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Cosmology

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.

Assumes Horizons, Expansion and Microwave background.

18 min read 8 figures Scales that defeat intuitionMeasuring the unreachable

The first rung of this anchor drew the standard picture — cosmic time upward, comoving distance sideways — and everything in it was a curve. The past light cone bulged outward. The Hubble sphere and the particle horizon both bent. The second rung added a third curve for the event horizon and found that it shrinks.

Those curves are not features of the universe. They are features of the clock, and there is another clock. On it every one of them is a straight line, and the arguments they were drawn to support become arguments about geometry rather than about integrals.

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.
Fig. 1 Conformal time upward against comoving distance sideways, for the same 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. The particle horizon is the 45° line from the origin and the event horizon is the 45° line back from the ceiling: the two horizons that were curves are the two edges of one light cone drawn twice. The universe has 46.1 gigalight-years of conformal time behind it and will ever accumulate only 62.8, and the finiteness of that ceiling is the whole reason an event horizon exists.

Nothing in that figure is a new calculation. It is the two integrals the earlier rungs already performed, replotted against a different vertical coordinate. What changed is that every statement about causality became a statement about a straight line.

What conformal time is

Cosmic time t is what a clock comoving with the expansion reads, and it is the natural coordinate for almost everything: the age of the universe, the duration of nucleosynthesis, the lifetime of a star. It is a bad coordinate for exactly one purpose, which is drawing causal relationships, and the reason is that the speed of light is constant in proper distance while the useful spatial coordinate is comoving.

A photon covers c dt of proper distance in cosmic time dt, and that is c dt/a of comoving distance. As the universe grows, the same interval of cosmic time buys less comoving ground — which is why a light cone drawn on comoving axes against cosmic time bends inward at late times, and why every horizon derived from it is a curve.

Conformal time removes exactly that. Define

η=∫0tc dt′a(t′)\eta = \int_0^t \frac{c\,dt'}{a(t')}

and the photon’s comoving position advances by dη in conformal time dη, at every epoch and in every cosmology, because dividing by the scale factor is precisely the conversion from proper to comoving. So a light ray is a 45° line, by construction and without approximation.

Two things follow immediately and neither is obvious in cosmic time.

The first is that the particle horizon is the conformal time. The comoving distance light has covered since the beginning is η, so the horizon that bounds what can be seen is a 45° line through the origin. That is why the observable universe is three times ct rather than ct: it is not a factor multiplying a distance, it is a different integral, and on these axes it is a straight edge.

The second is that the total conformal time is finite. The integral ∫dt/a converges as t → ∞ once Λ dominates, because a grows exponentially and the integrand dies. The universe has a last conformal moment. Everything after it is compressed into a diagram of finite height, which is what a conformal diagram is for and why the technique was invented for black-hole spacetimes before anybody drew a cosmology on one.

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.
Fig. 2 The same cosmology in cosmic time, for comparison — the picture the first two rungs of this ladder are drawn on. Everything that is straight in the figure above is bent here: the past light cone bulges out to the particle horizon at 46.1 Gly, the Hubble sphere curves through 14.5, and the event horizon at 16.7 curves the other way. The information content is identical, and the shape of an argument made on it is not. Nothing about the causal structure is easier to see here, and one thing — that the event horizon is a light cone — is invisible.

One more consequence is worth extracting before the horizons, because it is the reason the coordinate is worth using at all rather than merely correct. On these axes, causal contact is a ruler measurement. Two events are causally connected if and only if the comoving distance between them is less than the conformal time between them, which on a diagram whose two axes are both in gigalight-years is the question of whether a segment is steeper than 45 degrees. Nothing has to be integrated, nothing has to be looked up, and the answer does not depend on which epoch the two events are in.

The event horizon as the other half of a light cone

The event horizon on the ordinary diagram is a curve that shrinks with time, and its shrinking is what puts galaxies permanently out of reach. Its shape is not memorable and its behaviour has to be argued for.

In conformal coordinates it is a 45° line running down and outward from the ceiling. That is not a coincidence of drawing; it is the definition read differently. The comoving distance a signal sent at conformal time η will ever cover is η_max − η, because a photon covers one unit of comoving distance per unit of conformal time and there are η_max − η units left. The event horizon is therefore the future light cone of the here-and-now, extended to the end of conformal time.

Two lines, both at 45°, one from the beginning and one to the end. That is the entire causal structure of a Λ-dominated universe, and it fits in a sentence only because of the coordinate.

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.
Fig. 3 The consequence, in the coordinates of the second rung: for a galaxy at each redshift, the last event on it that will ever be visible from here. Beyond redshift 1.87 the curve is below the present, which is the statement that what such a galaxy is doing today will never be seen. That number is a crossing on a curve here; on the conformal diagram it is the redshift at which a galaxy’s worldline leaves the 45° line down from the ceiling, and the two are the same statement about the same integral.

Three times ct, read off a straight line

The first rung of this ladder spent its length explaining why the observable universe is 46 billion light years across when the universe is 13.8 billion years old, and the explanation was an integral: the comoving distance is ∫c dt/a, and dividing by a scale factor that was small early on multiplies the answer.

In conformal coordinates the same fact is a ratio of two axis readings. Conformal time now is 46.1 gigalight-years and cosmic time now is 13.80 billion years, and the factor of 3.3 between those two numbers is the answer. There is nothing further to derive. The particle horizon is η, and η exceeds ct because the early universe contributed conformal time out of proportion to the cosmic time it occupied — which is the same statement as before, expressed as a length on an axis rather than as an integral to be performed.

The distribution of that contribution is worth a number. Of the 46.1 gigalight-years of conformal time behind the present, 0.914 was accumulated before recombination, which is two per cent — accumulated in 380,000 years, which is 0.003 per cent of the elapsed cosmic time. The radiation era is therefore about seven hundred times over-represented on the conformal axis relative to the cosmic one, and that over-representation is the whole reason a diagram of the observable universe can show recombination at all.

It also explains an asymmetry in the diagram that is otherwise puzzling. The past occupies 46.1 units of the vertical axis and the entire infinite future occupies 16.7, so on this clock the universe is already nearly three quarters over. That is a statement about a coordinate rather than about the universe, and it is a statement with real content: the amount of comoving distance any signal will ever cover from here is less than a third of what light has already covered getting here.

The horizon problem as a question about triangles

The reason this coordinate is worth an essay rather than a paragraph is what it does to the argument inflation was invented to answer.

Two points on the last scattering surface, on opposite sides of the sky, are at the same temperature to one part in a hundred thousand. Each has a past light cone. In cosmic time, establishing that those cones do not intersect requires an integral: how far could light have travelled by recombination, converted to comoving distance, compared with the comoving separation of the two points. It is a calculation with an answer, and the answer is that they do not intersect.

In conformal time it is a drawing. Each point’s past light cone is a 45° wedge of height η_rec, so its comoving half-width is η_rec. Two points separated by more than 2η_rec have disjoint wedges. Full stop.

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.
Fig. 4 The early universe at a scale that shows it, with the conformal axis cut at 2.4 Gly instead of 62.8 and the comoving axis at 8 instead of 52. Last scattering is the horizontal line at η = 0.914. The two shaded wedges are the past light cones of two points on it, and they meet at the origin only because the separation drawn is smaller than 2η_rec. Move them further apart and they part company: the wedge is a triangle with a fixed apex angle, so the criterion is a comparison of two lengths and there is nothing else in it.

The angle that separation subtends on the sky is about 2.3 degrees, so the microwave sky holds roughly ten thousand patches with no common past. Every one of them is at the same temperature. That count is what the horizon problem is, and it is worth having as a number rather than as an adjective, because “the sky is uniform” is compatible with a great deal and “ten thousand causally independent regions agree to one part in a hundred thousand” is not.

The number is also the reason a coincidence is not an available answer. Ten thousand independent draws from any distribution wide enough to be interesting would show a spread of order the distribution’s own width divided by a hundred, and the observed spread is a hundred-thousandth of the mean. Either the patches were in contact, or the initial conditions were laid down by something that acted across all of them, or the uniformity is a brute fact with no explanation. Inflation is a proposal about the first of those. It is worth noticing that the third is not incoherent — a theory of initial conditions could simply assert uniformity — and that the reason it is unpopular is aesthetic rather than empirical.

There is a corollary about the size of the patches that is easy to miss. The wedge’s half-width is the particle horizon at recombination, not the sound horizon, and the two differ by about a factor of three because sound in the photon–baryon fluid travels at c/√3. The acoustic scale that sets the first peak in the microwave power spectrum is about one degree; the causal patch is about two and a third. Both are drawn on the same diagram and they are different lengths, and a great deal of confusion in popular accounts comes from treating the acoustic scale as the horizon.

What inflation is, in this picture

The conformal diagram also makes the standard resolution look like what it is: an assertion that the vertical axis does not start where the drawing starts.

The particle horizon is a 45° line through η = 0, and η = 0 is the beginning only if the integral ∫dt/a converges at the lower limit. It does for radiation and for matter. It does not for an epoch of accelerated expansion — during inflation a is exponential in t and ∫dt/a diverges at early times — so the conformal time before the end of inflation is unbounded below, and the origin of the diagram is not the origin of the light cones.

Extend the vertical axis to negative conformal time and the wedges get taller. Make it tall enough and every wedge overlaps every other. That is the whole geometric content of inflation as a solution to the horizon problem: the diagram was cut off too early.

One length, leaving and returning. The comoving Hubble radius c/aH against the scale factor, both logarithmic. Everything to the right of the kink is exact and is the same cosmology as every other figure here: after inflation the comoving Hubble radius grows, as a in the radiation era and as √a in the matter era, then turns over once Λ takes hold. Everything to the left is a schematic — the energy scale of inflation is unmeasured, so neither the height of the plateau nor the 62 e-folds drawn is a number anybody has — but the shape is not negotiable: in any accelerated expansion H is nearly constant, so c/aH falls as 1/a. The consequence is the mechanism. The solid horizontal line is a fixed comoving length of 209 Mpc: it starts inside the Hubble radius, is carried outside during inflation, sits frozen while nothing can act across it, and re-enters at z = 1090, which is last scattering — it is the scale the first acoustic peak is made of. The dashed line above it is the comoving size of the whole observable universe, 14148 Mpc, and the figure's quiet second finding is that it is still outside the Hubble radius: the very largest angular scales in the microwave background have never re-entered, which is why they show the primordial spectrum with no acoustic processing on it at all. One crossing does two jobs: it makes the sky uniform, because everything now visible was once in causal contact, and it makes it not quite uniform, because a quantum fluctuation stretched beyond the horizon has nothing left that can smooth it out.
Fig. 5 The same statement in the variable the mechanism is usually taught in: the comoving Hubble radius against the scale factor, falling during inflation and rising afterwards. The horizontal line is a fixed comoving length that starts inside the Hubble radius, is carried outside during inflation, and re-enters at last scattering. The comoving Hubble radius is not the particle horizon — it is the distance over which causal processes are efficient now rather than the distance light has ever covered — and conflating the two is the standard confusion in this subject. What inflation does to each is different: it makes the second large and the first small.

That distinction deserves stating plainly, because the conformal diagram makes one of the two obvious and hides the other. The particle horizon is cumulative and can only grow; the comoving Hubble radius is instantaneous and can shrink. During inflation the second shrinks by sixty e-folds while the first grows enormously, and it is the ratio of the two that decides whether a region that was in causal contact is currently able to act on itself. On the conformal diagram the particle horizon is the 45° line and the Hubble radius is not drawn at all.

The beginning is a surface, not a point

The conformal diagram corrects one thing that almost every popular account gets wrong, and it corrects it by making the correction visible rather than by asserting it.

On these axes the beginning of the universe is the horizontal line η = 0. It runs across the whole width of the diagram, and the diagram is infinite in width — the comoving axis is cut at 52 gigalight-years because that is where the drawing stops, not because the universe does. Every galaxy’s worldline is a vertical line, and every one of them reaches down to η = 0. So the initial singularity is a spacelike surface, present everywhere at once, and not an event that occurred somewhere.

That is worth saying flatly because the other picture is so persistent. There was no point from which the expansion proceeded. There is no location where it happened, no direction to look toward it, and no sense in which the material now in a distant galaxy started closer to some centre than the material here did. What the diagram shows is that every worldline shares a lower boundary, which is a statement about a common past and not about a common place.

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.
Fig. 6 The same diagram with the conformal axis cut at 20 gigalight-years and the comoving axis at 24 — an intermediate zoom between the early universe and the whole history. The vertical grey lines are galaxy worldlines and every one of them reaches the bottom edge, which is the beginning. It is a surface across the diagram rather than a point in it, and the diagram continues indefinitely to left and right. Last scattering is the horizontal line near the bottom; the light cone converging on the present is the pair of 45° lines above it.

The technique itself came from somewhere else. Penrose introduced conformal diagrams in 1963 to make the causal structure of black-hole spacetimes finite and drawable, which requires compactifying the coordinates as well as rescaling them, and cosmology adopted them afterwards. What survives the adoption is the property that matters: a conformal rescaling of the metric leaves light cones exactly where they are, so two cosmologies with utterly different expansion histories can be drawn on the same axes and their causal structures compared directly. That is not true of any diagram in cosmic time.

The scale a degree corresponds to, and the one it does not

The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has.
Fig. 7 The angular power spectrum of the microwave background, whose first peak sits near one degree. That degree is the sound horizon at recombination — the distance a pressure wave in the photon–baryon fluid could cross, at c/√3 — projected onto the sky through the angular-diameter distance to last scattering. The causal patch of the previous figure is the particle horizon at the same epoch and is about two and a third degrees, larger by roughly the same factor of √3 times a correction for the fluid’s early radiation domination. The two lengths are drawn on the same diagram, differ by a factor of a few, and are routinely conflated.

The conflation matters because the two lengths do different jobs. The sound horizon is a ruler: it has a computed physical size, it is seen at a measured angle, and the ratio is a distance — which is how the acoustic peaks measure the geometry and, through the same ruler at low redshift, how the baryon oscillations measure the expansion history. The particle horizon is a causal bound: it says what could have been in contact and computes no distance. One is a measurement and the other is an argument, and the conformal diagram draws the second and not the first.

A caution about the picture

Two things about conformal diagrams are worth flagging, because both are places where a reader can take more from the drawing than is in it.

The first is that the vertical axis is not a duration. Conformal time is measured here in gigalight-years — it is a comoving length divided by c — and the interval from the beginning to last scattering is 0.914 of those against 46.1 to the present. Read as elapsed time that is nonsense: recombination is at 380,000 years and the present at 13.8 billion, a ratio of 36,000 rather than 50. The stretching is the point of the coordinate and it is why the early universe is visible at all on a diagram that also holds the present.

The second is that the finite ceiling is a statement about Λ and not about the universe ending. Nothing happens at η = 62.8. Cosmic time runs on forever; conformal time approaches a limit because the integrand vanishes, in the same way that an infinite series can converge. What is finite is the amount of comoving distance any signal can ever cover, and that is exactly the event horizon.

A third caution belongs with those two and is about what the diagram omits rather than what it distorts. Every figure here is drawn for a homogeneous, isotropic universe, so a galaxy’s worldline is a vertical line and stays one. Real galaxies have peculiar velocities of a few hundred kilometres a second, and bound structures — the Local Group, a cluster — do not expand at all, so their members’ worldlines converge rather than staying parallel. On the scale of the whole diagram that is invisible: a thousand kilometres a second for the age of the universe moves a galaxy by about fourteen million light years, which is a third of a pixel at 52 gigalight-years across. It is nonetheless the reason nothing in these diagrams says anything about what happens to a galaxy, a star or a person as the expansion accelerates, and the popular inference that everything is eventually torn apart does not follow from any of them. That inference requires w below −1, which is the next rung.

The same diagram in proper distance. The same worldlines and light cone as the comoving diagram, replotted in proper distance — the separation that would be measured by a chain of rulers laid end to end at that instant. The galaxy worldlines splay apart because that is what expansion is, and the past light cone becomes a teardrop: it widens for the first 4.0 billion years and then narrows to zero at the present. The narrowing is the part worth stopping on. Light approaching us from far enough away spends its early life moving away in proper distance, because the space it is crossing expands faster than it can cross it, and only later — once it has crossed inside the Hubble sphere — does it start making progress. Every photon from a galaxy beyond about z = 1.6 did that.
Fig. 8 And a third view, for contrast: the same worldlines and light cone in proper distance, which is what a chain of rulers laid end to end would measure at each instant. The galaxy worldlines splay apart, and the past light cone becomes a teardrop — it widens and then narrows to zero at the present, because light approaching from far enough away spends its early life moving away in proper distance. That behaviour is genuinely physical and it is invisible on both of the other two diagrams. Three coordinate choices, one history, and each hides something the others show.

The habit

Choosing coordinates so that the thing being argued about becomes straight is not a cosmological technique. It is the general move, and this collection uses it repeatedly without always naming it.

The harmonic law becomes a straight line on logarithmic axes, and its slope is then a number to read rather than an exponent to fit. A curve of growth is drawn in the logarithm of the column density because the three regimes are three straight segments there and three unrelated shapes anywhere else. The Hertzsprung–Russell diagram runs backwards in temperature for a historical reason and stayed that way because the main sequence is a line on it. In each case a transformation removes a distortion that was in the description rather than in the thing.

What the three share is that the transformation is chosen after knowing what the argument is about. A logarithmic axis is right for the harmonic law because the claim is about an exponent; it would be wrong for a figure about the residual after the exponent is removed. Conformal time is right for a claim about causality and wrong for a claim about ages, which is why this collection’s essay on the age of the universe is drawn in cosmic time and this one is not. Choosing the coordinate is part of making the argument rather than a presentational step after it.

The cosmological case is the strongest of them because the transformation is exact and universal: light is at 45° in conformal coordinates in any Friedmann universe, whatever a(t) does, so a conformal diagram of a completely different cosmology can be compared with this one directly. That is why the technique survived from Penrose’s black-hole diagrams into cosmology, and why a conformal diagram is the first thing drawn when somebody proposes a bouncing universe, a cyclic one, or an eternally inflating one — the causal structure is the part of the proposal that has to make sense first.

Where this ladder goes next

Everything above takes the ceiling for granted. Conformal time has a finite total because the expansion accelerates, and it accelerates because there is a component with an equation of state near −1. That number is measured, and it is measured to a few per cent rather than exactly.

The next rung asks what the measurement still allows. Whether an event horizon exists at all is not a smooth function of the acceleration: the integral that defines it converges for w below −1/3 and diverges above, so the horizon’s size runs away as the equation of state approaches that value from below and shrinks toward zero as it goes the other way. Two sigma in one direction and two in the other are two quite different futures, and the reachable fraction of the sky is not a robust number in the way the particle horizon is.

Beyond it: the de Sitter horizon’s temperature, and the exact sense in which it is the same object as a black hole’s horizon with the observer moved to the other side.

Where this goes next

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. 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. One length, leaving and returning. The comoving Hubble radius c/aH against the scale factor, both logarithmic. Everything to the right of the kink is exact and is the same cosmology as every other figure here: after inflation the comoving Hubble radius grows, as a in the radiation era and as √a in the matter era, then turns over once Λ takes hold. Everything to the left is a schematic — the energy scale of inflation is unmeasured, so neither the height of the plateau nor the 62 e-folds drawn is a number anybody has — but the shape is not negotiable: in any accelerated expansion H is nearly constant, so c/aH falls as 1/a. The consequence is the mechanism. The solid horizontal line is a fixed comoving length of 209 Mpc: it starts inside the Hubble radius, is carried outside during inflation, sits frozen while nothing can act across it, and re-enters at z = 1090, which is last scattering — it is the scale the first acoustic peak is made of. The dashed line above it is the comoving size of the whole observable universe, 14148 Mpc, and the figure's quiet second finding is that it is still outside the Hubble radius: the very largest angular scales in the microwave background have never re-entered, which is why they show the primordial spectrum with no acoustic processing on it at all. One crossing does two jobs: it makes the sky uniform, because everything now visible was once in causal contact, and it makes it not quite uniform, because a quantum fluctuation stretched beyond the horizon has nothing left that can smooth it out. Two coincidences with one mechanism Opposite sides of the microwave sky were never in causal contact and have the same temperature to one part in a hundred thousand. The total density sits on the knife edge of flatness, which is an unstable equilibrium. Two fine-tunings of quite different kinds, and one epoch of accelerated expansion removes both.

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Causal contactComoving distanceConformal timeEvent horizonHorizon problemHubble sphereInflationLast scatteringLight coneObservable universeParticle horizonScale factor