The galaxies that are already out of reach

The rung below dealt with the horizon that looks backwards: light has been travelling for 13.8 billion years, the material that emitted the oldest of it is now 46 billion light years away, and nothing has outrun anything. That is the particle horizon, and it grows.

There is a second horizon that runs the other way, and it does not grow. It asks not what can be seen but what can be reached — whether a photon leaving here now ever arrives anywhere in particular — and the answer, with a cosmological constant in the model, is that most of the visible universe is already beyond it.

Three horizons, and only one of them is finite by accident #

Set them out together, because they are constantly confused and only one is subtle.

The particle horizon is the comoving distance light has covered since the beginning: $\chi_p = \int_0^{t_0} c\,dt/a$. It is finite because the integral converges at the lower limit — the scale factor goes to zero fast enough — and it grows without limit as $t_0$ increases. Today it is 46.1 billion light years.

The Hubble sphere is where the recession speed equals $c$: $\chi_H = c/(aH)$. It is not a horizon at all, and treating it as one is the commonest mistake in the subject.

The event horizon is the comoving distance light leaving now will ever cover: $\chi_e = \int_{t_0}^{\infty} c\,dt/a$. It is finite only if that integral converges — and whether it does depends entirely on how the universe expands in the far future.

In a matter-only universe, $a \propto t^{2/3}$, the integrand goes as $t^{-2/3}$, and the integral diverges. There is no event horizon: given long enough, a signal reaches everything.

With a cosmological constant the late-time expansion is exponential, $a \propto e^{Ht}$, the integrand falls exponentially, and the integral converges to $c/H$. The event horizon exists because $\Lambda$ is not zero, and for no other reason.

The horizon shrinks #

This is the part that decides everything and is the least obvious.

The comoving event horizon at time $t$ is $\int_t^\infty c\,dt'/a$. As $t$ increases the lower limit rises and the integral gets smaller. In the late-time exponential phase it becomes $c/(aH)$ with $H$ constant, so it falls as $1/a$ — the comoving event horizon shrinks towards zero.

The number of galaxies inside it therefore falls with time. Every galaxy currently just inside will one day be outside, and once outside it never comes back.

That is not a statement about galaxies moving. In comoving coordinates the galaxies do not move at all. It is a statement about how far a signal can get, and that distance is shrinking because the expansion is accelerating.

What “beyond the horizon” actually means #

It does not mean invisible. This is the point most worth being careful about.

A galaxy beyond the event horizon is still seen, brightly, in whatever telescope was already looking at it. What has happened is that the light it is emitting now will never arrive. What arrives from now on is light it emitted in the past — earlier and earlier fractions of its history, arriving more and more slowly and more and more redshifted, but arriving.

So the picture is not one of galaxies winking out. It is one of galaxies freezing: their apparent evolution slows asymptotically, their light reddens without limit, and the last event ever seen on each is a specific moment in its past.

For a galaxy exactly at today’s event horizon, that moment is today. For a galaxy at redshift 5 it is much earlier. For a galaxy comfortably inside the horizon it is far in the future, and there is no last event at all — the whole of its future arrives, spread over infinite time.

The one place the arithmetic is worth doing #

The claim that a galaxy at redshift 1.87 is on the horizon deserves a check that does not rely on the figure, because the two integrals involved run in opposite directions and their agreement is the whole geometry.

Take the galaxy at comoving distance $\chi$ equal to today’s event horizon, $\chi_e(t_0)$. The last event on it that will ever be seen is the one at the scale factor $a$ where $\chi_e(a) = \chi$ — and since $\chi_e(t_0)$ is $\chi$ by construction, that scale factor is $a = 1$, which is now.

So the galaxy at today’s event horizon is exactly the galaxy whose present is the last thing about it that will ever reach here. That is a consistency condition rather than a coincidence, and it is what the first figure is drawn to satisfy: the curve crosses the “now” line at the redshift whose comoving distance equals the event horizon, and it must, or the two integrals are not integrals of the same thing.

The redshift is then found by inverting the comoving distance, which is a different integral again — over the past rather than the future — and the answer is 1.87 for the Planck parameters.

Superluminal recession is not what does it #

Every galaxy beyond a redshift of about 1.5 is receding faster than light — a recession that is a change of scale rather than a speed through space, right now, and every one of them is perfectly visible. That fact is not a paradox and it is not the mechanism of the horizon.

A photon emitted towards here by a galaxy receding at $1.3c$ is, at that instant, losing ground: the space between it and here is expanding faster than it can cross it. But the Hubble sphere is itself growing in comoving terms — as long as the expansion decelerates or accelerates slowly enough — so the photon is eventually overtaken by it, at which point it starts making progress and arrives.

Whether that happens is precisely the question of whether the comoving Hubble radius is still growing. Under a cosmological constant it stops growing and begins to shrink, and a photon not already overtaken never will be.

The horizon is set by whether the Hubble sphere catches the photon, not by how fast the source is receding.

The numbers #

For the Planck cosmology:

The particle horizon is 46.1 billion light years comoving, corresponding to the microwave background at redshift 1090.

The event horizon is 16.7 billion light years comoving, and the redshift at which a galaxy sits on it today is 1.87.

The ratio of the volumes is $(16.7/46.1)^3 = 0.047$, so about ninety-five per cent of the galaxies now observable are already beyond reach. A signal sent today reaches five per cent of what can be seen.

And the far future: the event horizon tends to $c/H_\Lambda \approx 16$ billion light years in proper distance, permanently. Everything outside the Local Group — which is bound and does not expand — eventually crosses it and freezes.

What has already crossed #

It is worth putting a date on the crossing for a familiar object, because “already out of reach” reads as an abstraction and is not one.

A galaxy at redshift 1.87 is at a comoving distance of 16.7 billion light years and is being observed as it was about ten billion years ago. Everything further than that — which is the great majority of every deep field ever taken, and every galaxy that any survey has found beyond redshift two — is in that category.

The crossing itself is not an event that happened at the galaxy. Nothing occurs there. What happened is that at some cosmic time the comoving event horizon here shrank past that galaxy’s comoving distance, and the moment it did, the light the galaxy was emitting at that instant became the last that would ever arrive. For a galaxy at redshift 5 that moment was several billion years ago.

Nothing about this is observable from here, and nothing about it is observable from there. A horizon crossing is a fact about a pair of worldlines and about the whole future of the universe between them, which is a different kind of fact from anything else this collection measures.

What is observable about any of this #

Almost nothing directly, and it is worth saying so.

The event horizon is not an observable. It is a statement about the infinite future of a model, and it follows from the model’s late-time behaviour rather than from anything measured. Its existence rests on the assumption that dark energy behaves as a cosmological constant for ever — which is an extrapolation from a few billion years of data to infinity.

If the dark energy density decays, the horizon opens up again. If it grows, the horizon closes faster and eventually everything is beyond it. The measurement that bears on this is the equation-of-state parameter $w$, and the current constraint of $-1.03 \pm 0.03$ is consistent with a constant and does not settle the far future.

There is one effect that would be a genuine test, and it is nearly measurable. Redshift drift: a galaxy’s redshift changes with time, because the expansion rate changes, and the sign of the change differs between a decelerating and an accelerating universe. The magnitude is about a centimetre per second per decade, and instruments capable of it are being built.

The horizon problem, which is the same integral run backwards #

There is a well-known puzzle that uses exactly this machinery in the other direction, and setting the two side by side makes both clearer.

Two patches of the microwave sky separated by more than about two degrees have particle horizons, at the time the light left them, that do not overlap. Nothing that happened in one could have influenced the other, and yet their temperatures agree to one part in a hundred thousand. That is the horizon problem, and one epoch of accelerated expansion removes it and a second fine-tuning at once.

Both puzzles are statements about the integral $\int c\,dt/a$: the horizon problem says it was too small in the past, and the event horizon says it is too small in the future. And both are resolved — or created — by a period of accelerated expansion, which makes the comoving Hubble radius shrink.

Inflation shrinks it early, which lets a small causally connected patch grow to contain everything now observable. The cosmological constant shrinks it late, which lets what is now observable disperse beyond reach. The mechanism is the same and the two epochs differ by fifty-odd orders of magnitude in energy scale.

What the horizon does to the far future #

Follow it out and the consequences are stark, and they are the reason the rung is worth writing rather than merely stating.

The observable universe empties. In about $10^{11}$ years everything outside the Local Group has crossed the horizon and frozen — and the sky is dark for a reason that has nothing to do with a finite age. What remains visible is one merged elliptical galaxy, and a sky with nothing in it beyond.

The evidence for the expansion disappears with it. An observer then has no external galaxies to measure a redshift against, no microwave background — the most perfect blackbody ever measured will have redshifted to wavelengths longer than the horizon can support — and no way to determine that the universe is expanding at all. The cosmology inferred would be a static island, and it would be a correct description of everything measurable.

And that is a statement about now as much as about then. The evidence available at any epoch is a function of the epoch, and the present is not a privileged moment except that it happens to be one with a lot of evidence in it.

What could be reached rather than signalled #

The horizon computed here is the limit for light. Anything with mass travels slower, and the volume it can reach is smaller — by a great deal more than the modest difference in speed suggests.

The reason is that the shrinking is exponential in the late universe. A photon leaving now covers a comoving distance of 16.7 billion light years in the whole of the future; a probe travelling at nine tenths of light speed does not cover nine tenths of that, because it falls behind the photon early and the horizon is closing while it does so.

Working the integral for a constant fraction of light speed gives a reachable comoving distance that falls off steeply as the speed drops: at half of light speed it is roughly a third of the photon’s reach, and at a tenth it is under a twentieth.

A chemical rocket does not appear on that scale at all. Even a probe travelling at a hundredth of light speed — far beyond anything proposed — reaches a comoving distance of a few hundred million light years, which is a few tens of galaxies rather than a few hundred billion.

The comparison worth carrying is between the two numbers. Five per cent of the observable universe can be signalled to. A very small fraction of one per cent of it could ever be visited by anything launched from here, and the fraction shrinks every year.

That gap between what can be seen, what can be reached by light, and what can be reached by anything slower is entirely a consequence of one term in the Friedmann equation, and it did not exist as a limitation in a universe without it.

The horizon a black hole has, and how this one differs #

The phrase event horizon is borrowed, and it is worth setting the two uses side by side because the differences are as instructive as the similarity.

A black hole’s horizon is a surface in space. It is at a definite place, every observer agrees where it is, and an object crossing it is inside for everybody. It is a property of the spacetime rather than of anybody looking at it.

A cosmological event horizon is not a surface in space and is not agreed on. It is defined relative to one observer’s worldline — it is the boundary of the set of events that observer will ever see — so a different observer, elsewhere, has a different one, centred on themselves. A galaxy beyond my horizon is not beyond its own.

What the two share is the thermodynamics, and the shared part is what makes the analogy more than a name. A horizon of either kind has an associated temperature: an observer in an accelerating universe measures a faint bath of radiation coming from their horizon, at a temperature proportional to the expansion rate, in the same way that a black hole radiates at a temperature set by its surface gravity.

For the observed cosmological constant that temperature is about $10^{-30}$ kelvin, which is thirty orders of magnitude below the microwave background and will never be measured. It becomes the dominant temperature in the universe only once everything else has cooled past it, which is very much later than the epoch at which the last galaxy crosses the horizon.

Two horizons that arise from opposite geometries — one from mass concentrated, one from space expanding — turn out to have the same thermodynamic character, and that coincidence is one of the standing arguments that horizon thermodynamics is a statement about gravity rather than about black holes.

One more reading shows how much of the answer depends on the one constant that is currently disputed.

That is an unusual thing for a physical science to have to say: the sample is finite, its boundary is set by the expansion rather than by any instrument, and no future observation will enlarge it by more than a few per cent.

Where this ladder goes next #

This rung has separated two horizons that share a word, found that only one of them exists at all and that it exists only because the expansion accelerates, and counted what is on each side of it.

The rung above is the causal structure the two horizons together define. Drawn in conformal time, where light travels on forty-five degree lines, both horizons become straight and the whole history is a finite diagram — which is also where the horizon problem that inflation exists to solve becomes visible as a geometric fact rather than an arithmetic one.

Beside it lies the question of whether the extrapolation is legitimate: what the far future looks like for the equations of state that current data allow, from a decaying scalar field to a divergent one, and what each does to the horizon.

And below it, the habit: a limit is a property of an integral, not of a speed. Nothing in this essay is about anything travelling faster than light. Everything in it is about whether a particular integral converges, and the answer changed when a quarter of a magnitude turned up in a supernova survey.