# Going the distance: Mapping host galaxies of LIGO and Virgo sources in three dimensions using local cosmography and targeted follow-up

GW150914 claimed the title of many firsts—it was the first direct observation of gravitational waves, the first observation of a binary black hole system, the first observation of two black holes merging, the first time time we’ve tested general relativity in such extreme conditions… However, there are still many firsts for gravitational-wave astronomy yet to come (hopefully, some to be accompanied by cake). One of the most sought after, is the first is signal to have a clear electromagnetic counterpart—a glow in some part of the spectrum of light (from radio to gamma-rays) that we can observe with telescopes.

Identifying a counterpart is challenging, as it is difficult to accurately localise a gravitational-wave source. electromagnetic observers must cover a large area of sky before any counterparts fade. Then, if something is found, it can be hard to determine if that is from the same source as the gravitational waves, or some thing else…

To help the search, it helps to have as much information as possible about the source. Especially useful is the distance to the source. This can help you plan where to look. For nearby sources, you can cross-reference with galaxy catalogues, and perhaps pick out the biggest galaxies as the most likely locations for the source [bonus note]. Distance can also help plan your observations: you might want to start with regions of the sky where the source would be closer and so easiest to spot, or you may want to prioritise points where it is further and so you’d need to observe longer to detect it (I’m not sure there’s a best strategy, it depends on the telescope and the amount of observing time available). In this paper we describe a method to provide easy-to-use distance information, which could be supplied to observers to help their search for a counterpart.

### Going the distance

This work is the first spin-off from the First 2 Years trilogy of papers, which looked a sky localization and parameter estimation for binary neutron stars in the first two observing runs of the advance-detector era. Binary neutron star coalescences are prime candidates for electromagnetic counterparts as we think there should be a bigger an explosion as they merge. I was heavily involved in the last two papers of the trilogy, but this study was led by Leo Singer: I think I mostly annoyed Leo by being a stickler when it came to writing up the results.

Three-dimensional localization showing the 20%, 50%, and 90% credible levels for a typical two-detector early Advanced LIGO event. The Earth is shown at the centre, marked by $\oplus$. The true location is marked by the cross. Leo poetically described this as looking like the seeds of the jacaranda tree, and less poetically as potato chips. Figure 1 of Singer et al. (2016).

The idea is to provide a convenient means of sharing a 3D localization for a gravitational wave source. The full probability distribution is rather complicated, but it can be made more manageable if you break it up into pixels on the sky. Since astronomers need to decide where to point their telescopes, breaking up the 3D information along different lines of sight, should be useful for them.

Each pixel covers a small region of the sky, and along each line of sight, the probability distribution for distance $D$ can be approximated using an ansatz

$\displaystyle p(D|\mathrm{data}) \propto D^2\exp\left[-\frac{(D - \mu)^2}{2\sigma}\right]$,

where $\mu$ and $\sigma$ are calculated for each pixel individually.  The form of this ansatz can be understood as the posterior probability distribution is proportional to the product of the prior and the likelihood. Our prior is that sources are uniformly distributed in volume, which means $\propto D^2$, and the likelihood can often be well approximated as a Gaussian distribution, which gives the other piece [bonus note].

The ansatz doesn’t always fit perfectly, but it performs well on average. Considering the catalogue of binary neutron star signals used in the earlier papers, we find that roughly 50% of the time sources are found within the 50% credible volume, 90% are found in the 90% volume, etc. We looked at a more sophisticated means of constructing the localization volume in a companion paper.

The 3D localization is easy to calculate, and Leo has worked out a cunning way to evaluate the ansatz with BAYESTAR, our rapid sky-localization code, meaning that we can produce it on minute time-scales. This means that observers should have something to work with straight-away, even if we’ll need to wait a while for the full, final results. We hope that this will improve prospects for finding counterparts—some potential examples are sketched out in the penultimate section of the paper.

If you are interested in trying out the 3D information, there is a data release and the supplement contains a handy Python tutorial. We are hoping that the Collaboration will use the format for alerts for LIGO and Virgo’s upcoming observing run (O2).

arXiv: 1603.07333 [astro-ph.HE]; 1605.04242 [astro-ph.IM]
Journal: Astrophysical Journal Letters; 829(1):L15(7); 2016; Astrophysical Journal Supplement Series; 226(1):10(8); 2016
Data release: Going the distance
Favourite crisp flavour: Salt & vinegar
Favourite jacaranda: Jacaranda mimosifolia

### Bonus notes

#### Catalogue shopping

The Event’s source has a luminosity distance of around 250–570 Mpc. This is sufficiently distant that galaxy catalogues are incomplete and not much use when it comes to searching. GW151226 and LVT151012 have similar problems, being at around the same distance or even further.

#### The gravitational-wave likelihood

For the professionals interested in understanding more about the shape of the likelihood, I’d recommend Cutler & Flanagan (1994). This is a fantastic paper which contains many clever things [bonus bonus note]. This work is really the foundation of gravitational-wave parameter estimation. From it, you can see how the likelihood can be approximated as a Gaussian. The uncertainty can then be evaluated using Fisher matrices. Many studies have been done using Fisher matrices, but it important to check that this is a valid approximation, as nicely explained in Vallisneri (2008). I ran into a case when it didn’t during my PhD.

#### Mergin’

As a reminder that smart people make mistakes, Cutler & Flanagan have a typo in the title of arXiv posting of their paper. This is probably the most important thing to take away from this paper.

# Comprehensive all-sky search for periodic gravitational waves in the sixth science run LIGO data

The most recent, and most sensitive, all-sky search for continuous gravitational waves shows no signs of a detection. These signals from rotating neutron stars remain elusive. New data from the advanced detectors may change this, but we will have to wait a while to find out. This at least gives us time to try to figure out what to do with a detection, should one be made.

### New years and new limits

The start of the new academic year is a good time to make resolutions—much better than wet and windy January. I’m trying to be tidier and neater in my organisation. Amid cleaning up my desk, which is covered in about an inch of papers, I uncovered this recent Collaboration paper, which I had lost track of.

The paper is the latest in the continuous stream of non-detections of continuous gravitational waves. These signals could come from rotating neutron stars which are deformed or excited in some way, and the hope that from such an observation we could learn something about the structure of neutron stars.

The search uses old data from initial LIGO’s sixth science run. Searches for continuous waves require lots of computational power, so they can take longer than even our analyses of binary neutron star coalescences. This is a semi-coherent search, like the recent search of the Orion spur—somewhere between an incoherent search, which looks for signal power of any form in the detectors, and a fully coherent search, which looks for signals which exactly match the way a template wave evolves [bonus note]. The big difference compared to the Orion spur search, is that this one looks at the entire sky. This makes it less sensitive in those narrow directions, but means we are not excluding the possibility of sources from other locations.

Artist’s impression of the local part of the Milky Way. The yellow cones mark the extent of the Orion Spur spotlight search, and the pink circle shows the equivalent sensitivity of this all-sky search. Green stars indicate known pulsars. Original image: NASA/JPL-Caltech/ESO/R. Hurt.

The search identified 16 outliers, but an examination of all of these showed they could be explained either as an injected signal or as detector noise. Since no signals were found, we can instead place some upper limits on the strength of signals.

The plot below translates the calculated upper limits (above which there would have been a ~75%–95% chance of us detected the signal) into the size of neutron star deformations. Each curve shows the limits on detectable signals at different distance, depending upon their frequency and the rate of change of their frequency. The dotted lines show limits on ellipticity $\varepsilon$, a measure of how bumpy the neutron star is. Larger deformations mean quicker changes of frequency and produce louder signals, therefore they can can be detected further away.

Range of the PowerFlux search for rotating neutron stars assuming that spin-down is entirely due to gravitational waves. The solid lines show the upper limits as a function of the gravitational-wave frequency and its rate of change; the dashed lines are the corresponding limits on ellipticity, and the dotted line marks the maximum searched spin-down. Figure 6 of Abbott et al. (2016).

Neutron stars are something like giant atomic nuclei. Figuring the properties of the strange matter that makes up neutron stars is an extremely difficult problem. We’ll never be able to recreate such exotic matter in the laboratory. Gravitational waves give us a rare means of gathering experimental data on how this matter behaves. However, exactly how we convert a measurement of a signal into constraints on the behaviour of the matter is still uncertain. I think that making a detection might only be the first step in understanding the sources of continuous gravitational waves.

arXiv: 1605.03233 [gr-qc]
Journal: Physical Review D; 94(4):042002(14); 2016
To attempt to grow a beard. Beard stroking helps you think, right?

### Bonus note

#### The semi-coherent search

As the first step of this search, the PowerFlux algorithm looks for power that changes in frequency as expected for a rotating neutron star: it factors in Doppler shifting due to the motion of the Earth and a plausible spin down (slowing of the rotation) of the neutron star. As a follow up, the Loosely Coherent algorithm is used, which checks for signals which match short stretches of similar templates. Any candidates to make it through all stages of refinement are then examined in more detail. This search strategy is described in detail for the S5 all-sky search.

# Parameter estimation on gravitational waves from neutron-star binaries with spinning components

blIn gravitation-wave astronomy, some parameters are easier to measure than others. We are sensitive to properties which change the form of the wave, but sometimes the effect of changing one parameter can be compensated by changing another. We call this a degeneracy. In signals for coalescing binaries (two black holes or neutron stars inspiralling together), there is a degeneracy between between the masses and spins. In this recently published paper, we look at what this means for observing binary neutron star systems.

### History

This paper has been something of an albatross, and I’m extremely pleased that we finally got it published. I started working on it when I began my post-doc at Birmingham in 2013. Back then I was sharing an office with Ben Farr, and together with others in the Parameter Estimation Group, we were thinking about the prospect of observing binary neutron star signals (which we naively thought were the most likely) in LIGO’s first observing run.

One reason that this work took so long is that binary neutron star signals can be computationally expensive to analyse [bonus note]. The signal slowly chirps up in frequency, and can take up to a minute to sweep through the range of frequencies LIGO is sensitive to. That gives us a lot of gravitational wave to analyse. (For comparison, GW150914 lasted 0.2 seconds). We need to calculate waveforms to match to the observed signals, and these can be especially complicated when accounting for the effects of spin.

A second reason is shortly after submitting the paper in August 2015, we got a little distracted

This paper was the third of a trilogy look at measuring the properties of binary neutron stars. I’ve written about the previous instalment before. We knew that getting the final results for binary neutron stars, including all the important effects like spin, would take a long time, so we planned to follow up any detections in stages. A probable sky location can be computed quickly, then we can have a first try at estimating other parameters like masses using waveforms that don’t include spin, then we go for the full results with spin. The quicker results would be useful for astronomers trying to find any explosions that coincided with the merger of the two neutron stars. The first two papers looked at results from the quicker analyses (especially at sky localization); in this one we check what effect neglecting spin has on measurements.

### What we did

We analysed a population of 250 binary neutron star signals (these are the same as the ones used in the first paper of the trilogy). We used what was our best guess for the sensitivity of the two LIGO detectors in the first observing run (which was about right).

The simulated neutron stars all have small spins of less than 0.05 (where 0 is no spin, and 1 would be the maximum spin of a black hole). We expect neutron stars in these binaries to have spins of about this range. The maximum observed spin (for a neutron star not in a binary neutron star system) is around 0.4, and we think neutron stars should break apart for spins of 0.7. However, since we want to keep an open mind regarding neutron stars, when measuring spins we considered spins all the way up to 1.

### What we found

Our results clearly showed the effect of the mass–spin degeneracy. The degeneracy increases the uncertainty for both the spins and the masses.

Even though the true spins are low, we find that across the 250 events, the median 90% upper limit on the spin of the more massive (primary) neutron star is 0.70, and the 90% limit on the less massive (secondary) neutron star is 0.86. We learn practically nothing about the spin of the secondary, but a little more about the spin of the primary, which is more important for the inspiral. Measuring spins is hard.

The effect of the mass–spin degeneracy for mass measurements is shown in the plot below. Here we show a random selection of events. The banana-shaped curves are the 90% probability intervals. They are narrow because we can measure a particular combination of masses, the chirp mass, really well. The mass–spin degeneracy determines how long the banana is. If we restrict the range of spins, we explore less of the banana (and potentially introduce an offset in our results).

Rough outlines for 90% credible regions for component masses for a random assortments of signals. The circles show the true values. The coloured lines indicate the extent of the distribution with different limits on the spins. The grey area is excluded from our convention on masses $m_1 \geq m_2$. Figure 5 from Farr et al. (2016).

Although you can’t see it in the plot above, including spin does also increase the uncertainty in the chirp mass too. The plots below show the standard deviation (a measure width of the posterior probability distribution), divided by the mean for several mass parameters. This gives a measure of the fractional uncertainty in our measurements. We show the chirp mass $\mathcal{M}_\mathrm{c}$, the mass ratio $q = m_2/m_1$ and the total mass $M = m_1 + m_2$, where $m_1$ and $m_2$ are the masses of the primary and secondary neutron stars respectively. The uncertainties are small for louder signals (higher signal-to-noise ratio). If we neglect the spin, the true chirp mass can lie outside the posterior distribution, the average is about 5 standard deviations from the mean, but if we include spin, the offset is just 0.7 from the mean (there’s still some offset as we’re allowing for spins all the way up to 1).

Fractional statistical uncertainties in chirp mass (top), mass ratio (middle) and total mass (bottom) estimates as a function of network signal-to-noise ratio for both the fully spinning analysis and the quicker non-spinning analysis. The lines indicate approximate power-law trends to guide the eye. Figure 2 of Farr et al. (2016).

We need to allow for spins when measuring binary neutron star masses in order to explore for the possible range of masses.

Sky localization and distance, however, are not affected by the spins here. This might not be the case for sources which are more rapidly spinning, but assuming that binary neutron stars do have low spin, we are safe using the easier-to-calculate results. This is good news for astronomers who need to know promptly where to look for explosions.

arXiv: 1508.05336 [astro-ph.HE]
Journal: Astrophysical Journal825(2):116(10); 2016
Authorea [bonus note]: Parameter estimation on gravitational waves from neutron-star binaries with spinning components
Conference proceedings:
Early Advanced LIGO binary neutron-star sky localization and parameter estimation
Favourite albatross:
Wilbur

### Bonus notes

#### How long?

The plot below shows how long it took to analyse each of the binary neutron star signals.

Distribution of run times for binary neutron star signals. Low-latency sky localization is done with BAYESTAR; medium-latency non-spinning parameter estimation is done with LALInference and TaylorF2 waveforms, and high-latency fully spinning parameter estimation is done with LALInference and SpinTaylorT4 waveforms. The LALInference results are for 2000 posterior samples. Figure 9 from Farr et al. (2016).

BAYESTAR provides a rapid sky localization, taking less than ten seconds. This is handy for astronomers who want to catch a flash caused by the merger before it fades.

Estimates for the other parameters are computed with LALInference. How long this takes to run depends on which waveform you are using and how many samples from the posterior probability distribution you want (the more you have, the better you can map out the shape of the distribution). Here we show times for 2000 samples, which is enough to get a rough idea (we collected ten times more for GW150914 and friends). Collecting twice as many samples takes (roughly) twice as long. Prompt results can be obtained with a waveform that doesn’t include spin (TaylorF2), these take about a day at most.

For this work, we considered results using a waveform which included the full effects of spin (SpinTaylorT4). These take about twenty times longer than the non-spinning analyses. The maximum time was 172 days. I have a strong suspicion that the computing time cost more than my salary.

Waiting for LALInference runs to finish gives you some time to practise hobbies. This is a globe knitted by Hannah. The two LIGO sites marked in red, and a typical gravitational-wave sky localization stitched on.

In order to get these results, we had to add check-pointing to our code, so we could stop it and restart it; we encountered a new type of error in the software which manages jobs running on our clusters, and Hannah Middleton and I got several angry emails from cluster admins (who are wonderful people) for having too many jobs running.

In comparison, analysing GW150914, LVT151012 and GW151226 was a breeze. Grudgingly, I have to admit that getting everything sorted out for this study made us reasonably well prepared for the real thing. Although, I’m not looking forward to that first binary neutron star signal…

#### Authorea

Authorea is an online collaborative writing service. It allows people to work together on documents, editing text, adding comments, and chatting with each other. By the time we came to write up the paper, Ben was no longer in Birmingham, and many of our coauthors are scattered across the globe. Ben thought Authorea might be useful for putting together the paper.

Writing was easy, and the ability to add comments on the text was handy for getting feedback from coauthors. The chat was going for quickly sorting out issues like plots. Overall, I was quite pleased, up to the point we wanted to get the final document. Extracted a nicely formatted PDF was awkward. For this I switched to using the Github back-end. On reflection, a simple git repo, plus a couple of Skype calls might have been a smoother way of writing, at least for a standard journal article.

Authorea promises to be an open way of producing documents, and allows for others to comment on papers. I don’t know if anyone’s looked at our Authorea article. For astrophysics, most people use the arXiv, which is free to everyone, and I’m not sure if there’s enough appetite for interaction (beyond the occasional email to authors) to motivate people to look elsewhere. At least, not yet.

In conclusion, I think Authorea is a nice idea, and I would try out similar collaborative online writing tools again, but I don’t think I can give it a strong recommendation for your next paper unless you have a particular idea in mind of how to make the most of it.