Parameter estimation for binary neutron-star coalescences with realistic noise during the Advanced LIGO era

The first observing run (O1) of Advanced LIGO is nearly here, and with it the prospect of the first direct detection of gravitational waves. That’s all wonderful and exciting (far more exciting than a custard cream or even a chocolate digestive), but there’s a lot to be done to get everything ready. Aside from remembering to vacuum the interferometer tubes and polish the mirrors, we need to see how the data analysis will work out. After all, having put so much effort into the detector, it would be shame if we couldn’t do any science with it!

Parameter estimation

Since joining the University of Birmingham team, I’ve been busy working on trying to figure out how well we can measure things using gravitational waves. I’ve been looking at binary neutron star systems. We expect binary neutron star mergers to be the main source of signals for Advanced LIGO. We’d like to estimate how massive the neutron stars are, how fast they’re spinning, how far away they are, and where in the sky they are. Just published is my first paper on how well we should be able to measure things. This took a lot of hard work from a lot of people, so I’m pleased it’s all done. I think I’ve earnt a celebratory biscuit. Or two.

When we see something that looks like it could be a gravitational wave, we run code to analyse the data and try to work out the properties of the signal. Working out some properties is a bit trickier than others. Sadly, we don’t have an infinite number of computers, so it means it can take a while to get results. Much longer than the time to eat a packet of Jaffa Cakes…

The fastest algorithm we have for binary neutron stars is BAYESTAR. This takes the same time as maybe eating one chocolate finger. Perhaps two, if you’re not worried about the possibility of choking. BAYESTAR is fast as it only estimates where the source is coming from. It doesn’t try to calculate a gravitational-wave signal and match it to the detector measurements, instead it just looks at numbers produced by the detection pipeline—the code that monitors the detectors and automatically flags whenever something interesting appears. As far as I can tell, you give BAYESTAR this information and a fresh cup of really hot tea, and it uses Bayes’ theorem to work out how likely it is that the signal came from each patch of the sky.

To work out further details, we need to know what a gravitational-wave signal looks like and then match this to the data. This is done using a different algorithm, which I’ll refer to as LALInference. (As names go, this isn’t as cool as SKYNET). This explores parameter space (hopping between different masses, distances, orientations, etc.), calculating waveforms and then working out how well they match the data, or rather how likely it is that we’d get just the right noise in the detector to make the waveform fit what we observed. We then use another liberal helping of Bayes’ theorem to work out how probable those particular parameter values are.

It’s rather difficult to work out the waveforms, but some our easier than others. One of the things that makes things trickier is adding in the spins of the neutron stars. If you made a batch of biscuits at the same time you started a LALInference run, they’d still be good by the time a non-spinning run finished. With a spinning run, the biscuits might not be quite so appetising—I generally prefer more chocolate than penicillin on my biscuits. We’re working on speeding things up (if only to prevent increased antibiotic resistance).

In this paper, we were interested in what you could work out quickly, while there’s still chance to catch any explosion that might accompany the merging of the neutron stars. We think that short gamma-ray bursts and kilonovae might be caused when neutron stars merge and collapse down to a black hole. (I find it mildly worrying that we don’t know what causes these massive explosions). To follow-up on a gravitational-wave detection, you need to be able to tell telescopes where to point to see something and manage this while there’s still something that’s worth seeing. This means that using spinning waveforms in LALInference is right out, we just use BAYESTAR and the non-spinning LALInference analysis.

What we did

To figure out what we could learn from binary neutron stars, we generated a large catalogue of fakes signals, and then ran the detection and parameter-estimation codes on this to see how they worked. This has been done before in The First Two Years of Electromagnetic Follow-Up with Advanced LIGO and Virgo which has a rather delicious astrobites write-up. Our paper is the sequel to this (and features most of the same cast). One of the differences is that The First Two Years assumed that the detectors were perfectly behaved and had lovely Gaussian noise. In this paper, we added in some glitches. We took some real data™ from initial LIGO’s sixth science run and stretched this so that it matches the sensitivity Advanced LIGO is expected to have in O1. This process is called recolouring [bonus note]. We now have fake signals hidden inside noise with realistic imperfections, and can treat it exactly as we would real data. We ran it through the detection pipeline, and anything which was flagged as probably being a signal (we used a false alarm rate of once per century), was analysed with the parameter-estimation codes. We looked at how well we could measure the sky location and distance of the source, and the masses of the neutron stars. It’s all good practice for O1, when we’ll be running this analysis on any detections.

What we found

1. The flavour of noise (recoloured or Gaussian) makes no difference to how well we can measure things on average.
2. Sky-localization in O1 isn’t great, typically hundreds of square degrees (the median 90% credible region is 632 deg2), for comparison, the Moon is about a fifth of a square degree. This’ll make things interesting for the people with telescopes.

Probability that of a gravitational-wave signal coming from different points on the sky. The darker the red, the higher the probability. The star indicates the true location. This is one of the worst localized events from our study for O1. You can find more maps in the data release (including 3D versions), this is Figure 6 of Berry et al. (2015).

3. BAYESTAR does just as well as LALInference, despite being about 2000 times faster.

Sky localization (the size of the patch of the sky that we’re 90% sure contains the source location) varies with the signal-to-noise ratio (how loud the signal is). The approximate best fit is $\log_{10}(\mathrm{CR}_{0.9}/\mathrm{deg^2}) \approx -2 \log_{10}(\varrho) +5.06$, where $\mathrm{CR}_{0.9}$ is the 90% sky area and $\varrho$ is the signal-to-noise ratio. The results for BAYESTAR and LALInference agree, as do the results with Gaussian and recoloured noise. This is Figure 9 of Berry et al. (2015).

4. We can’t measure the distance too well: the median 90% credible interval divided by the true distance (which gives something like twice the fractional error) is 0.85.
5. Because we don’t include the spins of the neutron stars, we introduce some error into our mass measurements. The chirp mass, a combination of the individual masses that we’re most sensitive to [bonus note], is still reliably measured (the median offset is 0.0026 of the mass of the Sun, which is tiny), but we’ll have to wait for the full spinning analysis for individual masses.

Fraction of events with difference between the mean estimated and true chirp mass smaller than a given value. There is an error because we are not including the effects of spin, but this is small. Again, the type of noise makes little difference. This is Figure 15 of Berry et al. (2015).

There’s still some work to be done before O1, as we need to finish up the analysis with waveforms that include spin. In the mean time, our results are all available online for anyone to play with.

arXiv: 1411.6934 [astro-ph.HE]
Journal: Astrophysical Journal; 904(2):114(24); 2015
Data release: The First Two Years of Electromagnetic Follow-Up with Advanced LIGO and Virgo
Favourite colour: Blue. No, yellow…

Notes

The colour of noise: Noise is called white if it doesn’t have any frequency dependence. We made ours by taking some noise with initial LIGO’s frequency dependence (coloured noise), removing the frequency dependence (making it white), and then adding in the frequency dependence of Advanced LIGO (recolouring it).

The chirp mass: Gravitational waves from a binary system depend upon the masses of the components, we’ll call these $m_1$ and $m_2$. The chirp mass is a combination these that we can measure really well, as it determines the most significant parts of the shape of the gravitational wave. It’s given by

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

We get lots of good information on the chirp mass, unfortunately, this isn’t too useful for turning back into the individual masses. For that we next extra information, for example the mass ratio $m_2/m_1$. We can get this from less dominant parts of the waveform, but it’s not typically measured as precisely as the chirp mass, so we’re often left with big uncertainties.

Gravitational-wave sensitivity curves

Differing weights and differing measures—
the LORD detests them both. — Proverbs 20:10

As a New Year’s resolution, I thought I would try to write a post on each paper I have published. (I might try to go back and talk about my old papers too, but that might be a little too optimistic.)  Handily, I have a paper that was published in Classical & Quantum Gravity on Thursday, so let’s get on with it, and hopefully 2015 will deliver those hoverboards soon.

This paper was written in collaboration with my old officemates, Chris Moore and Rob Cole, and originates from my time in Cambridge. We were having a weekly group meeting (surreptitiously eating cake—you’re not meant to eat in the new meeting rooms) and discussing what to do for the upcoming open afternoon. Posters are good as you can use them to decorate your office afterwards, so we decided on making one on gravitational-wave astronomy. Gravitational waves come in a range of frequencies, just like light (electromagnetic radiation). You can observe different systems with different frequencies, but you need different instruments to do so. For light, the range is from high frequency gamma rays (observed with satellites like Fermi) to low frequency radio waves (observed with telescopes like those at Jodrell Bank or Arecibo), with visible light (observed with Hubble or your own eyes) in the middle. Gravitational waves also have a spectrum, ground-based detectors like LIGO measure the higher frequencies, pulsar timing arrays measure the lower frequencies, and space-borne detectors like eLISA measure stuff in the middle. We wanted a picture that showed the range of each instrument and the sources they could detect, but we couldn’t find a good up-to-date one. Chris is not one to be put off by a challenge (especially if it’s a source of procrastination), so he decided to have a go at making one himself. How hard could it be? We never made that poster, but we did end up with a paper.

When talking about gravitational-wave detectors, you normally use a sensitivity curve. This shows how sensitive it is at a given frequency: you plot a graph with the sensitivity curve on, and then plot the spectrum of the source you’re interested in on the same graph. If your source is above the sensitivity curve, you can detect it (yay), but if it lies below it, then you can’t pick it out from the noise (boo). Making a plot with lots of sensitivity curves on sounds simple: you look up the details for lots of detectors, draw them together and add a few sources. However, there are lots of different conventions for how you actually measure sensitivity, and they’re frequently muddled up! We were rather confused by the whole thing, but eventually (after the open afternoon had flown by), we figured things out and made our picture. So we wouldn’t forget, we wrote up the different conventions, why you might want to use each, and how to convert between them; these notes became the paper. We also thought it would be handy to have a website where you could make your own plot, picking which detectors and sources you wanted to include. Rob also likes a challenge (especially if it’s a source of procrastination), so he set about making such a thing. I think it turned out rather well!

That’s the story of the paper. It explains different conventions for characterising gravitational-wave detectors and sources, and gives some examples. If you’d actually like to know some of the details, I’ll give a little explanation now, if not, just have a look at the pretty plots below (or, if looking for your own source of procrastination, have a go at Space Time Quest, a game where you try to build the most sensitive detector).

There are three common conventions in use for sensitivity-curve plots: the characteristic strain, the amplitude spectral density and the energy density.

You might wonder why we don’t just directly use the amplitude of the wave? Gravitational waves are a stretching and squashing of spacetime, so you can characterise how much they stretch and squeeze things and use that to describe the size of your waves. The sensitivity of your detector is then how much various sources of noise cause a similar wibbling. The amplitude of the wave is really, really small, so it’s difficult to detect, but if you were to consider observations over a time interval instead of just one moment, it’s easier to spot a signal: hints that there might be a signal add up until you’re certain that it’s there. The characteristic strain is a way of modifying the amplitude to take into account how we add up the signal. It’s especially handy, as if you make a log–log plot (such that the space between 1 and 10 is the same as between 10 and 100, etc.), then the area between the characteristic strain of your source and the detector sensitivity curve gives you a measure of the signal-to-noise ratio, a measure of how loud (how detectable) a signal is.

Gravitational-wave sensitivity-curve plot using characteristic strain. The area between the detector’s curve and the top of the box for a source indicates how loud that signal would be.

The characteristic strain is handy for quickly working out how loud a signal is, but it’s not directly related to anything we measure. The noise in a detector is usually described by its power spectral density or PSD. This tells you how much wibbling there is on average. Actually, it tells you the average amount of wibbling squared. The square root of the PSD is the amplitude spectral density or ASD. This gives a handy indication of the sensitivity of your detector, which is actually related to what you measure.

Gravitational-wave sensitivity-curve plot using the square root of the power spectral density (the amplitude spectral density).

The PSD is tied to the detector, but isn’t too relevant to the actual waves. An interesting property of the waves is how much energy they carry. We talk about this in terms of the energy density, the energy per unit volume. Cosmologists love this, and to make things easy for themselves, they like to divide energy densities by the amount that would make the Universe flat. (If you’ve ever wondered what astrophysicists mean when they say the Universe is about 70% dark energy and about 25% dark matter, they’re using these quantities). To make things even simpler, they like to multiply this quantity by something related to the Hubble constant (which measures the expansion rate of the Universe), as this means things don’t change if you tweak the numbers describing how the Universe evolves. What you’re left with is a quantity $\Omega h_{100}^2$ that is really convenient if you’re a cosmologist, but a pain for anyone else. It does have the advantage of making the pulsar timing arrays look more sensitive though.

Gravitational-wave sensitivity-curve plot using the energy density that cosmologists love. The proper name of the plotted quantity is the critical energy density per logarithmic frequency interval multiplied by the reduced Hubble constant squared. I prefer Bob.

We hope that the paper will be useful for people (like us), who can never remember what the conventions are (and why). There’s nothing new (in terms of results) in this paper, but I think it’s the first time all this material has been collected together in one place. If you ever need to make a poster about gravitational waves, I know where you can find a good picture.

arXiv: 1408.0740 [gr-qc]
Journal: Classical & Qunatum Gravity32(1):015014(25); 2015
Website: Gravitational Wave Sensitivity Curve Plotter
Procrastination score: TBC