Astronomy & Celestial Mechanics
the sky, computed
Concept

Particle horizon — where it appears

How far light has travelled since the beginning, in comoving distance, which comes to three times the age multiplied by the speed of light. It is larger than the naive product of the speed of light and the age because the space the light crossed has expanded since, and it is what makes the microwave background's uniformity a problem.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Every shell contributes the same. Concentric shells of equal thickness around an observer, with stars scattered uniformly in volume. The number of stars in a shell grows as its radius squared and the flux from each falls as the radius squared, so the two cancel exactly and every shell delivers the same total light — the drawn counts are 6, 15, 31, 55, 88, in proportion to r³ − r₀³, and the drawn sizes fall as 1/r. That cancellation is the paradox, and it is why no amount of dust helps: dust absorbs the light and then re-radiates it, and in a universe old enough for the sum to converge it would come to the same temperature as the stars. The sum diverges linearly with radius, so something has to stop it — and the only two candidates are that the shells eventually overlap, or that there are no shells beyond a certain distance because there has not been time for their light to arrive.

Why the sky is dark

In an infinite universe of stars every sight line ends on a stellar surface, so the whole sky should be as bright as the Sun's disc. It is not, and the usual answer — that the expansion redshifts the light away — accounts for a factor of six out of a hundred trillion.

cosmology · Olbers's paradox Three horizons, and a light cone that bulges. Cosmic time upwards, comoving distance sideways, for the Planck 2018 cosmology. Galaxies sit still in these coordinates, so their worldlines are the vertical grey lines: comoving distance is defined to take the expansion out. The solid inner curve is the past light cone — the set of events whose light reaches here and now — and it reaches out to the particle horizon at 46.1 billion light years, which is the diagram's central number and the one that sounds impossible. The universe is 13.80 billion years old and light has travelled for 13.80 billion years, and yet the material that emitted the oldest light is now 46 billion light years away. Nothing has outrun light: the comoving distance covered is ∫c dt/a, and dividing by a scale factor that was small early on makes the integral three times ct. The dashed curve is the Hubble sphere, where the recession speed equals c, at 14.5 Gly today — and the light cone lies outside it for most of its length, which is exactly why a galaxy can be observed while receding faster than light. The outer dot-dash curve is the event horizon, at 16.7 Gly: a signal sent from here today never reaches anything beyond it.

A horizon three times larger than the age allows

The universe is 13.8 billion years old and light travels one light year a year, so the observable universe should be 13.8 billion light years across. It is 46.1, nothing has outrun light, and the discrepancy is a piece of arithmetic rather than a paradox.

cosmology · Horizons Beyond redshift 1.87, what a galaxy does today will never be seen. For a galaxy at each redshift, the cosmic time of the last event on it that will ever be visible from here — not the last that has arrived, but the last that ever arrives, integrated to infinite future time. The horizontal line is the present, 13.8 billion years. A galaxy below redshift 1.87 has its curve above that line: its entire future will be seen from here, arriving ever more slowly and ever more redshifted, so it never quite disappears. A galaxy above redshift 1.87 has its curve below the line, and that is the whole content of an event horizon: what such a galaxy is doing today will never be seen, ever, by anybody here. Only a finite slice of its history is coming, and when the last of that light arrives the object stops changing. The redshift at which the curve crosses is 1.87, the comoving distance there is 16.7 billion light years against a particle horizon of 46.1, and the ratio of the volumes says that 95 per cent of the galaxies now observable are already beyond reach. Superluminal recession is not what does this. Everything past about redshift 1.5 has always been receding faster than light and is seen perfectly well, because the Hubble sphere grows to meet the photon; what closes the horizon is that with a cosmological constant the comoving Hubble radius stops growing and begins to shrink, so a photon that has not already been overtaken never will be.

The galaxies that are already out of reach

With a cosmological constant the comoving distance a photon can ever cover converges, so there is a redshift beyond which light leaving today never arrives. It is 1.87, and about ninety-five per cent of the galaxies now visible are past it — which superluminal recession has nothing to do with.

cosmology · Horizons The same history, on the clock that straightens light. Conformal time upward against comoving distance sideways, for the Planck 2018 cosmology. Conformal time is ∫dt/a, which is exactly the comoving distance light covers, so on these axes every photon moves at forty-five degrees — at every epoch, whatever the expansion is doing. That single property turns every curved thing in the ordinary space-time diagram into a straight one. The universe began at η = 0 and is now at η = 46.1 Gly of conformal time; it will ever accumulate only 62.8, because the integral ∫dt/a converges once Λ dominates, and that finite ceiling is the whole reason an event horizon exists. The particle horizon is the 45° line from the origin and the event horizon is the 45° line back from the ceiling, so the two horizons that were curves are now the two edges of one light cone drawn twice. The shaded wedges are the past light cones of two points on the last scattering surface, at conformal time 0.914 Gly and comoving distance 45.2 Gly from here. They do not overlap. Two points on that surface separated by more than 2η_rec were never in causal contact, which subtends 2.31° on the sky, and the microwave sky therefore contains about 9,805 patches that have no common past and the same temperature to one part in a hundred thousand. That is the horizon problem, and in these coordinates it is a statement about whether two triangles intersect.

The clock on which light travels in straight lines

Cosmic time makes light cones bulge and horizons curve. There is another time coordinate on which a photon's worldline is a forty-five degree line at every epoch, and on it the horizon problem stops being a piece of arithmetic and becomes a question about whether two triangles overlap.

cosmology · Horizons Whether an event horizon exists at all, against one number. The comoving event horizon today — the distance a signal sent now will ever cover — against the equation of state of the dark energy, for a flat universe with the measured matter density. The curve runs away at w = −1/3 and does not exist above it: that is where the expansion stops accelerating, and in a universe that does not accelerate the integral ∫da/a²E diverges and every galaxy is eventually reachable, however far away. Below −1/3 the horizon is finite and shrinks as w falls, because a more negative equation of state makes the dark energy density grow with time rather than stay constant. At the cosmological constant's w = −1 the horizon is 16.7 billion light years against a particle horizon of 46.1, so 4.7 per cent of the volume now observable is still reachable. The band is the measured −1.03 ± 0.03. What the figure is for is the asymmetry in what the measurement still allows: two sigma toward zero puts the horizon at 17.5 Gly and two sigma the other way at 14.7, and the shape of the curve means that the closer the true value sits to −1/3 the more violently the answer moves. The reachable fraction is not a robust number in the way the particle horizon is.

Whether there is a horizon at all

An event horizon exists precisely when the expansion accelerates, and its size is not a smooth function of how much. The integral that defines it runs away as the equation of state approaches minus a third, so two sigma either way on a measured number are two very different futures.

cosmology · Horizons 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

Named alongside it

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

Comoving distanceConformal timeEvent horizonCausal contactDe sitter spaceHubble sphereLight coneObservable universeCosmological constantLast scatteringMean free pathOlbers's paradox

All concepts