Cosmology · Conditional futures of the Universe

Recollapse and other conditional possibilities

Compare three explicitly different classical expansion models, retaining the contracting branch and the assumptions behind a formal Big Rip.

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 geometry and a positive reference H do not alone determine fate.
Continue a supplied recollapsing solution through its turnaround.
State the persistent phantom assumption behind a finite-time divergence.

Three alternatives share one reference state

Supply three spatially flat classical general-relativistic models with a = 1 and H = H_ref > 0 at u = 0, where u = H_ref(t−t_ref). They agree at that reference, rather than throughout a measured cosmic history.

The first contains only a positive cosmological constant. The second contains pressureless matter and a negative cosmological constant. The third contains only a positive-density constant-w phantom fluid with w = −4/3.

These are separate backgrounds. Their energy and pressure assumptions differ, so their future behaviour can differ despite a common reference rate and spatial geometry.

No probability, empirical fit, present cosmological parameter or actual cosmic ending date is assigned. The comparison shows consequences of clearly stated mathematical continuations.

A flat model can turn around and contract

Define ε_* = 3c²H_ref²/(8πG). In the second model, supply ε_m = 2ε_*a⁻³ and ε_Λ = −ε_*, with P_m = 0 and P_Λ = +ε_*. The Friedmann equation is H²/H_ref² = 2a⁻³−1.

A signed solution is a = (2sin²θ)¹ᐟ³ and H/H_ref = cot θ, where θ = π/4+3u/2. Its future classical domain is 0 ≤ u < π/2.

At u = π/6, θ reaches π/2, H vanishes and a reaches 2¹ᐟ³ ≈ 1.259921. The acceleration remains negative, so the solution continues with H < 0 onto contraction.

Taking only the positive square root of H² after the turnaround would lose the contracting branch. The formal a → 0 limit at u → π/2 is a classical model boundary, rather than a validated description of the final physical regime.

A Big Rip needs a particular continuing stress law

For the positive-Λ-only model, a = exp(u) and H/H_ref = 1. Under its stipulated continuation, expansion persists with no finite-u divergence of a.

For the separate constant-w = −4/3 positive fluid, conservation gives ε/ε_* = a. Its expanding solution is a = (1−u/2)⁻² and H/H_ref = (1−u/2)⁻¹ on 0 ≤ u < 2.

That supplied phantom continuation has a formal finite-time divergence at u → 2, commonly called a Big Rip. Persistent constant phantom behaviour and the specified gravitational/component model are essential to this conclusion.

A temporary episode of w < −1 does not by itself establish this endpoint. No stable microscopic phantom theory or quantum completion is demonstrated by these homogeneous equations.

Separate supplied backgrounds; negative H indicates contraction in the dust-plus-negative-Λ model
u	Positive-Λ a	Dust + negative-Λ a	Dust + negative-Λ H/H_ref	Phantom a	Phantom H/H_ref
0	1	1	1	1	1
0.5	1.648721	1.259395	0.035413	1.777778	1.333333
0.75	2.117000	1.211481	−0.353289	2.56	1.6
1	2.718282	1.044986	−0.867562	4	2
1.5	4.481689	0.282185	−9.381260	16	4

Optional mathematical path: use more than the squared rate

For dust plus negative Λ, let y = a³ᐟ². Integrating the classical equations gives y = √2 sin θ on the stated branch. Its derivative changes sign when θ passes π/2.

The acceleration is (ä/a)/H_ref² = −a⁻³−1, negative at the turnaround as well as before it. This supplies the dynamical continuation through H = 0.

For the phantom alternative, H/H_ref = √a gives da/du = a³ᐟ². Integrating from a = 1 yields a⁻¹ᐟ² = 1−u/2, with the positive-a expanding branch ending formally at u = 2.

These derivations use different components and pressure laws. One cannot splice their endpoints into a single physical history without specifying a new consistent evolution model.

Three separate flat classical-GR backgrounds agree only at reference a = 1 and H/H_ref = 1. Dimensionless time u = H_ref(t−t_ref) is shown from 0 to 1.5. Pure positive Λ is solid blue with circles: a = exp(u), H/H_ref = 1. Dust plus negative Λ is dashed orange with squares: ε_m/ε_* = 2a^−3, ε_Λ/ε_* = −1, P_Λ/ε_* = +1, θ = π/4+3u/2, a = (2sin²θ)^(1/3) and H/H_ref = cot θ. Separate positive-density persistent phantom w = −4/3 is dotted green with diamonds: a = (1−u/2)^−2 and H/H_ref = (1−u/2)^−1. Panel A has a zero-based scale-factor ordinate from 0 to 17. Panel B has a separate signed expansion-rate ordinate from −10 to 5, with an explicit zero line. Violet triangles mark dust turnaround at u = π/6 ≈ 0.523599, a = 2^(1/3) ≈ 1.259921 and H = 0. Negative acceleration continues onto H < 0 contraction. At u = 1.5, dust has a ≈ 0.282185 and H/H_ref ≈ −9.381260, while phantom has a = 16 and H/H_ref = 4. Taking only the positive square root of the dust H² equation would lose its contracting branch. Formal dust collapse at u → π/2 and phantom divergence at u → 2 lie outside the displayed interval and are excluded singular limits. No infinity or singular-endpoint marker is drawn. A temporary w < −1 episode does not establish the persistent phantom result. No physical ending date, empirical fit, probability or microscopic viability is asserted. Original Ark teaching model.

State the assumptions behind the proposed fate

Recollapse is not determined by spatial curvature alone when additional stress-energy components are allowed. The negative constant in the flat example changes the future despite its zero spatial curvature.

Likewise, a positive H at one epoch does not prove unlimited future expansion. It specifies an instantaneous expanding state, while subsequent stress and dynamics determine the continuation.

A Big Crunch or Big Rip in these equations marks where the classical extrapolation becomes singular. Deciding what physics replaces that description requires information beyond the solution itself.

Compare scenarios through their assumptions and testable consequences, keeping finite-domain predictions separate from unlimited extrapolations. Agreement with a tested epoch does not establish that a chosen stress law persists forever.

Worked reasoning

Follow the negative-Λ turnaround

Does the classical solution end merely because H reaches zero?

At u = π/6, the scale factor is 2¹ᐟ³ and H = 0.
The acceleration remains negative: (ä/a)/H_ref² = −1/2−1 = −3/2.
The signed solution H/H_ref = cot(π/4+3u/2) becomes negative afterward.

No: it continues onto contraction; the later classical singular limit is a separate boundary.

Try the reasoning

What are a and H/H_ref at u = 1 for the supplied phantom model?
Show the answer and reasoning
a = 4 and H/H_ref = 2.
Substitute 1−u/2 = 1/2 into the two stipulated expressions.
Does flat spatial geometry guarantee endless expansion in these examples?
Show the answer and reasoning
No.
The supplied flat dust-plus-negative-Λ model turns around and contracts.
Does any finite episode with w < −1 establish the displayed Big Rip?
Show the answer and reasoning
No.
The finite endpoint follows from persistent constant phantom stress and the specified classical dynamics.

What to carry forward

Fate depends on future stress and dynamics as well as a reference state.
A recollapsing solution needs a signed contracting branch after turnaround.
Formal singular endpoints are conditional limits of the supplied models, rather than established cosmic forecasts.

References

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