Astronomy & Celestial Mechanics
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Horizons — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · cosmology
  2. 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.

    part 2 · cosmology
  3. 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.

    part 3 · cosmology
  4. 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.

    part 4 · cosmology
  5. Two horizons, one formula, and a factor of two. Horizon temperature against horizon radius, on logarithmic axes, for the two kinds of horizon this collection has. The upper line is a de Sitter horizon at T = ħc/2πk_BR and the lower is a Schwarzschild horizon at T = ħc/4πk_BR — the same expression with the same constants, differing by exactly two, and both falling as one over the radius so that a bigger horizon is a colder one. The cosmological horizon today has a radius of 4,451 megaparsecs and a temperature of 2.65e-30 K, which is thirty orders of magnitude below the microwave background and will never be measured by anything. A solar-mass black hole sits at 6.2e-8 K, and a black hole as cold as the sky would weigh 2.32e+22 solar masses — of the order of the mass inside the observable universe, which is not a coincidence, since both numbers are c³/GH up to factors of order one. The factor of two between the two lines is the one place the analogy is not exact, and it is not a convention: it comes from the periodicity of the Euclidean time coordinate, which is 8πGM/c³ for a black hole and 2π/H for de Sitter space. The entropies, by contrast, agree exactly.

    Two horizons that differ only in who is inside

    A black hole's horizon and the cosmological one share an entropy formula exactly and differ in temperature by precisely a factor of two. The factor of two is the whole of the difference, and what it encodes is which side of the surface the observer stands on.

    part 5 · cosmology

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