# Second star to the right and straight on ’til morning—Astrophysics white papers

What will be the next big thing in astronomy? One of the hard things about research is that you often don’t know what you will discover before you embark on an investigation. An idea might work out, or it might not, or along the way you might discover something unexpected which is far more interesting. As you might imagine, this can make laying definite plans difficult…

However, it is important to have plans for research. While you might not be sure of the outcome, it is necessary to weigh the risks and rewards associated with the probable results before you invest your time and taxpayers’ money!

To help with planning and prioritising, researchers in astrophysics often pull together white papers [bonus note]. These are sketches of ideas for future research, arguing why you think they might be interesting. These can then be discussed within the community to help shape the direction of the field. If other scientists find the paper convincing, you can build support which helps push for funding. If there are gaps in the logic, others can point these out to ave you heading the wrong way. This type of consensus building is especially important for large experiments or missions—you don’t want to spend a billion dollars on something unless you’re really sure it is a good idea and lots of people agree.

I have been involved with a few white papers recently. Here are some key ideas for where research should go.

### Ground-based gravitational-wave detectors: The next generation

We’ve done some awesome things with Advanced LIGO and Advanced Virgo. In just a couple of years we have revolutionized our understanding of binary black holes. That’s not bad. However, our current gravitational-wave observatories are limited in what they can detect. What amazing things could we achieve with a new generation of detectors?

It can take decades to develop new instruments, therefore it’s important to start thinking about them early. Obviously, what we would most like is an observatory which can detect everything, but that’s not feasible. In this white paper, we pick the questions we most want answered, and see what the requirements for a new detector would be. A design which satisfies these specifications would therefore be a solid choice for future investment.

Binary black holes are the perfect source for ground-based detectors. What do we most want to know about them?

1. How many mergers are there, and how does the merger rate change over the history of the Universe? We want to know how binary black holes are made. The merger rate encodes lots of information about how to make binaries, and comparing how this evolves compared with the rate at which the Universe forms stars, will give us a deeper understanding of how black holes are made.
2. What are the properties (masses and spins) of black holes? The merger rate tells us some things about how black holes form, but other properties like the masses, spins and orbital eccentricity complete the picture. We want to make precise measurements for individual systems, and also understand the population.
3. Where do supermassive black holes come from? We know that stars can collapse to produce stellar-mass black holes. We also know that the centres of galaxies contain massive black holes. Where do these massive black holes come from? Do they grow from our smaller black holes, or do they form in a different way? Looking for intermediate-mass black holes in the gap in-between will tells us whether there is a missing link in the evolution of black holes.

The detection horizon (the distance to which sources can be detected) for Advanced LIGO (aLIGO), its upgrade A+, and the proposed Cosmic Explorer (CE) and Einstein Telescope (ET). The horizon is plotted for binaries with equal-mass, nonspinning components. Adapted from Hall & Evans (2019).

What can we do to answer these questions?

1. Increase sensitivity! Advanced LIGO and Advanced Virgo can detect a $30 M_\odot + 30 M_\odot$ binary out to a redshift of about $z \approx 1$. The planned detector upgrade A+ will see them out to redshift $z \approx 2$. That’s pretty impressive, it means we’re covering 10 billion years of history. However, the peak in the Universe’s star formation happens at around $z \approx 2$, so we’d really like to see beyond this in order to measure how the merger rate evolves. Ideally we would see all the way back to cosmic dawn at $z \approx 20$ when the Universe was only 200 million years old and the first stars light up.
2. Increase our frequency range! Our current detectors are limited in the range of frequencies they can detect. Pushing to lower frequencies helps us to detect heavier systems. If we want to detect intermediate-mass black holes of $100 M_\odot$ we need this low frequency sensitivity. At the moment, Advanced LIGO could get down to about $10~\mathrm{Hz}$. The plot below shows the signal from a $100 M_\odot + 100 M_\odot$ binary at $z = 10$. The signal is completely undetectable at $10~\mathrm{Hz}$.

The gravitational wave signal from the final stages of inspiral, merger and ringdown of a two 100 solar mass black holes at a redshift of 10. The signal chirps up in frequency. The colour coding shows parts of the signal above different frequencies. Part of Figure 2 of the Binary Black Holes White Paper.

3. Increase sensitivity and frequency range! Increasing sensitivity means that we will have higher signal-to-noise ratio detections. For these loudest sources, we will be able to make more precise measurements of the source properties. We will also have more detections overall, as we can survey a larger volume of the Universe. Increasing the frequency range means we can observe a longer stretch of the signal (for the systems we currently see). This means it is easier to measure spin precession and orbital eccentricity. We also get to measure a wider range of masses. Putting the improved sensitivity and frequency range together means that we’ll get better measurements of individual systems and a more complete picture of the population.

How much do we need to improve our observatories to achieve our goals? To quantify this, lets consider the boost in sensitivity relative to A+, which I’ll call $\beta_\mathrm{A+}$. If the questions can be answered with $\beta_\mathrm{A+} = 1$, then we don’t need anything beyond the currently planned A+. If we need a slightly larger $\beta_\mathrm{A+}$, we should start investigating extra ways to improve the A+ design. If we need much larger $\beta_\mathrm{A+}$, we need to think about new facilities.

The plot below shows the boost necessary to detect a binary (with equal-mass nonspinning components) out to a given redshift. With a boost of $\beta_\mathrm{A+} = 10$ (blue line) we can survey black holes around $10 M_\odot$$30 M_\odot$ across cosmic time.

The boost factor (relative to A+) $\beta_\mathrm{A+}$ needed to detect a binary with a total mass $M$ out to redshift $z$. The binaries are assumed to have equal-mass, nonspinning components. The colour scale saturates at $\log_{10} \beta_\mathrm{A+} = 4.5$. The blue curve highlights the reach at a boost factor of $\beta_\mathrm{A+} = 10$. The solid and dashed white lines indicate the maximum reach of Cosmic Explorer and the Einstein Telescope, respectively. Part of Figure 1 of the Binary Black Holes White Paper.

The plot above shows that to see intermediate-mass black holes, we do need to completely overhaul the low-frequency sensitivity. What do we need to detect a $100 M_\odot + 100 M_\odot$ binary at $z = 10$? If we parameterize the noise spectrum (power spectral density) of our detector as $S_n(f) = S_{10}(f/10~\mathrm{Hz})^\alpha$ with a lower cut-off frequency of $f_\mathrm{min}$, we can investigate the various possibilities. The plot below shows the possible combinations of parameters which meet of requirements.

Requirements on the low-frequency noise power spectrum necessary to detect an optimally oriented intermediate-mass binary black hole system with two 100 solar mass components at a redshift of 10. Part of Figure 2 of the Binary Black Holes White Paper.

To build up information about the population of black holes, we need lots of detections. Uncertainties scale inversely with the square root of the number of detections, so you would expect few percent uncertainty after 1000 detections. If we want to see how the population evolves, we need these many per redshift bin! The plot below shows the number of detections per year of observing time for different boost factors. The rate starts to saturate once we detect all the binaries in the redshift range. This is as good as you’ll ever going to get.

Expected rate of binary black hole detections $R_\mathrm{det}$ per redshift bin as a function of A+ boost factor $\beta_\mathrm{A+}$ for three redshift bins. The merging binaries are assumed to be uniformly distributed with a constant merger rate roughly consistent with current observations: the solid line is about the current median, while the dashed and dotted lines are roughly the 90% bounds. Figure 3 of the Binary Black Holes White Paper.

Looking at the plots above, it is clear that A+ is not going to satisfy our requirements. We need something with a boost factor of $\beta_\mathrm{A+} = 10$: a next-generation observatory. Both the Cosmic Explorer and Einstein Telescope designs do satisfy our goals.

Title: Deeper, wider, sharper: Next-generation ground-based gravitational-wave observations of binary black holes
arXiv:
1903.09220 [astro-ph.HE]
Theme music: Daft Punk

### Extreme mass ratio inspirals are awesome

We have seen gravitational waves from a stellar-mass black hole merging with another stellar-mass black hole, can we observe a stellar-mass black hole merging with a massive black hole? Yes, these are a perfect source for a space-based gravitational wave observatory. We call these systems extreme mass-ratio inspirals (or EMRIs, pronounced em-rees, for short) [bonus note].

Having such an extreme mass ratio, with one black hole much bigger than the other, gives EMRIs interesting properties. The number of orbits over the course of an inspiral scales with the mass ratio: the more extreme the mass ratio, the more orbits there are. Each of these gives us something to measure in the gravitational wave signal.

A short section of an orbit around a spinning black hole. While inspirals last for years, this would represent only a few hours around a black hole of mass $M = 10^6 M_\odot$. The position is measured in terms of the gravitational radius $r_\mathrm{g} = GM/c^2$. The innermost stable orbit for this black hole would be about $r_\mathrm{g} = 2.3$. Part of Figure 1 of the EMRI White Paper.

As EMRIs are so intricate, we can make exquisit measurements of the source properties. These will enable us to:

• Measure the masses of both black holes to better than 10% precision
• Reconstruct the mass distribution of massive black holes out to redshift $z \gtrsim 4$
• Measure massive black hole spins to a precision of better than 0.001, giving us an insight into how they formed
• Perform precision tests of the no-hair theorem describing black holes in general relativity, and test alternative theories of gravity in the strong-field regime
• Cross-correlate locations with galaxy catalogues to measure the expansion of the Universe

Event rates for EMRIs are currently uncertain: there could be just one per year or thousands. From the rate we can figure out the details of what is going in in the nuclei of galaxies, and what types of objects you find there.

With EMRIs you can unravel mysteries in astrophysics, fundamental physics and cosmology.

Have we sold you that EMRIs are awesome? Well then, what do we need to do to observe them? There is only one currently planned mission which can enable us to study EMRIs: LISA. To maximise the science from EMRIs, we have to support LISA.

As an aspiring scientist, Lisa Simpson is a strong supporter of the LISA mission. Credit: Fox

Title: The unique potential of extreme mass-ratio inspirals for gravitational-wave astronomy
arXiv:
1903.03686 [astro-ph.HE]
Theme music: Muse

### Bonus notes

#### White paper vs journal article

Since white papers are proposals for future research, they aren’t as rigorous as usual academic papers. They are really attempts to figure out a good question to ask, rather than being answers. White papers are not usually peer reviewed before publication—the point is that you want everybody to comment on them, rather than just one or two anonymous referees.

Whilst white papers aren’t quite the same class as journal articles, they do still contain some interesting ideas, so I thought they still merit a blog post.

#### Recycling

I have blogged about EMRIs before, so I won’t go into too much detail here. It was one of my former blog posts which inspired the LISA Science Team to get in touch to ask me to write the white paper.

# The O2 Catalogue—It goes up to 11

The full results of our second advanced-detector observing run (O2) have now been released—we’re pleased to announce four new gravitational wave signals: GW170729, GW170809, GW170818 and GW170823 [bonus note]. These latest observations are all of binary black hole systems. Together, they bring our total to 10 observations of binary black holes, and 1 of a binary neutron star. With more frequent detections on the horizon with our third observing run due to start early 2019, the era of gravitational wave astronomy is truly here.

The population of black holes and neutron stars observed with gravitational waves and with electromagnetic astronomy. You can play with an interactive version of this plot online.

The new detections are largely consistent with our previous findings. GW170809, GW170818 and GW170823 are all similar to our first detection GW150914. Their black holes have masses around 20 to 40 times the mass of our Sun. I would lump GW170104 and GW170814 into this class too. Although there were models that predicted black holes of these masses, we weren’t sure they existed until our gravitational wave observations. The family of black holes continues out of this range. GW151012, GW151226 and GW170608 fall on the lower mass side. These overlap with the population of black holes previously observed in X-ray binaries. Lower mass systems can’t be detected as far away, so we find fewer of these. On the higher end we have GW170729 [bonus note]. Its source is made up of black holes with masses $50.7^{+16.3}_{-10.2} M_\odot$ and $34.4^{+8.9}_{-10.2} M_\odot$ (where $M_\odot$ is the mass of our Sun). The larger black hole is a contender for the most massive black hole we’ve found in a binary (the other probable contender is GW170823’s source, which has a $39.5^{+10.0}_{-6.6} M_\odot$ black hole). We have a big happy family of black holes!

Of the new detections, GW170729, GW170809 and GW170818 were both observed by the Virgo detector as well as the two LIGO detectors. Virgo joined O2 for an exciting August [bonus note], and we decided that the data at the time of GW170729 were good enough to use too. Unfortunately, Virgo wasn’t observing at the time of GW170823. GW170729 and GW170809 are very quiet in Virgo, you can’t confidently say there is a signal there [bonus note]. However, GW170818 is a clear detection like GW170814. Well done Virgo!

Using the collection of results, we can start understand the physics of these binary systems. We will be summarising our findings in a series of papers. A huge amount of work went into these.

### The papers

#### The O2 Catalogue Paper

Title: GWTC-1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs
arXiv:
1811.12907 [astro-ph.HE]
Data: Catalogue; Parameter estimation results
LIGO science summary: GWTC-1: A new catalog of gravitational-wave detections

The paper summarises all our observations of binaries to date. It covers our first and second observing runs (O1 and O2). This is the paper to start with if you want any information. It contains estimates of parameters for all our sources, including updates for previous events. It also contains merger rate estimates for binary neutron stars and binary black holes, and an upper limit for neutron star–black hole binaries. We’re still missing a neutron star–black hole detection to complete the set.

More details: The O2 Catalogue Paper

#### The O2 Populations Paper

Title: Binary black hole population properties inferred from the first and second observing runs of Advanced LIGO and Advanced Virgo
arXiv:
1811.12940 [astro-ph.HE]

Using our set of ten binary black holes, we can start to make some statistical statements about the population: the distribution of masses, the distribution of spins, the distribution of mergers over cosmic time. With only ten observations, we still have a lot of uncertainty, and can’t make too many definite statements. However, if you were wondering why we don’t see any more black holes more massive than GW170729, even though we can see these out to significant distances, so are we. We infer that almost all stellar-mass black holes have masses less than $45 M_\odot$.

More details: The O2 Populations Paper

### The O2 Catalogue Paper

Synopsis: O2 Catalogue Paper
Read this if: You want the most up-to-date gravitational results
Favourite part: It’s out! We can tell everyone about our FOUR new detections

This is a BIG paper. It covers our first two observing runs and our main searches for coalescing stellar mass binaries. There will be separate papers going into more detail on searches for other gravitational wave signals.

#### The instruments

Gravitational wave detectors are complicated machines. You don’t just take them out of the box and press go. We’ll be slowly improving the sensitivity of our detectors as we commission them over the next few years. O2 marks the best sensitivity achieved to date. The paper gives a brief overview of the detector configurations in O2 for both LIGO detectors, which did differ, and Virgo.

During O2, we realised that one source of noise was beam jitter, disturbances in the shape of the laser beam. This was particularly notable in Hanford, where there was a spot on the one of the optics. Fortunately, we are able to measure the effects of this, and hence subtract out this noise. This has now been done for the whole of O2. It makes a big difference! Derek Davis and TJ Massinger won the first LIGO Laboratory Award for Excellence in Detector Characterization and Calibration™ for implementing this noise subtraction scheme (the award citation almost spilled the beans on our new detections). I’m happy that GW170104 now has an increased signal-to-noise ratio, which means smaller uncertainties on its parameters.

#### The searches

We use three search algorithms in this paper. We have two matched-filter searches (GstLAL and PyCBC). These compare a bank of templates to the data to look for matches. We also use coherent WaveBurst (cWB), which is a search for generic short signals, but here has been tuned to find the characteristic chirp of a binary. Since cWB is more flexible in the signals it can find, it’s slightly less sensitive than the matched-filter searches, but it gives us confidence that we’re not missing things.

The two matched-filter searches both identify all 11 signals with the exception of GW170818, which is only found by GstLAL. This is because PyCBC only flags signals above a threshold in each detector. We’re confident it’s real though, as it is seen in all three detectors, albeit below PyCBC’s threshold in Hanford and Virgo. (PyCBC only looked at signals found in coincident Livingston and Hanford in O2, I suspect they would have found it if they were looking at all three detectors, as that would have let them lower their threshold).

The search pipelines try to distinguish between signal-like features in the data and noise fluctuations. Having multiple detectors is a big help here, although we still need to be careful in checking for correlated noise sources. The background of noise falls off quickly, so there’s a rapid transition between almost-certainly noise to almost-certainly signal. Most of the signals are off the charts in terms of significance, with GW170818, GW151012 and GW170729 being the least significant. GW170729 is found with best significance by cWB, that gives reports a false alarm rate of $1/(50~\mathrm{yr})$.

Cumulative histogram of results from GstLAL (top left), PyCBC (top right) and cWB (bottom). The expected background is shown as the dashed line and the shaded regions give Poisson uncertainties. The search results are shown as the solid red line and named gravitational-wave detections are shown as blue dots. More significant results are further to the right of the plot. Fig. 2 and Fig. 3 of the O2 Catalogue Paper.

The false alarm rate indicates how often you would expect to find something at least as signal like if you were to analyse a stretch of data with the same statistical properties as the data considered, assuming that they is only noise in the data. The false alarm rate does not fold in the probability that there are real gravitational waves occurring at some average rate. Therefore, we need to do an extra layer of inference to work out the probability that something flagged by a search pipeline is a real signal versus is noise.

The results of this calculation is given in Table IV. GW170729 has a 94% probability of being real using the cWB results, 98% using the GstLAL results, but only 52% according to PyCBC. Therefore, if you’re feeling bold, you might, say, only wager the entire economy of the UK on it being real.

We also list the most marginal triggers. These all have probabilities way below being 50% of being real: if you were to add them all up you wouldn’t get a total of 1 real event. (In my professional opinion, they are garbage). However, if you want to check for what we might have missed, these may be a place to start. Some of these can be explained away as instrumental noise, say scattered light. Others show no obvious signs of disturbance, so are probably just some noise fluctuation.

#### The source properties

We give updated parameter estimates for all 11 sources. These use updated estimates of calibration uncertainty (which doesn’t make too much difference), improved estimate of the noise spectrum (which makes some difference to the less well measured parameters like the mass ratio), the cleaned data (which helps for GW170104), and our most currently complete waveform models [bonus note].

This plot shows the masses of the two binary components (you can just make out GW170817 down in the corner). We use the convention that the more massive of the two is $m_1$ and the lighter is $m_2$. We are now really filling in the mass plot! Implications for the population of black holes are discussed in the Populations Paper.

Estimated masses for the two binary objects for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817 (solid), GW170608 (dashed), GW151226 (solid), GW151012 (dashed), GW170104 (solid), GW170814 (dashed), GW170809 (dashed), GW170818 (dashed), GW150914 (solid), GW170823 (dashed), GW170729 (solid). The contours mark the 90% credible regions. The grey area is excluded from our convention on masses. Part of Fig. 4 of the O2 Catalogue Paper. The mass ratio is $q = m_2/m_1$.

As well as mass, black holes have a spin. For the final black hole formed in the merger, these spins are always around 0.7, with a little more or less depending upon which way the spins of the two initial black holes were pointing. As well as being probably the most most massive, GW170729’s could have the highest final spin! It is a record breaker. It radiated a colossal $4.8^{+1.7}_{-1.7} M_\odot$ worth of energy in gravitational waves [bonus note].

Estimated final masses and spins for each of the binary black hole events in O1 and O2. From lowest chirp mass (left; red–orange) to highest (right; purple): GW170608 (dashed), GW151226 (solid), GW151012 (dashed), GW170104 (solid), GW170814 (dashed), GW170809 (dashed), GW170818 (dashed), GW150914 (solid), GW170823 (dashed), GW170729 (solid). The contours mark the 90% credible regions. Part of Fig. 4 of the O2 Catalogue Paper.

There is considerable uncertainty on the spins as there are hard to measure. The best combination to pin down is the effective inspiral spin parameter $\chi_\mathrm{eff}$. This is a mass weighted combination of the spins which has the most impact on the signal we observe. It could be zero if the spins are misaligned with each other, point in the orbital plane, or are zero. If it is non-zero, then it means that at least one black hole definitely has some spin. GW151226 and GW170729 have $\chi_\mathrm{eff} > 0$ with more than 99% probability. The rest are consistent with zero. The spin distribution for GW170104 has tightened up for GW170104 as its signal-to-noise ratio has increased, and there’s less support for negative $\chi_\mathrm{eff}$, but there’s been no move towards larger positive $\chi_\mathrm{eff}$.

Estimated effective inspiral spin parameters for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817, GW170608, GW151226, GW151012, GW170104, GW170814, GW170809, GW170818, GW150914, GW170823, GW170729. Part of Fig. 5 of the O2 Catalogue Paper.

For our analysis, we use two different waveform models to check for potential sources of systematic error. They agree pretty well. The spins are where they show most difference (which makes sense, as this is where they differ in terms of formulation). For GW151226, the effective precession waveform IMRPhenomPv2 gives $0.20^{+0.18}_{-0.08}$ and the full precession model gives $0.15^{+0.25}_{-0.11}$ and extends to negative $\chi_\mathrm{eff}$. I panicked a little bit when I first saw this, as GW151226 having a non-zero spin was one of our headline results when first announced. Fortunately, when I worked out the numbers, all our conclusions were safe. The probability of $\chi_\mathrm{eff} < 0$ is less than 1%. In fact, we can now say that at least one spin is greater than $0.28$ at 99% probability compared with $0.2$ previously, because the full precession model likes spins in the orbital plane a bit more. Who says data analysis can't be thrilling?

Our measurement of $\chi_\mathrm{eff}$ tells us about the part of the spins aligned with the orbital angular momentum, but not in the orbital plane. In general, the in-plane components of the spin are only weakly constrained. We basically only get back the information we put in. The leading order effects of in-plane spins is summarised by the effective precession spin parameter $\chi_\mathrm{p}$. The plot below shows the inferred distributions for $\chi_\mathrm{p}$. The left half for each event shows our results, the right shows our prior after imposed the constraints on spin we get from $\chi_\mathrm{eff}$. We get the most information for GW151226 and GW170814, but even then it’s not much, and we generally cover the entire allowed range of values.

Estimated effective inspiral spin parameters for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817, GW170608, GW151226, GW151012, GW170104, GW170814, GW170809, GW170818, GW150914, GW170823, GW170729. The left (coloured) part of the plot shows the posterior distribution; the right (white) shows the prior conditioned by the effective inspiral spin parameter constraints. Part of Fig. 5 of the O2 Catalogue Paper.

One final measurement which we can make (albeit with considerable uncertainty) is the distance to the source. The distance influences how loud the signal is (the further away, the quieter it is). This also depends upon the inclination of the source (a binary edge-on is quieter than a binary face-on/off). Therefore, the distance is correlated with the inclination and we end up with some butterfly-like plots. GW170729 is again a record setter. It comes from a luminosity distance of $2.75^{+1.35}_{-1.32}~\mathrm{Gpc}$ away. That means it has travelled across the Universe for $3.2$$6.2$ billion years—it potentially started its journey before the Earth formed!

Estimated luminosity distances and orbital inclinations for each of the events in O1 and O2. From lowest chirp mass (left; red) to highest (right; purple): GW170817 (solid), GW170608 (dashed), GW151226 (solid), GW151012 (dashed), GW170104 (solid), GW170814 (dashed), GW170809 (dashed), GW170818 (dashed), GW150914 (solid), GW170823 (dashed), GW170729 (solid). The contours mark the 90% credible regions. An inclination of zero means that we’re looking face-on along the direction of the total angular momentum, and inclination of $\pi/2$ means we’re looking edge-on perpendicular to the angular momentum. Part of Fig. 7 of the O2 Catalogue Paper.

#### Waveform reconstructions

To check our results, we reconstruct the waveforms from the data to see that they match our expectations for binary black hole waveforms (and there’s not anything extra there). To do this, we use unmodelled analyses which assume that there is a coherent signal in the detectors: we use both cWB and BayesWave. The results agree pretty well. The reconstructions beautifully match our templates when the signal is loud, but, as you might expect, can resolve the quieter details. You’ll also notice the reconstructions sometimes pick up a bit of background noise away from the signal. This gives you and idea of potential fluctuations.

Time–frequency maps and reconstructed signal waveforms for the binary black holes. For each event we show the results from the detector where the signal was loudest. The left panel for each shows the time–frequency spectrogram with the upward-sweeping chip. The right show waveforms: blue the modelled waveforms used to infer parameters (LALInf; top panel); the red wavelet reconstructions (BayesWave; top panel); the black is the maximum-likelihood cWB reconstruction (bottom panel), and the green (bottom panel) shows reconstructions for simulated similar signals. I think the agreement is pretty good! All the data have been whitened as this is how we perform the statistical analysis of our data. Fig. 10 of the O2 Catalogue Paper.

I still think GW170814 looks like a slug. Some people think they look like crocodiles.

We’ll be doing more tests of the consistency of our signals with general relativity in a future paper.

#### Merger rates

Given all our observations now, we can set better limits on the merger rates. Going from the number of detections seen to the number merger out in the Universe depends upon what you assume about the mass distribution of the sources. Therefore, we make a few different assumptions.

For binary black holes, we use (i) a power-law model for the more massive black hole similar to the initial mass function of stars, with a uniform distribution on the mass ratio, and (ii) use uniform-in-logarithmic distribution for both masses. These were designed to bracket the two extremes of potential distributions. With our observations, we’re starting to see that the true distribution is more like the power-law, so I expect we’ll be abandoning these soon. Taking the range of possible values from our calculations, the rate is in the range of $9.7$$101~\mathrm{Gpc^{-3}\,yr^{-1}}$ for black holes between $5 M_\odot$ and $50 M_\odot$ [bonus note].

For binary neutron stars, which are perhaps more interesting astronomers, we use a uniform distribution of masses between $1 M_\odot$ and $2 M_\odot$, and a Gaussian distribution to match electromagnetic observations. We find that these bracket the range $97$$4440~\mathrm{Gpc^{-3}\,yr^{-1}}$. This larger than are previous range, as we hadn’t considered the Gaussian distribution previously.

90% upper limits for neutron star–black hole binaries. Three black hole masses were tried and two spin distributions. Results are shown for the two matched-filter search algorithms. Fig. 14 of the O2 Catalogue Paper.

Finally, what about neutron star–black holes? Since we don’t have any detections, we can only place an upper limit. This is a maximum of $610~\mathrm{Gpc^{-3}\,yr^{-1}}$. This is about a factor of 2 better than our O1 results, and is starting to get interesting!

We are sure to discover lots more in O3… [bonus note].

### The O2 Populations Paper

Synopsis: O2 Populations Paper
Read this if: You want the best family portrait of binary black holes
Favourite part: A maximum black hole mass?

Each detection is exciting. However, we can squeeze even more science out of our observations by looking at the entire population. Using all 10 of our binary black hole observations, we start to trace out the population of binary black holes. Since we still only have 10, we can’t yet be too definite in our conclusions. Our results give us some things to ponder, while we are waiting for the results of O3. I think now is a good time to start making some predictions.

We look at the distribution of black hole masses, black hole spins, and the redshift (cosmological time) of the mergers. The black hole masses tell us something about how you go from a massive star to a black hole. The spins tell us something about how the binaries form. The redshift tells us something about how these processes change as the Universe evolves. Ideally, we would look at these all together allowing for mixtures of binary black holes formed through different means. Given that we only have a few observations, we stick to a few simple models.

To work out the properties of the population, we perform a hierarchical analysis of our 10 binary black holes. We infer the properties of the individual systems, assuming that they come from a given population, and then see how well that population fits our data compared with a different distribution.

In doing this inference, we account for selection effects. Our detectors are not equally sensitive to all sources. For example, nearby sources produce louder signals and we can’t detect signals that are too far away, so if you didn’t account for this you’d conclude that binary black holes only merged in the nearby Universe. Perhaps less obvious is that we are not equally sensitive to all source masses. More massive binaries produce louder signals, so we can detect these further way than lighter binaries (up to the point where these binaries are so high mass that the signals are too low frequency for us to easily spot). This is why we detect more binary black holes than binary neutron stars, even though there are more binary neutron stars out here in the Universe.

#### Masses

When looking at masses, we try three models of increasing complexity:

• Model A is a simple power law for the mass of the more massive black hole $m_1$. There’s no real reason to expect the masses to follow a power law, but the masses of stars when they form do, and astronomers generally like power laws as they’re friendly, so its a sensible thing to try. We fit for the power-law index. The power law goes from a lower limit of $5 M_\odot$ to an upper limit which we also fit for. The mass of the lighter black hole $m_2$ is assumed to be uniformly distributed between $5 M_\odot$ and the mass of the other black hole.
• Model B is the same power law, but we also allow the lower mass limit to vary from $5 M_\odot$. We don’t have much sensitivity to low masses, so this lower bound is restricted to be above $5 M_\odot$. I’d be interested in exploring lower masses in the future. Additionally, we allow the mass ratio $q = m_2/m_1$ of the black holes to vary, trying $q^{\beta_q}$ instead of Model A’s $q^0$.
• Model C has the same power law, but now with some smoothing at the low-mass end, rather than a sharp turn-on. Additionally, it includes a Gaussian component towards higher masses. This was inspired by the possibility of pulsational pair-instability supernova causing a build up of black holes at certain masses: stars which undergo this lose extra mass, so you’d end up with lower mass black holes than if the stars hadn’t undergone the pulsations. The Gaussian could fit other effects too, for example if there was a secondary formation channel, or just reflect that the pure power law is a bad fit.

In allowing the mass distributions to vary, we find overall rates which match pretty well those we obtain with our main power-law rates calculation included in the O2 Catalogue Paper, higher than with the main uniform-in-log distribution.

The fitted mass distributions are shown in the plot below. The error bars are pretty broad, but I think the models agree on some broad features: there are more light black holes than heavy black holes; the minimum black hole mass is below about $9 M_\odot$, but we can’t place a lower bound on it; the maximum black hole mass is above about $35 M_\odot$ and below about $50 M_\odot$, and we prefer black holes to have more similar masses than different ones. The upper bound on the black hole minimum mass, and the lower bound on the black hole upper mass are set by the smallest and biggest black holes we’ve detected, respectively.

Binary black hole merger rate as a function of the primary mass ($m_1$; top) and mass ratio ($q$; bottom). The solid line and dark and lighter bands show the median, 50% interval and 90% interval. The dashed line shows the posterior predictive distribution: our expectation for future observations averaging over our uncertainties. Fig. 1 of the O2 Populations Paper.

That there does seem to be a drop off at higher masses is interesting. There could be something which stops stars forming black holes in this range. It has been proposed that there is a mass gap due to pair instability supernovae. These explosions completely disrupt their progenitor stars, leaving nothing behind. (I’m not sure if they are accompanied by a flash of green light). You’d expect this to kick for black holes of about $50$$60 M_\odot$. We infer that 99% of merging black holes have masses below $43.8 M_\odot$ with Model A, $42.8 M_\odot$ with Model B, and $41.8 M_\odot$ with Model C. Therefore, our results are not inconsistent with a mass gap. However, we don’t really have enough evidence to be sure.

We can compare how well each of our three models fits the data by looking at their Bayes factors. These naturally incorporate the complexity of the models: models with more parameters (which can be more easily tweaked to match the data) are penalised so that you don’t need to worry about overfitting. We have a preference for Model C. It’s not strong, but I think good evidence that we can’t use a simple power law.

#### Spins

To model the spins:

• For the magnitude, we assume a beta distribution. There’s no reason for this, but these are convenient distributions for things between 0 and 1, which are the limits on black hole spin (0 is nonspinning, 1 is as fast as you can spin). We assume that both spins are drawn from the same distribution.
• For the spin orientations, we use a mix of an isotropic distribution and a Gaussian centred on being aligned with the orbital angular momentum. You’d expect an isotropic distribution if binaries were assembled dynamically, and perhaps something with spins generally aligned with each other if the binary evolved in isolation.

We don’t get any useful information on the mixture fraction. Looking at the spin magnitudes, we have a preference towards smaller spins, but still have support for large spins. The more misaligned spins are, the larger the spin magnitudes can be: for the isotropic distribution, we have support all the way up to maximal values.

Inferred spin magnitude distributions. The left shows results for the parametric distribution, assuming a mixture of almost aligned and isotropic spin, with the median (solid), 50% and 90% intervals shaded, and the posterior predictive distribution as the dashed line. The right shows a binned reconstruction of the distribution for aligned and isotropic distributions, showing the median and 90% intervals. Fig. 7 of the O2 Populations Paper.

Since spins are harder to measure than masses, it is not surprising that we can’t make strong statements yet. If we were to find something with definitely negative $\chi_\mathrm{eff}$, we would be able to deduce that spins can be seriously misaligned.

#### Redshift evolution

As a simple model of evolution over cosmological time, we allow the merger rate to evolve as $(1+z)^\lambda$. That’s right, another power law! Since we’re only sensitive to relatively small redshifts for the masses we detect ($z < 1$), this gives a good approximation to a range of different evolution schemes.

Evolution of the binary black hole merger rate (blue), showing median, 50% and 90% intervals. For comparison, reference non-evolving rates (from the O2 Catalogue Paper) are shown too. Fig. 5 of the O2 Populations Paper.

We find that we prefer evolutions that increase with redshift. There’s an 88% probability that $\lambda > 0$, but we’re still consistent with no evolution. We might expect rate to increase as star formation was higher bach towards $z =2$. If we can measure the time delay between forming stars and black holes merging, we could figure out what happens to these systems in the meantime.

The local merger rate is broadly consistent with what we infer with our non-evolving distributions, but is a little on the lower side.

### Bonus notes

#### Naming

Gravitational waves are named as GW-year-month-day, so our first observation from 14 September 2015 is GW150914. We realise that this convention suffers from a Y2K-style bug, but by the time we hit 2100, we’ll have so many detections we’ll need a new scheme anyway.

Previously, we had a second designation for less significant potential detections. They were LIGO–Virgo Triggers (LVT), the one example being LVT151012. No-one was really happy with this designation, but it stems from us being cautious with our first announcement, and not wishing to appear over bold with claiming we’d seen two gravitational waves when the second wasn’t that certain. Now we’re a bit more confident, and we’ve decided to simplify naming by labelling everything a GW on the understanding that this now includes more uncertain events. Under the old scheme, GW170729 would have been LVT170729. The idea is that the broader community can decide which events they want to consider as real for their own studies. The current condition for being called a GW is that the probability of it being a real astrophysical signal is at least 50%. Our 11 GWs are safely above that limit.

The naming change has hidden the fact that now when we used our improved search pipelines, the significance of GW151012 has increased. It would now be a GW even under the old scheme. Congratulations LVT151012, I always believed in you!

Is it of extraterrestrial origin, or is it just a blurry figure? GW151012: the truth is out there!.

#### Burning bright

We are lacking nicknames for our new events. They came in so fast that we kind of lost track. Ilya Mandel has suggested that GW170729 should be the Tiger, as it happened on the International Tiger Day. Since tigers are the biggest of the big cats, this seems apt.

Carl-Johan Haster argues that LIGO+tiger = Liger. Since ligers are even bigger than tigers, this seems like an excellent case to me! I’d vote for calling the bigger of the two progenitor black holes GW170729-tiger, the smaller GW170729-lion, and the final black hole GW17-729-liger.

Suggestions for other nicknames are welcome, leave your ideas in the comments.

#### August 2017—Something fishy or just Poisson statistics?

The final few weeks of O2 were exhausting. I was trying to write job applications at the time, and each time I sat down to work on my research proposal, my phone went off with another alert. You may be wondering about was special about August. Some have hypothesised that it is because Aaron Zimmerman, my partner for the analysis of GW170104, was on the Parameter Estimation rota to analyse the last few weeks of O2. The legend goes that Aaron is especially lucky as he was bitten by a radioactive Leprechaun. I can neither confirm nor deny this. However, I make a point of playing any lottery numbers suggested by him.

A slightly more mundane explanation is that August was when the detectors were running nice and stably. They were observing for a large fraction of the time. LIGO Livingston reached its best sensitivity at this time, although it was less happy for Hanford. We often quantify the sensitivity of our detectors using their binary neutron star range, the average distance they could see a binary neutron star system with a signal-to-noise ratio of 8. If this increases by a factor of 2, you can see twice as far, which means you survey 8 times the volume. This cubed factor means even small improvements can have a big impact. The LIGO Livingston range peak a little over $100~\mathrm{Mpc}$. We’re targeting at least $120~\mathrm{Mpc}$ for O3, so August 2017 gives an indication of what you can expect.

Binary neutron star range for the instruments across O2. The break around week 3 was for the holidays (We did work Christmas 2015). The break at week 23 was to tune-up the instruments, and clean the mirrors. At week 31 there was an earthquake in Montana, and the Hanford sensitivity didn’t recover by the end of the run. Part of Fig. 1 of the O2 Catalogue Paper.

Of course, in the case of GW170817, we just got lucky.

#### Sign errors

GW170809 was the first event we identified with Virgo after it joined observing. The signal in Virgo is very quiet. We actually got better results when we flipped the sign of the Virgo data. We were just starting to get paranoid when GW170814 came along and showed us that everything was set up right at Virgo. When I get some time, I’d like to investigate how often this type of confusion happens for quiet signals.

#### SEOBNRv3

One of the waveforms, which includes the most complete prescription of the precession of the spins of the black holes, we use in our analysis goes by the technical name of SEOBNRv3. It is extremely computationally expensive. Work has been done to improve that, but this hasn’t been implemented in our reviewed codes yet. We managed to complete an analysis for the GW170104 Discovery Paper, which was a huge effort. I said then to not expect it for all future events. We did it for all the black holes, even for the lowest mass sources which have the longest signals. I was responsible for GW151226 runs (as well as GW170104) and I started these back at the start of the summer. Eve Chase put in a heroic effort to get GW170608 results, we pulled out all the stops for that.

#### Thanksgiving

I have recently enjoyed my first Thanksgiving in the US. I was lucky enough to be hosted for dinner by Shane Larson and his family (and cats). I ate so much I thought I might collapse to a black hole. Apparently, a Thanksgiving dinner can be 3000–4500 calories. That sounds like a lot, but the merger of GW170729 would have emitted about $5 \times 10^{40}$ times more energy. In conclusion, I don’t need to go on a diet.

#### Confession

We cheated a little bit in calculating the rates. Roughly speaking, the merger rate is given by

$\displaystyle R = \frac{N}{\langle VT\rangle}$,

where $N$ is the number of detections and $\langle VT\rangle$ is the amount of volume and time we’ve searched. You expect to detect more events if you increase the sensitivity of the detectors (and hence $V$), or observer for longer (and hence increase $T$). In our calculation, we included GW170608 in $N$, even though it was found outside of standard observing time. Really, we should increase $\langle VT\rangle$ to factor in the extra time outside of standard observing time when we could have made a detection. This is messy to calculate though, as there’s not really a good way to check this. However, it’s only a small fraction of the time (so the extra $T$ should be small), and for much of the sensitivity of the detectors will be poor (so $V$ will be small too). Therefore, we estimated any bias from neglecting this is smaller than our uncertainty from the calibration of the detectors, and not worth worrying about.

#### New sources

We saw our first binary black hole shortly after turning on the Advanced LIGO detectors. We saw our first binary neutron star shortly after turning on the Advanced Virgo detector. My money is therefore on our first neutron star–black hole binary shortly after we turn on the KAGRA detector. Because science…

# Accuracy of inference on the physics of binary evolution from gravitational-wave observations

Gravitational-wave astronomy lets us observing binary black holes. These systems, being made up of two black holes, are pretty difficult to study by any other means. It has long been argued that with this new information we can unravel the mysteries of stellar evolution. Just as a palaeontologist can discover how long-dead animals lived from their bones, we can discover how massive stars lived by studying their black hole remnants. In this paper, we quantify how much we can really learn from this black hole palaeontology—after 1000 detections, we should pin down some of the most uncertain parameters in binary evolution to a few percent precision.

### Life as a binary

There are many proposed ways of making a binary black hole. The current leading contender is isolated binary evolution: start with a binary star system (most stars are in binaries or higher multiples, our lonesome Sun is a little unusual), and let the stars evolve together. Only a fraction will end with black holes close enough to merge within the age of the Universe, but these would be the sources of the signals we see with LIGO and Virgo. We consider this isolated binary scenario in this work [bonus note].

Now, you might think that with stars being so fundamentally important to astronomy, and with binary stars being so common, we’d have the evolution of binaries figured out by now. It turns out it’s actually pretty messy, so there’s lots of work to do. We consider constraining four parameters which describe the bits of binary physics which we are currently most uncertain of:

• Black hole natal kicks—the push black holes receive when they are born in supernova explosions. We now the neutron stars get kicks, but we’re less certain for black holes [bonus note].
• Common envelope efficiency—one of the most intricate bits of physics about binaries is how mass is transferred between stars. As they start exhausting their nuclear fuel they puff up, so material from the outer envelope of one star may be stripped onto the other. In the most extreme cases, a common envelope may form, where so much mass is piled onto the companion, that both stars live in a single fluffy envelope. Orbiting inside the envelope helps drag the two stars closer together, bringing them closer to merging. The efficiency determines how quickly the envelope becomes unbound, ending this phase.
• Mass loss rates during the Wolf–Rayet (not to be confused with Wolf 359) and luminous blue variable phases–stars lose mass through out their lives, but we’re not sure how much. For stars like our Sun, mass loss is low, there is enough to gives us the aurora, but it doesn’t affect the Sun much. For bigger and hotter stars, mass loss can be significant. We consider two evolutionary phases of massive stars where mass loss is high, and currently poorly known. Mass could be lost in clumps, rather than a smooth stream, making it difficult to measure or simulate.

We use parameters describing potential variations in these properties are ingredients to the COMPAS population synthesis code. This rapidly (albeit approximately) evolves a population of stellar binaries to calculate which will produce merging binary black holes.

The question is now which parameters affect our gravitational-wave measurements, and how accurately we can measure those which do?

Binary black hole merger rate at three different redshifts $z$ as calculated by COMPAS. We show the rate in 30 different chirp mass bins for our default population parameters. The caption gives the total rate for all masses. Figure 2 of Barrett et al. (2018)

### Gravitational-wave observations

For our deductions, we use two pieces of information we will get from LIGO and Virgo observations: the total number of detections, and the distributions of chirp masses. The chirp mass is a combination of the two black hole masses that is often well measured—it is the most important quantity for controlling the inspiral, so it is well measured for low mass binaries which have a long inspiral, but is less well measured for higher mass systems. In reality we’ll have much more information, so these results should be the minimum we can actually do.

We consider the population after 1000 detections. That sounds like a lot, but we should have collected this many detections after just 2 or 3 years observing at design sensitivity. Our default COMPAS model predicts 484 detections per year of observing time! Honestly, I’m a little scared about having this many signals…

For a set of population parameters (black hole natal kick, common envelope efficiency, luminous blue variable mass loss and Wolf–Rayet mass loss), COMPAS predicts the number of detections and the fraction of detections as a function of chirp mass. Using these, we can work out the probability of getting the observed number of detections and fraction of detections within different chirp mass ranges. This is the likelihood function: if a given model is correct we are more likely to get results similar to its predictions than further away, although we expect their to be some scatter.

If you like equations, the from of our likelihood is explained in this bonus note. If you don’t like equations, there’s one lurking in the paragraph below. Just remember, that it can’t see you if you don’t move. It’s OK to skip the equation.

To determine how sensitive we are to each of the population parameters, we see how the likelihood changes as we vary these. The more the likelihood changes, the easier it should be to measure that parameter. We wrap this up in terms of the Fisher information matrix. This is defined as

$\displaystyle F_{ij} = -\left\langle\frac{\partial^2\ln \mathcal{L}(\mathcal{D}|\left\{\lambda\right\})}{\partial \lambda_i \partial\lambda_j}\right\rangle$,

where $\mathcal{L}(\mathcal{D}|\left\{\lambda\right\})$ is the likelihood for data $\mathcal{D}$ (the number of observations and their chirp mass distribution in our case), $\left\{\lambda\right\}$ are our parameters (natal kick, etc.), and the angular brackets indicate the average over the population parameters. In statistics terminology, this is the variance of the score, which I think sounds cool. The Fisher information matrix nicely quantifies how much information we can lean about the parameters, including the correlations between them (so we can explore degeneracies). The inverse of the Fisher information matrix gives a lower bound on the covariance matrix (the multidemensional generalisation of the variance in a normal distribution) for the parameters $\left\{\lambda\right\}$. In the limit of a large number of detections, we can use the Fisher information matrix to estimate the accuracy to which we measure the parameters [bonus note].

We simulated several populations of binary black hole signals, and then calculate measurement uncertainties for our four population uncertainties to see what we could learn from these measurements.

### Results

Using just the rate information, we find that we can constrain a combination of the common envelope efficiency and the Wolf–Rayet mass loss rate. Increasing the common envelope efficiency ends the common envelope phase earlier, leaving the binary further apart. Wider binaries take longer to merge, so this reduces the merger rate. Similarly, increasing the Wolf–Rayet mass loss rate leads to wider binaries and smaller black holes, which take longer to merge through gravitational-wave emission. Since the two parameters have similar effects, they are anticorrelated. We can increase one and still get the same number of detections if we decrease the other. There’s a hint of a similar correlation between the common envelope efficiency and the luminous blue variable mass loss rate too, but it’s not quite significant enough for us to be certain it’s there.

Fisher information matrix estimates for fractional measurement precision of the four population parameters: the black hole natal kick $\sigma_\mathrm{kick}$, the common envelope efficiency $\alpha_\mathrm{CE}$, the Wolf–Rayet mass loss rate $f_\mathrm{WR}$, and the luminous blue variable mass loss rate $f_\mathrm{LBV}$. There is an anticorrealtion between $f_\mathrm{WR}$ and $\alpha_\mathrm{CE}$, and hints at a similar anticorrelation between $f_|mathrm{LBV}$ and $\alpha_\mathrm{CE}$. We show 1500 different realisations of the binary population to give an idea of scatter. Figure 6 of Barrett et al. (2018)

Adding in the chirp mass distribution gives us more information, and improves our measurement accuracies. The fraction uncertainties are about 2% for the two mass loss rates and the common envelope efficiency, and about 5% for the black hole natal kick. We’re less sensitive to the natal kick because the most massive black holes don’t receive a kick, and so are unaffected by the kick distribution [bonus note]. In any case, these measurements are exciting! With this type of precision, we’ll really be able to learn something about the details of binary evolution.

Measurement precision for the four population parameters after 1000 detections. We quantify the precision with the standard deviation estimated from the Fisher inforamtion matrix. We show results from 1500 realisations of the population to give an idea of scatter. Figure 5 of Barrett et al. (2018)

The accuracy of our measurements will improve (on average) with the square root of the number of gravitational-wave detections. So we can expect 1% measurements after about 4000 observations. However, we might be able to get even more improvement by combining constraints from other types of observation. Combining different types of observation can help break degeneracies. I’m looking forward to building a concordance model of binary evolution, and figuring out exactly how massive stars live their lives.

arXiv: 1711.06287 [astro-ph.HE]
Journal: Monthly Notices of the Royal Astronomical Society; 477(4):4685–4695; 2018
Favourite dinosaur: Professor Science

### Bonus notes

#### Channel selection

In practise, we will need to worry about how binary black holes are formed, via isolated evolution or otherwise, before inferring the parameters describing binary evolution. This makes the problem more complicated. Some parameters, like mass loss rates or black hole natal kicks, might be common across multiple channels, while others are not. There are a number of ways we might be able to tell different formation mechanisms apart, such as by using spin measurements.

#### Kick distribution

We model the supernova kicks $v_\mathrm{kick}$ as following a Maxwell–Boltzmann distribution,

$\displaystyle p(v_\mathrm{kick}) = \sqrt{\frac{2}{\pi}} \frac{v_\mathrm{kick}^2}{\sigma_\mathrm{kick}^3} \exp\left(\frac{-v_\mathrm{kick}^2}{2\sigma_\mathrm{kick}^2}\right)$,

where $\sigma_\mathrm{kick}$ is the unknown population parameter. The natal kick received by the black hole $v^*_\mathrm{kick}$ is not the same as this, however, as we assume some of the material ejected by the supernova falls back, reducing the over kick. The final natal kick is

$v^*_\mathrm{kick} = (1-f_\mathrm{fb})v_\mathrm{kick}$,

where $f_\mathrm{fb}$ is the fraction that falls back, taken from Fryer et al. (2012). The fraction is greater for larger black holes, so the biggest black holes get no kicks. This means that the largest black holes are unaffected by the value of $\sigma_\mathrm{kick}$.

#### The likelihood

In this analysis, we have two pieces of information: the number of detections, and the chirp masses of the detections. The first is easy to summarise with a single number. The second is more complicated, and we consider the fraction of events within different chirp mass bins.

Our COMPAS model predicts the merger rate $\mu$ and the probability of falling in each chirp mass bin $p_k$ (we factor measurement uncertainty into this). Our observations are the the total number of detections $N_\mathrm{obs}$ and the number in each chirp mass bin $c_k$ ($N_\mathrm{obs} = \sum_k c_k$). The likelihood is the probability of these observations given the model predictions. We can split the likelihood into two pieces, one for the rate, and one for the chirp mass distribution,

$\mathcal{L} = \mathcal{L}_\mathrm{rate} \times \mathcal{L}_\mathrm{mass}$.

For the rate likelihood, we need the probability of observing $N_\mathrm{obs}$ given the predicted rate $\mu$. This is given by a Poisson distribution,

$\displaystyle \mathcal{L}_\mathrm{rate} = \exp(-\mu t_\mathrm{obs}) \frac{(\mu t_\mathrm{obs})^{N_\mathrm{obs}}}{N_\mathrm{obs}!}$,

where $t_\mathrm{obs}$ is the total observing time. For the chirp mass likelihood, we the probability of getting a number of detections in each bin, given the predicted fractions. This is given by a multinomial distribution,

$\displaystyle \mathcal{L}_\mathrm{mass} = \frac{N_\mathrm{obs}!}{\prod_k c_k!} \prod_k p_k^{c_k}$.

These look a little messy, but they simplify when you take the logarithm, as we need to do for the Fisher information matrix.

When we substitute in our likelihood into the expression for the Fisher information matrix, we get

$\displaystyle F_{ij} = \mu t_\mathrm{obs} \left[ \frac{1}{\mu^2} \frac{\partial \mu}{\partial \lambda_i} \frac{\partial \mu}{\partial \lambda_j} + \sum_k\frac{1}{p_k} \frac{\partial p_k}{\partial \lambda_i} \frac{\partial p_k}{\partial \lambda_j} \right]$.

Conveniently, although we only need to evaluate first-order derivatives, even though the Fisher information matrix is defined in terms of second derivatives. The expected number of events is $\langle N_\mathrm{obs} \rangle = \mu t_\mathrm{obs}$. Therefore, we can see that the measurement uncertainty defined by the inverse of the Fisher information matrix, scales on average as $N_\mathrm{obs}^{-1/2}$.

For anyone worrying about using the likelihood rather than the posterior for these estimates, the high number of detections [bonus note] should mean that the information we’ve gained from the data overwhelms our prior, meaning that the shape of the posterior is dictated by the shape of the likelihood.

#### Interpretation of the Fisher information matrix

As an alternative way of looking at the Fisher information matrix, we can consider the shape of the likelihood close to its peak. Around the maximum likelihood point, the first-order derivatives of the likelihood with respect to the population parameters is zero (otherwise it wouldn’t be the maximum). The maximum likelihood values of $N_\mathrm{obs} = \mu t_\mathrm{obs}$ and $c_k = N_\mathrm{obs} p_k$ are the same as their expectation values. The second-order derivatives are given by the expression we have worked out for the Fisher information matrix. Therefore, in the region around the maximum likelihood point, the Fisher information matrix encodes all the relevant information about the shape of the likelihood.

So long as we are working close to the maximum likelihood point, we can approximate the distribution as a multidimensional normal distribution with its covariance matrix determined by the inverse of the Fisher information matrix. Our results for the measurement uncertainties are made subject to this approximation (which we did check was OK).

Approximating the likelihood this way should be safe in the limit of large $N_\mathrm{obs}$. As we get more detections, statistical uncertainties should reduce, with the peak of the distribution homing in on the maximum likelihood value, and its width narrowing. If you take the limit of $N_\mathrm{obs} \rightarrow \infty$, you’ll see that the distribution basically becomes a delta function at the maximum likelihood values. To check that our $N_\mathrm{obs} = 1000$ was large enough, we verified that higher-order derivatives were still small.

Michele Vallisneri has a good paper looking at using the Fisher information matrix for gravitational wave parameter estimation (rather than our problem of binary population synthesis). There is a good discussion of its range of validity. The high signal-to-noise ratio limit for gravitational wave signals corresponds to our high number of detections limit.

# Science with the space-based interferometer LISA. V. Extreme mass-ratio inspirals

The space-based observatory LISA will detect gravitational waves from massive black holes (giant black holes residing in the centres of galaxies). One particularly interesting signal will come from the inspiral of a regular stellar-mass black hole into a massive black hole. These are called extreme mass-ratio inspirals (or EMRIs, pronounced emries, to their friends) [bonus note]. We have never observed such a system. This means that there’s a lot we have to learn about them. In this work, we systematically investigated the prospects for observing EMRIs. We found that even though there’s a wide range in predictions for what EMRIs we will detect, they should be a safe bet for the LISA mission.

Artistic impression of the spacetime for an extreme-mass-ratio inspiral, with a smaller stellar-mass black hole orbiting a massive black hole. This image is mandatory when talking about extreme-mass-ratio inspirals. Credit: NASA

### LISA & EMRIs

My previous post discussed some of the interesting features of EMRIs. Because of the extreme difference in masses of the two black holes, it takes a long time for them to complete their inspiral. We can measure tens of thousands of orbits, which allows us to make wonderfully precise measurements of the source properties (if we can accurately pick out the signal from the data). Here, we’ll examine exactly what we could learn with LISA from EMRIs [bonus note].

First we build a model to investigate how many EMRIs there could be.  There is a lot of astrophysics which we are currently uncertain about, which leads to a large spread in estimates for the number of EMRIs. Second, we look at how precisely we could measure properties from the EMRI signals. The astrophysical uncertainties are less important here—we could get a revolutionary insight into the lives of massive black holes.

### The number of EMRIs

To build a model of how many EMRIs there are, we need a few different inputs:

1. The population of massive black holes
2. The distribution of stellar clusters around massive black holes
3. The range of orbits of EMRIs

We examine each of these in turn, building a more detailed model than has previously been constructed for EMRIs.

We currently know little about the population of massive black holes. This means we’ll discover lots when we start measuring signals (yay), but it’s rather inconvenient now, when we’re trying to predict how many EMRIs there are (boo). We take two different models for the mass distribution of massive black holes. One is based upon a semi-analytic model of massive black hole formation, the other is at the pessimistic end allowed by current observations. The semi-analytic model predicts massive black hole spins around 0.98, but we also consider spins being uniformly distributed between 0 and 1, and spins of 0. This gives us a picture of the bigger black hole, now we need the smaller.

Observations show that the masses of massive black holes are correlated with their surrounding cluster of stars—bigger black holes have bigger clusters. We consider four different versions of this trend: Gültekin et al. (2009); Kormendy & Ho (2013); Graham & Scott (2013), and Shankar et al. (2016). The stars and black holes about a massive black hole should form a cusp, with the density of objects increasing towards the massive black hole. This is great for EMRI formation. However, the cusp is disrupted if two galaxies (and their massive black holes) merge. This tends to happen—it’s how we get bigger galaxies (and black holes). It then takes some time for the cusp to reform, during which time, we don’t expect as many EMRIs. Therefore, we factor in the amount of time for which there is a cusp for massive black holes of different masses and spins.

That’s a nice galaxy you have there. It would be a shame if it were to collide with something… Hubble image of The Mice. Credit: ACS Science & Engineering Team.

Given a cusp about a massive black hole, we then need to know how often an EMRI forms. Simulations give us a starting point. However, these only consider a snap-shot, and we need to consider how things evolve with time. As stellar-mass black holes inspiral, the massive black hole will grow in mass and the surrounding cluster will become depleted. Both these effects are amplified because for each inspiral, there’ll be many more stars or stellar-mass black holes which will just plunge directly into the massive black hole. We therefore need to limit the number of EMRIs so that we don’t have an unrealistically high rate. We do this by adding in a couple of feedback factors, one to cap the rate so that we don’t deplete the cusp quicker than new objects will be added to it, and one to limit the maximum amount of mass the massive black hole can grow from inspirals and plunges. This gives us an idea for the total number of inspirals.

Finally, we calculate the orbits that EMRIs will be on.  We again base this upon simulations, and factor in how the spin of the massive black hole effects the distribution of orbital inclinations.

Putting all the pieces together, we can calculate the population of EMRIs. We now need to work out how many LISA would be able to detect. This means we need models for the gravitational-wave signal. Since we are simulating a large number, we use a computationally inexpensive analytic model. We know that this isn’t too accurate, but we consider two different options for setting the end of the inspiral (where the smaller black hole finally plunges) which should bound the true range of results.

Number of EMRIs for different size massive black holes in different astrophysical models. M1 is our best estimate, the others explore variations on this. M11 and M12 are designed to be cover the extremes, being the most pessimistic and optimistic combinations. The solid and dashed lines are for two different signal models (AKK and AKS), which are designed to give an indication of potential variation. They agree where the massive black hole is not spinning (M10 and M11). The range of masses is similar for all models, as it is set by the sensitivity of LISA. We can detect higher mass systems assuming the AKK signal model as it includes extra inspiral close to highly spinning black holes: for the heaviest black holes, this is the only part of the signal at high enough frequency to be detectable. Figure 8 of Babak et al. (2017).

Allowing for all the different uncertainties, we find that there should be somewhere between 1 and 4200 EMRIs detected per year. (The model we used when studying transient resonances predicted about 250 per year, albeit with a slightly different detector configuration, which is fairly typical of all the models we consider here). This range is encouraging. The lower end means that EMRIs are a pretty safe bet, we’d be unlucky not to get at least one over the course of a multi-year mission (LISA should have at least four years observing). The upper end means there could be lots—we might actually need to worry about them forming a background source of noise if we can’t individually distinguish them!

### EMRI measurements

Having shown that EMRIs are a good LISA source, we now need to consider what we could learn by measuring them?

We estimate the precision we will be able to measure parameters using the Fisher information matrix. The Fisher matrix measures how sensitive our observations are to changes in the parameters (the more sensitive we are, the better we should be able to measure that parameter). It should be a lower bound on actual measurement precision, and well approximate the uncertainty in the high signal-to-noise (loud signal) limit. The combination of our use of the Fisher matrix and our approximate signal models means our results will not be perfect estimates of real performance, but they should give an indication of the typical size of measurement uncertainties.

Given that we measure a huge number of cycles from the EMRI signal, we can make really precise measurements of the the mass and spin of the massive black hole, as these parameters control the orbital frequencies. Below are plots for the typical measurement precision from our Fisher matrix analysis. The orbital eccentricity is measured to similar accuracy, as it influences the range of orbital frequencies too. We also get pretty good measurements of the the mass of the smaller black hole, as this sets how quickly the inspiral proceeds (how quickly the orbital frequencies change). EMRIs will allow us to do precision astronomy!

Distribution of (one standard deviation) fractional uncertainties for measurements of the  massive black hole (redshifted) mass $M_z$. Results are shown for the different astrophysical models, and for the different signal models.  The astrophysical model has little impact on the uncertainties. M4 shows a slight difference as it assumes heavier stellar-mass black holes. The results with the two signal models agree when the massive black hole is not spinning (M10 and M11). Otherwise, measurements are more precise with the AKK signal model, as this includes extra signal from the end of the inspiral. Part of Figure 11 of Babak et al. (2017).

Distribution of (one standard deviation) uncertainties for measurements of the massive black hole spin $a$. The results mirror those for the masses above. Part of Figure 11 of Babak et al. (2017).

Now, before you get too excited that we’re going to learn everything about massive black holes, there is one confession I should make. In the plot above I show the measurement accuracy for the redshifted mass of the massive black hole. The cosmological expansion of the Universe causes gravitational waves to become stretched to lower frequencies in the same way light is (this makes visible light more red, hence the name). The measured frequency is $f_z = (1 + z)f$ where $f$ is the frequency emitted, and $z$ is the redshift ($z= 0$ for a nearby source, and is larger for further away sources). Lower frequency gravitational waves correspond to higher mass systems, so it is often convenient to work with the redshifted mass, the mass corresponding to the signal you measure if you ignore redshifting. The redshifted mass of the massive black hole is $M_z = (1+z)M$ where $M$ is the true mass. To work out the true mass, we need the redshift, which means we need to measure the distance to the source.

Distribution of (one standard deviation) fractional uncertainties for measurements of the luminosity distance $D_\mathrm{L}$. The signal model is not as important here, as the uncertainty only depends on how loud the signal is. Part of Figure 12 of Babak et al. (2017).

The plot above shows the fractional uncertainty on the distance. We don’t measure this too well, as it is determined from the amplitude of the signal, rather than its frequency components. The situation is much as for LIGO. The larger uncertainties on the distance will dominate the overall uncertainty on the black hole masses. We won’t be getting all these to fractions of a percent. However, that doesn’t mean we can’t still figure out what the distribution of masses looks like!

One of the really exciting things we can do with EMRIs is check that the signal matches our expectations for a black hole in general relativity. Since we get such an excellent map of the spacetime of the massive black hole, it is easy to check for deviations. In general relativity, everything about the black hole is fixed by its mass and spin (often referred to as the no-hair theorem). Using the measured EMRI signal, we can check if this is the case. One convenient way of doing this is to describe the spacetime of the massive object in terms of a multipole expansion. The first (most important) terms gives the mass, and the next term the spin. The third term (the quadrupole) is set by the first two, so if we can measure it, we can check if it is consistent with the expected relation. We estimated how precisely we could measure a deviation in the quadrupole. Fortunately, for this consistency test, all factors from redshifting cancel out, so we can get really detailed results, as shown below. Using EMRIs, we’ll be able to check for really small differences from general relativity!

Distribution of (one standard deviation) of uncertainties for deviations in the quadrupole moment of the massive object spacetime $\mathcal{Q}$. Results are similar to the mass and spin measurements. Figure 13 of Babak et al. (2017).

In summary: EMRIS are awesome. We’re not sure how many we’ll detect with LISA, but we’re confident there will be some, perhaps a couple of hundred per year. From the signals we’ll get new insights into the masses and spins of black holes. This should tell us something about how they, and their surrounding galaxies, evolved. We’ll also be able to do some stringent tests of whether the massive objects are black holes as described by general relativity. It’s all pretty exciting, for when LISA launches, which is currently planned about 2034…

One of the most valuable traits a student or soldier can have: patience. Credit: Sony/Marvel

arXiv: 1703.09722 [gr-qc]
Journal: Physical Review D; 477(4):4685–4695; 2018
Conference proceedings: 1704.00009 [astro-ph.GA] (from when work was still in-progress)
Estimated number of Marvel films before LISA launch: 48 (starting with Ant-Man and the Wasp)

### Bonus notes

#### Hyphenation

Is it “extreme-mass-ratio inspiral”, “extreme mass-ratio inspiral” or “extreme mass ratio inspiral”? All are used in the literature. This is one of the advantage of using “EMRI”. The important thing is that we’re talking about inspirals that have a mass ratio which is extreme. For this paper, we used “extreme mass-ratio inspiral”, but when I first started my PhD, I was first introduced to “extreme-mass-ratio inspirals”, so they are always stuck that way in my mind.

I think hyphenation is a bit of an art, and there’s no definitive answer here, just like there isn’t for superhero names, where you can have Iron Man, Spider-Man or Iceman.

#### Science with LISA

This paper is part of a series looking at what LISA could tells us about different gravitational wave sources. So far, this series covers

1. Massive black hole binaries
2. Cosmological phase transitions
3. Standard sirens (for measuring the expansion of the Universe)
4. Inflation
5. Extreme-mass-ratio inspirals

You’ll notice there’s a change in the name of the mission from eLISA to LISA part-way through, as things have evolved. (Or devolved?) I think the main take-away so far is that the cosmology group is the most enthusiastic.

# GW170817—The papers

gAfter 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 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 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).

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 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 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. A list of the first wave of papers was compiled by Maria Drout, Stefano Valenti, and Iair Arcavi as a starting point for further reading.

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 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 seems 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.

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).

The story of the most ambitious cross-over of astronomical observations might now becoming 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)

#### 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 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 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 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%).

# GW170817—The pot of gold at the end of the rainbow

Advanced LIGO and Advanced Virgo have detected their first binary neutron star inspiral. Remarkably, this event was observed not just with gravitational waves, but also across the electromagnetic spectrum, from gamma-rays to radio. This discovery confirms the theory that binary neutron star mergers are the progenitors of short gamma-ray bursts and kilonovae, and may be the primary source of heavy elements like gold.

In this post, I’ll go through some of the story of GW170817. As for GW150914, I’ll write another post on the more technical details of our papers, once I’ve had time to catch up on sleep.

### Discovery

The second observing run (O2) of the advanced gravitational-wave detectors started on 30 November 2016. The first detection came in January—GW170104. I was heavily involved in the analysis and paper writing for this. We finally finished up in June, at which point I was thoroughly exhausted. I took some time off in July [bonus note], and was back at work for August. With just one month left in the observing run, it would all be downhill from here, right?

August turned out to be the lava-filled, super-difficult final level of O2. As we have now announced, on August 14, we detected a binary black hole coalescence—GW170814. This was the first clear detection including Virgo, giving us superb sky localization. This is fantastic for astronomers searching for electromagnetic counterparts to our gravitational-wave signals. There was a flurry of excitement, and we thought that this was a fantastic conclusion to O2. We were wrong, this was just the save point before the final opponent. On August 17, we met the final, fire-ball throwing boss.

At 1:58 pm BST my phone buzzed with a text message, an automated alert of a gravitational-wave trigger. I was obviously excited—I recall that my exact thoughts were “What fresh hell is this?” I checked our online event database and saw that it was a single-detector trigger, it was only seen by our Hanford instrument. I started to relax, this was probably going to turn out to be a glitch. The template masses, were low, in the neutron star range, not like the black holes we’ve been finding. Then I saw the false alarm rate was better than one in 9000 years. Perhaps it wasn’t just some noise after all—even though it’s difficult to estimate false alarm rates accurately online, as especially for single-detector triggers, this was significant! I kept reading. Scrolling down the page there was an external coincident trigger, a gamma-ray burst (GRB 170817A) within a couple of seconds…

We’re gonna need a bigger author list. Credit: Zanuck/Brown Productions

Short gamma-ray bursts are some of the most powerful explosions in the Universe. I’ve always found it mildly disturbing that we didn’t know what causes them. The leading theory has been that they are the result of two neutron stars smashing together. Here seemed to be the proof.

The rapid response call was under way by the time I joined. There was a clear chirp in Hanford, you could be see it by eye! We also had data from Livingston and Virgo too. It was bad luck that they weren’t folded into the online alert. There had been a drop out in the data transfer from Italy to the US, breaking the flow for Virgo. In Livingston, there was a glitch at the time of the signal which meant the data wasn’t automatically included in the search. My heart sank. Glitches are common—check out Gravity Spy for some examples—so it was only a matter of time until one overlapped with a signal [bonus note], and with GW170817 being such a long signal, it wasn’t that surprising. However, this would complicate the analysis. Fortunately, the glitch is short and the signal is long (if this had been a high-mass binary black hole, things might not have been so smooth). We were able to exorcise the glitch. A preliminary sky map using all three detectors was sent out at 12:54 am BST. Not only did we defeat the final boss, we did a speed run on the hard difficulty setting first time [bonus note].

Spectrogram of Livingston data showing part of GW170817’s chirp (which sweeps upward in frequncy) as well as the glitch (the big blip at about $-0.6~\mathrm{s}$). The lower panel shows how we removed the glitch: the grey line shows gating window that was applied for preliminary results, to zero the affected times, the blue shows a fitted model of the glitch that was subtracted for final results. You can clearly see the chirp well before the glitch, so there’s no danger of it being an artefect of the glitch. Figure 2 of the GW170817 Discovery Paper

The three-detector sky map provided a great localization for the source—this preliminary map had a 90% area of ~30 square degrees. It was just in time for that night’s observations. The plot below shows our gravitational-wave localizations in green—the long band is without Virgo, and the smaller is with all three detectors—as with GW170814, Virgo makes a big difference. The blue areas are the localizations from Fermi and INTEGRAL, the gamma-ray observatories which measured the gamma-ray burst. The inset is something new…

Localization of the gravitational-wave, gamma-ray, and optical signals. The main panel shows initial gravitational-wave 90% areas in green (with and without Virgo) and gamma-rays in blue (the IPN triangulation from the time delay between Fermi and INTEGRAL, and the Fermi GBM localization). The inset shows the location of the optical counterpart (the top panel was taken 10.9 hours after merger, the lower panel is a pre-merger reference without the transient). Figure 1 of the Multimessenger Astronomy Paper.

That night, the discoveries continued. Following up on our sky location, an optical counterpart (AT 2017gfo) was found. The source is just on the outskirts of galaxy NGC 4993, which is right in the middle of the distance range we inferred from the gravitational wave signal. At around 40 Mpc, this is the closest gravitational wave source.

After this source was reported, I think about every single telescope possible was pointed at this source. I think it may well be the most studied transient in the history of astronomy. I think there are ~250 circulars about follow-up. Not only did we find an optical counterpart, but there was emission in X-ray and radio. There was a delay in these appearing, I remember there being excitement at our Collaboration meeting as the X-ray emission was reported (there was a lack of cake though).

The figure below tries to summarise all the observations. As you can see, it’s a mess because there is too much going on!

The timeline of observations of GW170817’s source. Shaded dashes indicate times when information was reported in a Circular. Solid lines show when the source was observable in a band: the circles show a comparison of brightnesses for representative observations. Figure 2 of the Multimessenger Astronomy Paper.

The observations paint a compelling story. Two neutron stars insprialled together and merged. Colliding two balls of nuclear density material at around a third of the speed of light causes a big explosion. We get a jet blasted outwards and a gamma-ray burst. The ejected, neutron-rich material decays to heavy elements, and we see this hot material as a kilonova [bonus material]. The X-ray and radio may then be the afterglow formed by the bubble of ejected material pushing into the surrounding interstellar material.

### Science

What have we learnt from our results? Here are some gravitational wave highlights.

We measure several thousand cycles from the inspiral. It is the most beautiful chirp! This is the loudest gravitational wave signal yet found, beating even GW150914. GW170817 has a signal-to-noise ratio of 32, while for GW150914 it is just 24.

Time–frequency plots for GW170104 as measured by Hanford, Livingston and Virgo. 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. The Livingston data have the glitch removed. Figure 1 of the GW170817 Discovery Paper.

The signal-to-noise ratios in the Hanford, Livingston and Virgo were 19, 26 and 2 respectively. The signal is quiet in Virgo, which is why you can’t spot it by eye in the plots above. The lack of a clear signal is really useful information, as it restricts where on the sky the source could be, as beautifully illustrated in the video below.

While we measure the inspiral nicely, we don’t detect the merger: we can’t tell if a hypermassive neutron star is formed or if there is immediate collapse to a black hole. This isn’t too surprising at current sensitivity, the system would basically need to convert all of its energy into gravitational waves for us to see it.

From measuring all those gravitational wave cycles, we can measure the chirp mass stupidly well. Unfortunately, converting the chirp mass into the component masses is not easy. The ratio of the two masses is degenerate with the spins of the neutron stars, and we don’t measure these well. In the plot below, you can see the probability distributions for the two masses trace out bananas of roughly constant chirp mass. How far along the banana you go depends on what spins you allow. We show results for two ranges: one with spins (aligned with the orbital angular momentum) up to 0.89, the other with spins up to 0.05. There’s nothing physical about 0.89 (it was just convenient for our analysis), but it is designed to be agnostic, and above the limit you’d plausibly expect for neutron stars (they should rip themselves apart at spins of ~0.7); the lower limit of 0.05 should safely encompass the spins of the binary neutron stars (which are close enough to merge in the age of the Universe) we have estimated from pulsar observations. The masses roughly match what we have measured for the neutron stars in our Galaxy. (The combinations at the tip of the banana for the high spins would be a bit odd).

Estimated masses for the two neutron stars in the binary. We show results for two different spin limits, $\chi_z$ is the component of the spin aligned with the orbital angular momentum. The two-dimensional shows the 90% probability contour, which follows a line of constant chirp mass. The one-dimensional plot shows individual masses; the dotted lines mark 90% bounds away from equal mass. Figure 4 of the GW170817 Discovery Paper.

If we were dealing with black holes, we’d be done: they are only described by mass and spin. Neutron stars are more complicated. Black holes are just made of warped spacetime, neutron stars are made of delicious nuclear material. This can get distorted during the inspiral—tides are raised on one by the gravity of the other. These extract energy from the orbit and accelerate the inspiral. The tidal deformability depends on the properties of the neutron star matter (described by its equation of state). The fluffier a neutron star is, the bigger the impact of tides; the more compact, the smaller the impact. We don’t know enough about neutron star material to predict this with certainty—by measuring the tidal deformation we can learn about the allowed range. Unfortunately, we also didn’t yet have good model waveforms including tides, so for to start we’ve just done a preliminary analysis (an improved analysis was done for the GW170817 Properties Paper). We find that some of the stiffer equations of state (the ones which predict larger neutron stars and bigger tides) are disfavoured; however, we cannot rule out zero tides. This means we can’t rule out the possibility that we have found two low-mass black holes from the gravitational waves alone. This would be an interesting discovery; however, the electromagnetic observations mean that the more obvious explanation of neutron stars is more likely.

From the gravitational wave signal, we can infer the source distance. Combining this with the electromagnetic observations we can do some cool things.

First, the gamma ray burst arrived at Earth 1.7 seconds after the merger. 1.7 seconds is not a lot of difference after travelling something like 85–160 million years (that’s roughly the time since the Cretaceous or Late Jurassic periods). Of course, we don’t expect the gamma-rays to be emitted at exactly the moment of merger, but allowing for a sensible range of emission times, we can bound the difference between the speed of gravity and the speed of light. In general relativity they should be the same, and we find that the difference should be no more than three parts in $10^{15}$.

Second, we can combine the gravitational wave distance with the redshift of the galaxy to measure the Hubble constant, the rate of expansion of the Universe. Our best estimates for the Hubble constant, from the cosmic microwave background and from supernova observations, are inconsistent with each other (the most recent supernova analysis only increase the tension). Which is awkward. Gravitational wave observations should have different sources of error and help to resolve the difference. Unfortunately, with only one event our uncertainties are rather large, which leads to a diplomatic outcome.

Posterior probability distribution for the Hubble constant $H_0$ inferred from GW170817. The lines mark 68% and 95% intervals. The coloured bands are measurements from the cosmic microwave background (Planck) and supernovae (SHoES). Figure 1 of the Hubble Constant Paper.

Finally, we can now change from estimating upper limits on binary neutron star merger rates to estimating the rates! We estimate the merger rate density is in the range $1540^{+3200}_{-1220}~\mathrm{Gpc^{-3}\,yr^{-1}}$ (assuming a uniform of neutron star masses between one and two solar masses). This is surprisingly close to what the Collaboration expected back in 2010: a rate of between $10~\mathrm{Gpc^{-3}\,yr^{-1}}$ and $10000~\mathrm{Gpc^{-3}\,yr^{-1}}$, with a realistic rate of $1000~\mathrm{Gpc^{-3}\,yr^{-1}}$. This means that we are on track to see many more binary neutron stars—perhaps one a week at design sensitivity!

### Summary

Advanced LIGO and Advanced Virgo observed a binary neutron star insprial. The rest of the astronomical community has observed what happened next (sadly there are no neutrinos). This is the first time we have such complementary observations—hopefully there will be many more to come. There’ll be a huge number of results coming out over the following days and weeks. From these, we’ll start to piece together more information on what neutron stars are made of, and what happens when you smash them together (take that particle physicists).

Also: I’m exhausted, my inbox is overflowing, and I will have far too many papers to read tomorrow.

GW170817 Discovery Paper: GW170817: Observation of gravitational waves from a binary neutron star inspiral
Multimessenger Astronomy Paper: Multi-messenger observations of a binary neutron star merger
Data release:
LIGO Open Science Center

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

### Bonus notes

#### Inbox zero

Over my vacation I cleaned up my email. I had a backlog starting around September 2015.  I think there were over 6000 which I sorted or deleted. I had about 20 left to deal with when I got back to work. GW170817 undid that. Despite doing my best to keep up, there are over a 1000 emails in my inbox…

#### Worst case scenario

Around the start of O2, I was asked when I expected our results to be public. I said it would depend upon what we found. If it was only high-mass black holes, those are quick to analyse and we know what to do with them, so results shouldn’t take long, now we have the first few out of the way. In this case, perhaps a couple months as we would have been generating results as we went along. However, the worst case scenario would be a binary neutron star overlapping with non-Gaussian noise. Binary neutron stars are more difficult to analyse (they are longer signals, and there are matter effects to worry about), and it would be complicated to get everyone to be happy with our results because we were doing lots of things for the first time. Obviously, if one of these happened at the end of the run, there’d be quite a delay…

I think I got that half-right. We’re done amazingly well analysing GW170817 to get results out in just two months, but I think it will be a while before we get the full O2 set of results out, as we’ve been neglecting otherthings (you’ll notice we’ve not updated our binary black hole merger rate estimate since GW170104, nor given detailed results for testing general relativity with the more recent detections).

At the time of the GW170817 alert, I was working on writing a research proposal. As part of this, I was explaining why it was important to continue working on gravitational-wave parameter estimation, in particular how to deal with non-Gaussian or non-stationary noise. I think I may be a bit of a jinx. For GW170817, the glitch wasn’t a big problem, these type of blips can be removed. I’m more concerned about the longer duration ones, which are less easy to separate out from background noise. Don’t say I didn’t warn you in O3.

#### Parameter estimation rota

The duty of analysing signals to infer their source properties was divided up into shifts for O2. On January 4, the time of GW170104, I was on shift with my partner Aaron Zimmerman. It was his first day. Having survived that madness, Aaron signed back up for the rota. Can you guess who was on shift for the week which contained GW170814 and GW170817? Yep, Aaron (this time partnered with the excellent Carl-Johan Haster). Obviously, we’ll need to have Aaron on rota for the entirety of O3. In preparation, he has already started on paper drafting

Methods Section: Chained ROTA member to a terminal, ignored his cries for help. Detections followed swiftly.

The lightest elements (hydrogen, helium and lithium) we made during the Big Bang. Stars burn these to make heavier elements. Energy can be released up to around iron. Therefore, heavier elements need to be made elsewhere, for example in the material ejected from supernova or (as we have now seen) neutron star mergers, where there are lots of neutrons flying around to be absorbed. Elements (like gold and platinum) formed by this rapid neutron capture are known as r-process elements, I think because they are beloved by pirates.

A couple of weeks ago, the Nobel Prize in Physics was announced for the observation of gravitational waves. In December, the laureates will be presented with a gold (not chocolate) medal. I love the idea that this gold may have come from merging neutron stars.

Here’s one we made earlier. Credit: Associated Press/F. Vergara

# Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignments

Gravitational waves allow us to infer the properties of binary black holes (two black holes in orbit about each other), but can we use this information to figure out how the black holes and the binary form? In this paper, we show that measurements of the black holes’ spins can help us this out, but probably not until we have at least 100 detections.

### Black hole spins

Black holes are described by their masses (how much they bend spacetime) and their spins (how much they drag spacetime to rotate about them). The orientation of the spins relative to the orbit of the binary could tell us something about the history of the binary [bonus note].

We considered four different populations of spin–orbit alignments to see if we could tell them apart with gravitational-wave observations:

1. Aligned—matching the idealised example of isolated binary evolution. This stands in for the case where misalignments are small, which might be the case if material blown off during a supernova ends up falling back and being swallowed by the black hole.
2. Isotropic—matching the expectations for dynamically formed binaries.
3. Equal misalignments at birth—this would be the case if the spins and orbit were aligned before the second supernova, which then tilted the plane of the orbit. (As the binary inspirals, the spins wobble around, so the two misalignment angles won’t always be the same).
4. Both spins misaligned by supernova kicks, assuming that the stars were aligned with the orbit before exploding. This gives a more general scatter of unequal misalignments, but typically the primary (bigger and first forming) black hole is more misaligned.

These give a selection of possible spin alignments. For each, we assumed that the spin magnitude was the same and had a value of 0.7. This seemed like a sensible idea when we started this study [bonus note], but is now towards the upper end of what we expect for binary black holes.

### Hierarchical analysis

To measurement the properties of the population we need to perform a hierarchical analysis: there are two layers of inference, one for the individual binaries, and one of the population.

From a gravitational wave signal, we infer the properties of the source using Bayes’ theorem. Given the data $d_\alpha$, we want to know the probability that the parameters $\mathbf{\Theta}_\alpha$ have different values, which is written as $p(\mathbf{\Theta}_\alpha|d_\alpha)$. This is calculated using

$\displaystyle p(\mathbf{\Theta}_\alpha|d_\alpha) = \frac{p(d_\alpha | \mathbf{\Theta}_\alpha) p(\mathbf{\Theta}_\alpha)}{p(d_\alpha)}$,

where $p(d_\alpha | \mathbf{\Theta}_\alpha)$ is the likelihood, which we can calculate from our knowledge of the noise in our gravitational wave detectors, $p(\mathbf{\Theta}_\alpha)$ is the prior on the parameters (what we would have guessed before we had the data), and the normalisation constant $p(d_\alpha)$ is called the evidence. We’ll use the evidence again in the next layer of inference.

Our prior on the parameters should actually depend upon what we believe about the astrophysical population. It is different if we believed that Model 1 were true (when we’d only consider aligned spins) than for Model 2. Therefore, we should really write

$\displaystyle p(\mathbf{\Theta}_\alpha|d_\alpha, \lambda) = \frac{p(d_\alpha | \mathbf{\Theta}_\alpha,\lambda) p(\mathbf{\Theta}_\alpha,\lambda)}{p(d_\alpha|\lambda)}$,

where  $\lambda$ denotes which model we are considering.

This is an important point to remember: if you our using our LIGO results to test your theory of binary formation, you need to remember to correct for our choice of prior. We try to pick non-informative priors—priors that don’t make strong assumptions about the physics of the source—but this doesn’t mean that they match what would be expected from your model.

We are interested in the probability distribution for the different models: how many binaries come from each. Given a set of different observations $\{d_\alpha\}$, we can work this out using another application of Bayes’ theorem (yay)

$\displaystyle p(\mathbf{\lambda}|\{d_\alpha\}) = \frac{p(\{d_\alpha\} | \mathbf{\lambda}) p(\mathbf{\lambda})}{p(\{d_\alpha\})}$,

where $p(\{d_\alpha\} | \mathbf{\lambda})$ is just all the evidences for the individual events (given that model) multiplied together, $p(\mathbf{\lambda})$ is our prior for the different models, and $p(\{d_\alpha\})$ is another normalisation constant.

Now knowing how to go from a set of observations to the probability distribution on the different channels, let’s give it a go!

#### Results

To test our approach made a set of mock gravitational wave measurements. We generated signals from binaries for each of our four models, and analysed these as we would for real signals (using LALInference). This is rather computationally expensive, and we wanted a large set of events to analyse, so using these results as a guide, we created a larger catalogue of approximate distributions for the inferred source parameters $p(\mathbf{\Theta}_\alpha|d_\alpha)$. We then fed these through our hierarchical analysis. The GIF below shows how measurements of the fraction of binaries from each population tightens up as we get more detections: the true fraction is marked in blue.

Probability distribution for the fraction of binaries from each of our four spin misalignment populations for different numbers of observations. The blue dot marks the true fraction: and equal fraction from all four channels.

The plot shows that we do zoom in towards the true fraction of events from each model as the number of events increases, but there are significant degeneracies between the different models. Notably, it is difficult to tell apart Models 1 and 3, as both have strong support for both spins being nearly aligned. Similarly, there is a degeneracy between Models 2 and 4 as both allow for the two spins to have very different misalignments (and for the primary spin, which is the better measured one, to be quite significantly misaligned).

This means that we should be able to distinguish aligned from misaligned populations (we estimated that as few as 5 events would be needed to distinguish the case that all events came from either Model 1  or Model 2 if those were the only two allowed possibilities). However, it will be more difficult to distinguish different scenarios which only lead to small misalignments from each other, or disentangle whether there is significant misalignment due to big supernova kicks or because binaries are formed dynamically.

The uncertainty of the fraction of events from each model scales roughly with the square root of the number of observations, so it may be slow progress making these measurements. I’m not sure whether we’ll know the answer to how binary black hole form, or who will sit on the Iron Throne first.

arXiv: 1703.06873 [astro-ph.HE]
Journal: Monthly Notices of the Royal Astronomical Society471(3):2801–2811; 2017
Birmingham science summary: Hierarchical analysis of gravitational-wave measurements of binary black hole spin–orbit misalignment (by Simon)
If you like this you might like: Farr et al. (2017)Talbot & Thrane (2017), Vitale et al. (2017), Trifirò et al. (2016), Minogue (2000)

### Bonus notes

#### Spin misalignments and formation histories

If you have two stars forming in a binary together, you’d expect them to be spinning in roughly the same direction, rotating the same way as they go round in their orbit (like our Solar System). This is because they all formed from the same cloud of swirling gas and dust. Furthermore, if two stars are to form a black hole binary that we can detect gravitational waves from, they need to be close together. This means that there can be tidal forces which gently tug the stars to align their rotation with the orbit. As they get older, stars puff up, meaning that if you have a close-by neighbour, you can share outer layers. This transfer of material will tend to align rotate too. Adding this all together, if you have an isolated binary of stars, you might expect that when they collapse down to become black holes, their spins are aligned with each other and the orbit.

Unfortunately, real astrophysics is rarely so clean. Even if the stars were initially rotating the same way as each other, they doesn’t mean that their black hole remnants will do the same. This depends upon how the star collapses. Massive stars explode as supernova, blasting off their outer layers while their cores collapse down to form black holes. Escaping material could carry away angular momentum, meaning that the black hole is spinning in a different direction to its parent star, or material could be blasted off asymmetrically, giving the new black hole a kick. This would change the plane of the binary’s orbit, misaligning the spins.

Alternatively, the binary could be formed dynamically. Instead of two stars living their lives together, we could have two stars (or black holes) come close enough together to form a binary. This is likely to happen in regions where there’s a high density of stars, such as a globular cluster. In this case, since the binary has been randomly assembled, there’s no reason for the spins to be aligned with each other or the orbit. For dynamically assembled binaries, all spin–orbit misalignments are equally probable.

This project was led by Simon Stevenson. It was one of the first things we started working on at the beginning of his PhD. He has now graduated, and is off to start a new exciting life as a postdoc in Australia. We got a little distracted by other projects, most notably analysing the first detections of gravitational waves. Simon spent a lot of time developing the COMPAS population code, a code to simulate the evolution of binaries. Looking back, it’s impressive how far he’s come. This paper used a simple approximation to to estimate the masses of our black holes: we called it the Post-it note model, as we wrote it down on a single Post-it. Now Simon’s writing papers including the complexities of common-envelope evolution in order to explain LIGO’s actual observations.

# GW170104 and me

On 4 January 2017, Advanced LIGO made a new detection of gravitational waves. The signal, which we call GW170104 [bonus note], came from the coalescence of two black holes, which inspiralled together (making that characteristic chirp) and then merged to form a single black hole.

On 4 January 2017, I was just getting up off the sofa when my phone buzzed. My new year’s resolution was to go for a walk every day, and I wanted to make use of the little available sunlight. However, my phone informed me that PyCBC (one or our search algorithms for signals from coalescing binaries) had identified an interesting event. I sat back down. I was on the rota to analyse interesting signals to infer their properties, and I was pretty sure that people would be eager to see results. They were. I didn’t leave the sofa for the rest of the day, bringing my new year’s resolution to a premature end.

Since 4 January, my time has been dominated by working on GW170104 (you might have noticed a lack of blog posts). Below I’ll share some of my war stories from life on the front line of gravitational-wave astronomy, and then go through some of the science we’ve learnt. (Feel free to skip straight to the science, recounting the story was more therapy for me).

Time–frequency plots for GW170104 as measured by Hanford (top) and Livingston (bottom). The signal is clearly visible as the upward sweeping chirp. The loudest frequency is something between E3 and G♯3 on a piano, and it tails off somewhere between D♯4/E♭4 and F♯4/G♭4. Part of Fig. 1 of the GW170104 Discovery Paper.

### The story

In the second observing run, the Parameter Estimation group have divided up responsibility for analysing signals into two week shifts. For each rota shift, there is an expert and a rookie. I had assumed that the first slot of 2017 would be a quiet time. The detectors were offline over the holidays, due back online on 4 January, but the instrumentalists would probably find some extra tinkering they’d want to do, so it’d probably slip a day, and then the weather would be bad, so we’d probably not collect much data anyway… I was wrong. Very wrong. The detectors came back online on time, and there was a beautifully clean detection on day one.

My partner for the rota was Aaron Zimmerman. 4 January was his first day running parameter estimation on live signals. I think I would’ve run and hidden underneath my duvet in his case (I almost did anyway, and I lived through the madness of our first detection GW150914), but he rose to the occasion. We had first results after just a few hours, and managed to send out a preliminary sky localization to our astronomer partners on 6 January. I think this was especially impressive as there were some difficulties with the initial calibration of the data. This isn’t a problem for the detection pipelines, but does impact the parameters which we infer, particularly the sky location. The Calibration group worked quickly, and produced two updates to the calibration. We therefore had three different sets of results (one per calibration) by 6 January [bonus note]!

Producing the final results for the paper was slightly more relaxed. Aaron and I conscripted volunteers to help run all the various permutations of the analysis we wanted to double-check our results [bonus note].

Recovered gravitational waveforms from analysis of GW170104. The broader orange band shows our estimate for the waveform without assuming a particular source (wavelet). The narrow blue bands show results if we assume it is a binary black hole (BBH) as predicted by general relativity. The two match nicely, showing no evidence for any extra features not included in the binary black hole models. Figure 4 of the GW170104 Discovery Paper.

I started working on GW170104 through my parameter estimation duties, and continued with paper writing.

Ahead of the second observing run, we decided to assemble a team to rapidly write up any interesting binary detections, and I was recruited for this (I think partially because I’m not too bad at writing and partially because I was in the office next to John Veitch, one of the chairs of the Compact Binary Coalescence group,so he can come and check that I wasn’t just goofing off eating doughnuts). We soon decided that we should write a paper about GW170104, and you can decide whether or not we succeeded in doing this rapidly…

Being on the paper writing team has given me huge respect for the teams who led the GW150914 and GW151226 papers. It is undoubtedly one of the most difficult things I’ve ever done. It is extremely hard to absorb negative remarks about your work continuously for months [bonus note]—of course people don’t normally send comments about things that they like, but that doesn’t cheer you up when you’re staring at an inbox full of problems that need fixing. Getting a collaboration of 1000 people to agree on a paper is like herding cats while being a small duckling.

On of the first challenges for the paper writing team was deciding what was interesting about GW170104. It was another binary black hole coalescence—aren’t people getting bored of them by now? The signal was quieter than GW150914, so it wasn’t as remarkable. However, its properties were broadly similar. It was suggested that perhaps we should title the paper “GW170104: The most boring gravitational-wave detection”.

One potentially interesting aspect was that GW170104 probably comes from greater distance than GW150914 or GW151226 (but perhaps not LVT151012) [bonus note]. This might make it a good candidate for testing for dispersion of gravitational waves.

Dispersion occurs when different frequencies of gravitational waves travel at different speeds. A similar thing happens for light when travelling through some materials, which leads to prisms splitting light into a spectrum (and hence the creation of Pink Floyd album covers). Gravitational waves don’t suffered dispersion in general relativity, but do in some modified theories of gravity.

It should be easier to spot dispersion in signals which have travelled a greater distance, as the different frequencies have had more time to separate out. Hence, GW170104 looks pretty exciting. However, being further away also makes the signal quieter, and so there is more uncertainty in measurements and it is more difficult to tell if there is any dispersion. Dispersion is also easier to spot if you have a larger spread of frequencies, as then there can be more spreading between the highest and lowest frequencies. When you throw distance, loudness and frequency range into the mix, GW170104 doesn’t always come out on top, depending upon the particular model for dispersion: sometimes GW150914’s loudness wins, other times GW151226’s broader frequency range wins. GW170104 isn’t too special here either.

Even though GW170104 didn’t look too exciting, we started work on a paper, thinking that we would just have a short letter describing our observations. The Compact Binary Coalescence group decided that we only wanted a single paper, and we wouldn’t bother with companion papers as we did for GW150914. As we started work, and dug further into our results, we realised that actually there was rather a lot that we could say.

I guess the moral of the story is that even though you might be overshadowed by the achievements of your siblings, it doesn’t mean that you’re not awesome. There might not be one outstanding feature of GW170104, but there are lots of little things that make it interesting. We are still at the beginning of understanding the properties of binary black holes, and each new detection adds a little more to our picture.

I think GW170104 is rather neat, and I hope you do too.

As we delved into the details of our results, we realised there was actually a lot of things that we could say about GW170104, especially when considered with our previous observations. We ended up having to move some of the technical details and results to Supplemental Material. With hindsight, perhaps it would have been better to have a companion paper or two. However, I rather like how packed with science this paper is.

The paper, which Physical Review Letters have kindly accommodated, despite its length, might not be as polished a classic as the GW150914 Discovery Paper, but I think they are trying to do different things. I rarely ever refer to the GW150914 Discovery Paper for results (more commonly I use it for references), whereas I think I’ll open up the GW170104 Discovery Paper frequently to look up numbers.

Although perhaps not right away, I’d quite like some time off first. The weather’s much better now, perfect for walking…

Success! The view across Lac d’Annecy. Taken on a stroll after the Gravitational Wave Physics and Astronomy Workshop, the weekend following the publication of the paper.

### The science

Advanced LIGO’s first observing run was hugely successful. Running from 12 September 2015 until 19 January 2016, there were two clear gravitational-wave detections, GW1501914 and GW151226, as well as a less certain candidate signal LVT151012. All three (assuming that they are astrophysical signals) correspond to the coalescence of binary black holes.

The second observing run started 30 November 2016. Following the first observing run’s detections, we expected more binary black hole detections. On 4 January, after we had collected almost 6 days’ worth of coincident data from the two LIGO instruments [bonus note], there was a detection.

#### The searches

The signal was first spotted by an online analysis. Our offline analysis of the data (using refined calibration and extra information about data quality) showed that the signal, GW170104, is highly significant. For both GstLAL and PyCBC, search algorithms which use templates to search for binary signals, the false alarm rate is estimated to be about 1 per 70,000 years.

The signal is also found in unmodelled (burst) searches, which look for generic, short gravitational wave signals. Since these are looking for more general signals than just binary coalescences, the significance associated with GW170104 isn’t as great, and coherent WaveBurst estimates a false alarm rate of 1 per 20,000 years. This is still pretty good! Reconstructions of the waveform from unmodelled analyses also match the form expected for binary black hole signals.

The search false alarm rates are the rate at which you’d expect something this signal-like (or more signal-like) due to random chance, if you data only contained noise and no signals. Using our knowledge of the search pipelines, and folding in some assumptions about the properties of binary black holes, we can calculate a probability that GW170104 is a real astrophysical signal. This comes out to be greater than $1 - (3\times10^5) = 0.99997$.

#### The source

As for the previous gravitational wave detections, GW170104 comes from a binary black hole coalescence. The initial black holes were $31.2^{+8.4}_{-6.0} M_\odot$ and $19.4^{+5.3}_{-5.9} M_\odot$ (where $1 M_\odot$ is the mass of our Sun), and the final black hole was $48.7^{+5.7}_{-4.6} M_\odot$. The quoted values are the median values and the error bars denote the central 90% probable range. The plot below shows the probability distribution for the masses; GW170104 neatly nestles in amongst the other events.

Estimated masses for the two black holes in the binary $m_1 \geq m_2$. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours for all events. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 2 of the GW170104 Discovery Paper.

GW150914 was the first time that we had observed stellar-mass black holes with masses greater than around $25 M_\odot$. GW170104 has similar masses, showing that our first detection was not a fluke, but there really is a population of black holes with masses stretching up into this range.

Black holes have two important properties: mass and spin. We have good measurements on the masses of the two initial black holes, but not the spins. The sensitivity of the form of the gravitational wave to spins can be described by two effective spin parameters, which are mass-weighted combinations of the individual spins.

• The effective inspiral spin parameter $\chi_\mathrm{eff}$ qualifies the impact of the spins on the rate of inspiral, and where the binary plunges together to merge. It ranges from +1, meaning both black holes are spinning as fast as possible and rotate in the same direction as the orbital motion, to −1, both black holes spinning as fast as possible but in the opposite direction to the way that the binary is orbiting. A value of 0 for $\chi_\mathrm{eff}$ could mean that the black holes are not spinning, that their rotation axes are in the orbital plane (instead of aligned with the orbital angular momentum), or that one black hole is aligned with the orbital motion and the other is antialigned, so that their effects cancel out.
• The effective precession spin parameter $\chi_\mathrm{p}$ qualifies the amount of precession, the way that the orbital plane and black hole spins wobble when they are not aligned. It is 0 for no precession, and 1 for maximal precession.

We can place some constraints on $\chi_\mathrm{eff}$, but can say nothing about $\chi_\mathrm{p}$. The inferred value of the effective inspiral spin parameter is $-0.12^{+0.21}_{-0.30}$. Therefore, we disfavour large spins aligned with the orbital angular momentum, but are consistent with small aligned spins, misaligned spins, or spins antialigned with the angular momentum. The value is similar to that for GW150914, which also had a near-zero, but slightly negative $\chi_\mathrm{eff}$ of $-0.06^{+0.14}_{-0.14}$.

Estimated effective inspiral spin parameter $\chi_\mathrm{eff}$ and effective precession spin $\chi_\mathrm{p}$ parameter. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models, as well as the prior probability distribution. The dotted lines mark the edge of our 90% probability intervals. We learn basically nothing about precession. Part of Figure 3 of the GW170104 Discovery Paper.

Converting the information about $\chi_\mathrm{eff}$, the lack of information about $\chi_\mathrm{p}$, and our measurement of the ratio of the two black hole masses, into probability distributions for the component spins gives the plots below [bonus note]. We disfavour (but don’t exclude) spins aligned with the orbital angular momentum, but can’t say much else.

Estimated orientation and magnitude of the two component spins. The distribution for the more massive black hole is on the left, and for the smaller black hole on the right. The probability is binned into areas which have uniform prior probabilities, so if we had learnt nothing, the plot would be uniform. Part of Figure 3 of the GW170104 Discovery Paper.

One of the comments we had on a draft of the paper was that we weren’t making any definite statements about the spins—we would have if we could, but we can’t for GW170104, at least for the spins of the two inspiralling black holes. We can be more definite about the spin of the final black hole. If two similar mass black holes spiral together, the angular momentum from the orbit is enough to give a spin of around $0.7$. The spins of the component black holes are less significant, and can make it a bit higher of lower. We infer a final spin of $0.64^{+0.09}_{-0.20}$; there is a tail of lower spin values on account of the possibility that the two component black holes could be roughly antialigned with the orbital angular momentum.

Estimated mass $M_\mathrm{f}$ and spin$a_\mathrm{f}$ for the final black hole. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 6 of the GW170104 Supplemental Material (Figure 11 of the arXiv version).

If you’re interested in parameter describing GW170104, make sure to check out the big table in the Supplemental Material. I am a fan of tables [bonus note].

#### Merger rates

Adding the first 11 days of coincident data from the second observing run (including the detection of GW170104) to the results from the first observing run, we find merger rates consistent with those from the first observing run.

To calculate the merger rates, we need to assume a distribution of black hole masses, and we use two simple models. One uses a power law distribution for the primary (larger) black hole and a uniform distribution for the mass ratio; the other uses a distribution uniform in the logarithm of the masses (both primary and secondary). The true distribution should lie somewhere between the two. The power law rate density has been updated from $31^{+42}_{-21}~\mathrm{Gpc^{-3}\,yr^{-1}}$ to $32^{+33}_{-20}~\mathrm{Gpc^{-3}\,yr^{-1}}$, and the uniform in log rate density goes from $97^{+135}_{-67}~\mathrm{Gpc^{-3}\,yr^{-1}}$ to $103^{+110}_{-63}~\mathrm{Gpc^{-3}\,yr^{-1}}$. The median values stay about the same, but the additional data have shrunk the uncertainties a little.

#### Astrophysics

The discoveries from the first observing run showed that binary black holes exist and merge. The question is now how exactly they form? There are several suggested channels, and it could be there is actually a mixture of different formation mechanisms in action. It will probably require a large number of detections before we can make confident statements about the the probable formation mechanisms; GW170104 is another step towards that goal.

There are two main predicted channels of binary formation:

• Isolated binary evolution, where a binary star system lives its life together with both stars collapsing to black holes at the end. To get the black holes close enough to merge, it is usually assumed that the stars go through a common envelope phase, where one star puffs up so that the gravity of its companion can steal enough material that they lie in a shared envelope. The drag from orbiting inside this then shrinks the orbit.
• Dynamical evolution where black holes form in dense clusters and a binary is created by dynamical interactions between black holes (or stars) which get close enough to each other.

It’s a little artificial to separate the two, as there’s not really such a thing as an isolated binary: most stars form in clusters, even if they’re not particularly large. There are a variety of different modifications to the two main channels, such as having a third companion which drives the inner binary to merge, embedding the binary is a dense disc (as found in galactic centres), or dynamically assembling primordial black holes (formed by density perturbations in the early universe) instead of black holes formed through stellar collapse.

All the channels can predict black holes around the masses of GW170104 (which is not surprising given that they are similar to the masses of GW150914).

The updated rates are broadly consistent with most channels too. The tightening of the uncertainty of the rates means that the lower bound is now a little higher. This means some of the channels are now in tension with the inferred rates. Some of the more exotic channels—requiring a third companion (Silsbee & Tremain 2017; Antonini, Toonen & Hamers 2017) or embedded in a dense disc (Bartos et al. 2016; Stone, Metzger & Haiman 2016; Antonini & Rasio 2016)—can’t explain the full rate, but I don’t think it was ever expected that they could, they are bonus formation mechanisms. However, some of the dynamical models are also now looking like they could predict a rate that is a bit low (Rodriguez et al. 2016; Mapelli 2016; Askar et al. 2017; Park et al. 2017). Assuming that this result holds, I think this may mean that some of the model parameters need tweaking (there are more optimistic predictions for the merger rates from clusters which are still perfectly consistent), that this channel doesn’t contribute all the merging binaries, or both.

The spins might help us understand formation mechanisms. Traditionally, it has been assumed that isolated binary evolution gives spins aligned with the orbital angular momentum. The progenitor stars were probably more or less aligned with the orbital angular momentum, and tides, mass transfer and drag from the common envelope would serve to realign spins if they became misaligned. Rodriguez et al. (2016) gives a great discussion of this. Dynamically formed binaries have no correlation between spin directions, and so we would expect an isotropic distribution of spins. Hence it sounds quite simple: misaligned spins indicates dynamical formation (although we can’t tell if the black holes are primordial or stellar), and aligned spins indicates isolated binary evolution. The difficulty is the traditional assumption for isolated binary evolution potentially ignores a number of effects which could be important. When a star collapses down to a black hole, there may be a supernova explosion. There is an explosion of matter and neutrinos and these can give the black hole a kick. The kick could change the orbital plane, and so misalign the spin. Even if the kick is not that big, if it is off-centre, it could torque the black hole, causing it to rotate and so misalign the spin that way. There is some evidence that this can happen with neutron stars, as one of the pulsars in the double pulsar system shows signs of this. There could also be some instability that changes the angular momentum during the collapse of the star, possibly with different layers rotating in different ways [bonus note]. The spin of the black hole would then depend on how many layers get swallowed. This is an area of research that needs to be investigated further, and I hope the prospect of gravitational wave measurements spurs this on.

For GW170104, we know the spins are not large and aligned with the orbital angular momentum. This might argue against one variation of isolated binary evolution, chemically homogeneous evolution, where the progenitor stars are tidally locked (and so rotate aligned with the orbital angular momentum and each other). Since the stars are rapidly spinning and aligned, you would expect the final black holes to be too, if the stars completely collapse down as is usually assumed. If the stars don’t completely collapse down though, it might still be possible that GW170104 fits with this model. Aside from this, GW170104 is consistent with all the other channels.

Estimated effective inspiral spin parameter $\chi_\mathrm{eff}$ for all events. To indicate how much (or little) we’ve learnt, the prior probability distribution for GW170104 is shown (the other priors are similar).All of the events have $|\chi_\mathrm{eff}| < 0.35$ at 90% probability. Figure 5 of the GW170104 Supplemental Material (Figure 10 of the arXiv version). This is one of my favourite plots [bonus note].

If we start looking at the population of events, we do start to notice something about the spins. All of the inferred values of $\chi_\mathrm{eff}$ are close to zero. Only GW151226 is inconsistent with zero. These values could be explained if spins are typically misaligned (with the orbital angular momentum or each other) or if the spins are typically small (or both). We know that black holes spins can be large from observations of X-ray binaries, so it would be odd if they are small for binary black holes. Therefore, we have a tentative hint that spins are misaligned. We can’t say why the spins are misaligned, but it is intriguing. With more observations, we’ll be able to confirm if it is the case that spins are typically misaligned, and be able to start pinning down the distribution of spin magnitudes and orientations (as well as the mass distribution). It will probably take a while to be able to say anything definite though, as we’ll probably need about 100 detections.

#### Tests of general relativity

As well as giving us an insight into the properties of black holes, gravitational waves are the perfect tools for testing general relativity. If there are any corrections to general relativity, you’d expect them to be most noticeable under the most extreme conditions, where gravity is strong and spacetime is rapidly changing, exactly as in a binary black hole coalescence.

For GW170104 we repeated tests previously performed. Again, we found no evidence of deviations.

We added extra terms to to the waveform and constrained their potential magnitudes. The results are pretty much identical to at the end of the first observing run (consistent with zero and hence general relativity). GW170104 doesn’t add much extra information, as GW150914 typically gives the best constraints on terms that modify the post-inspiral part of the waveform (as it is louder), while GW151226 gives the best constraint on the terms which modify the inspiral (as it has the longest inspiral).

We also chopped the waveform at a frequency around that of the innermost stable orbit of the remnant black hole, which is about where the transition from inspiral to merger and ringdown occurs, to check if the low frequency and high frequency portions of the waveform give consistent estimates for the final mass and spin. They do.

We have also done something slightly new, and tested for dispersion of gravitational waves. We did something similar for GW150914 by putting a limit on the mass of the graviton. Giving the graviton mass is one way of adding dispersion, but we consider other possible forms too. In all cases, results are consistent with there being no dispersion. While we haven’t discovered anything new, we can update our gravitational wave constraint on the graviton mass of less than $7.7 \times 10^{-23}~\mathrm{eV}/c^2$.

#### The search for counterparts

We don’t discuss observations made by our astronomer partners in the paper (they are not our results). A number (28 at the time of submission) of observations were made, and I expect that there will be a series of papers detailing these coming soon. So far papers have appeared from:

• AGILE—hard X-ray and gamma-ray follow-up. They didn’t find any gamma-ray signals, but did identify a weak potential X-ray signal occurring about 0.46 s before GW170104. It’s a little odd to have a signal this long before the merger. The team calculate a probability for such a coincident to happen by chance, and find quite a small probability, so it might be interesting to follow this up more (see the INTEGRAL results below), but it’s probably just a coincidence (especially considering how many people did follow-up the event).
• ANTARES—a search for high-energy muon neutrinos. No counterparts are identified in a ±500 s window around GW170104, or over a ±3 month period.
• AstroSat-CZTI and GROWTH—a collaboration of observations across a range of wavelengths. They don’t find any hard X-ray counterparts. They do follow up on a bright optical transient ATLASaeu, suggested as a counterpart to GW170104, and conclude that this is a likely counterpart of long, soft gamma-ray burst GRB 170105A.
• ATLAS and Pan-STARRS—optical follow-up. They identified a bright optical transient 23 hours after GW170104, ATLAS17aeu. This could be a counterpart to GRB 170105A. It seems unlikely that there is any mechanism that could allow for a day’s delay between the gravitational wave emission and an electromagnetic signal. However, the team calculate a small probability (few percent) of finding such a coincidence in sky position and time, so perhaps it is worth pondering. I wouldn’t put any money on it without a distance estimate for the source: assuming it’s a normal afterglow to a gamma-ray burst, you’d expect it to be further away than GW170104’s source.
• Borexino—a search for low-energy neutrinos. This paper also discusses GW150914 and GW151226. In all cases, the observed rate of neutrinos is consistent with the expected background.
• CALET—a gamma-ray search. This paper includes upper limits for GW151226, GW170104, GW170608, GW170814 and GW170817.
• DLT40—an optical search designed for supernovae. This paper covers the whole of O2 including GW170608, GW170814, GW170817 plus GW170809 and GW170823.
• Fermi (GBM and LAT)—gamma-ray follow-up. They covered an impressive fraction of the sky localization, but didn’t find anything.
• INTEGRAL—gamma-ray and hard X-ray observations. No significant emission is found, which makes the event reported by AGILE unlikely to be a counterpart to GW170104, although they cannot completely rule it out.
• The intermediate Palomar Transient Factory—an optical survey. While searching, they discovered iPTF17cw, a broad-line type Ic supernova which is unrelated to GW170104 but interesting as it an unusual find.
• Mini-GWAC—a optical survey (the precursor to GWAC). This paper covers the whole of their O2 follow-up including GW170608.
• The Owens Valley Radio Observatory Long Wavelength Array—a search for prompt radio emission.
• TOROS—optical follow-up. They identified no counterparts to GW170104 (although they did for GW170817).

If you are interested in what has been reported so far (no compelling counterpart candidates yet to my knowledge), there is an archive of GCN Circulars sent about GW170104.

### Summary

Advanced LIGO has made its first detection of the second observing run. This is a further binary black hole coalescence. GW170104 has taught us that:

• The discoveries of the first observing run were not a fluke. There really is a population of stellar mass black holes with masses above $25 M_\odot$ out there, and we can study them with gravitational waves.
• Binary black hole spins may be typically misaligned or small. This is not certain yet, but it is certainly worth investigating potential mechanisms that could cause misalignment.
• General relativity still works, even after considering our new tests.
• If someone asks you to write a discovery paper, run. Run and do not look back.

Title: GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2
Journal:
Physical Review Letters; 118(22):221101(17); 2017 (Supplemental Material)
arXiv: 1706.01812 [gr-qc]
Data release: LIGO Open Science Center
Science summary:
GW170104: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2

If you’re looking for the most up-to-date results regarding GW170104, check out the O2 Catalogue Paper.

### Bonus notes

#### Naming

Gravitational wave signals (at least the short ones, which are all that we have so far), are named by their detection date. GW170104 was discovered 2017 January 4. This isn’t too catchy, but is at least better than the ID number in our database of triggers (G268556) which is used in corresponding with our astronomer partners before we work out if the “GW” title is justified.

Previous detections have attracted nicknames, but none has stuck for GW170104. Archisman Ghosh suggested the Perihelion Event, as it was detected a few hours before the Earth reached its annual point closest to the Sun. I like this name, its rather poetic.

More recently, Alex Nitz realised that we should have called GW170104 the Enterprise-D Event, as the USS Enterprise’s registry number was NCC-1701. For those who like Star Trek: the Next Generation, I hope you have fun discussing whether GW170104 is the third or fourth (counting LVT151012) detection: “There are four detections!

#### The 6 January sky map

I would like to thank the wi-fi of Chiltern Railways for their role in producing the preliminary sky map. I had arranged to visit London for the weekend (because my rota slot was likely to be quiet… ), and was frantically working on the way down to check results so they could be sent out. I’d also like to thank John Veitch for putting together the final map while I was stuck on the Underground.

#### Binary black hole waveforms

The parameter estimation analysis works by matching a template waveform to the data to see how well it matches. The results are therefore sensitive to your waveform model, and whether they include all the relevant bits of physics.

In the first observing run, we always used two different families of waveforms, to see what impact potential errors in the waveforms could have. The results we presented in discovery papers used two quick-to-calculate waveforms. These include the effects of the black holes’ spins in different ways

• SEOBNRv2 has spins either aligned or antialigned with the orbital angular momentum. Therefore, there is no precession (wobbling of orientation, like that of a spinning top) of the system.
• IMRPhenomPv2 includes an approximate description of precession, packaging up the most important information about precession into a single parameter $\chi_\mathrm{p}$.

For GW150914, we also performed a follow-up analysis using a much more expensive waveform SEOBNRv3 which more fully includes the effect of both spins on precession. These results weren’t ready at the time of the announcement, because the waveform is laborious to run.

For GW170104, there were discussions that using a spin-aligned waveform was old hat, and that we should really only use the two precessing models. Hence, we started on the endeavour of producing SEOBNRv3 results. Fortunately, the code has been sped up a little, although it is still not quick to run. I am extremely grateful to Scott Coughlin (one of the folks behind Gravity Spy), Andrea Taracchini and Stas Babak for taking charge of producing results in time for the paper, in what was a Herculean effort.

I spent a few sleepless nights, trying to calculate if the analysis was converging quickly enough to make our target submission deadline, but it did work out in the end. Still, don’t necessarily expect we’ll do this for a all future detections.

Since the waveforms have rather scary technical names, in the paper we refer to IMRPhenomPv2 as the effective precession model and SEOBNRv3 as the full precession model.

#### On distance

Distance measurements for gravitational wave sources have significant uncertainties. The distance is difficult to measure as it determined from the signal amplitude, but this is also influences by the binary’s inclination. A signal could either be close and edge on or far and face on-face off.

Estimated luminosity distance $D_\mathrm{L}$ and binary inclination angle $\theta_{JN}$. The two-dimensional shows the probability distribution for GW170104 as well as 50% and 90% contours. The one-dimensional plot shows results using different waveform models. The dotted lines mark the edge of our 90% probability intervals. Figure 4 of the GW170104 Supplemental Material (Figure 9 of the arXiv version).

The uncertainty on the distance rather awkwardly means that we can’t definitely say that GW170104 came from a further source than GW150914 or GW151226, but it’s a reasonable bet. The 90% credible intervals on the distances are 250–570 Mpc for GW150194, 250–660 Mpc for GW151226, 490–1330 Mpc for GW170104 and 500–1500 Mpc for LVT151012.

Translating from a luminosity distance to a travel time (gravitational waves do travel at the speed of light, our tests of dispersion are consistent wit that!), the GW170104 black holes merged somewhere between 1.3 and 3.0 billion years ago. This is around the time that multicellular life first evolved on Earth, and means that black holes have been colliding longer than life on Earth has been reproducing sexually.

#### Time line

A first draft of the paper (version 2; version 1 was a copy-and-paste of the Boxing Day Discovery Paper) was circulated to the Compact Binary Coalescence and Burst groups for comments on 4 March. This was still a rough version, and we wanted to check that we had a good outline of the paper. The main feedback was that we should include more about the astrophysical side of things. I think the final paper has a better balance, possibly erring on the side of going into too much detail on some of the more subtle points (but I think that’s better than glossing over them).

A first proper draft (version 3) was released to the entire Collaboration on 12 March in the middle of our Collaboration meeting in Pasadena. We gave an oral presentation the next day (I doubt many people had read the paper by then). Collaboration papers are usually allowed two weeks for people to comment, and we followed the same procedure here. That was not a fun time, as there was a constant trickle of comments. I remember waking up each morning and trying to guess how many emails would be in my inbox–I normally low-balled this.

I wasn’t too happy with version 3, it was still rather rough. The members of the Paper Writing Team had been furiously working on our individual tasks, but hadn’t had time to look at the whole. I was much happier with the next draft (version 4). It took some work to get this together, following up on all the comments and trying to address concerns was a challenge. It was especially difficult as we got a series of private comments, and trying to find a consensus probably made us look like the bad guys on all sides. We released version 4 on 14 April for a week of comments.

The next step was approval by the LIGO and Virgo executive bodies on 24 April. We prepared version 5 for this. By this point, I had lost track of which sentences I had written, which I had merely typed, and which were from other people completely. There were a few minor changes, mostly adding technical caveats to keep everyone happy (although they do rather complicate the flow of the text).

The paper was circulated to the Collaboration for a final week of comments on 26 April. Most comments now were about typos and presentation. However, some people will continue to make the same comment every time, regardless of how many times you explain why you are doing something different. The end was in sight!

The paper was submitted to Physical Review Letters on 9 May. I was hoping that the referees would take a while, but the reports were waiting in my inbox on Monday morning.

The referee reports weren’t too bad. Referee A had some general comments, Referee B had some good and detailed comments on the astrophysics, and Referee C gave the paper a thorough reading and had some good suggestions for clarifying the text. By this point, I have been staring at the paper so long that some outside perspective was welcome. I was hoping that we’d have a more thorough review of the testing general relativity results, but we had Bob Wald as one of our Collaboration Paper reviewers (the analysis, results and paper are all reviewed internally), so I think we had already been held to a high standard, and there wasn’t much left to say.

We put together responses to the reports. There were surprisingly few comments from the Collaboration at this point. I guess that everyone was getting tired. The paper was resubmitted and accepted on 20 May.

One of the suggestions of Referee A was to include some plots showing the results of the searches. People weren’t too keen on showing these initially, but after much badgering they were convinced, and it was decided to put these plots in the Supplemental Material which wouldn’t delay the paper as long as we got the material submitted by 26 May. This seemed like plenty of time, but it turned out to be rather frantic at the end (although not due to the new plots). The video below is an accurate representation of us trying to submit the final version.

I have an email which contains the line “Many Bothans died to bring us this information” from 1 hour and 18 minutes before the final deadline.

After this, things were looking pretty good. We had returned the proofs of the main paper (I had a fun evening double checking the author list. Yes, all of them). We were now on version 11 of the paper.

Of course, there’s always one last thing. On 31 May, the evening before publication, Salvo Vitale spotted a typo. Nothing serious, but annoying. The team at Physical Review Letters were fantastic, and took care of it immediately!

There’ll still be one more typo, there always is…

Looking back, it is clear that the principal bottle-neck in publishing the results is getting the Collaboration to converge on the paper. I’m not sure how we can overcome this… Actually, I have some ideas, but none that wouldn’t involve some form of doomsday device.

#### Detector status

The sensitivities of the LIGO Hanford and Livinston detectors are around the same as they were in the first observing run. After the success of the first observing run, the second observing run is the difficult follow up album. Livingston has got a little better, while Hanford is a little worse. This is because the Livingston team concentrate on improving low frequency sensitivity whereas the Hanford team focused on improving high frequency sensitivity. The Hanford team increased the laser power, but this introduces some new complications. The instruments are extremely complicated machines, and improving sensitivity is hard work.

The current plan is to have a long commissioning break after the end of this run. The low frequency tweaks from Livingston will be transferred to Hanford, and both sites will work on bringing down other sources of noise.

While the sensitivity hasn’t improved as much as we might have hoped, the calibration of the detectors has! In the first observing run, the calibration uncertainty for the first set of published results was about 10% in amplitude and 10 degrees in phase. Now, uncertainty is better than 5% in amplitude and 3 degrees in phase, and people are discussing getting this down further.

#### Spin evolution

As the binary inspirals, the orientation of the spins will evolve as they precess about. We always quote measurements of the spins at a point in the inspiral corresponding to a gravitational wave frequency of 20 Hz. This is most convenient for our analysis, but you can calculate the spins at other points. However, the resulting probability distributions are pretty similar at other frequencies. This is because the probability distributions are primarily determined by the combination of three things: (i) our prior assumption of a uniform distribution of spin orientations, (ii) our measurement of the effective inspiral spin, and (iii) our measurement of the mass ratio. A uniform distribution stays uniform as spins evolve, so this is unaffected, the effective inspiral spin is approximately conserved during inspiral, so this doesn’t change much, and the mass ratio is constant. The overall picture is therefore qualitatively similar at different moments during the inspiral.

#### Footnotes

I love footnotes. It was challenging for me to resist having any in the paper.

#### Gravity waves

It is possible that internal gravity waves (that is oscillations of the material making up the star, where the restoring force is gravity, not gravitational waves, which are ripples in spacetime), can transport angular momentum from the core of a star to its outer envelope, meaning that the two could rotate in different directions (Rogers, Lin & Lau 2012). I don’t think anyone has studied this yet for the progenitors of binary black holes, but it would be really cool if gravity waves set the properties of gravitational wave sources.

I really don’t want to proof read the paper which explains this though.

#### Colour scheme

For our plots, we use a consistent colour coding for our events. GW150914 is blue; LVT151012 is green; GW151226 is red–orange, and GW170104 is purple. The colour scheme is designed to be colour blind friendly (although adopting different line styles would perhaps be more distinguishable), and is implemented in Python in the Seaborn package as colorblind. Katerina Chatziioannou, who made most of the plots showing parameter estimation results is not a fan of the colour combinations, but put a lot of patient effort into polishing up the plots anyway.

# Search for transient gravitational waves in coincidence with short-duration radio transients during 2007–2013

Gravitational waves give us a new way of observing the Universe. This raises the possibility of multimessenger astronomy, where we study the same system using different methods: gravitational waves, light or neutrinos. Each messenger carries different information, so by using them together we can build up a more complete picture of what’s going on. This paper looks for gravitational waves that coincide with radio bursts. None are found, but we now have a template for how to search in the future.

On a dark night, there are two things which almost everyone will have done: wondered at the beauty of the starry sky and wondered exactly what was it that just went bump… Astronomers do both. Transient astronomy is about figuring out what are the things which go bang in the night—not the things which make suspicious noises, but objects which appear (and usually disappear) suddenly in the sky.

Most processes in astrophysics take a looooong time (our Sun is four-and-a-half billion years old and is just approaching middle age). Therefore, when something happens suddenly, flaring perhaps over just a few seconds, you know that something drastic must be happening! We think that most transients must be tied up with a violent event such as an explosion. However, because transients are so short, it can difficult to figure out exactly where they come from (both because they might have faded by the time you look, and because there’s little information to learn from a blip in the first place).

Radio transients are bursts of radio emission of uncertain origin. We’ve managed to figure out that some come from microwave ovens, but the rest do seem to come from space. This paper looks at two types: rotating radio transients (RRATs) and fast radio bursts (FRBs). RRATs look like the signals from pulsars, except that they don’t have the characteristic period pattern of pulsars. It may be that RRATs come from dying pulsars, flickering before they finally switch off, or it may be that they come from neutron stars which are not normally pulsars, but have been excited by a fracturing of their crust (a starquake). FRBs last a few milliseconds, they could be generated when two neutron stars merge and collapse to form a black hole, or perhaps from a highly-magnetised neutron star. Normally, when astronomers start talking about magnetic fields, it means that we really don’t know what’s going on [bonus note]. That is the case here. We don’t know what causes radio transients, but we are excited to try figuring it out.

This paper searches old LIGO, Virgo and GEO data for any gravitational-wave signals that coincide with observed radio transients. We use a catalogue of RRATs and FRBs from the Green Bank Telescope and the Parkes Observatory, and search around these times. We use a burst search, which doesn’t restrict itself to any particular form of gravitational-wave; however, the search was tuned for damped sinusoids and sine–Gaussians (generic wibbles), cosmic strings (which may give an indication of how uncertain we are of where radio transients could come from), and coalescences of binary neutron stars or neutron star–black hole binaries. Hopefully the search covers all plausible options. Discovering a gravitational wave coincident with a radio transient would give us much welcomed information about the source, and perhaps pin down their origin.

Search results for gravitational waves coincident with radio transients. The probabilities for each time containing just noise (blue) match the expected background distribution (dashed). This is consistent with a non-detection.

The search discovered nothing. Results match what we would expect from just noise in the detectors. This is not too surprising since we are using data from the first-generation detectors. We’ll be repeating the analysis with the upgraded detectors, which can find signals from larger distances. If we are lucky, multimessenger astronomy will allow us to figure out exactly what needs to go bump to create a radio transient.

arXiv: 1605.01707 [astro-ph.HE]
Journal: Physical Review D; 93(12):122008(14); 2016
Science summary: Searching for gravitational wave bursts in coincidence with short duration radio bursts
Favourite thing that goes bump in the night: Heffalumps and Woozles [probably not the cause of radio transients]

### Bonus note

#### Magnetism and astrophysics

Magnetic fields complicate calculations. They make things more difficult to model and are therefore often left out. However, we know that magnetic fields are everywhere and that they do play important roles in many situations. Therefore, they are often invoked as an explanation of why models can’t explain what’s going on. I learnt early in my PhD that you could ask “What about magnetic fields?” at the end of almost any astrophysics seminar (it might not work for some observational talks, but then you could usually ask “What about dust?” instead). Handy if ever you fall asleep…

# A black hole Pokémon

The world is currently going mad for Pokémon Go, so it seems like the perfect time to answer the most burning of scientific questions: what would a black hole Pokémon be like?

Type: Dark/Ghost

Black holes are, well, black. Their gravity is so strong that if you get close enough, nothing, not even light, can escape. I think that’s about as dark as you can get!

After picking Dark as a primary type, I thought Ghost was a good secondary type, since black holes could be thought of as the remains of dead stars. This also fit well with black holes not really being made of anything—they are just warped spacetime—and so are ethereal in nature. Of course, black holes’ properties are grounded in general relativity and not the supernatural.

In the games, having a secondary type has another advantage: Dark types are weak against Fighting types. In reality, punching or kicking a black hole is a Bad Idea™: it will not damage the black hole, but will certainly cause you some difficulties. However, Ghost types are unaffected by Fighting-type moves, so our black hole Pokémon doesn’t have to worry about them.

Height: 0’04″/0.1 m

Real astrophysical black holes are probably a bit too big for Pokémon games.  The smallest Pokémon are currently the electric bug Joltik and fairy Flabébé, so I’ve made our black hole Pokémon the same size as these. It should comfortably fit inside a Pokéball.

Measuring the size of a black hole is actually rather tricky, since they curve spacetime. When talking about the size of a black hole, we normally think in terms of the Schwarzschild radius. Named after Karl Schwarzschild, who first calculated the spacetime of a black hole (although he didn’t realise that at the time), the Schwarzschild radius correspond to the event horizon (the point of no return) of a non-spinning black hole. It’s rather tricky to measure the distance to the centre of a black hole, so really the Schwarzschild radius gives an idea of the circumference (the distance around the edge) of the event horizon: this is 2π times the Schwarschild radius. We’ll take the height to really mean twice the Schwarzschild radius (which would be the Schwarzschild diameter, if that were actually a thing).

Weight: 7.5 × 1025 lbs/3.4 × 1025 kg

Although we made our black hole pocket-sized, it is monstrously heavy. The mass is for a black hole of the size we picked, and it is about 6 times that of the Earth. That’s still quite small for a black hole (it’s 3.6 million times less massive than the black hole that formed from GW150914’s coalescence). With this mass, our Pokémon would have a significant effect on the tides as it would quickly suck in the Earth’s oceans. Still, Pokémon doesn’t need to be too realistic.

Our black hole Pokémon would be by far the heaviest Pokémon, despite being one of the smallest. The heaviest Pokémon currently is the continent Pokémon Primal Groudon. This is 2,204.4 lbs/999.7 kg, so about 34,000,000,000,000,000,000,000 times lighter.

Within the games, having such a large weight would make our black hole Pokémon vulnerable to Grass Knot, a move which trips a Pokémon. The heavier the Pokémon, the more it is hurt by the falling over, so the more damage Grass Knot does. In the case of our Pokémon, when it trips it’s not so much that it hits the ground, but that the Earth hits it, so I think it’s fair that this hurts.

Gender: Unknown

Black holes are beautifully simple, they are described just by their mass, spin and electric charge. There’s no other information you can learn about them, so I don’t think there’s any way to give them a gender. I think this is rather fitting as the sun-like Solrock is also genderless, and it seems right that stars and black holes share this.

Ability: Sticky Hold
Hidden ability:
Soundproof

Sticky Hold prevents a Pokémon’s item from being taken. (I’d expect wild black hole Pokémon to be sometimes found holding Stardust, from stars they have consumed). Due to their strong gravity, it is difficult to remove an object that is orbiting a black hole—a common misconception is that it is impossible to escape the pull of a black hole, this is only true if you cross the event horizon (if you replaced the Sun with a black hole of the same mass, the Earth would happily continue on its orbit as if nothing had happened).

Soundproof is an ability that protects Pokémon from sound-based moves. I picked it as a reference to sonic (or acoustic) black holes. These are black hole analogues—systems which mimic some of the properties of black holes. A sonic black hole can be made in a fluid which flows faster than its speed of sound. When this happens, sound can no longer escape this rapidly flowing region (it just gets swept away), just like light can’t escape from the event horizon or a regular black hole.

Sonic black holes are fun, because you can make them in the lab. You can them use them to study the properties of black holes—there is much excitement about possibly observing the equivalent of Hawking radiation. Predicted by Stephen Hawking (as you might guess), Hawking radiation is emitted by black holes, and could cause them to evaporate away (if they didn’t absorb more than they emit). Hawking radiation has never been observed from proper black holes, as it is very weak. However, finding the equivalent for sonic black holes might be enough to get Hawking his Nobel Prize…

Moves:

Start — Gravity
Start — Crunch

The starting two moves are straightforward. Gravity is the force which governs black holes; it is gravity which pulls material in and causes the collapse  of stars. I think Crunch neatly captures the idea of material being squeezed down by intense gravity.

Level 16 — Vacuum Wave

Vacuum Wave sounds like a good description of a gravitational wave: it is a ripple in spacetime. Black holes (at least when in a binary) are great sources of gravitational waves (as GW150914 and GW151226 have shown), so this seems like a sensible move for our Pokémon to learn—although I may be biased. Why at level 16? Because Einstein first predicted gravitational waves from his theory of general relativity in 1916.

Level 18 — Discharge

Black holes can have an electric charge, so our Pokémon should learn an Electric-type move. Charged black holes can have some weird properties. We don’t normally worry about charged black holes for two reasons. First, charged black holes are difficult to make: stuff is usually neutral overall, you don’t get a lot of similarly charged material in one place that can collapse down, and even if you did, it would quickly attract the opposite charge to neutralise itself. Second, if you did manage to make a charged black hole, it would quickly lose its charge: the strong electric and magnetic fields about the black hole would lead to the creation of charged particles that would neutralise the black hole. Discharge seems like a good move to describe this process.

Why level 18? The mathematical description of charged black holes was worked out by Hans Reissner and Gunnar Nordström, the second paper was published in 1918.

Level 19 —Light Screen

In general relativity, gravity bends spacetime. It is this warping that causes objects to move along curved paths (like the Earth orbiting the Sun). Light is affected in the same way and gets deflected by gravity, which is called gravitational lensing. This was the first experimental test of general relativity. In 1919, Arthur Eddington led an expedition to measure the deflection of light around the Sun during a solar eclipse.

Black holes, having strong gravity, can strongly lens light. The graphics from the movie Interstellar illustrate this beautifully. Below you can see how the image of the disc orbiting the black hole is distorted. The back of the disc is visible above and below the black hole! If you look closely, you can also see a bright circle inside the disc, close to the black hole’s event horizon. This is known as the light ring. It is where the path of light gets so bent, that it can orbit around and around the black hole many times. This sounds like a Light Screen to me.

Light-bending around the black hole Gargantua in Interstellar. The graphics use proper simulations of black holes, but they did fudge a couple of details to make it look extra pretty. Credit: Warner Bros./Double Negative.

Level 29 — Dark Void
Level 36 — Hyperspace Hole

These are three moves which with the most black hole-like names. Dark Void might be “black hole” after a couple of goes through Google Translate. Hyperspace Hole might be a good name for one of the higher dimensional black holes theoreticians like to play around with. (I mean, they like to play with the equations, not actually the black holes, as you’d need more than a pair of safety mittens for that). Shadow Ball captures the idea that a black hole is a three-dimensional volume of space, not just a plug-hole for the Universe. Non-rotating black holes are spherical (rotating ones bulge out at the middle, as I guess many of us do), so “ball” fits well, but they aren’t actually the shadow of anything, so it falls apart there.

I’ve picked the levels to be the masses of the two black holes which inspiralled together to produce GW150914, measured in units of the Sun’s mass, and the mass of the black hole that resulted from their merger. There’s some uncertainty on these measurements, so it would be OK if the moves were learnt a few levels either way.

Level 63 — Whirlpool
Level 63 — Rapid Spin

When gas falls into a black hole, it often spirals around and forms into an accretion disc. You can see an artistic representation of one in the image from Instellar above. The gas swirls around like water going down the drain, making Whirlpool and apt move. As it orbits, the gas closer to the black hole is moving quicker than that further away. Different layers rub against each other, and, just like when you rub your hands together on a cold morning, they heat up. One of the ways we look for black holes is by spotting the X-rays emitted by these hot discs.

As the material spirals into a black hole, it spins it up. If a black hole swallows enough things that were all orbiting the same way, it can end up rotating extremely quickly. Therefore, I thought our black hole Pokémon should learn Rapid Spin as the same time as Whirlpool.

I picked level 63, as the solution for a rotating black hole was worked out by Roy Kerr in 1963. While Schwarzschild found the solution for a non-spinning black hole soon after Einstein worked out the details of general relativity in 1915, and the solution for a charged black hole came just after these, there’s a long gap before Kerr’s breakthrough. It was some quite cunning maths! (The solution for a rotating charged black hole was quickly worked out after this, in 1965).

Level 77 — Hyper Beam

Another cool thing about discs is that they could power jets. As gas sloshes around towards a black hole, magnetic fields can get tangled up. This leads to some of the material to be blasted outwards along the axis of the field. We’ve some immensely powerful jets of material, like the one below, and it’s difficult to imagine anything other than a black hole that could create such high energies! Important work on this was done by Roger Blandford and Roman Znajek in 1977, which is why I picked the level. Hyper Beam is no exaggeration in describing these jets.

Jets from Centaurus A are bigger than the galaxy itself! This image is a composite of X-ray (blue), microwave (orange) and visible light. You can see the jets pushing out huge bubbles above and below the galaxy. We think the jets are powered by the galaxy’s central supermassive black hole. Credit: ESO/WFI/MPIfR/APEX/NASA/CXC/CfA/A.Weiss et al./R.Kraft et al.

After using Hyper Beam, a Pokémon must recharge for a turn. It’s an exhausting move. A similar thing may happen with black holes. If they accrete a lot of stuff, the radiation produced by the infalling material blasts away other gas and dust, cutting off the black hole’s supply of food. Black holes in the centres of galaxies may go through cycles of feeding, with discs forming, blowing away the surrounding material, and then a new disc forming once everything has settled down. This link between the black hole and its environment may explain why we see a trend between the size of supermassive black holes and the properties of their host galaxies.

Level 100 — Spacial Rend
Level 100 — Roar of Time

To finish off, since black holes are warped spacetime, a space move and a time move. Relativity say that space and time are two aspects of the same thing, so these need to be learnt together.

It’s rather tricky to imagine space and time being linked. Wibbly-wobbly, timey-wimey, spacey-wacey stuff gets quickly gets befuddling. If you imagine just two space dimension (forwards/backwards and left/right), then you can see how to change one to the other by just rotating. If you turn to face a different way, you can mix what was left to become forwards, or to become a bit of right and a bit of forwards. Black holes sort of do the same thing with space and time. Normally, we’re used to the fact that we a definitely travelling forwards in time, but if you stray beyond the event horizon of a black hole, you’re definitely travelling towards the centre of the black hole in the same inescapable way. Black holes are the masters when it comes to manipulating space and time.

There we have it, we can now sleep easy knowing what a black hole Pokémon would be like. Well almost, we still need to come up with a name. Something resembling a pun would be traditional. Suggestions are welcome. The next games in the series are Pokémon Sun and Pokémon Moon. Perhaps with this space theme Nintendo might consider a black hole Pokémon too?