The lens a long wave walks round

The lens with no focal length traced rays past a point mass and found that it focuses nothing in particular: every ray is bent by $4GM/bc^2$, an angle that falls with distance from the axis, so rays from a distant source cross the axis everywhere beyond a certain distance rather than at a focus. For a source directly behind the mass the rays converge into a ring, the Einstein ring, and the magnification of the source, worked out from how the lens stretches its image, is infinite. Move the source a little off the axis and the ring breaks into two images, one bright and one faint, with a combined magnification that falls as the source moves out: ten at a tenth of an Einstein radius, 1.34 at one.

Every number in that account is a statement about rays, and rays are an approximation to waves that holds when the wavelength is small compared with everything else in the problem. Light passing a star satisfies it by a factor of $10^{20}$. Gravitational waves passing a star do not, and what they do instead is worth working out, because it is both a clean exercise in wave optics and a measurement that the detectors are now equipped to make.

The number a wave compares with the lens #

The figure plots the exact result of letting a wave pass a point mass, and nearly all of it is controlled by one number. Call it $w$:

$$w = \frac{8\pi G M f}{c^3} = 4\pi\,\frac{r_s}{\lambda},$$

where $f$ is the wave’s frequency, $\lambda$ its wavelength, and $r_s = 2GM/c^2$ the Schwarzschild radius of the lens. When $w$ is large, the curves do what ray optics says — they climb to the geometric magnifications and, off the axis, oscillate about them. When $w$ is small, every curve lies on the line at 1. The lens is not there.

The surprise is which length appears. The region a lensed ray passes through is the Einstein radius, which for a star half-way to a source a few billion light-years away is about two hundred billion kilometres — fifteen hundred times the distance from the Earth to the Sun. The Schwarzschild radius of the same star is three kilometres. One might have expected a wave to be lensed once its wavelength is small compared with the two hundred billion kilometres, and the figure says it is the three that matter. A gravitational wave of a hundred hertz has a wavelength of three thousand kilometres. It is tiny beside the Einstein radius and large beside the Schwarzschild radius, and it is the second comparison that decides.

The reason is that a wave is not affected by where its paths go but by how long they take. Lensing works because the waves from the source reach the observer along a family of paths, and they reinforce where those paths take nearly the same time. A point mass delays a path passing it by an amount set by its mass alone: of the order of $4GM/c^3$, just under twenty microseconds per solar mass, varying only logarithmically with how close the path goes. The two images of a lensed source arrive separated by a delay of exactly that order — 2.08 units of $4GM/c^3$ for a source one Einstein radius off the axis. If the wave’s period is longer than that delay, the paths cannot be told apart by their timing and the wave behaves as though nothing had bent it. The geometry of the bending is enormous, but the timing difference it creates is tiny, and the timing is what a wave reads. The ratio of the two is $w$, up to a factor of $4\pi$ that comes from the conventions of the calculation.

The delay in question is an old acquaintance. The delay that is not a bend measured it for radar echoes grazing the Sun: 233 microseconds added to a round trip to Venus, accumulated mostly far from the Sun, where the bending was negligible. The same logarithm of distance appears in both places, and for the same reason — the delay from a mass grows only slowly with how far away the path passes, so paths that differ enormously in geometry differ only slightly in time. That is why the size of the lens is the wrong comparison. Two paths a billion kilometres apart, one on each side of a star, arrive within a few tens of microseconds of each other, and a wave whose period is a hundred times longer cannot tell that they were different paths at all.

The same consideration explains a feature of the curves that ray optics cannot. In the bend Newton got half right, the deflection of a ray was the derivative of exactly this delay with respect to where the ray passes: rays bend towards the side on which they would be delayed more, as light bends towards glass. In the ray picture only that derivative matters, and it is large however light the lens is, provided the ray passes close. In the wave picture the delay itself matters, and for a light lens it is small. Bending is the slope of a landscape; a long wave feels only the landscape’s height, and a lens of a few solar masses is a slope without enough height for a wave of a few thousand kilometres to notice.

A focus that is a diffraction pattern #

The second figure looks at the same result the other way, across the shadow behind the lens rather than along the frequency. Geometric optics draws a spike: magnification falling away from infinity on the axis as one over the offset. The waves draw something familiar from optics instead. At low $w$ the profile is flat. At $w = 1$ a broad, gentle hump of 3.3 sits where the spike was. At $w = 4$ the hump is taller and narrower, 12.6 on the axis, and beyond it rings appear, the bright and dark fringes where the two geometric images interfere.

That is a diffraction pattern, and where rays stop being enough found its optical original: the bright spot that appears at the centre of the shadow of a disc, which rays forbid and waves require. Here the lens concentrates rather than blocks, but the logic is the same. A converging wave cannot be squeezed into a region smaller than about its own wavelength, so the infinite peak is replaced by a finite one whose height grows with frequency and whose width shrinks as $1/w$. On the axis the height has an exact closed form:

$$|F|^2 = \frac{\pi w}{1 - e^{-\pi w}},$$

which rises from 1 at low frequency and approaches $\pi w$ at high frequency. Geometric optics is the limit $w \to \infty$ of this, and the limit is infinite only because ray optics treats a wave as having no wavelength at all. A real focus, of light or of anything else, is always a pattern of finite size, and the image that is a diffraction pattern twice found the same rule setting the sharpness of every telescope.

Two images that cannot be seen separately #

The third figure fixes the source one Einstein radius off the axis and sweeps the frequency. At high frequency the exact result settles into a steady oscillation between 0.45 and 2.24, centred on the geometric 1.34. That oscillation is interference between the two images. Geometric optics adds their brightnesses, as it would for two separate stars; a wave adds their amplitudes, and the result depends on how far out of step they arrive. The two copies of the wave are delayed relative to each other by a fixed time, so as the frequency rises they drift in and out of phase, and the combined signal swings between the sum and the difference of their amplitudes.

The swing is a signature a telescope never sees in starlight. The fifth image a galaxy hides counted images that optical telescopes resolve separately on the sky; the images of a star lens are separated by milliarcseconds and are not resolved, but for light the interference between them washes out, because the light’s wavelength is so short that any spread of frequencies or any motion of the lens scrambles the phase. A gravitational wave is coherent over its whole frequency sweep and the detectors record its phase as well as its amplitude, so the beat would be there in the data, a regular rippling of the signal with frequency whose period gives the delay between images that cannot be seen apart. The rule is the one the fringe and the spectrum are one measurement found for an interferometer: fringes in frequency are the transform of a delay, and their spacing, $\Delta f = 1/\Delta t$, reads the delay directly. Here the interferometer is a lens of a few hundred solar masses, and its two arms are the two sides of a star.

What a lens writes on a chirp #

A merging pair of black holes emits a chirp, a wave whose frequency sweeps upward through the detectors’ band over its last seconds, and what the instrument actually hears described how the detectors fit such a signal against templates computed from general relativity. A lens between the merger and the Earth multiplies the signal by a factor that depends on frequency, so it writes a distortion along the sweep. The fourth figure shows three cases for a source half an Einstein radius off the axis. A lens of three solar masses is a weak perturbation: at the bottom of the band it does nothing, and the factor creeps up towards the top as $w$ grows. A lens of thirty solar masses is in the interesting regime, with $w$ passing through 1 inside the band, and the amplification rises smoothly from almost nothing at ten hertz to nearly doubling the amplitude at a few hundred. A lens of three hundred solar masses is close to the geometric limit, and the factor ripples rapidly between 0.49 and 2.03 as the two images beat.

None of these distortions changes the signal’s frequency sweep, which is set by the masses of the merging black holes, so none can be absorbed into a different pair of masses. That is what makes the search possible. A template that includes a lens of a given mass and offset fits a lensed signal better than any unlensed template, and the improvement measures the lens. The searches run on the detectors’ catalogues have so far found no event that is generally accepted as lensed, which is in line with the expected rate: a source has to pass within about an Einstein radius of a compact mass, and for the numbers of black holes and stars thought to lie along typical sight lines that happens to a small fraction of a per cent of detected events.

Which lenses each kind of wave can see #

Because $w$ is a product of the lens mass and the frequency, each kind of detector has a mass below which lenses become invisible to it, and the last figure draws the line. The ground detectors, at around a hundred hertz, cross over at eighty solar masses: stars and stellar black holes in the way would distort the chirps they hear rather than simply magnify them, and a population of such objects along the line of sight would show up as a frequency-dependent scatter in the signals. A detector in space, listening at millihertz frequencies, crosses over at millions of solar masses, the scale of the black holes at the centres of galaxies; one listening with pulsars, at nanohertz, at the mass of a galaxy. Each regime, for its own band, offers a census of lenses that geometric optics cannot provide: a measure of how much compact mass lies between the source and the Earth in each range.

Light sits at the far right. Its frequencies are so high that only a lens lighter than about $3 \times 10^{19}$ kilograms — a large asteroid — would reach $w \approx 1$, and nothing that light has ever been lensed by is that small. For gamma rays the threshold is lower still, and Andrew Gould proposed in 1992 that a population of asteroid-mass dark objects could be detected through the interference fringes they would imprint on the spectra of gamma-ray bursts behind them, an effect called femtolensing after the femtoarcsecond separation of its images. The searches have set limits rather than found anything, but they are the optical version of exactly this figure.

The equation behind the curves #

All five figures come from a single expression. A wave of frequency $f$ passing a point mass reaches the observer with its amplitude multiplied by a complex factor $F(w, y)$, found by adding up the contributions of all paths through the lens plane, each weighted by the phase its delay gives it. For a point mass the sum can be done exactly, and its magnitude is

$$|F(w,y)| = e^{\pi w/4}\,|\Gamma(1 - \tfrac{i w}{2})|\;|{}_1F_1(\tfrac{iw}{2}, 1; \tfrac{i w y^2}{2})|,$$

with $y$ the source’s offset in Einstein radii, $\Gamma$ the gamma function and ${}_1F_1$ the confluent hypergeometric function, first written down for this purpose by Takashi Nakamura and Shuji Deguchi in 1999. Its two limits are the two pictures. As $w \to 0$ the factor tends to 1: no lens. As $w \to \infty$ the integral is dominated by the two paths of stationary delay, which are the two geometric images, and it reproduces their magnifications together with the interference between them. The figures evaluate the expression by its power series, which converges for every argument but loses accuracy to cancellation when $w y^2$ is large; the ranges drawn stay where it is accurate, and the agreement with geometric optics at the upper end is checked rather than assumed.

Where the point mass stops being the lens #

The formula is for a single point mass in an otherwise empty universe, with the source and the observer far away and the lens’s gravity weak. Real lenses are rarely so clean. A star in a galaxy is lensing against the background of the galaxy’s own smooth mass, which adds shear and changes the caustic structure, and a lensed gravitational wave would usually be lensed by a galaxy first and by the stars within it second. The combination produces patterns of interference far richer than the single beat drawn here, and computing them for realistic configurations is an active problem. The formula also ignores the lens’s motion. Over the few seconds a chirp lasts, nothing moves appreciably, but for a source that radiates for months, as the sources of a space-based detector will, the source’s offset from the lens changes during the observation and the amplification changes with it.

Finally, the frequency $w$ contains the lens’s mass multiplied by $(1+z)$, the factor by which the expansion of the universe has stretched the wave since it passed the lens. What is measured is a redshifted mass, and separating the lens’s true mass from its distance needs other information, as it does for every other measurement of mass from a gravitational wave.

The picture a ray diagram never draws #

The figures show magnitudes, the factor by which a wave’s amplitude or power is multiplied, and leave out the phase. The phase is where the delay between images is written, and a detector that records the waveform records it too: a lensed chirp is shifted in arrival time and carries, in the strong-lensing regime, a phase jump between images that the geometry fixes. Nor do the figures show the wave itself — the curved fronts converging behind the lens and spreading past it — because in every case drawn the lens is so much smaller in delay than in size that a drawing at true scale would show nothing but straight lines.

Still open: how much dark mass lies between the sources and the Earth #

The census that wave-optics lensing could take has not been taken. Whether a significant fraction of the universe’s dark matter is in compact objects of stellar mass, as was briefly suggested when the first merging black holes turned out heavier than expected, is constrained by microlensing of stars in nearby galaxies and by several other arguments, and lensed gravitational waves would add an independent test along sight lines billions of light-years long. That needs a few thousand detected events with good signal, which the detectors are expected to reach over the coming decade; until then the frequency at which lenses of each mass become visible is known, and the number of such lenses is not.

The habit worth carrying away is to ask what a wave actually compares with. A point mass lenses a wave only to the extent that $w = 8\pi GMf/c^3$ is large: the wave reads the delay between paths, of order $GM/c^3$, not the size of the region those paths cross, so a star whose Einstein radius is two hundred billion kilometres is invisible to a wave longer than a few times its three-kilometre Schwarzschild radius — and a focus that geometric optics makes infinite is, on the axis, a finite $\pi w/(1 - e^{-\pi w})$. The same question, asked of any lens, separates the cases where rays are enough from the ones where they say something that no wave will do.