Cosmology · Horizons and the observable Universe

The event horizon depends on the future history

Compare two supplied expansion futures with the same reference scale and Hubble rate, and calculate why a late signal can arrive in one but never in the other.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Define an event horizon using the remaining future light-path integral.
Distinguish a finite-time arrival from an asymptotic limiting boundary.
Explain why a reference expansion rate alone does not fix future causal reach.

An event horizon asks a different question

A particle horizon uses the available past. A cosmological event horizon instead asks which signals emitted now, or at a specified later time, can ever reach an observer along the assumed future history. Its existence depends on that history extending over the stated domain.

For a noncompact FLRW example with a length-valued radial comoving coordinate r, the remaining comoving light reach is r_e(t) = c∫ from t to t_f of dt′/a(t′), when the integral converges. The upper boundary t_f belongs to the model. In the examples here it is infinite future cosmic time.

Multiplying the remaining comoving radius by a(t) gives its proper radius D_e(t) on that same cosmic-time slice. A finite remaining comoving reach and its corresponding proper radius are different quantities; neither is a physical wall.

This lesson considers one-way reception by an observer at the origin, with freely propagating test light and comoving emitters. A successful incoming signal does not by itself establish the conditions for a round-trip exchange or a journey with a massive spacecraft.

Equal reference rates need not imply equal futures

Choose a positive time unit T and future coordinate v = (t − t_ref)/T ≥ 0. Normalize both scale factors to 1 at the reference time. Future A is a_A(v) = 1 + v. Future B is a_B(v) = (1 + v/2)².

At v = 0 both have a = 1 and da/dt = 1/T, so both have reference Hubble rate H_ref = 1/T. They are separate supplied kinematic futures in spatially flat, noncompact FLRW geometries. No common observed past, matter model or fitted cosmological parameter set is asserted.

The dimensionless comoving distance light accumulates from v = 0 to v = V is ∫ from 0 to V of dv/a(v). For A this is ln(1 + V), which grows without bound as V grows. For B it is 2 − 2/(1 + V/2), which approaches 2.

Every finite V gives a reach smaller than 2 in B. Approaching a limiting distance as V tends to infinity is different from arriving there at a finite time. The endpoint must be interpreted using the integral, not simply the edge of a plotted curve.

Comoving and proper horizon radii can evolve differently

For future B, the remaining comoving radius at time v is r_e/(cT) = 2/(1 + v/2). Multiplying by a_B gives D_e/(cT) = 2 + v. The remaining comoving radius shrinks while the proper radius grows.

At v = 0 the remaining radii are 2 cT in both conventions because a = 1. At v = 2 the comoving radius is cT but the proper radius is 4 cT. At v = 6 they are cT/2 and 8 cT. These are statements on three different cosmic-time slices.

The light reach already accumulated since the reference time plus the remaining comoving reach equals 2 cT throughout B. This accounting uses one fixed comoving convention. Adding proper radii from different slices would not express the same identity.

A source’s comoving separation should be compared with the comoving future reach at its emission time. Alternatively use both proper quantities on that emission-time slice. Mixing conventions can reverse an apparent conclusion about whether a signal can arrive.

Two supplied futures for v ≥ 0; all light-path distances are divided by cT and are not observed cosmological radii
v	a_A	a_B	A accumulated comoving reach	B accumulated comoving reach	B remaining comoving radius	B proper horizon radius
0	1	1	0	0	2	2
2	3	4	ln 3 ≈ 1.09861	1	1	4
6	7	16	ln 7 ≈ 1.94591	1.5	0.5	8

A future assumption changes the answer

Let a comoving emitter at r = 3 cT send an inward photon at v = 0. Future A allows arrival when ln(1 + V) = 3, so V = e³ − 1 ≈ 19.08554. The time is long in the chosen units but finite.

In future B the total reference-time comoving reach is only 2 cT. The signal from 3 cT never reaches the origin within that specified future. More observing time within B cannot make the integral exceed its limiting value.

A reference-time signal from exactly r = 2 cT in B also does not arrive at any finite V; the limiting reach equals its separation only at infinite V. Reference-time emitters with smaller positive comoving separations do have finite arrival times in this ideal transparent background.

The two futures share a and H at the reference event yet answer this causal question differently. Measuring a local-in-time expansion rate alone cannot determine the complete future integral. Predictions about an actual event horizon require an explicit extrapolated future model.

Two separate supplied noncompact spatially flat kinematic FLRW futures have v = (t−t_ref)/T ≥ 0, positive arbitrary T, a_A = 1+v and a_B = (1+v/2)^2. Both have a_ref = 1 and H_ref T = 1, without asserting a shared observed past or a fitted matter model. The upper graph displays 0 ≤ v ≤ 20 and one common comoving length unit cT. Solid blue square-marked A accumulates ln(1+v), while dashed orange circle-marked B accumulates 2−2/(1+v/2). A dotted separation guide at 3 represents one emitter that sends an inward signal at v = 0. A reaches it at v = exp(3)−1 approximately 19.08554. B stays below its total asymptotic reach 2 at every finite v; the guide at 2 is a limit rather than a finite-time arrival. Future A is unbounded if continued as supplied, and B never covers separation 3 under its supplied infinite future. The lower panel independently displays B radii only for 0 ≤ v ≤ 6. Its dashed blue square-marked remaining comoving radius is r_e/(cT)=2/(1+v/2), while its solid orange circle-marked same-slice proper radius is D_e/(cT)=2+v. At v = 0,2,6 the pairs are (2,2), (1,4), (0.5,8). Adding accumulated and remaining comoving reach gives 2, but radii on different slices must not be mixed. Neither plot endpoint ends the future, and no currently measured event horizon, infinite-future observation, round-trip communication or actual future history is claimed. Original Ark teaching model.

Visibility today is not the same as future communication

An image received now concerns an earlier emission event. An emitter can therefore appear in an old image even when sufficiently late signals from it will fail to reach us under the assumed future. A statement about future reception should specify the emission time.

The event-horizon construction does not imply that every past emission from a source beyond a present horizon radius has always been invisible. It also does not imply that an object changes its locally measured speed when a horizon comparison changes.

The examples establish conditional mathematical consequences of two declared histories. They do not measure our Universe’s event-horizon radius, verify infinite future time or select the correct long-term expansion law.

When a horizon claim is made, identify the observer, emission time, spatial convention, scale-factor history, future endpoint and propagation assumptions. Those details turn an ambiguous phrase such as “forever beyond reach” into a question with a calculable meaning.

Worked reasoning

Compare the same emitted signal in two futures

A photon is emitted inward at v = 0 from r = 3 cT. Does it arrive in the two declared futures?

Future A accumulates ln(1 + V) cT, so set ln(1 + V) = 3.
The solution is V = e³ − 1 ≈ 19.08554, a finite arrival time.
Future B accumulates less than 2 cT at every finite V and has limiting reach 2 cT, so it cannot cover 3 cT.

It arrives in A at V ≈ 19.08554, and never arrives in the specified B future.

Try the reasoning

Does a reference-time signal from r = 2 cT arrive at a finite time in future B?
Show the answer and reasoning
No.
The accumulated reach approaches 2 cT only as V tends to infinity.
At v = 2 in future B, what are the remaining comoving and proper horizon radii?
Show the answer and reasoning
The comoving radius is cT; the proper radius is 4 cT.
The scale factor is 4, so the same-slice proper radius is four times the comoving radius.
Do equal a and H at one reference time force equal future event horizons?
Show the answer and reasoning
No.
The event-horizon integral depends on the entire assumed future scale-factor history.

What to carry forward

An event horizon is conditional on a specified future history and endpoint.
A comoving horizon can shrink while its same-time proper radius grows.
Current images, finite-time future arrivals and asymptotic boundary limits answer different questions.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.