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.

Black hole and neutron star masses

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}).

Inverse false alarm rates

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.

All binary masses

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].

All final black hole masses and spins

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}.

All effective inspiral spin parameters

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.

All effective precession spin parameters

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 3.26.2 billion years—it potentially started its journey before the Earth formed!

All distances and inclinations

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.

Spectrograms and waveforms

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 9.7101~\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 974440~\mathrm{Gpc^{-3}\,yr^{-1}}. This larger than are previous range, as we hadn’t considered the Gaussian distribution previously.

NSBH rate upper limits

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.

Population vs black hole mass

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 5060 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.

Parametric and binned spin magnitude distributions

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.

Rate versus redshift

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!

Trust LIGO

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.

Detector sensitivity across O2

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…

GW150914—The papers II

GW150914, The Event to its friends, was our first direct observation of gravitational waves. To accompany the detection announcement, the LIGO Scientific & Virgo Collaboration put together a suite of companion papers, each looking at a different aspect of the detection and its implications. Some of the work we wanted to do was not finished at the time of the announcement; in this post I’ll go through the papers we have produced since the announcement.

The papers

I’ve listed the papers below in an order that makes sense to me when considering them together. Each started off as an investigation to check that we really understood the signal and were confident that the inferences made about the source were correct. We had preliminary results for each at the time of the announcement. Since then, the papers have evolved to fill different niches [bonus points note].

13. The Basic Physics Paper

Title: The basic physics of the binary black hole merger GW150914
arXiv:
 1608.01940 [gr-qc]
Journal:
 Annalen der Physik529(1–2):1600209(17); 2017

The Event was loud enough to spot by eye after some simple filtering (provided that you knew where to look). You can therefore figure out some things about the source with back-of-the-envelope calculations. In particular, you can convince yourself that the source must be two black holes. This paper explains these calculations at a level suitable for a keen high-school or undergraduate physics student.

More details: The Basic Physics Paper summary

14. The Precession Paper

Title: Improved analysis of GW150914 using a fully spin-precessing waveform model
arXiv:
 1606.01210 [gr-qc]
Journal:
 Physical Review X; 6(4):041014(19); 2016

To properly measure the properties of GW150914’s source, you need to compare the data to predicted gravitational-wave signals. In the Parameter Estimation Paper, we did this using two different waveform models. These models include lots of features binary black hole mergers, but not quite everything. In particular, they don’t include all the effects of precession (the wibbling of the orbit because of the black holes spins). In this paper, we analyse the signal using a model that includes all the precession effects. We find results which are consistent with our initial ones.

More details: The Precession Paper summary

15. The Systematics Paper

Title: Effects of waveform model systematics on the interpretation of GW150914
arXiv:
 1611.07531 [gr-qc]
Journal: 
Classical & Quantum Gravity; 34(10):104002(48); 2017
LIGO science summary: Checking the accuracy of models of gravitational waves for the first measurement of a black hole merger

To check how well our waveform models can measure the properties of the source, we repeat the parameter-estimation analysis on some synthetic signals. These fake signals are calculated using numerical relativity, and so should include all the relevant pieces of physics (even those missing from our models). This paper checks to see if there are any systematic errors in results for a signal like GW150914. It looks like we’re OK, but this won’t always be the case.

More details: The Systematics Paper summary

16. The Numerical Relativity Comparison Paper

Title: Directly comparing GW150914 with numerical solutions of Einstein’s equations for binary black hole coalescence
arXiv:
 1606.01262 [gr-qc]
Journal:
 Physical Review D; 94(6):064035(30); 2016
LIGO science summary: Directly comparing the first observed gravitational waves to supercomputer solutions of Einstein’s theory

Since GW150914 was so short, we can actually compare the data directly to waveforms calculated using numerical relativity. We only have a handful of numerical relativity simulations, but these are enough to give an estimate of the properties of the source. This paper reports the results of this investigation. Unsurprisingly, given all the other checks we’ve done, we find that the results are consistent with our earlier analysis.

If you’re interested in numerical relativity, this paper also gives a nice brief introduction to the field.

More details: The Numerical Relativity Comparison Paper summary

The Basic Physics Paper

Synopsis: Basic Physics Paper
Read this if: You are teaching a class on gravitational waves
Favourite part: This is published in Annalen der Physik, the same journal that Einstein published some of his monumental work on both special and general relativity

It’s fun to play with LIGO data. The LIGO Open Science Center (LOSC), has put together a selection of tutorials to show you some of the basics of analysing signals. I wouldn’t blame you if you went of to try them now, instead of reading the rest of this post. Even though it would mean that no-one read this sentence. Purple monkey dishwasher.

The LOSC tutorials show you how to make your own version of some of the famous plots from the detection announcement. This paper explains how to go from these, using the minimum of theory, to some inferences about the signal’s source: most significantly that it must be the merger of two black holes.

GW150914 is a chirp. It sweeps up from low frequency to high. This is what you would expect of a binary system emitting gravitational waves. The gravitational waves carry away energy and angular momentum, causing the binary’s orbit to shrink. This means that the orbital period gets shorter, and the orbital frequency higher. The gravitational wave frequency is twice the orbital frequency (for circular orbits), so this goes up too.

The rate of change of the frequency depends upon the system’s mass. To first approximation, it is determined by the chirp mass,

\displaystyle \mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}},

where m_1 and m_2 are the masses of the two components of the binary. By looking at the signal (go on, try the LOSC tutorials), we can estimate the gravitational wave frequency f_\mathrm{GW} at different times, and so track how it changes. You can rewrite the equation for the rate of change of the gravitational wave frequency \dot{f}_\mathrm{GW}, to give an expression for the chirp mass

\displaystyle \mathcal{M} = \frac{c^3}{G}\left(\frac{5}{96} \pi^{-8/3} f_\mathrm{GW}^{-11/3} \dot{f}_\mathrm{GW}\right)^{3/5}.

Here c and G are the speed of light and the gravitational constant, which usually pop up in general relativity equations. If you use this formula (perhaps fitting for the trend f_\mathrm{GW}) you can get an estimate for the chirp mass. By fiddling with your fit, you’ll see there is some uncertainty, but you should end up with a value around 30 M_\odot [bonus note].

Next, let’s look at the peak gravitational wave frequency (where the signal is loudest). This should be when the binary finally merges. The peak is at about 150~\mathrm{Hz}. The orbital frequency is half this, so f_\mathrm{orb} \approx 75~\mathrm{Hz}. The orbital separation R is related to the frequency by

\displaystyle R = \left[\frac{GM}{(2\pi f_\mathrm{orb})^2}\right]^{1/3},

where M = m_1 + m_2 is the binary’s total mass. This formula is only strictly true in Newtonian gravity, and not in full general relativity, but it’s still a reasonable approximation. We can estimate a value for the total mass from our chirp mass; if we assume the two components are about the same mass, then M = 2^{6/5} \mathcal{M} \approx 70 M_\odot. We now want to compare the binary’s separation to the size of black hole with the same mass. A typical size for a black hole is given by the Schwarzschild radius

\displaystyle R_\mathrm{S} = \frac{2GM}{c^2}.

If we divide the binary separation by the Schwarzschild radius we get the compactness \mathcal{R} = R/R_\mathrm{S} \approx 1.7. A compactness of \sim 1 could only happen for black holes. We could maybe get a binary made of two neutron stars to have a compactness of \sim2, but the system is too heavy to contain two neutron stars (which have a maximum mass of about 3 M_\odot). The system is so compact, it must contain black holes!

What I especially like about the compactness is that it is unaffected by cosmological redshifting. The expansion of the Universe will stretch the gravitational wave, such that the frequency gets lower. This impacts our estimates for the true orbital frequency and the masses, but these cancel out in the compactness. There’s no arguing that we have a highly relativistic system.

You might now be wondering what if we don’t assume the binary is equal mass (you’ll find it becomes even more compact), or if we factor in black hole spin, or orbital eccentricity, or that the binary will lose mass as the gravitational waves carry away energy? The paper looks at these and shows that there is some wiggle room, but the signal really constrains you to have black holes. This conclusion is almost as inescapable as a black hole itself.

There are a few things which annoy me about this paper—I think it could have been more polished; “Virgo” is improperly capitalised on the author line, and some of the figures are needlessly shabby. However, I think it is a fantastic idea to put together an introductory paper like this which can be used to show students how you can deduce some properties of GW150914’s source with some simple data analysis. I’m happy to be part of a Collaboration that values communicating our science to all levels of expertise, not just writing papers for specialists!

During my undergraduate degree, there was only a single lecture on gravitational waves [bonus note]. I expect the topic will become more popular now. If you’re putting together such a course and are looking for some simple exercises, this paper might come in handy! Or if you’re a student looking for some project work this might be a good starting reference—bonus points if you put together some better looking graphs for your write-up.

If this paper has whetted your appetite for understanding how different properties of the source system leave an imprint in the gravitational wave signal, I’d recommend looking at the Parameter Estimation Paper for more.

The Precession Paper

Synopsis: Precession Paper
Read this if: You want our most detailed analysis of the spins of GW150914’s black holes
Favourite part: We might have previously over-estimated our systematic error

The Basic Physics Paper explained how you could work out some properties of GW150914’s source with simple calculations. These calculations are rather rough, and lead to estimates with large uncertainties. To do things properly, you need templates for the gravitational wave signal. This is what we did in the Parameter Estimation Paper.

In our original analysis, we used two different waveforms:

  • The first we referred to as EOBNR, short for the lengthy technical name SEOBNRv2_ROM_DoubleSpin. In short: This includes the spins of the two black holes, but assumes they are aligned such that there’s no precession. In detail: The waveform is calculated by using effective-one-body dynamics (EOB), an approximation for the binary’s motion calculated by transforming the relevant equations into those for a single object. The S at the start stands for spin: the waveform includes the effects of both black holes having spins which are aligned (or antialigned) with the orbital angular momentum. Since the spins are aligned, there’s no precession. The EOB waveforms are tweaked (or calibrated, if you prefer) by comparing them to numerical relativity (NR) waveforms, in particular to get the merger and ringdown portions of the waveform right. While it is easier to solve the EOB equations than full NR simulations, they still take a while. To speed things up, we use a reduced-order model (ROM), a surrogate model constructed to match the waveforms, so we can go straight from system parameters to the waveform, skipping calculating the dynamics of the binary.
  • The second we refer to as IMRPhenom, short for the technical IMRPhenomPv2. In short: This waveform includes the effects of precession using a simple approximation that captures the most important effects. In detail: The IMR stands for inspiral–merger–ringdown, the three phases of the waveform (which are included in in the EOBNR model too). Phenom is short for phenomenological: the waveform model is constructed by tuning some (arbitrary, but cunningly chosen) functions to match waveforms calculated using a mix of EOB, NR and post-Newtonian theory. This is done for black holes with (anti)aligned spins to first produce the IMRPhenomD model. This is then twisted up, to include the dominant effects of precession to make IMRPhenomPv2. This bit is done by combining the two spins together to create a single parameter, which we call \chi_\mathrm{p}, which determines the amount of precession. Since we are combining the two spins into one number, we lose a bit of the richness of the full dynamics, but we get the main part.

The EOBNR and IMRPhenom models are created by different groups using different methods, so they are useful checks of each other. If there is an error in our waveforms, it would lead to systematic errors in our estimated paramters

In this paper, we use another waveform model, a precessing EOBNR waveform, technically known as SEOBNRv3. This model includes all the effects of precession, not just the simple model of the IMRPhenom model. However, it is also computationally expensive, meaning that the analysis takes a long time (we don’t have a ROM to speed things up, as we do for the other EOBNR waveform)—each waveform takes over 20 times as long to calculate as the IMRPhenom model [bonus note].

Our results show that all three waveforms give similar results. The precessing EOBNR results are generally more like the IMRPhenom results than the non-precessing EOBNR results are. The plot below compares results from the different waveforms [bonus note].

Comparison of results from non-precessing EOBNR, precessing IMRPhenom and precessing EOBNR waveforms

Comparison of parameter estimates for GW150914 using different waveform models. The bars show the 90% credible intervals, the dark bars show the uncertainty on the 5%, 50% and 95% quantiles from the finite number of posterior samples. The top bar is for the non-precessing EOBNR model, the middle is for the precessing IMRPhenom model, and the bottom is for the fully precessing EOBNR model. Figure 1 of the Precession Paper; see Figure 9 for a comparison of averaged EOBNR and IMRPhenom results, which we have used for our overall results.

We had used the difference between the EOBNR and IMRPhenom results to estimate potential systematic error from waveform modelling. Since the two precessing models are generally in better agreement, we have may have been too pessimistic here.

The main difference in results is that our new refined analysis gives tighter constraints on the spins. From the plot above you can see that the uncertainty for the spin magnitudes of the heavier black hole a_1, the lighter black hole a_2 and the final black hole (resulting from the coalescence) a_\mathrm{f}, are slightly narrower. This makes sense, as including the extra imprint from the full effects of precession gives us a bit more information about the spins. The plots below show the constraints on the spins from the two precessing waveforms: the distributions are more condensed with the new results.

Black hole spins estimated using precessing IMRPhenom and EOBNR waveforms

Comparison of orientations and magnitudes of the two component spins. The spin is perfectly aligned with the orbital angular momentum if the angle is 0. The left disk shows results using the precessing IMRPhenom model, the right using the precessing EOBNR model. In each, the distribution for the more massive black hole is on the left, and for the smaller black hole on the right. Adapted from Figure 5 of the Parameter Estimation Paper and Figure 4 of the Precession Paper.

In conclusion, this analysis had shown that included the full effects of precession do give slightly better estimates of the black hole spins. However, it is safe to trust the IMRPhenom results.

If you are looking for the best parameter estimates for GW150914, these results are better than the original results in the Parameter Estimation Paper. However, the O2 Catalogue Paper includes results using improved calibration and noise power spectral density estimation, as well as using precessing waveforms!

The Systematics Paper

Synopsis: Systematics Paper
Read this if: You want to know how parameter estimation could fare for future detections
Favourite part: There’s no need to panic yet

The Precession Paper highlighted how important it is to have good waveform templates. If there is an error in our templates, either because of modelling or because we are missing some physics, then our estimated parameters could be wrong—we would have a source of systematic error.

We know our waveform models aren’t perfect, so there must be some systematic error, the question is how much? From our analysis so far (such as the good agreement between different waveforms in the Precession Paper), we think that systematic error is less significant than the statistical uncertainty which is a consequence of noise in the detectors. In this paper, we try to quantify systematic error for GW150914-like systems.

To asses systematic errors, we analyse waveforms calculated by numerical relativity simulations into data around the time of GW150914. Numerical relativity exactly solves Einstein’s field equations (which govern general relativity), so results of these simulations give the most accurate predictions for the form of gravitational waves. As we know the true parameters for the injected waveforms, we can compare these to the results of our parameter estimation analysis to check for biases.

We use waveforms computed by two different codes: the Spectral Einstein Code (SpEC) and the Bifunctional Adaptive Mesh (BAM) code. (Don’t the names make them sound like such fun?) Most waveforms are injected into noise-free data, so that we know that any offset in estimated parameters is dues to the waveforms and not detector noise; however, we also tried a few injections into real data from around the time of GW150914. The signals are analysed using our standard set-up as used in the Parameter Estimation Paper (a couple of injections are also included in the Precession Paper, where they are analysed with the fully precessing EOBNR waveform to illustrate its accuracy).

The results show that in most cases, systematic errors from our waveform models are small. However, systematic errors can be significant for some orientations of precessing binaries. If we are looking at the orbital plane edge on, then there can be errors in the distance, the mass ratio and the spins, as illustrated below [bonus note]. Thankfully, edge-on binaries are quieter than face-on binaries, and so should make up only a small fraction of detected sources (GW150914 is most probably face off). Furthermore, biases are only significant for some polarization angles (an angle which describes the orientation of the detectors relative to the stretch/squash of the gravitational wave polarizations). Factoring this in, a rough estimate is that about 0.3% of detected signals would fall into the unlucky region where waveform biases are important.

Inclination dependence of parameter recovery

Parameter estimation results for two different GW150914-like numerical relativity waveforms for different inclinations and polarization angles. An inclination of 0^\circ means the binary is face on, 180^\circ means it face off, and an inclination around 90^\circ is edge on. The bands show the recovered 90% credible interval; the dark lines the median values, and the dotted lines show the true values. The (grey) polarization angle \psi = 82^\circ was chosen so that the detectors are approximately insensitive to the h_+ polarization. Figure 4 of the Systematics Paper.

While it seems that we don’t have to worry about waveform error for GW150914, this doesn’t mean we can relax. Other systems may show up different aspects of waveform models. For example, our approximants only include the dominant modes (spherical harmonic decompositions of the gravitational waves). Higher-order modes have more of an impact in systems where the two black holes are unequal masses, or where the binary has a higher total mass, so that the merger and ringdown parts of the waveform are more important. We need to continue work on developing improved waveform models (or at least, including our uncertainty about them in our analysis), and remember to check for biases in our results!

The Numerical Relativity Comparison Paper

Synopsis: Numerical Relativity Comparison Paper
Read this if: You are really suspicious of our waveform models, or really like long tables or numerical data
Favourite part: We might one day have enough numerical relativity waveforms to do full parameter estimation with them

In the Precession Paper we discussed how important it was to have accurate waveforms; in the Systematics Paper we analysed numerical relativity waveforms to check the accuracy of our results. Since we do have numerical relativity waveforms, you might be wondering why we don’t just use these in our analysis? In this paper, we give it a go.

Our standard parameter-estimation code (LALInference) randomly hops around parameter space, for each set of parameters we generate a new waveform and see how this matches the data. This is an efficient way of exploring the parameter space. Numerical relativity waveforms are too computationally expensive to generate one each time we hop. We need a different approach.

The alternative, is to use existing waveforms, and see how each of them match. Each simulation gives the gravitational waves for a particular mass ratio and combination of spins, we can scale the waves to examine different total masses, and it is easy to consider what the waves would look like if measured at a different position (distance, inclination or sky location). Therefore, we can actually cover a fair range of possible parameters with a given set of simulations.

To keep things quick, the code averages over positions, this means we don’t currently get an estimate on the redshift, and so all the masses are given as measured in the detector frame and not as the intrinsic masses of the source.

The number of numerical relativity simulations is still quite sparse, so to get nice credible regions, a simple Gaussian fit is used for the likelihood. I’m not convinced that this capture all the detail of the true likelihood, but it should suffice for a broad estimate of the width of the distributions.

The results of this analysis generally agree with those from our standard analysis. This is a relief, but not surprising given all the other checks that we have done! It hints that we might be able to get slightly better measurements of the spins and mass ratios if we used more accurate waveforms in our standard analysis, but the overall conclusions are  sound.

I’ve been asked if since these results use numerical relativity waveforms, they are the best to use? My answer is no. As well as potential error from the sparse sampling of simulations, there are several small things to be wary of.

  • We only have short numerical relativity waveforms. This means that the analysis only goes down to a frequency of 30~\mathrm{Hz} and ignores earlier cycles. The standard analysis includes data down to 20~\mathrm{Hz}, and this extra data does give you a little information about precession. (The limit of the simulation length also means you shouldn’t expect this type of analysis for the longer LVT151012 or GW151226 any time soon).
  • This analysis doesn’t include the effects of calibration uncertainty. There is some uncertainty in how to convert from the measured signal at the detectors’ output to the physical strain of the gravitational wave. Our standard analysis fold this in, but that isn’t done here. The estimates of the spin can be affected by miscalibration. (This paper also uses the earlier calibration, rather than the improved calibration of the O1 Binary Black Hole Paper).
  • Despite numerical relativity simulations producing waveforms which include all higher modes, not all of them are actually used in the analysis. More are included than in the standard analysis, so this will probably make negligible difference.

Finally, I wanted to mention one more detail, as I think it is not widely appreciated. The gravitational wave likelihood is given by an inner product

\displaystyle L \propto \exp \left[- \int_{-\infty}^{\infty}  \mathrm{d}f  \frac{|s(f) - h(f)|^2}{S_n(f)}  \right],

where s(f) is the signal, h(f) is our waveform template and S_n(f) is the noise spectral density (PSD). These are the three things we need to know to get the right answer. This paper, together with the Precession Paper and the Systematics Paper, has been looking at error from our waveform models h(f). Uncertainty from the calibration of s(f) is included in the standard analysis, so we know how to factor this in (and people are currently working on more sophisticated models for calibration error). This leaves the noise PSD S_n(f)

The noise PSD varies all the time, so it needs to be estimated from the data. If you use a different stretch of data, you’ll get a different estimate, and this will impact your results. Ideally, you would want to estimate from the time span that includes the signal itself, but that’s tricky as there’s a signal in the way. The analysis in this paper calculates the noise power spectral density using a different time span and a different method than our standard analysis; therefore, we expect some small difference in the estimated parameters. This might be comparable to (or even bigger than) the difference from switching waveforms! We see from the similarity of results that this cannot be a big effect, but it means that you shouldn’t obsess over small differences, thinking that they could be due to waveform differences, when they could just come from estimation of the noise PSD.

Lots of work is currently going into making sure that the numerator term |s(f) - h(f)|^2 is accurate. I think that the denominator S_n(f) needs attention too. Since we have been kept rather busy, including uncertainty in PSD estimation will have to wait for a future set papers.

Bonus notes

Finches

100 bonus points to anyone who folds up the papers to make beaks suitable for eating different foods.

The right answer

Our current best estimate for the chirp mass (from the O2 Catalogue Paper) would be 31.2_{-1.5}^{+1.7} M_\odot. You need proper templates for the gravitational wave signal to calculate this. If you factor in the the gravitational wave gets redshifted (shifted to lower frequency by the expansion of the Universe), then the true chirp mass of the source system is 28.6_{-1.5}^{+1.6} M_\odot.

Formative experiences

My one undergraduate lecture on gravitational waves was the penultimate lecture of the fourth-year general relativity course. I missed this lecture, as I had a PhD interview (at the University of Birmingham). Perhaps if I had sat through it, my research career would have been different?

Good things come…

The computational expense of a waveform is important, as when we are doing parameter estimation, we calculate lots (tens of millions) of waveforms for different parameters to see how they match the data. Before O1, the task of using SEOBNRv3 for parameter estimation seemed quixotic. The first detection, however, was enticing enough to give it a try. It was a truly heroic effort by Vivien Raymond and team that produced these results—I am slightly suspicious the Vivien might actually be a wizard.

GW150914 is a short signal, meaning it is relatively quick to analyse. Still, it required us using all the tricks at our disposal to get results in a reasonable time. When it came time to submit final results for the Discovery Paper, we had just about 1,000 samples from the posterior probability distribution for the precessing EOBNR waveform. For comparison, we had over 45,000 sample for the non-precessing EOBNR waveform. 1,000 samples isn’t enough to accurately map out the probability distributions, so we decided to wait and collect more samples. The preliminary results showed that things looked similar, so there wouldn’t be a big difference in the science we could do. For the Precession Paper, we finally collected 2,700 samples. This is still a relatively small number, so we carefully checked the uncertainty in our results due to the finite number of samples.

The Precession Paper has shown that it is possible to use the precessing EOBNR for parameter estimation, but don’t expect it to become the norm, at least until we have a faster implementation of it. Vivien is only human, and I’m sure his family would like to see him occasionally.

Parameter key

In case you are wondering what all the symbols in the results plots stand for, here are their usual definitions. First up, the various masses

  • m_1—the mass of the heavier black hole, sometimes called the primary black hole;
  • m_2—the mass of the lighter black hole, sometimes called the secondary black hole;
  • M—the total mass of the binary, M = m_1 + m_2;
  • M_\mathrm{f}—the mass of the final black hole (after merger);
  • \mathcal{M}—the chirp mass, the combination of the two component masses which sets how the binary inspirals together;
  • q—the mass ratio, q = m_1/m_2 \leq 1. Confusingly, numerical relativists often use the opposite  convention q = m_2/m_1 \geq 1 (which is why the Numerical Relativity Comparison Paper discusses results in terms of 1/q: we can keep the standard definition, but all the numbers are numerical relativist friendly).

A superscript “source” is sometimes used to distinguish the actual physical masses of the source from those measured by the detector which have been affected by cosmological redshift. The measured detector-frame mass is m = (1 + z) m^\mathrm{source}, where m^\mathrm{source} is the true, redshift-corrected source-frame mass and z is the redshift. The mass ratio q is independent of the redshift. On the topic of redshift, we have

  • z—the cosmological redshift (z = 0 would be now);
  • D_\mathrm{L}—the luminosity distance.

The luminosity distance sets the amplitude of the signal, as does the orientation which we often describe using

  • \iota—the inclination, the angle between the line of sight and the orbital angular momentum (\boldsymbol{L}). This is zero for a face-on binary.
  • \theta_{JN}—the angle between the line of sight (\boldsymbol{N}) and the total angular momentum of the binary (\boldsymbol{J}); this is approximately equal to the inclination, but is easier to use for precessing binaries.

As well as masses, black holes have spins

  • a_1—the (dimensionless) spin magnitude of the heavier black hole, which is between 0 (no spin) and 1 (maximum spin);
  • a_2—the (dimensionless) spin magnitude of the lighter black hole;
  • a_\mathrm{f}—the (dimensionless) spin magnitude of the final black hole;
  • \chi_\mathrm{eff}—the effective inspiral spin parameter, a combinations of the two component spins which has the largest impact on the rate of inspiral (think of it as the spin equivalent of the chirp mass);
  • \chi_\mathrm{p}—the effective precession spin parameter, a combination of spins which indicate the dominant effects of precession, it’s 0 for no precession and 1 for maximal precession;
  • \theta_{LS_1}—the primary tilt angle, the angle between the orbital angular momentum and the heavier black holes spin (\boldsymbol{S_1}). This is zero for aligned spin.
  • \theta_{LS_2}—the secondary tilt angle, the angle between the orbital angular momentum and the lighter black holes spin (\boldsymbol{S_2}).
  • \phi_{12}—the angle between the projections of the two spins on the orbital plane.

The orientation angles change in precessing binaries (when the spins are not perfectly aligned or antialigned with the orbital angular momentum), so we quote values at a reference time corresponding to when the gravitational wave frequency is 20~\mathrm{Hz}. Finally (for the plots shown here)

  • \psi—the polarization angle, this is zero when the detector arms are parallel to the h_+ polarization’s stretch/squash axis.

For more detailed definitions, check out the Parameter Estimation Paper or the LALInference Paper.