long-lived neutron star is possible, even given the observations. Therefore, we should remain open minded.

Margalit & Metzger (2017) and Bauswein et al. (2017) note that the relatively large amount of ejecta inferred from observations [citation note] is easier to explain when there is delayed (on timescales of $> 10~\mathrm{ms}$). This is difficult to resolve unless neutron star radii are small ($\lesssim 11~\mathrm{km}$). Metzger, Thompson & Quataert (2018) derive how this tension could be resolved if the remnant was a rapidly spinning magnetar with a life time of $0.1$$1~\mathrm{s}$Matsumoto et al. (2018), suggest that the optical emission is powered by the jet and material accreting onto the central object, rather than r-process decay, and this permits much smaller amounts of ejecta, which could also solve the issue. Yu & Dai (2017) suggest that accretion onto a long-lived neutron star could power the emission, and would only require a single opacity for the ejecta. Li et al. (2018) put forward a similar theory, arguing that both the high ejecta mass and low opacity are problems for the standard r-process explanation, but fallback onto a neutron star could work. However, Margutti et al. (2018) say that X-ray emission powered by a central engine is disfavoured at all times.

In conclusion, it seems probable that we ended up with a black hole, and we had an a unstable neutron star for a short time after merger, but I don’t think it’s yet settled how long this was around.

Gill, Nathanail & Rezzolla (2019) considered how long it would take to produce the observed amount of ejecta, and the relative amounts of red and blue eject, as well as the delay time between the gravitational-wave measurement of the merger and the observation of the gamma-ray burst, to estimate how long it took the remnant to collapse to a black hole. They find a lifetime of $= 0.98^{+0.31}_{-0.26}~\mathrm{s}$.

#### Twin stars

We might not have two neutron stars with the same equation of state if they can undergo a phase transition. This would be kind of of like if one one made up of fluffer marshmallow, and the other was made up of gooey toasted marshmallow: they have the same ingredient, but in one the type of stuff has changed, giving it different physical properties. Standard neutron stars could be made of hadronic matter, kind of like a giant nucleus, but we could have another type where the hadrons break down into their component quarks. We could therefore have two neutron stars with similar masses but with very different equations of state. This is referred to as the twin star scenario. Hybrid stars which have quark cores surrounded by hadronic outer layers are often discussed in this context.

#### Neutron star equation of state

Several papers have explored what we can deduce about the nature of neutron star stuff™ from gravitational wave or electromagnetic observations the neutron star coalescence. It is quite a tricky problem. Below are some investigations into the radii of neutron stars and their tidal deformations; these seem compatible with the radii inferred in the GW170817 Equation-of-state Paper.

Bauswein et al. (2017) argue that the amount of ejecta inferred from the kilonova is too large for there to have been a prompt collapse to a black hole [citation note]. Using this, they estimate that the radius of a non-rotating neutron star of mass $1.6~\mathrm{M_\odot}$ has a radius of at least $10.68_{-0.04}^{+0.15}~\mathrm{km}$. They also estimate that the radius for the maximum mass nonrotating neutron star must be greater than $9.60_{-0.03}^{+0.14}~\mathrm{km}$. Köppel, Bovard & Rezzolla (2019) calculate a similar, updated analysis, using a new approach to fit for the maximum mass of a neutron star, and they find a radius for $1.6~\mathrm{M_\odot}$ is greater than  $10.90~\mathrm{km}$, and for $1.4~\mathrm{M_\odot}$  is greater than $10.92~\mathrm{km}$.

Annala et al. (2018) combine our initial measurement of the tidal deformation, with the requirement hat the equation of state supports a $2 M_\odot$ neutron star (which they argue requires that the tidal deformation of a $1.4 M_\odot$ neutron star is at least $120$). They argue that the latter condition implies that the radius of a $1.4 M_\odot$ neutron star is at least $9.9~\mathrm{km}$ and the former that it is less than $13.6~\mathrm{km}$.

Radice et al. (2018) combine together observations of the kilonova (the amount of ejecta inferred) with gravitational-wave measurements of the masses to place constraints on the tidal deformation. From their simulations, they argue that to explain the ejecta, the combined dimensionless tidal deformability must be $\tilde{\Lambda} > 400$. This is consistent with results in the GW170817 Properties Paper, but would eliminate the main peak of the distribution we inferred from gravitational waves alone. However, Kuichi et al. (2019) show that it is possible to get the required ejecta for smaller tidal deformations, depending upon assumptions about the maximum neutron star mass (higher masses allow smaller tidal deformations)mand asymmetry of the binary components.

Lim & Holt (2018) perform some equation-of-state calculations. They find that their particular method (chiral effective theory) is already in good agreement with estimates of the maximum neutron star mass and tidal deformations. Which is nice. Using their models, they predict that for GW170817’s chirp mass $\tilde{\Lambda} = 532^{+106}_{-119}$.

Raithel, Özel & Psaltis (2018) argue that for a given chirp mass, $\tilde{\Lambda}$ is only a weak function of component masses, and depends mostly on the radii. Therefore, from our initial inferred value, they put a 90% upper limit on the radii of $13~\mathrm{km}$.

Most et al. (2018) consider a wide range of parametrised equations of state. They consider both hadronic (made up of particles like neutrons and protons) equation of states, and ones where they undergo phase transitions (with hadrons breaking into quarks), which could potentially mean that the two neutron stars have quite different properties [citation note]. A number of different constraints are imposed, to give a selection of potential radius ranges. Combining the requirement that neutron stars can be up to $2.01 M_\odot$ (Antoniadis et al. 2013), the maximum neutron star mass of $2.17 M_\odot$ inferred by Margalit & Metzger (2017), our initial gravitational-wave upper limit on the tidal deformation and the lower limit from Radice et al. (2018), they estimate that the radius of a $1.4 M_\odot$ neutron star is $12.00$$13.45~\mathrm{km}$ for the hadronic equation of state. For the equation of state with the phase transition, they do the same, but without the tidal deformation from Radice et al. (2018), and find the radius of a $1.4 M_\odot$ neutron star is $8.53$$13.74~\mathrm{km}$.

Paschalidis et al. (2018) consider in more detail the idea equations of state with hadron–quark phase transitions, and the possibility that one of the components of GW170817’s source was a hadron–quark hybrid star. They find that the initial tidal measurements are consistent with this.

Burgio et al. (2018) further explore the possibility that the two binary components have different properties. They consider both there being a hadron–quark phase transition, and also that one star is hadronic and the other is a quark star (made up of deconfined quarks, rather than ones packaged up inside hadrons). X-ray observations indicate that neutron stars have radii in the range $9.9$$11.2~\mathrm{km}$, whereas most of the radii inferred for GW170817’s components are larger. This paper argues that this can be resolved if one of the components of GW170817’s source was a hadron–quark hybrid star or a quark star.

De et al. (2018) perform their own analysis of the gravitational signal, with a variety of different priors on the component masses. They assume that the two neutron stars have the same radii. In the GW170817 Equation-of-state Paper we find that the difference can be up to about $2~\mathrm{km}$, which I think makes this an OK approximation; Zhao & Lattimer (2018) look at this in more detail. Within their approximation, they estimate the neutron stars to have a common radius of $8.9$$13.2~\mathrm{km}$.

Malik et al. (2018) use the initial gravitational-wave upper bound on tidal deformation and the lower bound from Radice et al. (2018) in combination with several equations of state (calculated using relativistic mean field and of Skyrme Hartree–Fock recipes, which sound delicious). For a $1.4 M_\odot$ neutron star, they obtain a tidal deformation in the range $344$$859$ and the radius in the range $11.82$$13.72~\mathrm{km}$.

Radice & Dai (2018) do their own analysis of our gravitational-wave data (using relative binning) and combine this with an analysis of the electromagnetic observations using models for the accretion disc. They find that the areal radius of a $1.4 M_\odot$ is $12.2^{+1.0}_{-0.8} \pm 0.2~\mathrm{km}$. These results are in good agreement with ours, their inclusion of electromagnetic data pushes their combined results towards larger values for the tidal deformation.

Montaña et al. (2018) consider twin star scenarios [citation note] where we have a regular hadronic neutron star and a hybrid hadron–quark star. They find the data are consistent with neutron star–neutron star, neutron star–hybrid star or hybrid star–hybrid star binaries. Their Table II is a useful collection of results for the radius of a  $1.4 M_\odot$ neutron star, including the possibility of phase transitions.

Coughlin et al. (2018) use our LIGO–Virgo results and combine them with constraints from the observation of the kilonova (combined with fits to numerical simulations) and the gamma-ray burst. The electromagnetic observations give some extra information of the tidal deformability, mass ratio and inclination. They use the approximation that the neutron stars have equal radii. They find that the tidal deformability $\tilde{\Lambda}$ has a 90% interval $279$$822$ and the neutron star radius is $11.1$$13.4~\mathrm{km}$.

Zhou, Chen & Zhang (2019) use data from heavy ion collider experiments, which constrains the properties of nuclear density stuff™ at one end of the spectrum, the existence of $2 M_\odot$ neutron stars, and our GW170817 Equation-of-state Paper constraints on the tidal deformation to determine that the radius of a $1.4 M_\odot$ neutron star is $11.1$$13.3~\mathrm{km}$.

Kumar & Landry (2019) use the GW170817 Equation-of-state Paper constraints, and combine these of electromagnetic constraints to get an overall tidal deformability measurement. They use of observations of X-ray bursters from Özel et al. (2016) which give mass and radius measurements, and translate these using universal relations. Their overall result is the tidal deformability of a $1.4 M_\odot$ neutron star is $112^{+46}_{-33}$.

Gamba, Read & Wade (2019) estimate the systematic error in the GW170817 Equation-of-state Paper results for the neutron star radius which may have been introduced from assumptions about the crust’s equation of state. They find that the error could be $0.3~\mathrm{km}$ (about 3%).