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.

Part of the Galaxy searched

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.

Limits on detectable signals and ellipticities

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
Other new academic year resolution:
 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.

Searches for continuous gravitational waves from nine young supernova remnants

The LIGO Scientific Collaboration is busy analysing the data we’re currently taking with Advanced LIGO at the moment. However, the Collaboration is still publishing results from initial LIGO too. The most recent paper is a search for continuous waves—signals that are an almost constant hum throughout the observations. (I expect they’d be quite annoying for the detectors). Searching for continuous waves takes a lot of computing power (you can help by signing up for Einstein@Home), and is not particularly urgent since the sources don’t do much, hence it can take a while for results to appear.

Supernova remnants

Massive stars end their lives with an explosion, a supernova. Their core collapses down and their outer layers are blasted off. The aftermath of the explosion can be beautiful, with the thrown-off debris forming a bubble expanding out into the interstellar medium (the diffuse gas, plasma and dust between stars). This structure is known as a supernova remnant.

The bubble of a supernova remnant

The youngest known supernova remnant, G1.9+0.3 (it’s just 150 years old), observed in X-ray and optical light. The ejected material forms a shock wave as it pushes the interstellar material out of the way. Credit: NASA/CXC/NCSU/DSS/Borkowski et al.

At the centre of the supernova remnant may be what is left following the collapse of the core of the star. Depending upon the mass of the star, this could be a black hole or a neutron star (or it could be nothing). We’re interested in the case it is a neutron star.

Neutron stars

Neutron stars are incredibly dense. One teaspoon’s worth would have about as much mass as 300 million elephants. Neutron stars are like giant atomic nuclei. We’re not sure how matter behaves in such extreme conditions as they are impossible to replicate here on Earth.

If a neutron star rotates rapidly (we know many do) and has an uneven or if there are waves in the the neutron star that moves lots of material around (like Rossby waves on Earth), then it can emit continuous gravitational waves. Measuring these gravitational waves would tell you about how bumpy the neutron star is or how big the waves are, and therefore something about what the neutron star is made from.

Neutron stars are most likely to emit loud gravitational waves when they are young. This is for two reasons. First, the supernova explosion is likely to give the neutron star a big whack, this could ruffle up its surface and set off lots of waves, giving rise to the sort of bumps and wobbles that emit gravitational waves. As the neutron star ages, things can quiet down, the neutron star relaxes, bumps smooth out and waves dissipate. This leaves us with smaller gravitational waves. Second, gravitational waves carry away energy, slowing the rotation of the neutron star. This also means that the signal gets quieter (and harder) to detect as the  neutron star ages.

Since young neutron stars are the best potential sources, this study looked at nine young supernova remnants in the hopes of finding continuous gravitational waves. Searching for gravitational waves from particular sources is less computationally expensive than searching the entire sky. The search included Cassiopeia A, which had been previously searched in LIGO’s fifth science run, and G1.9+0.3, which is only 150 years old, as discovered by Dave Green. The positions of the searched supernova remnants are shown in the map of the Galaxy below.

Galactic map of supernova remnants

The nine young supernova remnants searched for continuous gravitational waves. The yellow dot marks the position of the Solar System. The green markers show the supernova remnants, which are close to the Galactic plane. Two possible positions for Vela Jr (G266.2−1.2) were used, since we are uncertain of its distance. Original image: NASA/JPL-Caltech/ESO/R. Hurt.

Gravitational-wave limits

No gravitational waves were found. The search checks how well template waveforms match up with the data. We tested that this works by injecting some fake signals into the data.  Since we didn’t detect anything, we can place upper limits on how loud any gravitational waves could be. These limits were double-checked by injecting some more fake signals at the limit, to see if we could detect them. We quoted 95% upper limits, that is where we expect that if a signal was present we could see it 95% of the time. The results actually have a small safety margin built in, so the injected signals were typically found 96%–97% of the time. In any case, we are fairly sure that there aren’t gravitational waves at or above the upper limits.

These upper limits are starting to tell us interesting things about the size of neutron-star bumps and waves. Hopefully, with data from Advanced LIGO and Advanced Virgo, we’ll actually be able to make a detection. Then we’ll not only be able to say that these bumps and waves are smaller than a particular size, but they are this size. Then we might be able to figure out the recipe for making the stuff of neutron stars (I think it might be more interesting than just flour and water).

arXiv: 1412.5942 [astro-ph.HE]
Journal: Astrophysical Journal; 813(1):39(16); 2015
Science summary: Searching for the youngest neutron stars in the Galaxy
Favourite supernova remnant:
 Cassiopeia A

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.

    Sky localization map for O1.

    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 for binary neutron stars during O1.

    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.

    Mean offset in chirp-mass estimates when not including the effects of spin.

    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.

Directed search for gravitational waves from Scorpius X-1 with initial LIGO

new paper from the LIGO Scientific Collaboration has snuck out. It was actually published back in March but I didn’t notice it, nearly risking my New Year’s resolution. This is another paper on continuous waves from rotating neutron stars, so it’s a little outside my area of expertise. However, there is an official science summary written by people who do know what they’re talking about.

The paper looks at detecting gravitational waves from a spinning neutron star. We didn’t find any. However, we have slightly improved our limit for how loud they need to be before we would have detected them, which is nice.

Neutron stars can rotate rapidly. They can be spun up if they accrete material from a disc orbiting them. If they neutron star has an asymmetry, if it has a little bump, as it rotates it emits gravitational waves. The gravitational waves carry away angular momentum, which should spin down the neutron star. This becomes more effective as the angular velocity increases. At some point you expect that the spin-up effect from accretion balances the spin-down effect of gravitational waves and you are left with a neutron star spinning at pretty constant velocity. We have some evidence that this might happen, as low-mass X-ray binaries seem to have their spins clustered in a small range of frequencies. Assuming we do have this balance, we are looking for a continuous gravitational wave with constant frequency, a rather dull humming.

Scorpius X-1 is the brightest X-ray source in the sky. It contains a neutron star, so it’s a good place to check for gravitational waves from neutron stars. In this case, we’re using data from initial LIGO’s fifth science run (4 November 2005–1 October 2007). This has been done before, but this paper implements some new techniques. I expect that the idea is to test things out ahead of getting data with Advanced LIGO.

X-ray image of Scorpius X-1

Swift X-ray Telescope image of Scorpius X-1 and the X-ray nova J1745-26 (a stellar-mass black hole), along with the scale of moon, as they would appear in the field of view from Earth. Credit: NASA/Goddard Space Flight Center/S. Immler and H. Krimm.

A limit of 10 days’ worth of data is used, as this should be safely within the time taken for the rotational frequency to fluctuate by a noticeable amount due to variation in the amount of accretion. In human terms, that would be the time between lunch and dinner, where your energy levels change because of how much you’ve eaten. They picked data from 21–31 August 2007, as their favourite (it has the best noise performance over the frequency range of interest), and used two other segments to double-check their findings. We’d be able to use more data if we knew how the spin wandered with time.

We already know a lot about Scorpius X-1 from electromagnetic observations (like where it is and its orbital parameters). We don’t know its spin frequency, but we might have an idea about the orientation of its spin if this coincides with radio jets. The paper considers two cases: one where we don’t know anything about the spin orientation, and one where we use information from the jets. The results are similar in both cases.

As the neutron star orbits in its binary system, it moves back and forth which Doppler shifts the gravitational waves. This adds a little interest to the hum, spreading it out over a range of frequencies. The search looks for gravitational waves over this type of frequency range, which they refer to as sidebands.

There are a few events where it looks like there is something, but after carefully checking, these look like they are entirely consistent with noise. I guess this isn’t too surprising. Since they didn’t detect anything, they can only impose an upper limit. This is stronger than the previous upper limit, but only by a factor of about 1.4. This might not sound too great, but the previous analysis used a year of data, whereas this only used 10 days. This method therefore saves a lot on computational time.

The result of the paper is quite nice, but not too exciting. If it were a biscuit, it’d probably be a rich tea. It’s nice to have, but it’s not a custard cream.

arXiv: 1412.5942 [astro-ph.HE]
Journal: Physical Review D; 91(6):062008(20); 2015
Science summary: Combing Initial LIGO Data for the Potentially Strong Continuous Wave Emitter Scorpius X-1
Biscuit rating:
Rich tea

Advanced LIGO (the paper)

Continuing with my New Year’s resolution to write a post on every published paper, the start of March see another full author list LIGO publication. Appearing in Classical & Quantum Gravity, the minimalistically titled Advanced LIGO is an instrumental paper. It appears a part of a special focus issue on advanced gravitational-wave detectors, and is happily free to read (good work there). This is The Paper™ for describing how the advanced detectors operate. I think it’s fair to say that my contribution to this paper is 0%.

LIGO stands for Laser Interferometer Gravitational-wave Observatory. As you might imagine, LIGO tries to observe gravitational waves by measuring them with a laser interferometer. (It won’t protect your fencing). Gravitational waves are tiny, tiny stretches and squeezes of space. To detect them we need to measure changes in length extremely accurately. I had assumed that Advanced LIGO will achieve this supreme sensitivity through some dark magic invoked by sacrificing the blood, sweat, tears and even coffee of many hundreds of PhD students upon the altar of science. However, this paper actually shows it’s just really, really, REALLY careful engineering. And giant frickin’ laser beams.

The paper goes through each aspect of the design of the LIGO detectors. It starts with details of the interferometer. LIGO uses giant lasers to measure distances extremely accurately. Lasers are bounced along two 3994.5 m arms and interfered to measure a change in length between the two. In spirit, it is a giant Michelson interferometer, but it has some cunning extra features. Each arm is a Fabry–Pérot etalon, which means that the laser is bounced up and down the arms many times to build up extra sensitivity to any change in length. There are various extra components to make sure that the laser beam is as stable as possible, all in all, there are rather a lot of mirrors, each of which is specially tweaked to make sure that some acronym is absolutely perfect.

Advanced LIGO optical configuration. IT's a bit more complicated than a basic Michelson interferometer.

Fig. 1 from Aasi et al. (2015), the Advanced LIGO optical configuration. All the acronyms have to be carefully placed in order for things to work. The laser beam starts from the left, passing through subsystems to make sure it’s stable. It is split in two to pass into the interferometer arms at the top and right of the diagram. The laser is bounced many times between the mirrors to build up sensitivity. The interference pattern is read out at the bottom. Normally, the light should interfere destructively, so the output is dark. A change to this indicates a change in length between the arms. That could be because of a passing gravitational wave.

The next section deals with all the various types of noise that affect the detector. It’s this noise that makes it such fun to look for the signals. To be honest, pretty much everything I know about the different types of noise I learnt from Space-Time Quest. This is a lovely educational game developed by people here at the University of Birmingham. In the game, you have to design the best gravitational-wave detector that you can for a given budget. There’s a lot of science that goes into working out how sensitive the detector is. It takes a bit of practice to get into it (remember to switch on the laser first), but it’s very easy to get competitive. We often use the game as part of outreach workshops, and we’ve had some school groups get quite invested in the high-score tables. My tip is that going underground doesn’t seem to be worth the money. Of course, if you happen to be reviewing the proposal to build the Einstein Telescope, you should completely ignore that, and just concentrate how cool the digging machine looks. Space-Time Quest shows how difficult it can be optimising sensitivity. There are trade-offs between different types of noise, and these have been carefully studied. What Space-Time Quest doesn’t show, is just how much work it takes to engineer a detector.

The fourth section is a massive shopping list of components needed to build Advanced LIGO. There are rather more options than in Space-Time Quest, but many are familiar, even if given less friendly names. If this section were the list of contents for some Ikea furniture, you would know that you’ve made a terrible life-choice; there’s no way you’re going to assemble this before Monday. Highlights include the 40 kg mirrors. I’m sure breaking one of those would incur more than seven years bad luck. For those of you playing along with Space-Time Quest at home, the mirrors are fused silica. Section 4.8.4 describes how to get the arms to lock, one of the key steps in commissioning the detectors. The section concludes with details of how to control such a complicated instrument, the key seems to be to have so many acronyms that there’s no space for any component to move in an unwanted way.

The paper closes with on outlook for the detector sensitivity. With such a complicated instrument it is impossible to be certain how things will go. However, things seem to have been going smoothly so far, so let’s hope that this continues. The current plan is:

  • 2015 3 months observing at a binary neutron star (BNS) range of 40–80 Mpc.
  • 2016–2017 6 months observing at a BNS range of 80–120 Mpc.
  • 2017–2018 9 months observing at a BNS range of 120–170 Mpc.
  • 2019 Achieve full sensitivity of a BNS range of 200 Mpc.

The BNS range is the distance at which a typical binary made up of two 1.4 solar mass neutrons stars could be detected when averaging over all orientations. If you have a perfectly aligned binary, you can detect it out to a further distance, the BNS horizon, which is about 2.26 times the BNS range. There are a couple of things to note from the plan. First, the initial observing run (O1 to the cool kids) is this year! The second is how much the range will extend before hitting design sensitivity. This should significantly increase the number of possible detections, as each doubling of the range corresponds to a volume change of a factor of eight. Coupling this with the increasing length of the observing runs should mean that the chance of a detection increases every year. It will be an exciting few years for Advanced LIGO.

arXiv: 1411.4547 [gr-qc]
Journal: Classical & Quantum Gravity; 32(7):074001(41); 2015
Science summary: Introduction to LIGO & Gravitational Waves
Space-Time Quest high score: 34.859 Mpc

Narrow-band search of continuous gravitational-wave signals from Crab and Vela pulsars in Virgo VSR4 data

Collaboration papers

I’ve been a member of the LIGO Scientific Collaboration for just over a year now. It turns out that designing, building and operating a network of gravitational-wave detectors is rather tricky, maybe even harder than completing Super Mario Bros. 3, so it takes a lot of work. There are over 900 collaboration members, all working on different aspects of the project. Since so much of the research is inter-related, certain papers (such as those that use data from the instruments) written by collaboration members have to include the name of everyone who works (at least half the time) on LIGO-related things. After a year in the collaboration, I have now levelled up to be included in the full author list (if there was an initiation ritual, I’ve suppressed the memory). This is weird: papers appear with my name on that I’ve not actually done any work for. It seems sort of like having to bring cake into your office on your birthday: you do have to share your (delicious) cupcakes with everyone else, but in return you get cake even when your birthday is nowhere near. Perhaps all those motivational posters where right about the value of teamwork? I do feel a little guilty about all the extra trees that will die because of people printing out these papers.

My New Year’s resolution was to write a post about every paper I have published. I am going to try to do the LIGO papers too. This should at least make sure that I actually read them all. There are official science summaries written by the people who did actually do the work, which may be better if you actually want an accurate explanation. My first collaboration paper is a joint publication of the LIGO and Virgo collaborations (even more sharing).

Searching for gravitational waves from pulsars

Neutron stars are formed from the cores of dead stars. When a star’s nuclear fuel starts to run out, their core collapses. The most massive form black holes, the lightest (like our Sun) form white dwarfs, and the ones in the middle form neutron stars. These are really dense, they have about the same mass as our entire Sun (perhaps twice the Sun’s mass), but are just a few kilometres across. Pulsars are a type of neutron star, they emit a beam of radiation that sweeps across the sky as they rotate, sort of like a light-house. If one of these beams hits the Earth, we see a radio pulse. The pulses come regularly, so you can work out how fast the pulsar is spinning (and do some other cool things too).

A pulsar

The mandatory cartoon of a pulsar that everyone uses. The top part shows the pulsar and its beams rotating, and the bottom part shows the signal measured on Earth. We not really sure where the beams come from, it’ll be something to do with magnetic fields. Credit: M. Kramer

Because pulsars rotate really quickly, if they have a little bump on their surface, they can emit (potentially detectable) gravitational waves. This paper searches for these signals from the Crab and Vela pulsars. We know where these pulsars are, and how quickly they are rotating, so it’s possible to do a targeted search for gravitational waves (only checking the data for signals that are close to what we expect). Importantly, some wiggle room in the frequency is allowed just in case different parts of the pulsar slosh around at slightly different rates and so the gravitational-wave frequency doesn’t perfectly match what we’d expect from the frequency of pulses; the search is done in a narrow band of frequencies around the expected one. The data used is from Virgo’s fourth science run (VSR4). That was taken back in 2011 (around the time that Captain America was released). The search technique is new (Astone et al., 2014), it’s the first one that incorporates this searching in a narrow band of frequencies; I think the point was to test their search technique on real data before the advanced detectors start producing new data.

Composite Crab

Composite image of Hubble (red) optical observations and Chandra (blue) X-ray observations of the Crab pulsar. The pulsar has a mass of 1.4 solar masses and rotates every 30 ms. Credit: Hester et al.

The pulsars emit gravitational waves continuously, they just keep humming as they rotate. The frequency will slow gradually as the pulsar loses energy. As the Earth rotates, the humming gets louder and quieter because the sensitivity of gravitational-wave detectors depends upon where the source is in the sky. Putting this all together gives you a good template for what the signal should look like, and you can see how well it fits the data. It’s kind of like trying to find the right jigsaw piece by searching for the one that interlocks best with those around it. Of course, there is a lot of noise in our detectors, so it’s like if the jigsaw was actually made out of jelly: you could get many pieces to fit if you squeeze them the right way, but then people wouldn’t believe that you’ve actually found the right one. Some detection statistics (which I don’t particularly like, but probably give a sensible answer) are used to quantify how likely it is that they’ve found a piece that fits (that there is a signal). The whole pipeline is tested by analysing some injected signals (artificial signals made to see if things work made both by adding signals digitally to the data and by actually jiggling the mirrors of the interferometer). It seems to do OK here.

Turning to the actual data, they very carefully show that they don’t think they’ve detected anything for either Vela or Crab. Of course, all the cool kids don’t detect gravitational waves, so that’s not too surprising.

Zoidberg is an expert on crabs, pulsing or otherwise

This paper doesn’t claim a detection of gravitational waves, but it doesn’t stink like Zoidberg.

Having not detected anything, you can place an upper limit of the amplitude of any waves that are emitted (because if they were larger, you would’ve detected them). This amplitude can then be compared with what’s expected from the spin-down limit: the amplitude that would be required to explain the slowing of the pulsar. We know how the pulsars are slowing, but not why; it could be because of energy being lost to magnetic fields (the energy for the beams has to come from somewhere), it could be through energy lost as gravitational waves, it could be because of some internal damping, it could all be gnomes. The spin-down limit assumes that it’s all because of gravitational waves, you couldn’t have bigger amplitude waves than this unless something else (that would have to be gnomes) was pumping energy into the pulsar to keep it spinning. The upper limit for the Vela pulsar is about the same as the spin-down limit, so we’ve not learnt anything new. For the Crab pulsar, the upper limit is about half the spin-down limit, which is something, but not really exciting. Hopefully, doing the same sort of searches with data from the advanced detectors will be more interesting.

In conclusion, the contents of this paper are well described by its title:

  • Narrow-band search: It uses a new search technique that is not restricted to the frequency assumed from timing pulses
  • of continuous gravitational-wave signals: It’s looking for signals from rotating neutron stars (that just keep going) and so are always in the data
  • from Crab and Vela pulsars: It considers two particular sources, so we know where in parameter space to look for signals
  • in Virgo VSR4 data: It uses real data, but from the first generation detectors, so it’s not surprising it doesn’t see anything

It’s probably less fun that eating a jigsaw-shaped jelly, but it might be more useful in the future.

arXiv: 1410.8310 [gr-qc]
Journal: Physical Review D; 91(2):022004(15); 2015
Science summary: An Extended Search for Gravitational Waves from the Crab and Vela Pulsars
Percentage of paper that is author list: ~30%

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.

Characteristic strain plot

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.

ASD plot

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.

Energy density plot

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