# 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