A chirp, and a mass in the way
Two black holes spiralling into each other shake space itself. The wave they send out rises in pitch and loudness for a few seconds before they merge, which is why it is called a chirp. For two black holes of ten Suns each, a detector such as LIGO hears it from about 20 to about 220 vibrations per second, for roughly six seconds.
Now put a heavy object close to the line between the black holes and Earth: a star, a lone black hole or a clump of dark matter. Its gravity bends passing waves, much as a glass lens bends light. The wave can go round it on either side, so two paths reach the detector. They differ in length and in how deep they dip into the object's gravity, so one copy arrives later. The bending also changes how loud each copy is. The copy that passes farther from the mass is made louder; the one that squeezes past close to it is usually weaker.
Two copies of one sound
Picture two loudspeakers playing the same recording, one of them 20 thousandths of a second behind. For some notes the delay holds a whole number of vibrations, so the two copies push together and the note gets loud. For notes in between, a crest from one copy meets a trough from the other and they cancel. With a 20-thousandth-of-a-second delay, the loud notes sit 50 vibrations per second apart. Sound engineers call this a comb filter. You hear something like it when a microphone picks up a voice together with its echo from a nearby wall.
Gravity adds one twist. The later copy comes out shifted by a quarter of a vibration, so the loud notes slide a little. Their spacing stays one over the delay. A chirp runs through all the notes in turn, so the detector hears the sum swell and fade as the pitch rises.
How much louder each frequency gets can be written down. If μ+ and μ− are how much the bending brightens the two copies and Δt is the delay, the extra power at frequency f is
The last term swings between plus and minus as f grows; that swing is the fringe pattern. Take a lens of a thousand Suns placed half an Einstein radius from the line of sight. The Einstein radius is the size of the bright ring a perfectly aligned source would form around the lens. The delay is then about 20 thousandths of a second and the second copy is about 0.6 times as strong as the first. The wave's amplitude swings between roughly a half and twice its unlensed value. Ryuichi Takahashi and Takashi Nakamura worked out the full answer for a single point-like mass in 2003. Wave effects in gravitational lensing of gravitational waves from chirping binaries
When the waves are too long to split
Light has a wavelength far smaller than any star, so a star always splits it cleanly into separate images. Gravitational waves from merging black holes are enormous by comparison: at 100 vibrations per second, a wave is 3000 kilometres long. That happens to be the size of a thousand Suns squeezed into a black hole.
When the wave is much longer than that size, it no longer notices two paths. It rolls around the mass, the way a long ocean swell rolls past a wooden post without leaving a shadow. The two copies blur into one and the fringes disappear. Takahashi and Nakamura pointed out that the ratio of the lens's black-hole size to the wavelength decides which case applies. For ground-based detectors listening to merging neutron stars, they estimated that lenses between about ten and ten thousand Suns fall in the range where these wave effects matter.
A lens of fifteen Suns at the same offset shows the difference. The two-path formula above predicts a gain of about one and a half even at the lowest notes. The full wave calculation stays close to one there and rises only slowly with pitch, and the chirp keeps 99.75 percent of its original shape.
Has anyone heard it?
Not yet. The LIGO, Virgo and KAGRA teams searched every black-hole merger from their third observing run for these beats, modelling the lens as an isolated point mass, and found no compelling evidence. Search for gravitational-lensing signatures in the full third observing run
One event, GW200208_130117, mildly preferred the lensed model. The teams suspected that a short burst of detector noise had made the signal dip in the middle, which looks like a beat. That is the real difficulty. Noise, or a waveform model that is slightly wrong, can also make the loudness wobble. A lens can only be claimed when the lensed explanation beats every ordinary one by more than noise alone tends to manage.
Real lenses are seldom alone, either. A star that lenses a wave usually sits inside a galaxy, whose own gravity changes the pattern. Work on such cases suggests that for stars of about one Sun, the waves are too long and the extra brightening is largely washed out. Stellar-mass microlensing of gravitational waves
The next chapters derive the delay, the exact amplification and the test a search uses: why a lens delays a gravitational wave.