# What GW170729’s exceptional mass and spin tells us about its family tree

One of the great discoveries that came with our first observation of gravitational waves was that black holes can merge—two black holes in a binary can come together and form a bigger black hole. This had long been predicted, but never before witnessed. If black holes can merge once, can they go on to merge again? In this paper, we calculated how to identify a binary containing a second-generation black hole formed in a merger.

### Merging black holes

Black holes have two important properties: their mass and their spin. When two black holes merge, the resulting black hole has:

1. A mass which is almost as big as the sum of the masses of its two parents. It is a little less (about 5%) as some of the energy is radiated away as gravitational waves.
2. A spin which is around 0.7. This is set by the angular momentum of the two black holes as they plunge in together. For equal-mass black holes, the orbit of the two black holes will give about enough angular momentum for the final black hole to be about 0.7. The spins of the two parent black holes will cause a bit a variation around this, depending upon the orientations of their spins. For more unequal mass binaries, the spin of the larger parent black hole becomes more important.

To look for second-generation (or higher) black holes formed in mergers, we need to look for more massive black holes with spins of about 0.7 [bonus note].

Combining black holes. The result of a merger is a larger black hole with significant spin. From Dawn Finney.

The difficult bit here is that we don’t know the distribution of masses and spins of the initial first-generation black holes. What is they naturally form with spins of 0.7? How can you tell if a black hole is unexpectedly large if you don’t know what sizes to expect? With the discovery of the 10 binary black holes found in our first and second observing runs, we are able to start making inferences about the properties of black holes—using these measurements of the population, we can estimate how probable it is that a binary contains a second generation black hole versus containing two first generation black hole.

### GW170729

Amongst the black holes observed in O1 and O2, the source of GW170729 stands out. It is both the most massive, and one of only two systems (the other being GW151226) showing strong evidence for spin. This got me wondering if it could be a second-generation system? The high mass would be explained as we have a second-generation black hole, and the spin is larger than usual as a spin 0.7 sticks out.

Chase Kimball worked out the relative probability of getting a system with a given chirp mass and effective inspiral spin for a binary with a second-generation black hole verses a binary with only first-generation black holes. We worked in terms of chirp mass and effective inspiral spin, as these are the properties we measure well from a gravitational-wave signal.

Relative likelihood of a binary black hole being second-generation versus first-generation for different values of the chirp mass and the magnitude of the effective inspiral spin. The white contour gives the 90% credible area for GW170729. Figure 1 of Kimball et al. (2019).

The plot above shows the relative probabilities. Yellow indicate chirp mass and effective inspiral spins which are more likely with second-generation systems, while dark purple indicates values more likely with first-generation systems.. The first thing I realised was my idea about the spin was off. We expect binaries with second-generation black holes to be formed dynamically. Following the first merger, the black hole wander around until it gets close enough to form a new binary with a new black hole. For dynamically formed binaries the spins should be randomly distributed. This means that there’s only a small probability of having a spin aligned with the orbital angular momentum as measured for GW170729. Most of the time, you’d measure an effective inspiral spin of around zero.

Since we don’t know exactly the chirp mass and effective inspiral spin for GW170729, we have to average over our uncertainty. That gives the ratio of the probability of observing GW170729 given a second-generation source, verses given a first-generation source. Using different inferred black hole populations (for example, ones inferred including and excluding GW170729), we find ratios of between 0.2 (meaning the first-generation origin is more likely) and 16 (meaning second generation is more likely). The results change significantly as the result is sensitive to the maximum mass of a black hole. If we include GW170729 in our population inference for first-generation systems, the maximum mass goes up, and it’s easier to explain the system as first-generation (as you’d expect).

Before you place your bets, there is one more piece to the calculation. We have calculated the relative probabilities of the observed properties assuming either first-generation black holes or a second-generation black hole, but we have not folded in the relative rates of mergers [bonus note]. We expect first-generation only binaries to be more common than ones containing second generation black holes. In simulations of globular clusters, at most about 20% of merging binaries are with second-generation black holes. For binaries not in an environment like a globular cluster (where there are lots of nearby black holes to grab), we expect the fraction of second-generation black holes in binaries to be basically zero. Therefore, on balance we have at best a weak preference for a second-generation black hole and most probably just two first-generation black holes in GW170729’s source, despite its large mass.

### Verdict

What we have learnt from this calculation is that it seems that all of the first 10 binary black holes contain only first-generation black holes. It is safe to infer the properties of first-generation black holes from these observations. Detecting second-generation black holes requires knowledge of this distribution, and crucially if there is a maximum mass. As we get more detection, we’ll be able to pin this down. There is still a lot to learn about the full black hole family.

If you’d like to understand our calculation, the paper is extremely short. It is therefore an excellent paper to bring to journal club if you are a PhD student who forgot you were presenting this week…

arXiv: 1903.07813 [astro-ph.HE]
Journal: Research Notes of the AAS; 4(1):2; 2020 [bonus note]
Gizmodo story: The gravitational wave detectors are turning back on and we’re psyched
Theme music: Nice to see you!

### Bonus notes

#### Useful papers

Back in 2017 two papers hit the arXiv [bonus bonus note] at pretty much the same time addressing the expected properties of second-generation black holes: Fishbach, Holz & Farr (2017), and Gerosa & Berti (2017). Both are nice reads.

I was asked how we could tell if the black holes we were seeing were themselves the results of mergers back in 2016 when I was giving a talk to the Carolian Astronomical Society. It was a good question. I explained about the masses and spins, but I didn’t think about how to actually do the analysis to infer if we had a merger. I now make a note to remember any questions I’m asked, as they can be good inspiration for projects!

#### Bayes factor and odds ratio

The quantity we work out in the paper is the Bayes factor for a second-generation system verses a first-generation one

$\displaystyle \frac{P(\mathrm{GW170729}|\mathrm{Gen\ 2})}{P(\mathrm{GW170729}|\mathrm{Gen\ 1})}$.

What we want is the odds ratio

$\displaystyle \frac{P(\mathrm{Gen\ 2}|\mathrm{GW170729})}{P(\mathrm{Gen\ 1}|\mathrm{GW170729})}$,

which gives the betting odds for the two scenarios. The convert the Bayes factor into an odds ratio we need the prior odds

$\displaystyle \frac{P(\mathrm{Gen\ 2})}{P(\mathrm{Gen\ 1})}$.

We’re currently working on a better way to fold these pieces together.

#### 1000 words

As this was a quick calculation, we thought it would be a good paper to be a Research Note. Research Notes are limited to 1000 words, which is a tough limit. We carefully crafted the document, using as many word-saving measures (such as abbreviations), as we could. We made it to the limit by our counting, only to submit and find that we needed to share another 100 off! Fortunately, the arXiv [bonus bonus note] is more forgiving, so you can read our more relaxed (but still delightfully short) version there. It’s the one I’d recommend.

#### arXiv

For those reading who are not professional physicists, the arXiv (pronounced archive, as the X is really the Greek letter chi χ) is a preprint server. It where physicists can post version of their papers ahead of publication. This allows sharing of results earlier (both good as it can take a while to get a final published paper, and because you can get feedback before finalising a paper), and, vitally, for free. Most published papers require a subscription to read. Fine if you’re at a university, not so good otherwise. The arXiv allows anyone to read the latest research. Admittedly, you have to be careful, as not everything on the arXiv will make it through peer review, and not everyone will update their papers to reflect the published version. However, I think the arXiv is a very good thing™. There are few things I can think of which have benefited modern science as much. I would 100% support those behind the arXiv receiving a Nobel Prize, as I think it has had just as a significant impact on the development of the field as the discovery of dark matter, understanding nuclear fission, or deducing the composition of the Sun.