Precessions bought one hertz at a time

A black hole carries two numbers, its mass and its spin, and the gravitational waves from a merging pair are the only measurement that reaches both holes of a pair at once. The masses come out well, at least in combination: the chirp of the signal fixes one number where two masses were to a fraction of a per cent. The spins come out poorly. What the waveform records most clearly is a single mass-weighted projection of both spins onto the orbit’s axis — the effective spin — and the individual spins, their sizes and their tilts, are almost unconstrained in nearly every event so far.

The information that would separate them is in the precession. When either hole’s spin is tilted away from the orbit’s axis, the spins and the orbital angular momentum exert torques on one another, and the orbital plane wobbles about the total angular momentum like a tilted top. A detector fixed in space sees the wobble as a slow modulation of the signal’s amplitude and phase, and the shape and period of that modulation depend on the spins separately. One wobble is a hint. Several are a measurement.

How many wobbles a signal carries depends on how much of the inspiral the detector hears, and a ground-based detector cannot hear below about twenty hertz, because the ground itself moves too much. This essay counts.

A count with a formula #

The precession rate of an orbit whose spins are tilted can be written in a simple approximation that is accurate when one spin dominates or when the spins are small compared with the orbit’s angular momentum. The plane precesses about the total angular momentum at a rate proportional to that angular momentum divided by the cube of the separation. Early in the inspiral the total is dominated by the orbit’s own angular momentum, and expressing everything through the orbital speed $v$ gives a precession frequency that is a fixed fraction of the orbital frequency,

$$\frac{\Omega_p}{\omega} = \left(2 + \tfrac{3}{2}q\right)\eta\, v^2,$$

where $q$ is the ratio of the lighter mass to the heavier and $\eta$ the symmetric mass ratio, a quarter for equal masses. At large separation $v$ is small and the plane precesses once for every few hundred orbits; near merger $v$ approaches half the speed of light and it precesses once every few orbits.

The same physics makes the precession frequency itself a measurement. Its rate depends on the masses and on the spins’ sizes, and a signal long enough to follow the modulation through several cycles measures how the rate changes as the orbit shrinks, which constrains the individual spin magnitudes independently of their tilts. A single cycle gives a hint of a tilt; several give a rate; many give its evolution — three successively stronger statements, each needing a lower starting frequency than the last.

Counting precession cycles means integrating that rate over the time the signal spends in band, and the time spent near each frequency is set by the chirp — by how fast gravitational radiation drains the orbit. Both depend on the symmetric mass ratio, inversely in the chirp, and the dependence cancels. What remains is

$$N_p \simeq \frac{5\left(2 + \tfrac{3}{2}q\right)}{192\,\pi^2}\,\frac{1}{M f_{\mathrm{low}}},$$

in units where $G$ and $c$ are one, with $M$ the total mass and $f_{\mathrm{low}}$ the lowest gravitational-wave frequency the detector hears. The opening figure’s slopes, all within a few per cent of minus one, are that formula. The small departures come from the end of the band at the last stable orbit, where the integral stops.

A heavy pair barely wobbles #

The formula puts the total mass in the denominator, and the consequence is the first thing to notice in the opening figure. The heaviest pair, thirty-six and twenty-nine solar masses — the masses of the first merger ever detected — precesses less than once between twenty hertz and merger. The lightest, a black hole of eight solar masses with a neutron star’s mass beside it, precesses six times.

A heavier pair is a slower pair at every orbital separation; its whole inspiral is shifted down in frequency, so less of it lies above any fixed limit, and the count falls as the inverse of the mass. Above about sixty solar masses, a detector stopping at twenty hertz records less than one complete precession. That is most of the binary black holes detected so far: the population is dominated by heavy pairs, because heavy pairs are loud and can be heard from further away. The events that are easiest to detect are the ones whose spins are hardest to separate.

A pair of ten solar masses in total, by contrast, wobbles eight times in the same band. Such pairs are quieter and heard from a smaller volume, so there are fewer of them in the catalogues; for the ones that are loud enough, the individual spins are within reach.

Orbits are cheap, precessions are dear #

Lowering the detector’s lowest frequency adds signal from earlier in the inspiral, and the two kinds of cycle respond to it differently.

The number of orbits in band grows as the lowest frequency to the power minus five thirds, faster than the precessions. Going from twenty hertz to five multiplies the orbits by ten and the precessions by four. The reason is that precession is slow early — the ratio of precession to orbital frequency grows as $v^2$ — and early is exactly the part of the inspiral that a lower limit adds. Most of the orbits are at low frequency and most of the precessions are at high frequency, close to merger, where the signal is strongest anyway.

This changes what a better low-frequency limit buys. The orbits it adds sharpen the chirp mass and the effective spin, which depend on the phase accumulated over many cycles; each factor of ten in orbits improves them roughly tenfold. The precessions it adds are fewer, but they are the only source of the information about individual spins, and there is no substitute for them in the rest of the signal.

The mass ratio hardly enters #

The symmetric mass ratio cancels between the precession rate and the time in band, which leaves the mass ratio only in the factor $2 + \tfrac32 q$.

That factor runs from two to three and a half, so at a fixed total mass the mass ratio changes the count by less than a factor of two, and in the direction opposite to intuition: equal-mass pairs precess slightly more often than unequal ones. What does change strongly with the mass ratio is the depth of the wobble rather than its frequency. A lighter companion carries less orbital angular momentum, so the heavier hole’s spin is a larger fraction of the total and tilts the plane through a larger angle. Unequal pairs precess as often, and wobble harder.

That trade is why the events with clear precession signatures so far have been unequal ones. A pair with a mass ratio of a quarter and a strongly spinning heavier hole makes a deep modulation that is visible even with one or two cycles; an equal pair needs more cycles to show the same information, because each is shallower.

The cone opens toward merger #

The depth of the wobble depends on the angle of the cone the orbital plane sweeps, and that angle is set by the ratio of the spin to the orbital angular momentum.

The orbit’s angular momentum shrinks as the orbit does, as the inverse of the orbital speed, while the spins stay fixed. So the cone widens as the pair spirals in. The precessions a low-frequency detector adds are early ones, made in a narrow cone — many cycles, each a small wobble. For an equal pair, the cycles between one and five hertz sweep a cone of three to six degrees; the cycles a twenty-hertz detector records near merger sweep ten to sixteen.

The figure is a reminder that a precession count is not a signal-to-noise ratio. A wobble of three degrees modulates the amplitude by a few per cent, and a few per cent of a faint early signal is fainter still. Whether the early cycles constrain the spins depends on how loud the signal is at those frequencies, which depends on the detector’s sensitivity there — and a detector whose lowest frequency is five hertz is barely sensitive at five hertz. Counted cycles are an upper bound on what can be measured, not the measurement.

A wobble that also measures the distance #

The precession carries a second piece of information that has nothing to do with spin, and it may matter more for most uses of the signal. A merger’s amplitude at the detector depends on two things that the leading-order waveform cannot separate: the distance to the source, and the angle at which the orbit is seen. A face-on orbit at a large distance and an edge-on orbit closer by can produce the same signal, and a distance tangled with an angle is the dominant uncertainty in the distance to almost every merger heard so far — which is also the dominant uncertainty in using mergers as distance indicators that need no calibrated chain beneath them to measure the expansion of the universe.

Precession breaks the degeneracy. A wobbling orbit is seen at a changing angle, and the modulation it imposes on the signal’s amplitude and polarisation depends on the mean viewing angle: a nearly face-on orbit wobbles toward and away from face-on, changing the signal little; an inclined one swings through a wide range of apparent shapes. Measuring the modulation measures the inclination, and with it the distance. The same count of cycles that decides whether the spins can be separated decides how much of the distance uncertainty can be removed, and for the heavy pairs that dominate the catalogues the answer at twenty hertz is very little.

The first wobbles heard #

The catalogues so far hold close to a hundred binary black-hole mergers from the first three observing runs, and more from the fourth. In nearly all of them, the precession is unmeasured: the posterior distribution for the in-plane spin is close to the prior, which is to say the data say nothing about it. A handful show some evidence. The clearest have been unequal pairs — a thirty-solar-mass hole with an eight-solar-mass companion, heard in 2019, whose mass ratio made its orbit’s angular momentum small enough for even a modest spin to tilt it; and a 2020 event whose precession signature was the strongest yet, though the claim depends on how a burst of instrumental noise in one detector, overlapping the signal, was subtracted.

That is the pattern the figures predict. Heavy, nearly equal pairs show nothing, because they wobble less than once and shallowly. Unequal pairs show something, because their wobbles are deeper even though there are no more of them. And the one place where the counts are large — light pairs, at many cycles — is where the signals are quietest and the events fewest.

How long the signal lasts #

The same arithmetic sets the time each signal spends in band, which falls even faster with frequency.

The time in band falls as the lowest frequency to the power minus eight thirds. A heavy pair that a current detector hears for under a second would be heard for half a minute from five hertz and for most of an hour from one; a light pair heard for forty seconds now would be heard for half an hour from five hertz and for days from one. Long signals are a gift for the spins and a burden for the analysis — the pair’s orbit is still slightly eccentric that early, unless every pair arrives circular long before, and the detector moves with the Earth for long enough to change its antenna pattern — because the detectors’ noise is not stationary over hours and the signal has to be followed through changes in the instrument’s response.

The wall in the ground #

The limit at twenty hertz is not a design choice. A ground-based detector’s mirrors hang from seismic isolation systems that suppress the ground’s motion by many orders of magnitude, and the suppression falls off at low frequency. Below about ten hertz two effects that no isolation removes take over: the thermal motion of the suspensions, and the direct gravitational pull of the ground and the air moving around the detector — Newtonian noise, the gravity gradient of a surface wave passing under the building. A mirror can be isolated from what pushes it; it cannot be isolated from what pulls it.

The detectors planned for the next generation push the limit to about five hertz by going underground, where surface waves are weaker, and by longer arms that make every motion smaller relative to the signal. Below a hertz, a detector has to leave the ground. A space-based interferometer with arms of millions of kilometres works from a tenth of a millihertz to a tenth of a hertz — still far above the nanohertz band in which a detector the size of the Galaxy listens to supermassive pairs — and a stellar-mass black-hole pair passes through its band years before merging, precessing thousands of times and moving through the higher band of the ground detectors only at the end. The same pair seen by both would have its early precession measured from space and its merger from the ground — a combination that would separate the two spins almost completely, for the handful of pairs loud enough in both.

What the count leaves out #

The formula assumes simple precession, in which the total angular momentum’s direction is fixed and the plane rotates around it at one rate. When both holes spin with comparable tilts, the spins also precess about each other, and the motion is a combination of several frequencies; the count then means the dominant one. When the spin is comparable to the orbit’s angular momentum, near merger for a rapidly spinning hole with a light companion, the total angular momentum is no longer dominated by the orbit and the approximation fails. And a rare class of configurations, with the spin nearly opposite the orbit, can flip into a “transitional” precession in which the total angular momentum itself tumbles — not captured here at all.

None of these changes the scaling. The precession rate is a fixed multiple of the orbital frequency times $v^2$, the time in band is set by the chirp, and the count grows as the inverse of the lowest frequency for any configuration. The approximation sets the constant in front.

A spin inherited from a merger #

There is a second reason to want individual spins, beyond the formation channel. A black hole made by the merger of two others is born spinning at about seven-tenths of the maximum, almost regardless of what its parents were doing, because most of its angular momentum comes from the orbit they were on. In a dense cluster the merger product can stay, find a new partner, and merge again. A pair containing such a second-generation black hole has a heavier member spinning at 0.7 in a random direction — a configuration that precesses strongly and that a measured spin magnitude on the heavier hole would identify directly.

The effective spin cannot identify it, because a spin of 0.7 tilted at random projects onto the orbit’s axis as anything between minus 0.7 and plus 0.7, and averages to zero. The individual spin magnitude can. A population of heavy mergers whose heavier members all spin near 0.7 would be the signature of repeated mergers, and would say that some of the heaviest black holes heard — those above the mass at which stars are thought to be unable to leave a black hole at all — were built in two steps. Those are exactly the heavy pairs that precess least in the current band.

Still open: whether the spins were tilted at all #

The arithmetic says what a detector can hope for: for the heavy pairs that dominate the catalogues, about one precession at twenty hertz, five at five hertz, and tens below a hertz. The open question is what the precession would show if it were measured. A pair formed from two stars that evolved together in a binary should have spins nearly aligned with the orbit, since the same disc and the same tides shaped both; a pair assembled in a dense cluster, by the encounters that harden black-hole binaries, should have spins tilted at random. The effective spins measured so far cluster near zero with a tail to positive values, which is consistent with a mixture and with several other things. Individual spin tilts would separate the channels pair by pair rather than statistically, and that is the measurement a low-frequency detector exists to make — precessions bought one hertz at a time, at a cost set by how hard it is to stand still on a moving planet.