Cosmology · Horizons and the observable Universe

What can be inferred beyond the observable region?

Construct two exact test fields that agree throughout one observer’s present causal record but differ outside it, then distinguish conditional extrapolation from direct observation.

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

Separate direct observation from a model-dependent extension beyond an observed region.
Calculate a finite-speed example in which different exterior initial data produce the same current causal record.
Explain why current underdetermination does not mean every full model is equally supported or forever untestable.

A finite observed region does not specify an entire spatial slice

Observations received at one location probe a past light cone and its physically accessible records. Extending a cosmological description beyond that region requires assumptions about geometry, matter, initial conditions or statistical regularity.

Such assumptions can be useful and can have testable consequences within the observed region. A conditional extrapolation should still be reported as an inference rather than as a direct image or measurement of every distant location.

Underdetermination is a narrower statement than claiming that all conceivable global models are equally plausible. A proposal must satisfy its own equations, constraints, consistency requirements and the available evidence.

The following exact wave example isolates finite propagation. It is a test scalar on fixed flat spacetime with zero backreaction, not two complete general-relativistic Universes obtained by arbitrarily changing distant matter. It supplies no gravitational model for such a change.

Put the difference outside the present causal interval

Use dimensionless time u = t/T ≥ 0 and position s = X/(cT), where T is positive. A dimensionless scalar φ obeys the one-dimensional test-wave equation φ_uu = φ_ss on the entire real spatial line, with no walls or forcing.

In case A, the initial field and its initial time derivative vanish everywhere. In case B, the initial time derivative is also zero, while the initial field is b(s) = [(s − 2)(4 − s)]³ for 2 < s < 4 and zero elsewhere.

The profile b and its first two derivatives join continuously to zero at the endpoints. Its value is 1 at s = 3 and 0 throughout the interval −1 ≤ s ≤ 1. The profile is a supplied mathematical field, not a measured map of galaxies or unseen cosmic matter.

Choose the current observer event O at u = 1, s = 0. Its initial-data dependence interval for this finite-speed equation is [−1,1]. The two cases have identical initial data on that entire interval, including the initial time derivative.

The current causal record agrees exactly

For the stated zero initial velocity, the exact d’Alembert solution is φ_B(u,s) = [b(s − u) + b(s + u)]/2. Case A remains zero. The two translated terms propagate at the declared unit speed in these dimensionless coordinates.

Throughout O’s past domain 0 ≤ u ≤ 1 and |s| ≤ 1 − u, both arguments s − u and s + u lie in [−1,1]. Since b vanishes there, φ_B = 0 throughout that domain, just as in case A.

Thus even a complete record of this supplied field in the declared current past domain would not distinguish the two exterior profiles. The agreement follows from the equation and the positions of the initial data, rather than from measurement noise.

Differentiating the two translated terms verifies φ_uu = φ_ss; setting u = 0 gives φ_B = b and φ_B,u = 0. The smooth joining at the profile endpoints is sufficient for this classical second-derivative check. No numerical wave solver or fitted parameter is required.

Unknown now need not mean unknowable forever

At the origin, φ_B(u,0) = [b(−u) + b(u)]/2. For u ≥ 0 the first term is zero, so the observer receives a pulse only for 2 < u < 4. At u = 3 its value is 1/2, whereas case A remains zero.

At u = 2.5 and 3.5 the value is 0.2109375; at u = 2 and 4 it is zero. The difference that was outside the current causal record later enters the observer’s record by ordinary finite-speed propagation.

This example has no event horizon preventing the later pulse from arriving. A region outside one current causal domain is not automatically outside every future observable domain. A claim about permanent inaccessibility needs additional future-history conditions.

The wave calculation does not establish that actual cosmic exterior matter can be altered freely while keeping every gravitational observable unchanged. Full gravitational initial data obey constraints, and a proposed cosmological alternative requires its own consistent construction.

Origin values for two exact test fields on fixed Minkowski spacetime; u = 1 is the declared current observation time
Observer time u	Case B φ(u,0)	Interpretation
0	0	Identical initial local value
1	0	Identical current value
2	0	Pulse endpoint still zero
2.5	0.2109375	Later difference arrives
3	0.5	Pulse maximum at origin
3.5	0.2109375	Later difference continues
4	0	Pulse endpoint returns to zero

This is an exact one-dimensional scalar test wave on fixed flat Minkowski spacetime with zero backreaction, not a constructed alternative gravitational Universe or an observed sky map. Dimensionless coordinates are u=t/T≥0 and s=X/(cT), with arbitrary positive T and equation φ_uu=φ_ss. Case A has zero initial field and velocity everywhere. Case B also has zero initial velocity but initial b(s)=[(s−2)(4−s)]³ for 2<s<4 and zero elsewhere; its first two derivatives join continuously to zero. The upper plot displays −1.5≤s≤5 and shows A’s dashed blue zero profile and B’s solid orange compact bump, whose support endpoints are 2 and 4 and peak is 1 at s=3. Values at s=2.5 and 3.5 are 0.421875. Shading identifies the initial interval [−1,1] relevant to the current observer event O=(u=1,s=0), where both initial profiles and initial velocities agree. The exact solution is φ_B(u,s)=[b(s−u)+b(s+u)]/2. Both fields vanish throughout O’s past domain 0≤u≤1, |s|≤1−u. The lower plot displays the origin record for 0≤u≤4: A stays zero, while B=b(u)/2 remains zero through u=2, reaches 0.2109375 at u=2.5, peaks at 0.5 at u=3, returns through 0.2109375 at u=3.5 and reaches zero at u=4. Shading and a guide mark the current record only through u=1, not a permanent barrier. The translated pulse arrives by ordinary finite-speed evolution. Current agreement need not mean permanent unknowability, but this specified test-field degeneracy neither permits arbitrary full-GR initial-data changes nor establishes equal plausibility of all global cosmological models. Original Ark teaching model.

Make an extrapolation precise enough to assess

A global cosmological claim should state which observations support it, which assumptions extend the description, and which alternatives the observations can distinguish. Theoretical consistency and indirect evidence can constrain a proposal even when a distant region is not directly imaged.

Claims about global topology, total extent or conditions far beyond the observed region need not have the same evidence status. Some can produce observable consequences under stated conditions; others may remain underdetermined within a chosen model class.

The finite-speed example establishes one explicit degeneracy for a particular test equation and current record. It does not prove that every topology or origin story is equally likely, nor does it remove the need for quantitative comparisons among full models.

A useful conclusion names the boundary of the inference: “If these equations and initial conditions continue, this extension follows.” New accessible evidence can then test the assumptions or distinguish alternatives when their predictions differ.

Worked reasoning

Check whether the exterior bump can affect O

Can the initial bump at 2 < s < 4 alter the origin’s record by u = 1?

For the origin through u = 1, the relevant initial positions lie in [−1,1].
Both initial profiles and both initial time derivatives vanish throughout that interval.
Equivalently, φ_B(u,0) = b(u)/2 = 0 for 0 ≤ u ≤ 1, while φ_A is also zero. At u = 3 the B value instead becomes 1/2.

No. The current causal record agrees, but a later finite-speed pulse distinguishes the cases.

Try the reasoning

Does this test-field example show that every proposed global cosmological model is equally supported?
Show the answer and reasoning
No.
It proves a specific finite-data degeneracy under a stated wave equation. Complete cosmological models must also satisfy their equations, constraints and observations.
What is φ_B at the origin at u = 3?
Show the answer and reasoning
1/2.
b(−3) = 0 and b(3) = 1, so their half-sum is 1/2.
Should a model-dependent extension beyond the observed region be reported as direct observation?
Show the answer and reasoning
No.
State the extrapolation and its assumptions, and identify what evidence could test its consequences.

What to carry forward

Direct observations and conditional global extrapolations have different evidence roles.
Finite-speed evolution can leave distinct exterior initial data indistinguishable in a current causal record.
Present underdetermination neither makes all complete models equivalent nor guarantees permanent unknowability.

References

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