Cosmology · Conditional futures of the Universe

Uncertainty grows beyond the tested epoch

Construct different smooth continuations with an identical supplied past, distinguishing conditional extrapolation from a probability forecast or an impossibility theorem.

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

Explain why a past history alone may not choose among unspecified future rules.
Compare finite integrated expansion for three smooth continuation functions.
Separate model alternatives, statistical uncertainty and a complete physical theory.

Agreement can extend beyond one reference point

The earlier comparison matched only one scale factor and expansion rate. Now construct three examples that agree throughout an entire supplied past interval, −1 ≤ u ≤ 0.

Set u = H_ref(t−t_ref), with positive H_ref, and require H/H_ref = 1 and a = exp(u) throughout that interval. These are exact mathematical inputs rather than a measured history with error bars.

For u > 0, allow different specified continuation rules. The purpose is to expose the role of the rule itself, while retaining a smooth join to the common past.

The alternatives are phenomenological homogeneous effective-fluid constructions. They are not three established microscopic theories, and no probability distribution over them has been supplied.

Use a smooth expansion-rate family

Let α be 0, 1/16 or 1/8, and define h(u) = H/H_ref = 1/[1+α max(u,0)³] on −1 ≤ u ≤ 2. All members have positive expansion rate in that finite interval.

At u = 0, h and its first two derivatives are continuous and equal for the three cases. The differing future rules therefore introduce no jump in H or those two derivatives at the reference.

The scale factor is a(u) = exp[∫ from 0 to u of h(v)dv], with a(0) = 1. A later scale factor depends on the accumulated rate, not only the value of H at the final time.

The α = 0 member continues as exp(u). The positive-α members have decreasing future h and less accumulated expansion over the displayed future interval, despite their exact shared supplied past.

Three chosen finite continuation rules; the common past and curve span are not observational data or a confidence band
α	H/H_ref at u = 1	a at u = 1	w at u = 1	H/H_ref at u = 2	a at u = 2	w at u = 2
0	1	2.718282	−1	1	7.389056	−1
1/16	0.941176	2.677570	−0.875	0.666667	6.069575	−0.5
1/8	0.888889	2.640066	−0.75	0.5	5.319066	0

One can define a background without completing its microphysics

For each function, use flat classical GR and define ε_ref = 3c²H_ref²/(8πG). Let the effective energy density be ε = ε_ref h², positive throughout the displayed interval.

Define the pressure through the acceleration equation. Its ratio is w = −1−(2/3)h′/h², where the prime denotes differentiation with respect to u.

This gives w = −1 on the common past and w = −1+2αu² for u > 0. With this defined pressure and density, local homogeneous conservation is satisfied.

The construction is consistent as a background effective stress assignment. It does not specify a particle model, a field potential, sound speed, perturbation stability or quantum completion. Within an already fixed microscopic theory, pressure and H cannot generally be chosen independently.

Optional mathematical path: inspect the joined functions

For u > 0, h′ = −3αu²/(1+αu³)². Dividing by h² cancels the denominator and yields the stated w. The first and second derivatives of h tend to zero at the reference from either side.

The density derivative is ε′ = 2ε_ref hh′. The combination ε′+3h(ε+P) vanishes after substituting w = −1−2h′/(3h²), so the defined quantities obey the same continuity equation.

Numerically integrating h from 0 to 2 gives a ≈ 5.319066 for α = 1/8, compared with exp(2) ≈ 7.389056 for α = 0. The ratio is about 0.719857.

These finite results require the whole supplied h(u). Substituting only the final h = 1/2 into a constant-rate exponential would impose a different history and give a different calculation.

Three supplied flat-GR homogeneous effective continuations use α = 0, 1/16 and 1/8 over −1 ≤ u ≤ 2, where u = H_ref(t−t_ref). Their whole supplied past −1 ≤ u ≤ 0 is exactly H/H_ref = 1 and a = exp(u), not observed data. Future h = H/H_ref = 1/[1+α max(u,0)³] stays positive. H and its first two derivatives join continuously at zero; H need not be three-times continuously differentiable. Panel A has a zero-based h ordinate from 0 to 1.1. Solid blue circles identify α = 0, dashed orange squares 1/16, and dotted green diamonds 1/8. They overlap throughout the past, then separate. Panel B has its own linear ordinate from 0.68 to 1.04 for accumulated a/exp(u), not expansion rate. The scale factor is exp[∫ from 0 to u of h(v)dv]. At u = 2 the three scale factors are approximately 7.389056, 6.069575 and 5.319066; their ratios to exp(2) are 1, 0.821427595 and 0.719857284. Dotted vertical lines mark the arbitrary join u = 0. With ε_ref = 3c²H_ref²/(8πG), define ε = ε_ref h² and w = −1−2h′/(3h²), equal to −1+2αu² in the future and −1 in the past. This satisfies background conservation without providing a particle theory, field potential, sound speed, perturbation stability or quantum completion. Markers are finite rule samples. Curve spans are not confidence, credible or prediction bands. Further theory or later measurements may restrict these assumptions. Original Ark teaching model.

A family of curves is not a probability interval

The three continuations demonstrate underdetermination within a class where the future effective rule was not fixed by the common supplied past. They do not show that every fully specified physical theory has arbitrary future evolution.

No sampling model, likelihood, prior over α or forecast distribution was defined. The space between the curves therefore has no assigned confidence or credible probability.

Additional physical principles, a complete dynamical theory or later measurements can exclude continuation assumptions. The example does not prove that future physics is unlearnable in principle.

A useful forecast distinguishes uncertainty within a specified model from uncertainty about the continuation model. Report the tested domain, future assumptions, available discriminating evidence and the finite range where the calculation remains justified.

Worked reasoning

Compare α = 0 and α = 1/8 at u = 2

What differs after their identical supplied past?

For α = 0, h = 1, w = −1 and a = exp(2) ≈ 7.389056.
For α = 1/8, h = 1/2 and w = 0 at u = 2. Integrating its full rate gives a ≈ 5.319066.
The scale-factor ratio is about 0.719857; no probability has been attached to either rule.

The specified future stress and integrated expansion differ, even though the functions agree throughout the supplied past interval.

Try the reasoning

Do these functions jump in H, H′ or H″ at u = 0?
Show the answer and reasoning
No.
The max(u,0)³ dependence makes H twice continuously differentiable across the join.
Does the three-curve span represent a 95 percent prediction interval?
Show the answer and reasoning
No.
No probability model over continuation rules or sampling outcomes was supplied.
Does this example prove that no additional theory or observation can improve future prediction?
Show the answer and reasoning
No.
It demonstrates dependence on unspecified continuation assumptions, which additional physical information may restrict.

What to carry forward

An identical supplied past need not select a future rule within an unspecified effective model class.
The scale factor integrates the whole expansion-rate history.
Alternative curves, statistical forecast intervals and complete physical theories make different claims.

References

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