Limiting factors

Yesterday the search that I've spent most of the last two years performing went on arXiv (following on from the search described in this post). I'll reproduce the abstract here and then try and explain a bit more about it:

Searches for gravitational waves from known pulsars with S5 LIGO data

We present a search for gravitational waves from 116 known millisecond and young pulsars using data from the fifth science run of the LIGO detectors. For this search ephemerides overlapping the run period were obtained for all pulsars using radio and X-ray observations. We demonstrate an updated search method that allows for small uncertainties in the pulsar phase parameters to be included in the search. We report no signal detection from any of the targets and therefore interpret our results as upper limits on the gravitational wave signal strength. Our best (lowest) upper limit on gravitational wave amplitude is 2.3x10^-26 for J1603-7202 and our best (lowest) limit on the inferred pulsar ellipticity is 7.0x10^-8 for J2124-3358. Of the recycled millisecond pulsars several of the measured upper limits are only about an order of magnitude above their spin-down limits. For the young pulsars J1913+1011 and J1952+3252 we are only a factor of a few above the spin-down limit, and for the X-ray pulsar J0537-6910 we reach the spin-down limit under the assumption that any gravitational wave signal from it stays phase locked to the X-ray pulses over timing glitches. We also present updated limits on gravitational radiation from the Crab pulsar, where the measured limit is now a factor of seven below the spin-down limit. This limits the power radiated via gravitational waves to be less than ~2% of the available spin-down power.

Firstly, if you follow the link to the paper, you might notice that there are actually 679 authors on this paper! That's quite a few more than you'll see on most astrophysics papers (by a couple of orders of magnitude!) and I'm somewhere in the middle of it. The reason for this is that the work has been done as part of the LIGO Scientific Collaboration (LSC) and the Virgo Collaboration, and everyone who has put work into designing, building, commissioning and running the detectors and analysis infrastructure (like code libraries, computing resources, theoretical understanding) that we use is included. We also have a few extra authors from outside these two collaborations. These are a selection of electromagnetic pulsar astronomers who have provided vital information about the sources we're looking for in gravitational waves. Here we've used data from the fifth science run (S5) of the LIGO gravitational wave detectors, which took place from Novmeber 2005 to October 2007, and during which the detectors were at their design sensitivity.

The aim of this search is to use already known astronomical objects - pulsars (neutron stars) - as targets from which to look for gravitational wave emission - these are probably the most promising targets within our own galaxy, although there are many exciting extra-galatic sources. These objects are very dense (about the mass of the sun compressed into a sphere about 20km across) and rapidly rotating (they spin round 100 times or more every second - i.e. the surface of the star is rotating at about 20 million km/hr, or 2% of the speed of light) - both of which make these potentially good sources of gravitational waves. It requires a bit more than just being massive and fast for these objects to emit gravitational waves - they have to have some sort of deformation making them triaxial (like a rubgy ball, but with the difference between the axes being far smaller at less than around a millimetre). Are they actually deformed at this level (or a larger or smaller amount)? Well that's something we want to find out by looking for gravitational waves from them. And if we do see something we can make constraints on the exact make-up of the material in the pulsars.

The reason to go after known pulsars (rather than neutron stars that aren't currently observed [which we are also doing, and you can help with]) is that it's relatively computationally easy. If you don't know where to look to find a source (in, say, sky position and frequency) then you literally have to look everywhere, which requires huge computational resources. Whereas, if you have a known object you know precisely where to look. So that's what we do. We take our gravitational wave detector data and look at the frequency of a particular pulsar and see if there's a signal sticking up above the noise floor at that spot (it's not quite as simple as that because our data is produced by a detector sitting on the Earth, which is a moving platform, so any signal hitting the detector will be Doppler shifted [i.e. the frequency will change slightly as we're travelling towards and away from the source] and we have to take this into account - luckily we know the position of the source and the speed and position of the Earth and can calculate this shift - and also the pulsar spins-down, so it's frequency changes as it loses energy [and some young pulsars have messy frequency evolutions due to still unknown effects]). This signal is parameterised by four physical quantities of the source - the gravitational wave amplitude (directly related to the size of deformity on the star), the initial phase of the signal, the inclination of the pulsar, and the polarisation angle - so we try and estimate these parameters from our data. Unfortunately in this search we saw no signals from any of the pulsars we looked at, so rather than being able to estimate these parameters we instead set an upper limit for each - a limit at which we say that, at a 95% degree-of-belief, the gravitational wave amplitude from the pulsar must be less than.

For this search our best upper limit on amplitude (i.e. the smallest value) is 2.3x10^-26 (or 0.000000000000000000000000023) for a pulsar called J1603-7202 (the name gives the sky position of the pulsar in right ascension α, and declination δ, so this pulsar is at α = 16^h03^m and δ = -72^o02'). This value is obviously small, but it's a dimensionless quantity called strain, so what does it actually mean physically. Well given our detectors are 4km long (well two of them were 4km and one was 2km) it means that the signal from this pulsar cannot be changing the length of the detector by more that 2.3x10^-26 x 4000m = 9.2x10^-23 metres (or 0.000000000000000000000092 metres, or, in the favourite units of human hair widths, its about 1 billion billion^th of a hhw). If you look at the displacement sensitivity of the detectors then, at their best, between about 100-200 Hz, they would seem to show that they can reach 1x10^-19 m Hz^-1/2 - which is four orders of magnitude worse than our result! However, that value has the funny units of m Hz^-1/2, which is displacement spectral density - if we have a continuous signal that we can track over long time periods (like a known pulsars) we can get better sensitivity than this by integrating over time (basically just using long observations), which allows us to dig into the amplitude spectral density noise floor by the square root of the observation time (is seconds) - hence our better limits.

We can use our amplitude upper limit to set a limit on the ellipticity of the pulsar (basically the size of the deformation) - this requires us to assume that we know the star's moment of inertia (which in fact could be uncertain by up to a factor of three depending on the mass and make-up of the star i.e. it's equation of state), and that we accurately know the star's distance (which are generally uncertain by 10-20%, but could be uncertain by a factor of 2-3!). However, plugging in standard numbers for these values, our best upper limit on the ellipticity for any of the pulsars in our search was 7.0x10^-8 for J2124-3358. Given that the star has a diameter of about 20km this would mean that any deformations on it are less that about 1.5mm i.e. our direct observations rule out hills on this star, which is hundreds of light years away, being larger than 1.5mm!

Is an upper limit by itself interesting? Well that depends on your previous ideas about the source. If you think that the source could, or should, be emitting gravitational waves at a level above your upper limit then you can infer new information. In our case, for each pulsar, we have something called the "spin-down upper limit", which says that if we assume that all the kinetic energy that the pulsar loses as it spins-down is being radiated as gravitational waves then we can calculate a limit on the amplitude of those waves. We can compare our direct observational upper limits with these spin-down limits, and if we beat them we're entering a new regime of knowledge about the pulsar. The majority of our pulsars are actually spinning down rather slowly (the fast, millisecond, pulsars are generally slowly spinning down, thought to be due to them having weak magnetic fields) so their spin-down limits are small and therefore our results are still well above them (by about 10-100 times). But for a few pulsars (generally younger pulsars) we approach, and in the case of the Crab pulsar (and if you assume larger moments of inertia also for J0537-6910) beat, this spin-down limit. So, we're in a regime where we potentially could see gravitational waves from the Crab pulsar. However if we look at the ellipticity we can estimate for it, given our measured upper limit, we get a value of around 1x10^-4. Normal neutron star equations of state would suggest that the largest possible ellipticities that could be sustained by the star would likely be at least an order of magnitude (probably more) less than this, so you have to go to more exotic, and probably less likely, equations of state (things like quark stars) to get potential ellipticities at this level. Unfortunately these values are maximum possible deformations that the star might have, they could just be far smoother. For the Crab pulsar even though we've not seen a gravitational wave signal it's still nice to think about what this results means - in terms of the fraction of the star's energy it is losing when it spins-down, we can now say that less than 2% of it is being emitted via gravitational waves (magnetic dipole radiation and powering the accelerating expansion of the Crab nebula seem to dominate the energy loss).

These searches are not stopping because we've not seen anything yet. New data is currently being taken, which will soon hopefully surpass the S5 run in its sensitivity. We also have data from the Virgo detector, which will allow us to look for pulsars at lower frequencies than are accessible than LIGO - this is a frequency range where there are quite a few young rapidly spinning-down pulsars, which we hope to be able to surpass the spin-down limit for. New pulsars are also being discovered all the time (Fermi is has observed several new pulsars in γ-rays) and hopefully one of these could be a large gravitational wave emitter. We've only really just started these searches, so there's still a lot more to do and a detection could be just around the corner.