Astronomy & Celestial Mechanics
the sky, computed
Concept

Gravitational-wave background — where it appears

The superposition of gravitational waves from sources too numerous to resolve individually. The nanohertz band's background, detected in 2023 from correlated wander in pulsar arrival times, is made of supermassive black hole binaries and is a confusion limit rather than a wall.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Nothing visible in any pulsar, and a quadrupole in the angle between them. Above: 4 millisecond pulsars' timing residuals over 15 years, at the few hundred nanoseconds a good one reaches. Each wanders, and none of them shows anything a reader could call a signal; a gravitational-wave background of amplitude 2.4·10⁻¹⁵ at one cycle per year contributes a common part to all of them that is smaller than each pulsar's own red noise. Below: the correlation between pairs, against the angle on the sky between them. 2211 pairs out of 67 pulsars, binned into 15 angles, against three curves with no free parameters between them. A quadrupolar background gives the Hellings–Downs shape — positive for nearby pulsars, negative near 83°, and back up to exactly half its zero-separation value at 180° because a background looks the same in opposite directions. An error in the observatory clock would give a flat line, because it shifts every pulsar identically. An error in the solar-system ephemeris would give a cosine, because it moves the barycentre in one direction. The drawn points prefer the quadrupole over the flat line by Δχ² = 358. That is the detection: not a waveform, not an event, not a moment — a shape in an angle, accumulated over fifteen years, on data taken for another purpose entirely.

A detector the size of the galaxy

At a nanohertz no instrument can be built, so the clocks already in the sky are used instead. The signal is in no single pulsar's data — it is in the correlation between pairs as a function of the angle between them, and that curve has no free parameters at all.

gravitation · Gravitational waves Three messengers, and the three walls they end on. The redshift of the last opaque surface, for the three things that cross the universe, on one logarithmic axis spanning thirty-one decades. Olbers' argument compares how far a sight line runs before it ends on something against how far anything has had time to come — and for starlight the first is 1.7·10¹⁸ Mpc against a horizon of 1.4·10⁴ Mpc, which is why the optical sky is dark. That comparison is not what decides the other two. A relic neutrino's mean free path against ordinary matter works out at 7.7·10³⁷ Mpc — 4·10¹⁹ times starlight's, so on the mean-free-path argument alone the neutrino sky should be darker still. It is not, because the quantity that terminates a sight line is the WALL, and the walls are at z = 1,090, z ≈ 6×10⁹ and nowhere at all. Every neutrino sight line ends on a surface from one second after the beginning; every gravitational-wave sight line runs to the beginning, or to a binary. Both of those skies are saturated — which is what Olbers' argument predicted, and what the optical sky refuses to do. The three are also wildly unequal in brightness: the microwave sky is 996 nW m⁻² sr⁻¹, and the gravitational-wave background at Ω = 10⁻⁹ is 0.018 — a saturated sky five decades fainter than a dark one.

Two skies where the paradox comes out right

Olbers argued that every sight line should end on a source and the sky should blaze. In neutrinos and in gravitational waves it does — the walls are at one second and at no time at all — and both of those skies have now been detected.

cosmology · Olbers's paradox The stability of a pulsar clock against atomic clocks, over averaging times from a day to thirty years. Fractional frequency instability, on a logarithmic axis, against averaging time from a day to thirty years. A single millisecond pulsar timed to 300 nanoseconds every 14 days improves as τ^(−3/2) while its white timing noise averages down, and flattens where its slow spin noise, drawn at 10⁻¹⁵ at ten years, takes over. An ensemble of 20 such pulsars is 4.5 times better if their noises are independent. The atomic curves are representative: a hydrogen maser, excellent at hours and drifting at years, and a steered atomic time scale limited near 3·10⁻¹⁶ at long times. In this representative comparison the pulsar ensemble becomes the steadier of the two beyond about 3 years of averaging. At a day a pulsar is a billion times worse than an atomic clock; over decades it is competitive, because nothing inside a neutron star changes on the timescales on which laboratory clocks are rebuilt.

A clock made of neutron stars

The best atomic clocks are rebuilt every decade and averaged every month. A millisecond pulsar has kept its own time for a billion years. Averaged over years, an ensemble of pulsars is as steady as the world's atomic time scale — steady enough to find its errors — but it can never say how fast that time scale runs.

sky · Timescales One pulsar's stability, limited by the background every pulsar shares. The cubic-fit stability of a single millisecond pulsar, on logarithmic axes, against averaging time from 0.3 to 40 years. White timing noise of 100 ns every 14 days averages down as τ^(−3/2) (falling dashed line). The gravitational-wave background at the amplitude the timing arrays reported in 2023, A = 2.4·10⁻¹⁵, raises the curve as τ^(2/3) (rising dashed line). Their sum (solid) is lowest at 2.8 years, at 2.8·10⁻¹⁵; averaging longer than that makes this pulsar a worse clock, not a better one, because the background grows faster in the residuals than the white noise shrinks. At ten years the background alone is 5.5·10⁻¹⁵.

A clock whose best interval is three years

A pulsar time scale was expected to improve for as long as anyone kept timing it. The background the timing arrays found in 2023 says otherwise — a single pulsar is steadiest over about three years and worse over every longer interval, and how far an array of them can push that floor down depends less on how many pulsars it holds than on how evenly they cover the sky.

sky · Timescales

Named alongside it

The objects these essays reach for when they reach for this one.

Millisecond pulsarAllan deviationClock ensembleHellings downs curvePulsar time scaleA pulsar timing arrayRed noiseCharacteristic strainChirp massConfusion limitCorrelationCosmic neutrino background

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