Cosmology · Conditional futures of the Universe

Which assumptions control a future extrapolation

Compare three supplied universes with the same reference expansion rate and identify what must be assumed before extending a history into the future.

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

Distinguish a measured or specified state from an evolution rule.
Compare conditional futures sharing a reference scale factor and expansion rate.
Identify assumptions about components, conservation and gravity in an extrapolation.

A present state needs a rule for what comes next

An expansion rate describes how quickly the scale factor is changing at a specified epoch. Predicting a later scale factor also requires a rule for how that rate will evolve.

In a cosmological model, the rule depends on the gravitational equations, the components and their pressure, conservation or interactions, and the initial conditions. A component’s future behaviour is an assumption unless the theory and evidence establish its continuation.

Compare three supplied examples that agree only at a reference epoch. Set a = 1 and the same positive H_ref there, with spatial flatness and classical general relativity in each example.

These choices do not say that the examples fit the same complete observed past. They isolate why knowing one expansion rate is insufficient to choose a unique future.

Three separate fluid hypotheses

Each example contains one positive homogeneous separately conserved fluid, with a constant equation-of-state ratio w = P/ε. Choose w = 0, −2/3 or −1 as three alternatives.

All have the same reference energy density ε_ref = 3c²H_ref²/(8πG). Their pressures differ, and so do the gravitational accelerations even at that reference.

Conservation gives ε/ε_ref = a⁻³⁽¹⁺ʷ⁾. The positive Friedmann branch then determines how a evolves. These are separate universes, rather than three components added into one model.

Use dimensionless future time u = H_ref(t−t_ref). Its zero marks the supplied reference. No numerical cosmic age, date or measured present parameter is assigned.

The same initial rate produces different later expansion

For w > −1, define b = 3(1+w)/2. The conditional solution is a(u) = (1+bu)¹ᐟᵇ and H/H_ref = 1/(1+bu). For w = −1, a = exp(u) and H/H_ref = 1.

At u = 0, all three give a = 1 and H/H_ref = 1. At u = 1, the scale factors are about 1.842016, 2.25 and 2.718282.

The deceleration parameters q = −äa/ȧ² are 1/2, −1/2 and −1 respectively. The matter example decelerates, the other two accelerate, and the accelerating w = −2/3 example still has a decreasing H.

The table displays consequences of the selected fluid assumptions. Its alternatives are neither observed outcomes nor probabilities for the actual Universe.

Three supplied flat single-fluid futures, each normalized to a = 1 and H = H_ref at u = 0
w	a at u = 0.5	a at u = 1	a at u = 2	H/H_ref at u = 2	q
0	1.452196	1.842016	2.519842	0.25	0.5
−2/3	1.5625	2.25	4	0.5	−0.5
−1	1.648721	2.718282	7.389056	1	−1

Three separate flat classical-GR universes contain one positive homogeneous, separately conserved fluid each, with pressure ratio w = 0, −2/3 or −1. They agree only at reference a = 1 and H = H_ref, with energy density ε_ref = 3c²H_ref²/(8πG) in J/m³. Their pressures and histories need not agree. Dimensionless future time u = H_ref(t−t_ref) runs from 0 to 2. Panel A has a zero-based scale-factor axis from 0 to 8. Panel B has a separate zero-based H/H_ref axis from 0 to 1.1. Dust is a solid blue line with circles; w = −2/3 is dashed orange with squares; w = −1 is dotted green with diamonds. At u = 1 the scale factors are approximately 1.842016, 2.25 and 2.718282. At u = 2 they are approximately 2.519842, 4 and 7.389056, with expansion-rate ratios 0.25, 0.5 and 1. For w > −1, b = 3(1+w)/2 gives a = (1+bu)^(1/b) and H/H_ref = 1/(1+bu). For w = −1, a = exp(u) and H/H_ref = 1. The deceleration parameters are 0.5, −0.5 and −1; the w = −2/3 example accelerates while H decreases. Markers show supplied finite samples, not observations. These alternatives are neither fits to a shared complete past nor future probabilities. Changing pressure, interactions, components or gravity requires a new continuation. Original Ark teaching model.

Check the continuation before extending the curve

The constant-w power law must be reconsidered if pressure evolves, components exchange energy or a new component becomes relevant. The same is true if the assumed gravitational dynamics cease to apply.

Spatial geometry alone does not determine expansion’s fate for arbitrary stress-energy. In particular, a statement about flatness cannot replace the component and future-evolution assumptions.

Other remote-future questions need additional physics. Stellar populations depend on gas supply and stellar evolution; matter stability depends on particle properties; semiclassical black-hole emission has its own regime of validity.

A conditional calculation remains useful when these assumptions are visible. It can show which observations or theoretical results would distinguish alternatives and which prediction would change if an assumption failed.

A scenario is not an established endpoint

A forecast should specify its tested epoch, chosen evolution law, initial conditions and range of application. A formal mathematical limit should be distinguished from a validated description of its physical endpoint.

Parameter uncertainty within one model and uncertainty about the model family are different. Improving a parameter estimate does not automatically establish that its assumed law persists indefinitely.

These three examples differ in their stress already at the reference epoch. A later lesson constructs continuations that instead agree throughout an entire supplied past interval, making a separate point about model assumptions.

For now, carry a simple question into each future scenario: what must remain true for this conclusion to follow, and which piece of evidence supports that requirement?

Worked reasoning

Compare the scale factor at u = 1

Do identical reference a and H select a unique later scale factor?

For dust, b = 3/2, so a = (5/2)²ᐟ³ ≈ 1.842016.
For w = −2/3, b = 1/2 and a = (3/2)² = 2.25.
For w = −1, a = e ≈ 2.718282.

No: the stated stress assumptions produce different futures despite the common reference rate.

Try the reasoning

Does knowing H at one epoch determine w?
Show the answer and reasoning
No.
Pressure and component assumptions supply additional information about the rate’s evolution.
Can the w = −2/3 example accelerate while H decreases?
Show the answer and reasoning
Yes.
Its q is −1/2 while H/H_ref = 1/(1+u/2) decreases.
Have these three examples been shown to fit the same complete observed past?
Show the answer and reasoning
No.
They were normalized to the same a and H at one supplied reference epoch only.

What to carry forward

A reference expansion rate does not supply its future evolution law.
Components, pressure, conservation and gravity determine a conditional continuation.
A useful future scenario states both its conclusions and their assumptions.

References

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