A quieter harmonic that measures the angle the loud one hides

A gravitational-wave signal from a merging binary gives a distance with nothing calibrated beneath it: the amplitude of the wave falls as the inverse of the distance, the chirp fixes the intrinsic loudness, and their ratio is the distance in absolute units. The catch is the angle. The same wave is louder seen along the orbit’s axis than from its side, so the amplitude measures the distance divided by a function of the inclination, and a nearby binary seen obliquely is indistinguishable from a distant one seen face-on. For the binary neutron star whose light was also seen, the angle came from elsewhere. For the hundreds of binaries with no light, it has to come from the wave.

It can, because the wave is not a single tone. The dominant radiation is quadrupolar, at twice the orbital frequency, and its two polarisations depend on the inclination in a way that is almost — but only almost — the same for every angle within forty degrees of the axis. Underneath it are quieter harmonics, at the orbital frequency, at three times it, at four times it, each beamed in a different direction. One of them vanishes along the axis entirely. Its size relative to the dominant harmonic is a function of the angle and not of the distance, and measuring it measures the inclination directly.

This essay draws how that works and where it does not. The quiet harmonics are loudest for binaries whose two masses are very different; they are silent for equal masses; and they are weakest exactly where the binaries a detector finds most often lie.

Each harmonic is beamed in its own direction #

The radiation from a binary is expanded in harmonics labelled by two integers, as the electron orbitals of an atom are: $l$ is the multipole order, quadrupole for $l = 2$ and octupole for $l = 3$, and $m$ counts how many times the pattern repeats round the orbit, so that the harmonic $(l, m)$ oscillates at $m$ times the orbital frequency. Each harmonic carries its own angular pattern, the spin-weighted spherical harmonic appropriate to a wave with two transverse polarisations, and the pattern says how much of that harmonic reaches an observer at each inclination.

The (2,2) harmonic is the one detectors chiefly hear. Its pattern peaks along the axis of the orbit, where both polarisations are equal and the wave is circularly polarised, and falls to a little over a third edge-on, where only one polarisation survives. Between face-on and about forty degrees it changes slowly: the difference between those angles is a change in amplitude of a few tens of per cent, and a change in amplitude is exactly what a change in distance also produces.

The (3,3) harmonic has a different pattern. It is zero along the axis — a three-fold pattern cannot be seen from a direction that has no preferred azimuth — and it rises to a maximum about forty-eight degrees from the axis, staying strong through edge-on. The (2,1) harmonic, at the orbital frequency itself, is also zero along the axis and peaks edge-on. The (4,4) harmonic, at four times the orbital frequency, peaks about sixty degrees off. Each one, measured against the (2,2), says something about where the observer stands.

Why the loud harmonic is nearly blind to the angle #

The dominant harmonic does carry the inclination in principle, and it is worth seeing exactly why it does so poorly in practice. Its two polarisations have amplitudes proportional to $(1+\cos^2\iota)/2$ and $\cos\iota$. Face-on both are one, and the wave is circular. At thirty degrees they are 0.88 and 0.87; at forty-five, 0.75 and 0.71. Through the whole of the range from the axis to about forty-five degrees, the two polarisations stay nearly equal to each other and fall together, so the wave stays nearly circular and merely gets quieter. A wave that gets quieter is a wave that could be further away.

Only beyond about sixty degrees do the polarisations separate decisively, the cross polarisation falling towards zero edge-on while the plus polarisation levels at a half, and the wave becomes measurably elliptical and then linear. A network of detectors at different orientations can measure that ellipticity, and for an inclined binary it does. For a binary within forty degrees of face-on, the ellipticity is within the errors of being circular, and the dominant harmonic offers a distance and an inclination that slide along each other: the closer the binary, the more inclined it must be to be as quiet as it is. The posterior for a typical detection is a long curved ridge in the plane of distance against inclination, running from a face-on binary at the far end to an inclined one at the near end, and the prior on orientations — more binaries inclined than face-on — tilts the weight along it in a way the data cannot correct.

That is the ambiguity the quiet harmonics are needed for. They do not refine the dominant harmonic’s measurement of the polarisation; they add a separate observable, one that is zero along the ridge’s far end and grows along it towards the near end. A measured (3,3) amplitude is a point on that ridge, and it pins the binary there.

A harmonic whose size is an angle #

The ratio of the (3,3) amplitude to the (2,2) amplitude is the quantity that matters, because both are at the same distance and the distance cancels from it. For a binary of a given mass ratio, at a given stage of its inspiral, the ratio rises from zero face-on, through a tenth or so at thirty degrees, to a quarter or more edge-on. A measured ratio therefore picks out an inclination — and with the inclination fixed, the amplitude of the (2,2) harmonic gives the distance with no degeneracy left.

The ratio also grows through the inspiral. The (3,3) amplitude carries an extra factor of the orbital velocity over the speed of light relative to the (2,2), because it is the next order in the expansion of the radiation in powers of $v/c$. Early in the signal, when the binary is wide and slow, the harmonic is a few per cent of the dominant one at best; in the last orbits before merger, where the velocity is a third of the speed of light or more, it is several tens of per cent. The figures are drawn near the end of the inspiral, which is where most of the signal from a binary of black holes accumulates.

What the ratio cannot do is distinguish an inclination from its mirror image across the orbital plane: a binary seen from above at thirty degrees and one seen from below at thirty degrees from the other pole give the same ratio. That ambiguity, between a binary orbiting clockwise and counter-clockwise on the sky, is broken by the relative phase of the harmonics and by the polarisation measured across a network of detectors, and it does not affect the distance.

Equal masses keep their angle hidden #

The odd harmonics — those with odd $m$ — vanish for a binary of two equal masses, and the reason is a symmetry rather than a calculation. Exchanging two identical bodies changes nothing, and it is equivalent to rotating the binary by half a turn about its axis. A harmonic with odd $m$ changes sign under a half-turn, so it must be zero. The amplitude of each odd harmonic is proportional to the fractional mass difference $(m_1 - m_2)/M$, and the even harmonics beyond the quadrupole are proportional to a different combination that survives at equal masses but is smaller by another factor of $v/c$.

The consequence is a sharp division in which binaries can have their inclinations measured this way. A pair of black holes of nearly equal mass — the commonest kind detected — radiates almost no (3,3) harmonic at any angle, and the (4,4), which it does radiate, is a few per cent of the dominant. A pair with a mass ratio of a quarter radiates a (3,3) harmonic a fifth as loud as the (2,2) at sixty degrees. The same mass difference that kicks the merged black hole out of its host — asymmetric radiation carrying net momentum — is what makes the asymmetric harmonics loud, and for the same reason: a binary with a lighter partner is lopsided, and its radiation is lopsided with it.

The first two events in which a harmonic beyond the quadrupole was detected, in 2019, were the two most unequal binaries then seen. GW190412 had a mass ratio of about 0.28; GW190814, a black hole of about twenty-three solar masses with a companion of 2.6, had about 0.11. In both, the (3,3) harmonic was measured with a signal-to-noise of a few — about three in the first and more than six in the second — and in both, it narrowed the inclination and moved it away from face-on.

GW190814 shows the effect most clearly, because its mass ratio is extreme and its total signal was strong. Analysed with the quadrupole alone, its inclination was spread across most of the range consistent with its amplitude; with the higher harmonics included, it was confined to within a few tens of degrees of a value well off the axis, and its distance narrowed with it. The improvement came almost entirely from the (3,3) harmonic, whose presence at that strength is impossible for a face-on binary. It was a case of a detector hearing a note that one orientation cannot produce, and drawing the conclusion about the orientation.

The same events also tested the harmonic content itself. General relativity fixes the relative amplitudes of the harmonics for given masses and spins, with no free parameter, and a measured (3,3) harmonic whose strength disagreed with its prediction would be a sign of something wrong with the theory or the source model. The measured strengths agreed. A harmonic that measures an angle is also a harmonic whose loudness is a prediction, and the prediction held.

The same binary seen along its axis and from the side #

What the harmonic does to the signal is visible in the waveform itself. Scale two copies of the same binary’s wave to the same peak height — removing the difference in loudness that a distance could also explain — and compare their shapes. Seen face-on the wave is a pure sinusoid at twice the orbital frequency: one full cycle of strain every half-orbit, every crest the same. Seen at seventy degrees, with the (3,3) harmonic a third as loud as the (2,2), the crests alternate in height and the troughs are skewed, a pattern that repeats once per orbit rather than twice.

That shape difference is what a template bank of waveforms with higher harmonics is built to find, and it is something an inclination can produce and a distance cannot. The quadrupole alone would give two waves that are identical after scaling; the degeneracy between distance and angle is the statement that scaling is all there is. Any feature that survives the scaling breaks it.

The detected binaries are the ones that hide their angle #

There is a cost, and it comes from the detector’s preference. A face-on binary is louder than an edge-on one at the same distance by a factor of nearly three, and so it is detectable to nearly three times the distance, over a volume more than twenty times larger. Most binaries are inclined, because there are more directions near the equator of a sphere than near its poles, but the detected sample is weighted by that volume. More than a third of detections lie within thirty degrees of face-on, against an eighth of all binaries, and the median detected inclination is about thirty-six degrees.

Within thirty degrees of face-on the (3,3) harmonic is weak, a tenth of the dominant or less even for a mass ratio of a quarter. The detected binaries crowd into exactly the range of angles where the quiet harmonic is quietest, and where, separately, the dominant harmonic changes least with angle. Both routes to the inclination are at their worst where the events are. The same pattern is familiar from planets found by their stars’ reflex motion, whose masses come with the sine of an unknown angle: a detection method’s sensitivity is itself a function of the very angle it fails to measure, and the sample it yields is biased towards the cases in which it fails worst.

How well the harmonic pins the angle #

A rough idea of how much the harmonic buys comes from treating its measured ratio as known to an error set by the overall signal-to-noise and asking how sharply the ratio changes with angle. For a binary of mass ratio 0.1 at a signal-to-noise of 25, the inclination comes out to within a few degrees across most of the range; for a mass ratio of a half, within ten to fifteen. In either case that is a great improvement on the forty or sixty degrees of degeneracy the dominant harmonic leaves, and through the distance–angle relation it shrinks the distance uncertainty correspondingly.

The estimate is optimistic in two directions and both are worth saying. Near face-on the harmonic is nearly absent, and an error set by the overall signal overstates what the measured absence of a weak signal can say; a real analysis finds the inclination of a face-on binary poorly constrained even with higher harmonics in its model. And towards edge-on, the ratio stops changing with angle and the help vanishes, though there the dominant harmonic’s own polarisation content — nearly linear edge-on — carries information instead. The curves are a sketch of where the harmonic matters, not a forecast for any detector.

A distance that the harmonics can sharpen #

The point of the inclination is the distance, and the point of the distance, for many of these binaries, is the expansion rate of the universe. A binary with a measured distance and a redshift from its host galaxy gives one point on the relation between them, and the expansion rate is the slope. Without a counterpart the redshift must be guessed statistically from the galaxies in the binary’s sky area and distance range, and the narrower the distance, the fewer candidate galaxies lie in the volume, and the sharper the guess.

So the harmonics feed into cosmology through a chain: mass asymmetry makes the (3,3) harmonic loud; the harmonic fixes the inclination; the inclination fixes the distance; the distance narrows the list of host galaxies. The binaries that help most are the unequal ones seen obliquely, which are the ones detected least often. The equal-mass, face-on binaries that dominate catalogues contribute through numbers rather than through individual precision.

There is another route to the same information. A binary whose black holes are spinning with their spins tilted to the orbit has an orbital plane that precesses during the inspiral, so its inclination to the line of sight changes as it goes, and the waveform’s modulation records the angle. Precession and higher harmonics are both strongest for unequal masses, and both are absent for the commonest case. Measuring a black hole’s spin and measuring its binary’s orientation turn out to need the same thing from the wave: an asymmetry the equal-mass binary does not have.

What the curves leave out #

The harmonic amplitudes are the leading terms of the post-Newtonian expansion, evaluated at a single orbital velocity near the end of the inspiral rather than integrated over the signal and weighted by a detector’s sensitivity at each frequency; the (3,3) harmonic sits at one and a half times the dominant frequency, where the detectors’ noise is different, and that changes its effective loudness by tens of per cent. The merger and ringdown, where the harmonic content is largest, are not drawn. Spins, which modify the amplitudes and add precession, are set to zero. The detection-selected inclination distribution assumes a single detector with equal response to both polarisations and a threshold on amplitude alone. The inclination uncertainty is a one-parameter estimate that ignores the correlations a full analysis would find between inclination, mass ratio and spin. The values for GW190412 and GW190814 are the published mass ratios; their angles are not reproduced here.

Still open: how a distance with no host is turned into a redshift #

The higher harmonics shrink a binary’s distance range, and a distance with no counterpart still has to be paired with a redshift. That pairing is done statistically: every galaxy in a catalogue that lies in the binary’s sky area at a plausible distance is a candidate host, each is weighted, and the expansion rate is the value that best explains the whole set of binaries at once. It depends on the catalogue being complete to the distances involved, which at the distances of most detected black-hole binaries it is not, and on the probability that a galaxy hosts a merger scaling with its mass or star formation in a way that has to be assumed.

How much the inclinations recovered from harmonics improve that inference, given the incompleteness of the catalogues and the face-on bias of the detections, is the question the harmonics leave next. The answer depends on how many unequal-mass binaries the detectors find, which depends on how black holes pair in nature — a distribution that the mass gap and its impostors have already shown to be stranger than expected.