Whether there is a horizon at all

The second rung of this anchor found that ninety-five per cent of the galaxies now visible are already beyond reach, and attributed it to the cosmological constant. The third drew the same fact as a 45° line coming down from a finite ceiling in conformal time. Both took the acceleration as given and both used w = −1 without comment.

The measured value is not −1. It is about −1.03 with an uncertainty of a few per cent, and the number that goes into the integral is not the number that comes out of the measurement in any simple way.

A horizon is a convergence question #

Everything about this rung is one integral and whether it converges.

The comoving distance a signal sent at scale factor a will ever cover is ∫ from a to infinity of da′/(a′²E(a′)), where E is the expansion rate in units of its present value. Substituting u = 1/a′ turns that into ∫ from 0 to 1/a of du/E(1/u), which is a proper integral with a finite range and a possible singularity at the origin.

At the origin — which is the infinite future — the dark energy dominates and E(1/u) goes as u^(3(1+w)/2), so the integrand goes as u^(−3(1+w)/2). That converges when 3(1+w)/2 < 1, which is w < −1/3, and diverges otherwise. The boundary at −1/3 is exactly the boundary between acceleration and deceleration: the deceleration parameter for a single component is (1 + 3w)/2, which is zero there.

So the existence of an event horizon and the acceleration of the expansion are the same statement, and they are the same statement for a reason that is arithmetic rather than physical. There is no separate fact about causality to be established once the acceleration is known.

It is worth pausing on how recent that connection is as a piece of established fact rather than as a possibility. Until the supernova surveys of 1998 the expected answer was deceleration, in which case ∫da/a²E diverges, there is no event horizon, and every galaxy now visible is eventually reachable by a signal. The discovery that changed the sign of the acceleration also, in the same stroke and without anybody framing it that way at the time, closed the universe causally. A quarter of a magnitude in the brightness of distant supernovae is what put ninety-five per cent of the visible sky permanently out of reach.

What happens on either side of the boundary #

The divergence at −1/3 is logarithmic, and a logarithm is slow. That is why the horizon does not simply vanish as w rises: it grows continuously, past the particle horizon, past any finite number, and the growth is gentle enough that a per-cent measurement of w does not pin the horizon to a per cent.

Below −1 the horizon shrinks, and the reason is worth stating because it is the opposite of the intuition that more dark energy means more of it. Dark energy with w < −1 has an energy density that grows as the universe expands: the density scales as a^(−3(1+w)), and for w below −1 that exponent is positive. So the expansion rate grows without bound, the comoving Hubble radius collapses, and the distance a signal can cover shrinks toward zero.

There is a name for dark energy with w below −1 and the name carries a warning: phantom energy. Such a component violates the null energy condition, which is the weakest of the standard energy conditions in general relativity and the one that most theorems assume. A field theory that produces it generically has a kinetic term with the wrong sign, which makes the vacuum unstable to producing pairs of arbitrarily high-energy quanta. That is not a proof that it cannot exist — the energy conditions are assumptions rather than laws, and each has been violated in some quantum context — but it is the reason w = −1 is treated as a boundary rather than as a point in the middle of a range.

The measurement, awkwardly, sits just past it. Most published combinations put the best-fit w slightly below −1, by about one sigma. That is not evidence for phantom energy; it is what a distribution centred on −1 does half the time. It is worth knowing because a reader encountering a paper whose central value is −1.03 should not read the sign as a result.

The end that a constant w below minus one requires #

That growth has a consequence that is not optional. If w is genuinely constant and genuinely below −1, the scale factor reaches infinity at a finite cosmic time.

The word for the end is a big rip, and what happens is a consequence of the same growth. Any bound system is held together against the local expansion by whatever binds it, and the dark energy density inside it grows without limit; when it exceeds what holds the system together, the system comes apart. That happens in order of binding energy density, so clusters first, then galaxies, then planetary systems, then stars, then planets, then atoms, then nuclei, each at its own moment, and the whole sequence occupies the last fraction of the time to the singularity.

Two things about that should be said plainly. It is not a prediction: the data do not require w to be below −1, and a constant w is itself an assumption that no theory motivates. And it is the only future in this family that ends. Every other value of w gives a universe that runs forever — expanding forever if w < −1/3, or expanding and eventually recollapsing only if there is enough curvature or a component with positive pressure, which the measurements exclude.

Why the equation of state is measured so poorly #

A few per cent on a dimensionless number sounds like a good measurement, and by the standards of the quantities in this collection it is. The difficulty is what it is a measurement of.

That integration is the whole of the problem, and it compounds with a degeneracy. The distance to a given redshift depends on both the matter density and the equation of state, and the two trade against each other: a higher Ω_m and a more negative w produce nearly the same distance–redshift relation as a lower Ω_m and a less negative one. The published constraint on w is therefore never from supernovae alone. It comes from combining them with the acoustic scale, which fixes the matter density independently, and with the microwave background, which fixes it again at a different epoch. Three datasets and one number.

Two histories with different w(z) can also produce almost identical distance–redshift relations, because the distance is an integral and integrals lose information. The number that would say whether it is a constant is the derivative of w with respect to the scale factor, and it is essentially unconstrained: the published limits on it are of order unity, which is to say that w could have been anything within a factor of two over the range that has been observed.

So the horizon computed above assumes a constant w, and the assumption is not tested at the precision the answer is quoted to. What the figures show is what each constant value implies, and a universe whose w drifts is not on any of those curves.

Three lengths that are all called horizons #

Before the accounting, one distinction has to be repeated because the whole rung turns on it and it is the standard place to go wrong.

The Hubble sphere is where the recession speed equals c: it is c/H, today 14.5 billion light years, and it is an instantaneous property of the expansion rate. Galaxies cross it in both directions and nothing special happens to them when they do. The particle horizon is the comoving distance light has covered since the beginning, 46.1 billion light years, and it depends on the whole past expansion history. The event horizon is the comoving distance a signal sent now will ever cover, 16.7 billion light years at w = −1, and it depends on the whole future.

All three are lengths of order c/H. None of them is the same as either of the others. Only the third is what this rung is about, and only the third is a prediction rather than a measurement — the first two are computed from data and the third is computed from an extrapolation over infinite time.

The confusion between the first and the third is the most consequential, because it produces the standard false statement that galaxies receding faster than light cannot be seen. They can, and most of what is seen is doing it: everything beyond redshift about 1.5 has always been receding superluminally and is observed perfectly well, because the Hubble sphere grows to meet the photon. What closes the event horizon is not a speed limit but the failure of an integral to diverge.

What is already decided and what is not #

It is worth separating the parts of the future that survive the uncertainty from the parts that do not, because the two are usually presented together.

Decided, for any w below about −0.9: the expansion accelerates and continues to; the Local Group remains bound and merges into a single elliptical galaxy in a few billion years; everything not gravitationally bound to it recedes and reddens; the microwave background cools out of detectability; the observable universe eventually contains only the merged remnant and its own stars. That sequence follows from acceleration alone and is insensitive to the exact value.

The timescale of the last item is worth having, because it is the one number in this future that is short enough to be surprising. The nearest large galaxies outside the Local Group are in the Virgo cluster at about 50 million light years, receding at roughly 1,000 km/s. They cross today’s event horizon — in the sense that light they emit after that point never arrives here — in something like 150 billion years, which is ten times the present age of the universe. By about a trillion years an observer in the merged remnant sees no galaxy but their own and a microwave background at a fraction of a millikelvin, and the evidence for an expanding universe available to them is essentially nil. That situation is sometimes offered as a puzzle about whether cosmology is knowable; what it actually illustrates is that a great deal of what can be measured is a function of when.

Not decided: how much of what is now visible remains reachable, which varies from about one per cent to more than everything across the range of equations of state that have been drawn here. Whether there is an event horizon at all, if w is above −1/3 — excluded at high confidence, but by a measurement of the same kind. Whether the universe ends at a finite time. And whether “constant w” is a coherent description of the dark energy at all, which nothing yet tests.

What would change the answer #

The parameter that decides everything here is measured to a few per cent and is not going to be measured to a fraction of a per cent by more of the same.

What is under way is different in kind. Wide surveys measuring the acoustic scale at many redshifts constrain the expansion history point by point rather than as one average, which is what turns a constraint on w into a constraint on w(z). Weak lensing constrains the growth of structure, which depends on the dark energy differently from the way distances do — and a disagreement between a growth measurement and a distance measurement would be evidence that the dark energy is not a simple component at all, which is the outcome that would matter most. And a large sample of supernovae at redshift around 0.3, where the dark energy density is largest relative to matter, is worth more per object than one at redshift 1.

None of that will settle whether the equation of state is exactly −1, because no measurement establishes an exact value. What it can do is bound the derivative, and that is the quantity the constant-w assumption in every figure here stands or falls on.

What the picture cannot show #

Every curve here is drawn for a flat universe with a single dark-energy component of constant equation of state. Three assumptions, and each one is doing work.

Flatness is the safest: the curvature is measured to be zero to a few parts in a thousand, and a small curvature changes the integrals by less than the uncertainty in w. Constancy is the least safe, for the reasons above. And “a single component” is a modelling choice rather than a measurement — the observations constrain the total, and a mixture of two components with different equations of state produces an effective w that varies with redshift and is not any of the curves drawn.

One more assumption is hidden in the arithmetic rather than in the model, and it is the one most likely to be missed. Everything here is computed for an observer at rest with respect to the expansion, at this cosmic time. Both qualifications matter. The event horizon shrinks with time, so a signal sent in ten billion years reaches less than one sent today, and the reachable fraction quoted is a property of now. And the horizon is centred on the observer: there is no privileged place in the universe from which more is reachable, and every observer everywhere has their own 16.7-gigalight-year sphere, overlapping everybody else’s. The horizon is not a place. It is a relationship between two worldlines.

There is also a subtler limitation, and it is the reason a reader should be careful with the reachable-fraction numbers in particular. A comoving event horizon is a statement about signals, not about journeys. Reaching a galaxy requires travelling there, and a traveller moving at any speed below c covers less comoving distance than light does, so the volume actually reachable by anything material is strictly smaller than the numbers quoted and depends on when the journey starts. The figures answer the question they are drawn for — what can ever be signalled — and treating that as a statement about what could be visited overstates it substantially.

That last contrast is the honest summary of the rung. The particle horizon is a measurement: it integrates over an expansion history that supernovae, the acoustic peaks and the galaxy surveys have all mapped. The event horizon is a prediction, and it is a prediction whose input is the one parameter of the cosmological model that the data constrain least well and whose behaviour is least smooth. The two are usually quoted in the same sentence, in the same units, to the same number of decimal places, and only one of them is a number about the world.

The generalisation #

The structure worth taking out of this rung is that a quantity defined by an improper integral has a qualitative answer as well as a quantitative one, and the qualitative answer changes discontinuously while the parameter changes smoothly.

The event horizon does not shrink gradually to nothing as the acceleration weakens. It grows without bound and then ceases to exist. That is what a divergence does, and it means that no amount of precision on the parameter buys proportional precision on the answer near the boundary. The same shape appears in the deuterium residue, where an integrated destruction is exponential in a duration; in the escape condition, where a trajectory either returns or does not and the boundary is a single value of one number; and in the convergence of a comoving distance, which is this same integral read at a different limit.

The practical consequence is a habit rather than a result. When a quantity is an integral to infinity, the useful first question is not what it evaluates to but where it stops converging — because that boundary is usually a physical statement, and the value of the integral near it is usually not a robust number.

Where this ladder goes next #

A horizon that exists has a temperature. The de Sitter horizon radiates at ħH/2πk_B, which for the measured expansion rate is about 3 × 10⁻³⁰ K — thirty orders of magnitude below anything that will ever be measured, and not therefore an unimportant number.

The next rung is what that temperature is, and the exact sense in which a cosmological horizon and a black hole’s horizon are the same object. They share an entropy formula exactly and differ in temperature by precisely a factor of two, and the factor of two is the whole content of the difference: an observer is outside one and inside the other.