Finding echoes
In September 2015 the two LIGO detectors recorded, for the first time, a ripple in space itself: two black holes, each about thirty times the mass of the Sun, had spiralled into each other more than a billion years ago. The ripple stretched and squeezed the detectors' four-kilometre arms by less than a thousandth of the width of a proton. This page is about a question that came up soon after: how sure are we that what merged were black holes, and what would we hear if they were not?
I worked on that question for my master's thesis at Nikhef in Amsterdam, and the method became a paper with Ka Wa Tsang and colleagues. Below, the story in four animations.
What a gravitational wave does
Put a ring of free-floating masses in space and let a gravitational wave pass through, straight at you. The ring is stretched one way and squeezed the other, then the reverse, once per wave cycle. There are two ways to do this, called the plus and cross polarisations: the second is the first turned by 45 degrees. A real wave carries a mix of both. From a pair of black holes seen face-on the two arrive a quarter of a cycle apart, and the stretch direction spins round.
The four highlighted masses are where the ends of a detector's arms would sit. A LIGO detector is essentially two of those masses, one at the end of each arm, with a laser measuring the distance between them. The effect here is exaggerated by about twenty orders of magnitude; the real stretch was one part in 1021.
What a merger sounds like
The signal from two merging black holes has three parts. The inspiral: a rising whistle as the pair circles ever faster. The merger: the loudest moment, when they touch. And the ringdown: the newly formed black hole is lopsided at first and rings like a struck bell while it settles, for a few milliseconds. Then nothing. A black hole has a horizon, a surface of no return, so whatever falls in stays in and the bell goes quiet.
What if it is not quite a black hole?
Nobody has seen a horizon. We have seen objects that are dark, heavy and compact enough to have one, and the ringdown of the 2015 merger looks exactly like general relativity says a black hole's should. But theorists have proposed several kinds of object that would look the same from far away and yet have no horizon: stars made of exotic particles, bubbles of vacuum energy held in by a shell of matter, wormholes. Collectively, exotic compact objects. What they share is a surface of some kind sitting a hair's breadth outside where the horizon would have been.
That surface changes the ending of the story. The ringing happens at the light ring, an orbit at one and a half times the horizon radius where light can circle the hole, and it runs both ways: outward, as the ringdown we hear, and inward. A black hole swallows the inward part. An object with a surface bounces it back up to the light ring, which for waves acts as a barrier: a fraction of the energy leaks out, the rest is reflected back down, and so on. The result is a train of echoes.
The gap between echoes is the time a wave needs to go from the light ring down to the surface and back. The drawing measures distance the way the wave does, in travel time, and because time runs so slowly that deep in the gravity well, a surface closer to the horizon is further away: push the surface down and the wall moves off from the barrier. But the gap depends only on the logarithm of how close the surface sits to the would-be horizon: move the slider from a hundred-thousandth of the radius to a Planck length, thirty-five powers of ten, and the gap merely grows eightfold. For a remnant like the 2015 one, roughly sixty times the mass of the Sun, that is about a tenth of a second between echoes whatever the details. Long compared with the few milliseconds of ringdown, short enough to look for right after the main signal. The leak slider sets how much of the energy escapes at each bounce; the rest comes round again, quieter each time. Set the object to a black hole and the inward part falls through the horizon: one burst, then nothing.
A shape without a theory
The trouble with searching for echoes is that nobody can tell you exactly what one looks like. Each candidate object predicts a different waveform, most of them only roughly, and the right object may be one nobody has written down. What the pictures do agree on are four fingerprints: the echoes arrive at a constant time gap, each is quieter than the last because the barrier only leaks a fraction, each is longer and lower in pitchbecause the barrier lets low frequencies through more easily, and each may pick up a phase shift from the bounce off the surface. The first two are on solid ground; the widening is softer, and the phase is a guess.
So my thesis took a search that assumes no waveform at all and taught it those four fingerprints. BayesWave, an algorithm by Neil Cornish and Tyson Littenberg, describes whatever is in the data as a sum of small wave packets, sine-Gaussians, and lets Bayesian statistics decide how many are needed. We replaced the single packet by a train of packets, with the gap, damping, widening and phase step as knobs the algorithm turns while it fits. Here are those knobs.
The dashed lines are the envelopes of the individual echoes; the small ticks mark their centres. Switch on the spectrum and the echoes show up as a comb: a signal that repeats every Δt has frequency teeth 1/Δt apart, which is the second way an echo search can recognise a train. The listen button plays the train pitched up three times and slowed down five times, because a 170-hertz ping forty milliseconds apart is a low buzz rather than a series of echoes.
Pulling it out of the noise
In a real detector the echoes would be buried. How buried is measured by the signal-to-noise ratio, SNR: roughly, how many times louder than the noise a perfect filter would find the signal. The 2015 ringdown alone had an SNR of about 8. The thesis planted echo trains into simulated LIGO noise at SNRs of 8, 12, 18 and 25 and asked the algorithm to compare three explanations of the data: only noise, a glitch in one detector, or a real signal seen by both. The comparison comes out as a Bayes factor, the odds of the signal explanation against the noise explanation.
The trace is the data in units of the noise level, so drag the SNR slider and watch the echoes surface out of the same noise. The train buried here is whatever you built in the figure above. "Show fit" runs a small cousin of BayesWave on the data: it builds whatever stands out of the noise from wave packets, one at a time, keeps a packet only when it stands well above anything noise alone produces, and knows nothing about echoes. At SNR 8 it usually finds nothing. As the SNR rises it picks up the first burst, then the echoes one by one, and reads off their spacing. The gauge underneath places the SNR on the log Bayes factors the thesis measured for these injections. At SNR 8 the runs were indistinguishable from pure noise; from SNR 12 the signal explanation won clearly; from around 15 the algorithm also measured the echoes: at SNR 25 it read back a gap of 40 ± 7 ms, a damping of 0.69 ± 0.05 and a widening of 1.16 ± 0.09 from a train planted with 40 ms, 0.7 and 1.2. The phase step it could not measure at all. One finding worth being upfront about: the trains did not detect echoes any better than plain BayesWave with single packets did. What they added is the reading of the fingerprints.
Two honest caveats. The noise here is white, while real detector noise is coloured and full of glitches, which is exactly why the glitch hypothesis is in the comparison. And the gauge is an interpolation of four measured points for one family of injections, not a general law. Others have reported tentative hints of echoes in the real LIGO data, at a significance that did not survive closer scrutiny; no echo has been found. The point of a shape-free search is to be ready when a louder event comes, without betting on one theory of what a non-black-hole would be.
Read more
- A morphology-independent data analysis method for detecting and characterizing gravitational wave echoes
- BayesWave: Bayesian inference for gravitational wave bursts and instrument glitches
- Testing the nature of dark compact objects: a status report
- Echoes from the Abyss and Low significance of evidence for black hole echoes