A clock whose best interval is three years

A clock made of neutron stars ended on a hope with a definite shape. A millisecond pulsar’s measurement noise averages down steeply as the averaging interval grows, so a pulsar time scale ought to keep improving for as long as anyone keeps pointing telescopes at it — until it meets a floor set by something the pulsars themselves do. Where that floor lay was not known. If it was low, a pulsar ensemble would become the steadiest clock available for intervals of decades; if it was high, the pulsars would have found the limit of what a spinning neutron star can keep.

Part of that question has since been answered from an unexpected direction. In 2023 the pulsar timing arrays reported a gravitational-wave background: a slow, random stretching of space by the combined signal of supermassive black-hole binaries throughout the universe, at a characteristic strain of about $2.4\times10^{-15}$ at a frequency of one cycle a year. That background passes over every pulsar the arrays time. It does not care how quiet a neutron star’s interior is, and it cannot be improved by a better receiver. Whatever floor a pulsar sets for itself, the background sets one first, and since its amplitude has been measured, where that floor lies is now something that can be calculated rather than hoped about.

The calculation gives three answers, and none is quite what the hope implied. A single pulsar is steadiest over an interval of about three years and less steady over every interval longer than that. The interval barely moves when the pulsar is timed more precisely. And an array of pulsars can push the floor down a long way — but by how evenly its pulsars are spread over the sky, much more than by how many there are.

The measure a pulsar can be judged by #

Before the background can be compared with anything, the stability of a pulsar clock has to be stated in a form that suits it, and the familiar one does not. An atomic clock’s stability is quoted as an Allan deviation: average the clock’s fractional frequency over an interval, compare consecutive averages, and take the scatter. A pulsar cannot be treated that way, because its timing model already fits its spin frequency and the rate at which that frequency is slowing. Any error in the pulsar’s arrival times that is constant, linear or quadratic in time is absorbed into those fitted parameters, and is simply invisible. The first thing a pulsar can be said to measure about a clock is the part of its error that is cubic in time.

So the stability used for pulsars — written $\sigma_z$ and introduced by Matsakis, Taylor and Eubanks in 1997 — is built on that cubic. The residuals are cut into segments of length $\tau$, a cubic is fitted to each, and the scatter of the cubic coefficient, scaled by $\tau^2/2\sqrt5$, is the stability at that averaging time. It is normalised so that for the simplest kind of noise it agrees closely with the Allan deviation. For the steep, slow noises that matter here it is the honest measure, because it is blind to exactly the things the timing model would remove anyway.

Every curve that follows is that measure, computed as an expectation rather than simulated: the cubic coefficient is a fixed weighted sum of the residuals, so its variance under a noise of known spectrum is an integral of that spectrum against the weights. The pulsar is timed to 100 nanoseconds every fourteen days, which is what the best-timed millisecond pulsars reach. The background has the amplitude the arrays published and the spectral slope that a population of circular binaries driven by gravitational radiation would give: its residuals have a power spectrum falling as $f^{-13/3}$.

One pulsar, and the interval at which it is best #

The two contributions run in opposite directions. The pulsar’s white timing noise is independent from one fortnightly measurement to the next, so a longer segment contains more of them and the fitted cubic is pinned down better; together with the $\tau^2$ that converts a cubic coefficient into a frequency stability, that makes $\sigma_z$ fall as $\tau^{-3/2}$. The background does the opposite. Its power is concentrated at low frequencies, and a longer segment admits slower wanderings of larger amplitude into the cubic. Its $\sigma_z$ rises as $\tau^{2/3}$, an exponent fixed by the $-13/3$ of its spectrum through $(13/3-3)/2$.

The sum has a minimum, and with these numbers it sits at 2.8 years and $2.8\times10^{-15}$. Below that interval the pulsar is limited by how well it is measured; above it, by the background. At ten years the background alone contributes $5.5\times10^{-15}$, twice the minimum. A single pulsar, averaged over a decade, is a worse clock than the same pulsar averaged over three years — which is the reverse of the property that was supposed to make pulsars valuable, and it holds for every pulsar in the sky, because the background is everywhere.

That is not a failure of the pulsar. The neutron star may be turning as steadily as anyone hoped; its arrival times are being modulated on the way to the telescope by the space they cross. To the clock comparison it makes no difference where the wandering happens. What reaches the telescope is a clock with a wander in it, and the wander has a known spectrum.

Why a better telescope buys so little #

The obvious response is to time the pulsar better. A larger telescope, a wider receiver band and a better correction for the interstellar plasma all reduce the white noise, and a pulsar timed to ten nanoseconds would be a much better clock at a few months. It is worth asking what it does to the floor.

The answer comes out of the two power laws. Write the variance as a white part $a\,\tau^{-3}$, with $a$ proportional to the square of the timing noise $\sigma_t$, and a background part $b\,\tau^{4/3}$. The minimum is where the derivative vanishes:

$$\tau_\ast = \left(\frac{9a}{4b}\right)^{3/13} \propto \sigma_t^{\,6/13}\,A^{-6/13}.$$

Six thirteenths is a weak power. A thirtyfold improvement in timing shortens the best interval by a factor of $30^{6/13}$, about 4.8; a hundredfold improvement, by about 8. The best interval cannot be pushed down to weeks by any telescope that will be built, because to halve it the timing noise has to fall by a factor of $2^{13/6}$, about four and a half, and each further halving costs the same factor again.

The floor itself is the value of the sum at its minimum, and the same algebra gives

$$\sigma_{z,\min} \propto \sigma_t^{\,4/13}\,A^{9/13}.$$

The floor depends more strongly on the background than on the pulsar — nine thirteenths against four. Timing a pulsar ten times better lowers it by a factor of $10^{4/13}$, almost exactly two; a background twice as strong would raise it by $2^{9/13}$, about 1.6. The quality of a single pulsar as a clock over years is set mostly by the sky it sits in and only weakly by the instrument pointed at it.

None of this is peculiar to pulsars. Any clock whose white noise falls as one power of the averaging time and whose red noise rises as another has a best interval and a floor given by the same two exponents. An atomic clock has the same shape, with its own exponents: a hydrogen maser is superb at hours and drifts at weeks, and the reason a laboratory steers it with a caesium fountain is that the maser’s best interval is short. What is unusual about the pulsar is that its red noise is not in the clock at all, and so cannot be engineered away.

A shared noise that does not average to a shared number #

The single pulsar is not the case that matters, though. A pulsar time scale is an ensemble — an average over many pulsars, in the way the atomic time scale averages hundreds of clocks — and an ensemble improves on its members only as far as their errors are independent. The background is not independent. It passes over every pulsar, and the arrays detected it precisely because it correlates them. The natural expectation, and the one the earlier essay drew, is that a noise shared by every pulsar survives the average intact and sets a common floor no ensemble can go below.

That expectation treats the background as though it were shared in the way a clock error is shared, and it is not. A clock error at the observatory delays every pulsar’s pulses by the same amount at the same moment: the correlation between any two pulsars is one, whatever the angle between them. The background correlates two pulsars by an amount that depends on the angle between them, through the Hellings–Downs curve, and that curve is positive for close pairs and negative for pairs at right angles.

On a sky covered evenly, pairs separated by an angle $\theta$ occur in proportion to $\sin\theta$ — there are few pairs nearly in the same direction, few nearly opposite, and many near right angles. Weighting the Hellings–Downs curve by that share and adding up gives zero, not approximately but exactly. In the language of spherical harmonics the curve has no monopole and no dipole; its lowest component is a quadrupole, which is why it is the signature the arrays search for and why it cannot be confused with a clock error or an ephemeris error.

The consequence for a clock is direct. The average of $N$ pulsars’ residuals has a background variance equal to one pulsar’s variance times

$$\frac{1}{N^2}\Big(N + \sum_{a\neq b}\Gamma_{ab}\Big),$$

where $\Gamma_{ab}$ is the Hellings–Downs correlation of each pair. If the sum over pairs vanishes, what is left is $1/N$, exactly the factor an independent noise gives. On an evenly covered sky the background averages down in a pulsar ensemble as though every pulsar had its own — not because it is independent, but because its positive and negative correlations cancel in the mean.

How evenly the sky is covered decides the floor #

Real arrays are not evenly spread. Millisecond pulsars are found where surveys look and where the Galaxy puts them, and the timing arrays’ lists are weighted toward the inner Galaxy, and each array toward the part of the sky its own telescopes can follow. In a concentrated array many pairs are close together, close pairs correlate positively, and the sum over pairs is no longer zero. The figure models the concentration with a von Mises–Fisher distribution — the spherical counterpart of a normal distribution about one direction — of strength $\kappa$.

With moderate concentration, $\kappa = 2$, half the pulsars lie within 49° of the centre. The background left in the average still falls with $N$, but more slowly, reaching 1.6 per cent at two hundred pulsars rather than the half a per cent an even sky gives. With strong concentration, $\kappa = 6$, half lie within 28°, and the fraction levels off near 12 per cent: past a few dozen pulsars, adding more barely moves it, because each new pulsar lands among the others and correlates with them positively. The floor of a pulsar time scale is set by the shape of the array on the sky more than by its size. A hundred pulsars with half of them inside a circle 28° in radius are worth, against the background, about as much as eight spread evenly over the whole sky.

This has a direct practical reading. The pulsars that are hardest to add to a timing array — faint ones in the parts of the sky an array’s telescopes see poorly, pulsars far from the Galactic plane — are the ones that do most for its clock, because they fill the parts of the sky where the background’s negative correlations live. A timing campaign planned to maximise the number of pulsars and one planned to make the best clock would choose different targets.

The equal-weight average drawn here is also the crudest estimator of a clock. The arrays do not average; they fit, with a model of every noise each pulsar carries and of how the background correlates them, and a clock signal fitted alongside a background whose angular pattern is known can be separated from it better than any simple average separates them. Where the sum over pairs does not vanish, the fit is what recovers the difference. The curves are therefore the floor an array would reach without modelling the background, which is the right starting point, because modelling it requires knowing its spectrum and its amplitude — and those were measured only recently, with uncertainties of tens of per cent.

A record that lengthens without getting steadier #

The question the earlier essay asked — how a pulsar time scale improves with age — can now be drawn for a whole ensemble. Take a hundred pulsars timed from 2005, and ask each year what the time scale can say about the longest interval its data can test. A third of the span is a fair choice, since three segments are the fewest that give a scatter. Early on the longest interval is short, the white noise dominates, and the stability improves steeply as the record lengthens. That is the improvement the hope was built on.

For an evenly spread array it stops in 2014, nine years in, when the longest testable interval reaches the 2.8 years at which the background takes over. The best stability ever shown stays at $2.8\times10^{-16}$ from then on. The stability over the longest interval, the one the time scale exists to provide, rises again: $6.6\times10^{-16}$ over the decade-long intervals available in 2045, $10^{-15}$ over the quarter-century intervals of 2085. The record keeps getting longer, and the intervals it can speak about keep getting longer with it, but past the turn every longer interval is a less stable one.

The best interval for the ensemble is the same 2.8 years as for one pulsar. That is not a coincidence. Averaging $N$ evenly spread pulsars divides both the white variance and the background variance by $N$, so their ratio, which is all the best interval depends on, is unchanged; only the floor falls, by $\sqrt N$. A concentrated array divides the background by less, so its best interval is shorter and its floor higher. With $\kappa = 6$ it turns in 2010, when the longest interval is a year and a half, and climbs to $2.3\times10^{-15}$ by 2045.

The pulsar time scale does not improve with age past the point where its record reaches the background. It improves with breadth — with pulsars added where the array is thin — and with knowledge, as the background’s spectrum is measured well enough to be fitted out rather than averaged.

What the comparison with atomic time becomes #

None of this makes the pulsars useless as an outside witness to the atomic time scale. It changes what they witness. The atomic scale’s own long-term stability is estimated near $3\times10^{-16}$ at years, with nothing steadier inside the laboratory to check it against. An evenly spread array of a hundred pulsars, averaged over its best interval, reaches the same level. Over intervals of a decade and more it does more than twice as badly, and over a quarter of a century more than three times.

So the pulsars are a check on the atomic scale at the intervals of a few years where both are near their best, and a weaker check at the decades where the atomic scale most needs one. The weakness is the price of the signal the arrays were built to find: the same correlated wander that is a discovery to the astronomers looking for black-hole binaries is a noise to the metrologists looking for a clock. It is the same data, read for two different purposes, and the reading that wants the background sees it as signal and the reading that wants the clock sees it as the floor.

There is a symmetry in that worth noticing. As a detector the array wants its pulsars to agree on the background, and as a clock it wants them to disagree — to sit at angles where their background correlations cancel. An array laid out over an even sky serves both. A clustered one serves neither well: its close pairs carry little angular information about the Hellings–Downs shape, and its average keeps the background it cannot cancel.

The comparison with the redefinition of the second points the same way. The optical clocks that will realise the second are steadier than caesium by two orders of magnitude at every interval a laboratory can reach, and an optical time scale built from them will want an outside check at years and decades. The pulsars can provide one at a few parts in $10^{16}$ and no better unless the background is modelled out. At the level of $10^{-18}$, which is where the optical scale will sit, no pulsar ensemble now imaginable is a witness at all.

What the curves assume #

The background is a single power law at the published amplitude. The arrays’ measurement fits a characteristic strain near $2.4\times10^{-15}$ with uncertainties of tens of per cent, and its spectral slope is consistent with $-13/3$ in the residual spectrum but not tightly fixed by it. A steeper or flatter slope would change both exponents in the scaling law, and a background that turns over at low frequencies — as one would if the binaries were driven by their environments rather than by radiation — would weaken the rise at the longest intervals.

The pulsar has white noise only, apart from the background. Real millisecond pulsars carry their own red spin noise, of different strengths and slopes, and noise from the interstellar plasma, which multi-frequency timing removes imperfectly. An intrinsic red noise steeper than the background would make the turn come earlier and set the floor itself; one shallower would hide under it. Unlike the background, a pulsar’s own spin noise is independent between pulsars, and averages down as $1/N$ on any sky.

The ensemble is an equal-weight average. A real time scale weights its pulsars by their quality and, increasingly, fits the background with its known angular correlation alongside the clock signal. Both do better than the curves drawn, and the second is the route by which a clustered array can recover what its geometry costs it.

The array shapes are a model. The von Mises–Fisher distributions stand in for the real concentration of timing-array pulsars toward the inner Galaxy, which is neither symmetric about one direction nor fixed as the arrays grow.

And $\sigma_z$ is one choice of measure. Its normalisation matches the Allan deviation for white frequency noise; for the steep noises here the two differ by a factor of order one. The exponents — the $6/13$ and the $4/13$ — do not depend on the choice, since any measure of this kind sees each power law as a power of the averaging time.

Still open: whether the background can be subtracted from a clock #

The floor drawn here is the floor of an array that does not model the background. The arrays now measure its amplitude and its correlation pattern directly, and as their data lengthen they will measure its spectrum well enough to fit it out of every pulsar’s residuals jointly, leaving a clock signal that is in principle free of it. How well that subtraction can be done is a question about estimation rather than about pulsars: a background known only statistically can be removed only as far as each realisation of it is pinned down by the array’s own correlations, and that depends on the number of pulsar pairs at angles where the Hellings–Downs curve is steep. An array of a few hundred pulsars spread over the whole sky might cut the background’s contribution to the clock tenfold, which would bring a pulsar time scale to a few parts in $10^{17}$ over decades. That is far short of what an optical clock reaches in a day. But it is a level held over intervals that no laboratory ensemble can check itself over at all, and it would be held by clocks that were turning before there was anyone to build a laboratory.