Astronomy & Celestial Mechanics
the sky, computed
Cosmology

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.

Assumes Expansion and Hubble constant.

18 min read 9 figures Scales that defeat intuitionMeasuring the unreachable

The number quoted for the size of the observable universe is 93 billion light years across, or 46 billion in radius. The number quoted for its age is 13.8 billion years. Both are correct, and put side by side they look like a contradiction: light has been travelling for 13.8 billion years, so how is anything 46 billion light years away.

Nothing has travelled faster than light. The resolution is that the two numbers are answers to different questions, and the arithmetic that connects them is one integral.

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. 1 Comoving distance sideways, cosmic time upwards, for the Planck 2018 cosmology. Galaxies sit still in these coordinates, so their worldlines are the vertical grey lines — comoving distance is defined to divide the expansion out. The solid inner curve is the past light cone, the set of events whose light arrives here and now, and it reaches the particle horizon at 46.1 billion light years. The dashed curve is the Hubble sphere at 14.5, where the recession rate equals cc; the light cone lies outside it for most of its length. The outer dot-dashed curve is the event horizon at 16.7, beyond which a signal sent from here today never arrives.

The integral

The comoving distance light has covered since the beginning is not ct0ct_0. It is

χph=∫0t0c dta(t),\chi_{\rm ph} = \int_0^{t_0} \frac{c\,dt}{a(t)},

and the division by aa is the whole of it. A light ray that crossed a certain physical distance early on, when aa was small, crossed a much larger comoving distance — because the space it traversed has been stretching ever since, and comoving distance measures separations as they are now.

Put concretely: light emitted at z=1100z = 1100 crossed about 42 million light years of physical distance in its first million years. That patch of space is now a thousand times bigger. The integral adds up all such contributions, and for the measured cosmology it comes to 3.2 times c/H0c/H_0, or 46.1 billion light years.

Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround.
Fig. 2 Why the factor is three and not something else. The scale factor against time, for four cosmologies. The integral ∫c dt/a\int c\,dt/a is dominated by the early part of the history, where aa is smallest, and how much it is dominated depends on how fast aa rose there. A matter-only universe with a∝t2/3a \propto t^{2/3} gives exactly 3ct03ct_0; the measured cosmology, which decelerates then accelerates, gives 3.35 times its own ct0ct_0. An empty universe with a∝ta \propto t gives a logarithmically divergent integral, which is a warning that this quantity is more sensitive to the early history than to anything else.

Three surfaces, and they are not the same

The single largest source of confusion here is that three different lengths all come out near c/H0c/H_0 and are all called horizons in loose usage. They answer three different questions.

The Hubble sphere is where the recession rate equals cc: comoving radius c/aHc/aH, today 14.5 billion light years. It is not a horizon at all. It is a surface that moves, and things cross it in both directions.

The particle horizon is how far light has come: 46.1 billion light years, and growing. It bounds what can be seen. Every galaxy inside it is in principle observable now; nothing outside it has ever been.

The event horizon is how far a signal sent now will ever get: 16.7 billion light years, and shrinking in comoving terms. It bounds what can be influenced. It exists only because the expansion is accelerating; in a matter-only universe the corresponding integral diverges and everything is eventually reachable.

The three are easy to keep apart by asking what each one is an integral of, and over what range. The particle horizon integrates c dt/ac\,dt/a from the beginning to now; the event horizon integrates the same thing from now to the end of time; the Hubble sphere is not an integral at all but a local ratio, c/aHc/aH, evaluated at one instant. Two are cumulative and one is instantaneous, which is exactly why the Hubble sphere is the one that things cross in both directions and the only one that is not a horizon.

The light cone bulges outside the Hubble sphere

The most instructive feature of the hero figure is that the past light cone is not inside the Hubble sphere. It starts inside it near the present, swells outside it, and comes back in at the far end.

That has a consequence worth stating carefully. Every galaxy at a redshift above about 1.5 was receding faster than light when it emitted the light now arriving, and a great many of them still are. Nothing prevents this. A photon aimed this way always makes progress in comoving coordinates — its comoving speed is c/ac/a, which is positive — but whether it makes progress in proper distance depends on whether the recession of the space it is currently in exceeds cc. Outside the Hubble sphere it does not, and the photon is carried backwards even while pointed forwards.

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. 3 The same worldlines and light cone in proper distance — the separation a chain of rulers laid end to end would measure at each instant. The galaxy worldlines splay apart, because that is what expansion is. The light cone becomes a teardrop: it widens for the first few billion years and then narrows to zero at the present. The widening is the part worth stopping on. Light approaching from far enough away spends its early life moving away in proper distance, because the space it is crossing is expanding faster than it can cross it, and only after the Hubble sphere overtakes it does it start closing the gap.

The teardrop’s maximum is at about 5.8 billion light years, at a lookback time of some 5 billion years and a redshift near 1.6 — which is not a coincidence. It is the same redshift at which the angular-diameter distance turns over, because both are asking where the light was furthest away in proper terms.

There is a way of putting all of this that removes the strangeness without removing the content. In comoving coordinates a photon always makes progress and the picture is unremarkable: a light cone that opens as time runs backwards, exactly as a light cone in flat spacetime does, only wider. Every counter-intuitive statement in this essay is a statement about proper distance, and proper distance is a derived quantity — the comoving separation multiplied by a scale factor that is itself changing while the light is in flight. The oddity is not in the physics but in insisting on a coordinate that moves.

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. 4 The same diagram carried to sixty billion years instead of thirty. The three surfaces do what the algebra says they will and the picture is the argument: the particle horizon keeps growing but ever more slowly, the event horizon flattens to a constant comoving distance, and the two converge on each other from opposite sides. Everything outside that limit is permanently out of reach and everything inside it will eventually be seen in full. The future of this diagram is a horizon that stops moving, which is what a universe with a cosmological constant looks like from inside.

What is measured, and what is computed

None of the three numbers is a measurement. No length in this essay is observed. What is observed is a redshift; the horizon is ∫c dt/a\int c\,dt/a evaluated in a model whose parameters were fitted to other data. There is a further honesty required about the word “observable”. The particle horizon at 46.1 billion light years is the limit for light emitted at t=0t = 0, and no light was emitted at t=0t = 0 that anyone can detect: the universe was opaque until 380,000 years. So the practically observable universe is bounded by the surface of last scattering at 45.6 billion light years, a per cent inside the particle horizon. Neutrinos decoupled at one second and gravitational waves in principle at the Planck time, so the horizon for those messengers is larger — but neither has been detected from that epoch and one of them probably never will be.

The distinction shows up in an unexpected place, which is why the night sky is dark. That argument is an integral over shells cut off at the horizon, and which horizon is used changes the answer by nothing at all — a per cent of a factor of 101410^{14} — while changing whether there is a cutoff changes it by the whole fourteen orders of magnitude. The particle horizon is one of those quantities whose exact value matters much less than its existence.

What sets the horizon in the first place

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 comoving Hubble radius against the scale factor, over the whole history. Right of the kink it is exact: after inflation, c/aHc/aH grows as aa in the radiation era and as a\sqrt{a} in the matter era, then turns over once the cosmological constant dominates. Left of it is a schematic, because the energy scale of inflation is unmeasured — but the shape is not negotiable, since HH is nearly constant during accelerated expansion and c/aHc/aH therefore falls as 1/a1/a. The horizontal lines are two fixed comoving lengths: the scale of the first acoustic peak, which left the Hubble radius during inflation and re-entered at z=1090z = 1090, and the whole observable universe, which has never re-entered.

That figure carries the reason the particle horizon is finite at all, and it is not the reason a first pass suggests. If the universe began with a hot dense phase and nothing else, the integral ∫c dt/a\int c\,dt/a from zero is finite because a→0a \to 0 fast enough — and its finiteness is what creates the horizon problem, since regions on opposite sides of the sky would then never have been in causal contact. Inflation makes the integral much larger by inserting an epoch in which aa grows exponentially while HH stays fixed, so that a great deal of comoving distance is covered by light in a short time.

The horizon that matters for causality is therefore not the one drawn in the hero figure. That one is computed with the post-inflationary expansion history and answers “how far has light come since the hot phase began”. The causal horizon, which answers “what regions could ever have communicated”, is vastly larger if inflation happened, and the difference is the entire content of the horizon problem.

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. 6 The same sixty billion years in proper distance. The teardrop of the past light cone is unchanged — it is history and history does not extend — and what grows is the splay of the worldlines, exponentially, because a constant Hubble rate is exponential expansion. In these coordinates the horizons look like they are shrinking while in the comoving picture they look like they are growing, and neither is an illusion: the comoving statement is about how much material is enclosed and the proper statement is about how far away it is.

Where the three surfaces are going

The three lengths were separated above by what question each answers. Following them forward is worth doing, because their present near-agreement is temporary and the way it resolves says what kind of universe this is.

In a universe dominated by a cosmological constant the expansion rate tends to a constant, and everything about the geometry simplifies. The Hubble sphere stops moving in proper terms and settles at c/HΛc/H_\Lambda. The event horizon settles at the same proper radius, because in exact de Sitter space the two coincide — the surface at which recession reaches cc is also the surface beyond which a signal never arrives. And the particle horizon grows without bound, because light continues to arrive from material that was already inside it.

So two of the three converge and one runs away, and the converging pair is the one that bounds what can be affected while the diverging one bounds what can be seen. That is the geometry of the far future: an observer will be able to see an ever-growing comoving volume in principle, while the volume they can influence, or that can influence them, shrinks to a fixed proper sphere.

The apparent contradiction — seeing more while reaching less — dissolves once it is noticed that the light arriving from distant material was emitted long ago. The particle horizon grows because light emitted in the past is still in transit, not because anything new is becoming visible. In practice the galaxies now crossing the event horizon fade and redden out of detectability within a few tens of billions of years, so the usable observable universe shrinks while the formal one grows.

The present epoch is the one in which the three surfaces are within a factor of three of one another, and that is a statement about now rather than about the geometry. Early on, the Hubble sphere and the particle horizon were nearly the same and there was no event horizon at all; late on, the Hubble sphere and the event horizon coincide and the particle horizon is far outside both. This is the interval during which all three are separately meaningful and comparable in size, which is a large part of why the distinctions between them need making at all.

What the pictures cannot show

Nothing in the hero figure is to scale in time. Cosmic time runs linearly from 0 to 30 gigayears, so the whole of inflation, nucleosynthesis, recombination and the first galaxies are compressed into a band thinner than the axis. The interesting structure in the early universe — the epoch that dominates the horizon integral — is invisible on a linear time axis, which is why the inflation figure uses the scale factor instead.

The event horizon is drawn as a curve and it is a prediction. Its value depends on the dark energy remaining a cosmological constant for ever. If the equation of state is −0.9-0.9 rather than −1-1 the event horizon eventually disappears; if it is below −1-1 the universe tears itself apart in finite time. The current constraint is w=−1.03±0.03w = -1.03 \pm 0.03, so the drawn curve is the central case of a family and not a fact.

And the diagram shows a slice through a three-dimensional situation. The horizon is a sphere, and the two curves either side of the axis are one surface seen edge-on. A reader inclined to ask what is beyond the edges of the plot is asking a question the diagram cannot answer, and neither can any observation: the region beyond the particle horizon is unobservable by construction, and every statement about it is an extrapolation of the cosmological principle.

Where the microwave background actually is

It is worth following one shell through the arithmetic, because the numbers make the whole essay concrete and because the shell in question is the one every cosmological result is measured against.

The photons of the microwave background were emitted when the scale factor was about a thousandth of its present value. At that moment the matter that emitted them sat about 42 million light years away in proper distance — a separation smaller than the present distance to the Virgo cluster, and one that a reader can picture.

Those photons then travelled for 13.8 billion years and arrived here. The matter that emitted them, meanwhile, has been carried outward by the expansion, and it now sits about 45.6 billion light years away. It is the same matter, it has not moved relative to the space around it, and the factor of a thousand between the two figures is entirely the growth of the scale factor.

Three statements follow that sound contradictory and are not. The light travelled for 13.8 billion years. The source was 42 million light years away when it emitted. The source is 45.6 billion light years away now. All three are true, and the reason they can be is that each names a different time at which the distance is evaluated — and proper distance is a quantity that has to be evaluated at a time.

The lookback arithmetic makes the same point from the other side. Everything in the observable universe was, at the moment the background was emitted, inside a sphere 42 million light years in radius, which contained the material that has since become every galaxy anyone will ever see. The observable universe was small, and it was small in exactly the sense that its contents were close together rather than in the sense that anything bounded it.

The one number that has not changed through any of this is the comoving separation, which was 45.6 billion light years then and is 45.6 billion light years now, because that is what comoving coordinates are for.

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. 7 The event horizon read the other way: for a galaxy at each redshift, the last moment on its own clock that will ever be seen from here. The window has been widened to eighty gigalight-years of comoving distance so that the curve’s approach to the present can be seen — beyond redshift 1.87 the answer is a time already past, and the further out the galaxy the further back that moment lies. What the event horizon forbids is not the galaxy but its future, and this is that statement drawn as a curve rather than as a surface.

What it means to be inside an event horizon that is shrinking

The event horizon is the one with consequences for the far future, and they are worth stating because they are unusually definite.

As the cosmological constant takes over, HH tends to a constant HΛ=H0ΩΛ=0.83H0H_\Lambda = H_0\sqrt{\Omega_\Lambda} = 0.83 H_0, and the comoving event horizon shrinks towards a fixed proper radius of c/HΛc/H_\Lambda, about 17.5 billion light years. Galaxies at fixed comoving coordinates cross outward through it and never return. Every galaxy currently at a redshift above about 1.8 has already crossed: light it emits today will never reach the Milky Way.

What remains gravitationally bound stays bound — the Local Group is not expanding, any more than a galaxy’s rotation curve is — so the eventual picture is a single merged galaxy in an otherwise empty and dark sky, with the microwave background redshifted to invisibility. An observer then would have no evidence of an expansion and no evidence of a hot early phase: no external galaxies to plot against redshift, no background to measure a temperature from, and no deuterium left to weigh the baryons with, since ten billion years of star formation will have destroyed most of it. The interval during which the history of the universe is readable from inside it is finite, and this is roughly the middle of it, which is a fact about the present epoch worth as much attention as the coincidence between the matter and dark-energy densities.

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. 8 And the same construction stopped at the present. Nothing to the future of the horizontal line at 13.8 billion years is drawn, which is what a diagram of the observable universe as an observation rather than as a prediction looks like. The particle horizon is 46.5 gigalight-years at the top edge and the light cone below it is the whole of what has been seen; every statement in this essay about the horizons converging is a statement about the part of the previous figure that this one leaves out.

The generalisation

The trap this essay exists to dismantle is a specific and recurring one: multiplying a rate by a time when the rate is not constant.

ct0ct_0 is the distance light covers in t0t_0 at fixed scale factor, and the scale factor is not fixed. The same error in miniature produces every one of the misreadings this collection has met. The Hubble time is the age only if the expansion rate has been constant, and it has not. A recession velocity of czcz is a velocity only for small zz. Escape speed is 2GM/r\sqrt{2GM/r} and not the speed needed to coast to infinity at constant velocity, because the field weakens on the way out.

The general form is that a quantity defined as an integral is being estimated by a product, and the product is right only when the integrand is constant. The diagnostic is always the same: write the integral, look at where its contribution is concentrated, and check whether the integrand is anything like its present value there. For the horizon it is concentrated in the first thousandth of the history, where aa was three orders of magnitude smaller, and that is the entire discrepancy.

One more range shows the event horizon at the comoving distance a survey can actually reach.

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. 9 The event horizon over forty comoving gigaparsecs. The redshift beyond which nothing can ever reach an observer here is under two, so more than half the galaxies in any deep image are already out of causal reach — the observable universe and the reachable one are different objects.

Where the ladder goes next

The figure showing the comoving Hubble radius raised something it did not settle: that the horizon at recombination subtends about a degree on the sky, so patches further apart than that were never in causal contact and have no business being at the same temperature. They are, to one part in a hundred thousand. The next essay is about that.

Later rungs on this anchor: conformal time, which turns the light cones into straight lines and makes the horizon problem a statement about a diagram’s geometry; the de Sitter horizon and its temperature; whether the observable universe’s contents constitute a fair sample; the future visibility limit, and the finite number of galaxies that will ever be seen; and the difference between a horizon in cosmology and a horizon around a black hole, which share a name and a great deal of mathematics and differ in who is inside.

Where this goes next

Four expansion histories that agree exactly today. The scale factor against time, with the present at zero and every model normalised to a = 1 and an expansion rate of 67.36 km/s/Mpc there. That normalisation is the figure: four universes that are indistinguishable from a measurement made now, separated entirely by what is behind and ahead of them. Each curve is integrated from da/dt = aH₀E(a) rather than from its own closed form, so the four are compared through one routine. The age each implies is where its curve meets zero: empty (Ω = 0) 14.52 Gyr, matter only (Ω = 1) 9.68 Gyr, ΛCDM (Planck 2018) 13.80 Gyr, closed (Ω = 2) 8.29 Gyr. The empty universe's 14.52 Gyr is the Hubble time 1/H₀ exactly, which is what makes it the natural thing to measure an acceleration against. The closed model turns over and is stopped at its turnaround. One number that is an age, a size and a density The slope of the velocity–distance relation has units of inverse time, so it can be read as an age, multiplied by c to give a length, or squared to give a density. All three readings are natural, all three are quoted, and not one of them is the quantity it appears to be. Four distances to the same galaxy. Four quantities all called "the distance", against redshift, at a Hubble constant of 67.36 km/s/Mpc, a matter density parameter of 0.3153 and a dark-energy parameter of 0.6847. They agree below z ≈ 0.1 and then part company completely. Comoving distance is the separation now, and it is what a map of the universe is drawn in. Luminosity distance is what a brightness gives, and it is larger by (1+z) because the photons arrive both redshifted and spread out in time. Light-travel distance is the age difference times c, and it is bounded by the age of the universe. Angular-diameter distance is what an angle gives, and it is the odd one: it rises, turns over at z = 1.59 where it reaches 1.79 Gpc, and falls thereafter. Past that redshift a galaxy of fixed size looks bigger the further away it is, because the universe it is being seen across was smaller when the light left it. At z = 10 the four differ by a factor of 121 between the largest and the smallest, so a sentence quoting a cosmological distance without saying which one has not given a number. Four distances to the same galaxy Inside the Local Group the word "distance" has one meaning. Past a redshift of about a tenth it has four, they disagree by factors of a hundred by the time the light is old, and one of them stops increasing and starts coming back. The same recession law from two different galaxies. Twenty galaxies at fixed comoving positions after the whole picture has been multiplied by 1.34, with each galaxy's displacement drawn from where it was to where it is. Every arrow's tail sits at the old separation and its tip at the new one, so the arrow is exactly 0.34 times its tail's distance from the highlighted galaxy — proportional to separation for one reason and no other: a uniform scaling moves everything in proportion to its distance from whatever point the scaling is measured about. The right-hand panel measures from a different galaxy and gets the identical law with the identical constant. That is the content of a linear velocity–distance relation. It is the signature of an expansion with no centre, and the observation that every galaxy recedes is therefore not evidence that this one is the centre — it is evidence that none of them is. The shift that is not a Doppler shift Every galaxy beyond the Local Group has its lines shifted to the red, by an amount proportional to its distance. Read as a velocity that looks like a confession that everything is fleeing from here; read as a change of scale it says the opposite, because a uniform expansion produces exactly the same law measured from any galaxy in it.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Causal contactComoving distanceConformal timeDe sitter spaceEvent horizonHubble sphereLight coneObservable universeParticle horizonSuperluminal recession