Cosmology · Conditional futures of the Universe

Continued expansion and changing observability

Track light pulses in a specified eternally expanding geometry and distinguish an event-horizon boundary from the disappearance of earlier light.

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

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After reading, you can

Define the future light-travel question behind an event horizon.
Compute arrivals and redshifts for three supplied pulses.
Distinguish geometric reachability from source survival and detector sensitivity.

An event horizon concerns an emission event

Continued expansion can change which signals an observer can ever receive. A cosmological event horizon addresses light emitted at a specified event and the whole assumed remaining future.

It differs from the particle horizon, which concerns the past, and generally from the Hubble distance c/H. Whether a future event horizon exists depends on the full future expansion law.

Use a spatially flat de Sitter reference with constant positive H_Λ and a = exp(u), where u = H_Λt. Assume that expansion law continues for the entire future.

Define R = c/H_Λ as a length unit. Place a comoving observer at χ = 0 and a comoving source at χ = R/2. Their coordinate separation is fixed, while the source’s proper distance at emission is a_eR/2.

Later pulses need progressively later arrivals

The source emits inward radial light pulses at scale factors a_e = 1, 3/2 and 7/4. There is no intervening absorption or detector threshold in the model.

For an arriving pulse, integration of the light path gives exp(−u_o) = 1/a_e−1/2. Here u_o is the observer’s dimensionless arrival time.

Thus the first pulse arrives at u_o = ln 2, the second at ln 6 and the third at ln 14. Their redshift factors 1+z = a_o/a_e are 2, 4 and 8.

The finite arrivals show that continued expansion does not simply stop all incoming light. Which emission can arrive is a question about the emission time, separation and future light path together.

Comoving source at χ/R = 1/2 in eternal de Sitter; the boundary row has no finite arrival
Emission a_e	Source proper distance/R at emission	Arrival u_o	1+z	Status
1	0.5	ln 2 ≈ 0.693147	2	Finite arrival
1.5	0.75	ln 6 ≈ 1.791759	4	Finite arrival
1.75	0.875	ln 14 ≈ 2.639057	8	Finite arrival
2	1	No finite value	No finite arrival redshift	Asymptotic boundary

A limiting signal is not a finite arrival

At a_e = 2, the right side of the arrival equation is zero. Reaching the observer would require u_o to grow without bound. The pulse therefore has no finite arrival in the supplied model.

For a_e > 2, the right side is negative and cannot equal exp(−u_o). Such later emission events cannot send a light signal to this observer under the assumed eternal expansion law.

Earlier pulses are not erased when the source’s later events fall beyond the horizon. The source is not destroyed by a geometric boundary, and receiving earlier emission is different from reaching its later events.

No finite detector sensitivity has been specified. A pulse that can arrive may still be too faint for a particular instrument, while geometric nonarrival is a stronger and different statement.

Exact flat de Sitter expansion a = exp(u), with u = H_Λt, is assumed to continue forever. R = c/H_Λ is a length. The comoving source stays at χ/R = 0.5 and the observer at 0; a is dimensionless. The graph has linear time u from 0 to 4 and comoving distance χ/R from 0 to 0.6. The solid blue circle-ended pulse is emitted at a_e = 1, the dashed orange square-ended pulse at 1.5, and the dotted green diamond-ended pulse at 1.75. Their trajectories are χ/R = 0.5−1/a_e+exp(−u), from u_e = ln(a_e) to finite arrival u_o = −ln(1/a_e−0.5). The native table gives arrivals ln 2 ≈ 0.693147, ln 6 ≈ 1.791759 and ln 14 ≈ 2.639057, with redshift factors 1+z = 2, 4 and 8. The violet dash-dot pulse emitted at a_e = 2 coincides with the comoving event horizon χ/R = exp(−u). It is shown only from ln 2 to 4, has no terminal or arrival marker, and stays above the observer's zero baseline at every finite time. Its arrival and arrival redshift are not assigned. Later emission cannot reach this observer under the eternal assumed law. Earlier signals are not erased and the source is not destroyed. The proper event radius is R here; equality with c/H is not general. No luminosity, absorption or detector threshold is modeled. Original Ark teaching model.

Optional mathematical path: integrate the remaining conformal distance

For radial inward light in this flat geometry, dχ/dt = −c/a. The maximum remaining comoving travel from emission time u_e to the infinite future is R∫ from u_e to infinity of exp(−u)du = R exp(−u_e).

Finite arrival requires this remaining distance to be strictly greater than the source’s comoving separation R/2. Equality is the asymptotic case, rather than a finite crossing.

The comoving event-horizon radius is χ_event = R exp(−u), while its proper radius is aχ_event = R. Its equality with c/H_Λ is special to this constant-H reference.

A pulse trajectory is χ(u)/R = 1/2−exp(−u_e)+exp(−u). Setting χ = 0 gives the arrival equation used above and keeps every distance and time convention explicit.

Observability changes depend on the future law

The calculation assumes eternal de Sitter expansion, freely propagating light and comoving endpoints. Changing the future scale factor can change the remaining light-travel integral and the horizon conclusion.

Locally bound systems need not follow the comoving source trajectory used here. Their internal dynamics require a separate local model rather than multiplying every internal separation by a.

Redshift, arrival rate, source luminosity and detector response all matter for practical observation. This light-path exercise supplies geometric reachability and redshift, without a luminosity or detection model.

The useful distinction is between earlier light still arriving, later emission events becoming unreachable, and a real instrument’s diminishing ability to observe a signal. None of these implies that the source suddenly ceases to exist.

Worked reasoning

Follow the pulse emitted at a_e = 3/2

When does it arrive, and what is its redshift factor?

The arrival equation gives exp(−u_o) = 2/3−1/2 = 1/6.
Therefore u_o = ln 6 and a_o = 6.
The redshift factor is a_o/a_e = 6/(3/2) = 4.

It arrives at u_o = ln 6 with 1+z = 4 in the supplied eternal de Sitter geometry.

Try the reasoning

Does the pulse emitted at a_e = 2 arrive at a very large but finite u_o?
Show the answer and reasoning
No.
It is the asymptotic boundary; exp(−u_o) reaches zero only in the infinite-time limit.
Are already emitted earlier pulses erased when later source events become unreachable?
Show the answer and reasoning
No.
Each emission event has its own light path; the horizon does not destroy previously emitted light.
Is a cosmological event horizon always at c/H?
Show the answer and reasoning
No.
An event horizon uses the whole assumed remaining expansion history; the equality here is special to constant-H de Sitter.

What to carry forward

An event horizon is a conditional whole-future light-travel boundary.
Earlier light and later emission events can have different arrival possibilities.
Geometric reachability, source survival and detector sensitivity are distinct questions.

References

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