A distance tangled with an angle

A merging binary is the only astronomical object that announces its own distance. The frequency and its rate of change give the masses; the masses give the intrinsic amplitude of the wave; comparing that against the amplitude measured gives the distance, in metres, with no calibrator, no standard candle and no ladder under it.

That is a remarkable thing and it is very nearly useless on its own, because the amplitude does not depend on the distance alone. It depends on the distance divided by a function of the angle between the orbit’s axis and the line of sight, and a single detector measures the quotient.

The degeneracy is not a defect of the instruments and cannot be engineered away. It is a statement about what a quadrupole radiates, and it would be there for a perfect detector with infinite bandwidth and no noise at all. That distinguishes it sharply from every other limitation on these measurements, all of which improve as the detectors do.

Why the wave knows about the angle #

A binary’s two masses orbit in a plane, and the wave they radiate is not isotropic. Along the axis the two mass quadrupoles rotate face-on and the emission is circularly polarised at full strength. In the plane of the orbit, the motion is seen edge-on and only one polarisation survives, at half the amplitude.

Written out, the two polarisation amplitudes are

$$h_+ \propto \frac{1 + \cos^2\iota}{2}\cdot\frac{1}{D},\qquad h_\times \propto \cos\iota\cdot\frac{1}{D},$$

with $\iota$ the inclination of the orbital axis to the line of sight. A detector responds to a linear combination of the two set by its antenna pattern, so what it measures is an effective distance — the real one divided by a factor between one half and one.

The factor of two is not large by the standards of this subject, but it is large compared with everything else in a siren measurement. The masses come out to a per cent or better; the sky position, with three detectors, to tens of square degrees; the distance carries a factor-of-two ambiguity that no amount of signal-to-noise removes, because it is a degeneracy rather than a noise.

Two features of that expression are worth pausing on. First, the plus amplitude never vanishes: even exactly edge-on, a binary is visible at half strength, which is why the degeneracy is a factor of two rather than a factor of infinity. Second, both amplitudes carry the distance in exactly the same way, so no combination of them measures the distance without also measuring the angle. The information about orientation is entirely in their ratio, and the information about distance is entirely in their overall scale, and a single detector measures one number where two are needed.

There is a further complication that is easy to miss: an interferometer’s antenna pattern depends on where the source is on the sky, and the sky position is being fitted at the same time. So the combination of polarisations a given detector measures is itself uncertain, and the distance, the inclination, the sky position and the polarisation angle enter as a tangle of four parameters rather than as two. The pairwise degeneracy in the first figure is a slice through a four-dimensional ridge, and the ridge is what the sampler has to explore. This is why a siren’s distance posterior is computed rather than read off a formula.

The prior makes it worse before it makes it better #

There is a second effect, and it acts in the opposite direction to the one intuition suggests.

Orientations are isotropic, so the probability of an inclination is proportional to $\sin\iota$: half of all binaries lie beyond sixty degrees, and edge-on is far more common than face-on. A prior that prefers edge-on prefers small distances, because an edge-on binary producing the measured amplitude is nearer.

But the detection process pulls the other way. A face-on binary is louder, so it is detectable further away, and the volume surveyed goes as the cube of that distance. Selection therefore favours face-on systems by a factor of about $2^3 = 8$ in volume, and the detected population is not isotropic even though the underlying one is.

The two effects partly cancel and neither is negligible, so the posterior on the distance is a genuinely two-sided object that depends on both. This is the reason a siren’s distance is quoted with an asymmetric error bar rather than a symmetric one, and the reason the asymmetry is large: a typical single-event distance posterior runs from about 0.7 to 1.6 times its median.

It is worth stating the selection effect carefully, because it is a genuine bias in the population rather than an artefact of any one measurement. If the underlying binaries are isotropically oriented, the detected ones are not: the detection horizon is a factor of two further for face-on systems, so the detected sample over-represents them by a factor of eight in volume. That is a large distortion, and any statement about the population — how eccentric they are, what the mass distribution is, whether the spins are aligned — has to be corrected for it, in the same way an exoplanet survey’s yield has to be corrected for what it could have seen.

What the counterpart is worth #

The neutron-star merger of August 2017 is the case where the degeneracy was broken from outside, and it is worth following in detail because each step contributed a different amount.

The gravitational-wave data alone gave a distance of $40^{+8}_{-14}$ megaparsecs — a fractional uncertainty of about a quarter, dominated entirely by the inclination.

An electromagnetic counterpart was found, so the host galaxy was identified and its recession velocity was known. That is the other half of a Hubble measurement and it is what makes a siren useful at all: a distance in megaparsecs and a velocity in kilometres a second, with no rung of any ladder between them.

The counterpart also constrained the inclination independently. The event produced a short gamma-ray burst, and a burst is beamed; the afterglow’s rise and decline, monitored in radio and X-rays for two years, is fitted with a structured jet whose viewing angle is a parameter. That fit gave roughly fifteen to twenty-five degrees from the axis — face-on, in the language of the first figure — and imposing it collapsed the distance posterior.

The published Hubble constant from that event went from about fifteen per cent uncertain on the gravitational-wave data alone to under seven per cent with the jet constraint. Half the error bar was orientation.

The lesson from that sequence is not that counterparts are necessary. It is that the constraint came from a completely different physical process — the geometry of a relativistic jet, measured by watching an afterglow brighten and fade over two years — that happens to share one parameter with the gravitational-wave problem. The inclination is the orbital axis in one measurement and the jet axis in the other, and identifying the two is an assumption, though a good one: the jet is launched along the angular momentum axis of the remnant, which is the orbital axis of what made it.

That assumption is the kind worth flagging. It is almost certainly right, it is not a measurement, and it is carrying a substantial share of the error bar on the best independent Hubble constant available.

The rest of the signal is not degenerate at all #

It is worth being clear that the ambiguity is confined. Almost everything else a merger measures is measured extremely well, and the contrast is instructive.

The masses come from the phase evolution, which is a frequency measurement, and frequency measurements are immune to everything the amplitude is vulnerable to. A detector whose calibration is ten per cent wrong measures the masses correctly and the distance ten per cent wrong.

That is the cleanest statement of the difference between the two kinds of degeneracy this collection keeps meeting. A degeneracy that a better measurement breaks is a nuisance. A degeneracy that a better measurement does not break is a structural feature of what the observable is, and it can only be broken by a different observable.

The same contrast appears in the sky localisation. A single detector localises a source to a ring; two detectors localise it to two arcs on that ring, from the arrival-time difference; three narrow it to a patch. Each addition buys geometry rather than sensitivity, and the pattern is general — a detector the size of a galaxy faces the identical problem at a wavelength eleven orders of magnitude longer, and solves it the same way, by having many independent lines of sight rather than one good one.

What was actually measured #

Three numbers anchor the argument.

The factor of two is exact. It is not an empirical statement about detectors: $(1+\cos^2\iota)/2$ takes the value one at $\iota = 0$ and one half at $\iota = 90°$, and that is the whole of it. Any single interferometer sensitive to only one polarisation carries exactly this ambiguity.

Three detectors are worth roughly one polarisation ratio. The two LIGO instruments are deliberately near-aligned, so they measure almost the same combination and add signal-to-noise rather than information about the orientation. Virgo is oriented differently, and its addition to the 2017 event is what gave the sky localisation and a weak inclination constraint — enough to exclude edge-on, not enough to pin the angle.

The dark-siren route works and is expensive. Where no counterpart is found, the host can be marginalised over the galaxies in the localisation volume, each weighted by its own redshift. That recovers a Hubble constant without an electromagnetic detection, and it converges as roughly the inverse square root of the number of events. The current dark-siren constraints from tens of events are weaker than the single event with a counterpart.

And there is a fourth number, and it is the one that decides how the field will look in ten years. A single event’s orientation ambiguity contributes about fifteen per cent to a Hubble constant, and independent events average that down as the inverse square root of their number: a hundred events with counterparts would give one and a half per cent, which is decisive against a five-sigma disagreement between an early-universe value and a late-universe one. The rate of neutron-star mergers with detected counterparts is the quantity that decides when, and it is currently about one every few years.

Where the picture stops #

There are three, and the second is the one likely to change first.

It assumes the orbit is circular. An eccentric binary radiates in harmonics of the orbital frequency with a different angular pattern, so in principle the eccentricity supplies extra orientation information. In practice every pair arrives circular, because radiation reaction circularises the orbit long before the signal enters the band, so this route is unavailable for exactly the systems being observed.

Higher-order emission breaks it partially, and will break it better. The quadrupole formula is the leading term; the sub-dominant multipoles have their own inclination dependences, and they are detectable in systems with unequal masses. Events with mass ratios far from one already show them, and for those the inclination is measured from the waveform alone. As the detectors improve, this becomes the standard route and the counterpart becomes a bonus.

And the degeneracy is with the luminosity distance, which is not a distance. What a siren measures is the luminosity distance, which in an expanding universe differs from every other distance by factors of the redshift — the same distinction that runs through the whole distance ladder. At forty megaparsecs the difference is negligible and at a gigaparsec it is not, so a siren used for cosmology is measuring a quantity whose relation to the thing wanted is itself cosmology-dependent.

One more limit belongs here because it is about the word rather than the physics. “Standard siren” is an analogy with “standard candle”, and the analogy is inexact in a way that matters. A candle is standard because its intrinsic brightness is assumed known from a calibrated population; a siren’s intrinsic amplitude is computed from the masses that the same signal measures, so nothing is assumed and nothing is calibrated. What the siren does share with the candle is that the observed quantity is an amplitude, and every amplitude in astronomy is a product of a source property and a geometric one. The masses that come out of an orbit as a lower bound are the same trade seen in a different observable: a quantity multiplied by an unknown angle, reported as though the angle were known.

Why the ambiguity is worth understanding rather than deploring #

A standard siren is the most direct distance measurement in astronomy. It has no calibration, no metallicity dependence, no extinction correction, and no reliance on any other measurement. Those are exactly the properties that make it valuable in the standoff over the Hubble constant, where the two existing answers disagree by five standard deviations and both are built on long chains of calibration.

An independent method with a factor-of-two ambiguity that averages down is worth a great deal more than a precise method with an unknown systematic. The orientation ambiguity is statistical: it is different for every event, it is understood exactly, and it shrinks as the square root of the number of events. Nothing about it can conspire to shift the answer in one direction.

That is the property that makes the method decisive in the end, and it is worth stating plainly because it is the opposite of how the error bar looks. A siren’s uncertainty is large and honest. The competing methods’ uncertainties are small and rest on assumptions that would move the answer if they were wrong.

It is also worth noticing that the ambiguity has a preferred direction with respect to the thing being measured. Because detected events skew face-on and face-on means further, ignoring the degeneracy entirely — taking the effective distance as the distance — systematically underestimates distances and therefore overestimates the Hubble constant. The size of that error is a factor approaching two in the worst case and around thirty per cent for a typical detected orientation. Nobody makes that mistake deliberately, but it is the shape of the error that any incomplete treatment of the inclination will produce, and knowing its sign is most of what is needed to catch one.

Put the selection effect alongside the degeneracy, because the two combine in a way that biases a population. A face-on binary emits more strongly along the line of sight than an edge-on one, so at a fixed distance a face-on source is louder and more likely to be detected — by a factor that makes the detected population strongly weighted towards face-on orientations. Since face-on is also the orientation that makes a source look nearer than it is, the detected sample is systematically biased towards inferred distances shorter than the true ones unless the selection is modelled. The correction is straightforward in principle: the prior on inclination is not uniform in the angle but weighted by the detection probability. It is easy to get wrong, and getting it wrong shifts every distance in one direction, which is exactly the sort of error that a measurement of the expansion rate cannot absorb.

A final point about what the degeneracy is not. It is not a limitation of the detectors’ sensitivity, and it does not narrow as the instruments improve: a louder signal is measured more precisely along the degenerate direction and no better across it. Only a second observable — a counterpart, a network geometry, a higher harmonic — changes the shape of the answer rather than its size.

Where the ladder goes next #

The rung directly above is the sub-dominant multipoles: what a waveform with unequal masses says about its own orientation, and how much of the ambiguity disappears when the emission is no longer purely quadrupolar. The rung after it is the dark-siren statistics — how a distance with no host galaxy is combined with a catalogue of candidate hosts, and what that inference assumes about the completeness of the catalogue.