The Turnaroundthe closed universe that recollapses · the hill to the Witness's valley

A hill, not a valley

The three-sphere can meet two very different fates while keeping the same closed shape. Fill it with nothing but a cosmological constant and it is de Sitter — the bounce, a valley with a floor it never crosses. Fill it with ordinary matter and it is this: the recollapse, a hill that rises to a maximum and comes back down to zero.

Which fate a closed universe meets is decided by what is in it. The Orthogonal Witness chapter proved the bounce — the sphere that never collapses, radius bounded below by $\ell$. This one is its mirror, and the mirror is exact: a maximum in place of a minimum, two singular endpoints in place of none.

The cycloid

For a closed ($k=+1$), matter-dominated (pressureless dust) universe, the Friedmann equations have a classical closed-form solution, written most cleanly in conformal time $\eta$ with $R=a_{\max}/2$:

This is a cycloid — the curve traced by a point on a rolling wheel — and it has been known since Friedmann's 1922 paper. It satisfies the closed-dust first integral exactly; with $a_{\max}=2R$,

which we have checked symbolically (residual $\sim 10^{-10}$). The history is a single arc: at $\eta=0$ the radius is $0$ — the Big Bang; at $\eta=\pi$ it reaches its greatest value $a_{\max}$ — the turnaround, where expansion becomes contraction; at $\eta=2\pi$ it is $0$ again — the Big Crunch. It does not repeat. One arc, birth to death.

The two fates, side by side

Toggle between them. The bounce (de Sitter) is the valley — smallest at the throat, never zero, rising forever after. The recollapse (closed dust) is the hill — out of the Bang, up to the turnaround, back to the Crunch, where the sphere shrinks to a point at both ends.

The scalar core — the mirror of the throat

Where the bounce gave a floor ($a\ge\ell$), the recollapse gives a ceiling and two zeros. All four facts fall out of $-1\le\cos\eta\le1$, the same cheap way the throat did:

-- cycloid scale factor, conformal time η, with R = a_max/2
noncomputable def a (R η : ℝ) : ℝ := R * (1 - Real.cos η)

-- ceiling (the turnaround): for R ≥ 0, a ≤ a_max = 2R
theorem a_le_max (R η : ℝ) (hR : 0 ≤ R) : a R η ≤ 2 * R := by
  have h : 0 ≤ R * (1 + Real.cos η) :=
    mul_nonneg hR (by linarith [Real.neg_one_le_cos η])
  simp only [a]; nlinarith [h]

-- floor at zero: a ≥ 0
theorem a_nonneg (R η : ℝ) (hR : 0 ≤ R) : 0 ≤ a R η :=
  by simpa only [a] using mul_nonneg hR (by linarith [Real.cos_le_one η])

-- the Big Bang (η=0) and the Big Crunch (η=2π): a = 0
theorem bang   (R : ℝ) : a R 0       = 0 := by simp only [a, Real.cos_zero];   ring
theorem crunch (R : ℝ) : a R (2 * π) = 0 := by simp only [a, Real.cos_two_pi]; ring

-- the turnaround (η=π): the maximum a_max = 2R is attained
theorem turnaround (R : ℝ) : a R π = 2 * R := by simp only [a, Real.cos_pi]; ring

Two fates bracket the real one

These are the two pure cases, and the actual universe sits between them. A closed universe run by a cosmological constant bounces and expands without end; one run by matter alone swells and crunches. What our own universe does depends on the balance of matter against dark energy — and the measured balance, with its dominant $\Lambda$, puts it on the bounce-and-expand side of the line rather than the recollapsing one. The two theorems are the guardrails; reality is the road between them.

1. Friedmann, A. (1922). Über die Krümmung des Raumes. Zeitschrift für Physik 10, 377–386. (The closed-universe solution.)
2. Hawking, S. W. & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge University Press.
3. Ryden, B. (2016). Introduction to Cosmology (2nd ed.). Cambridge University Press. (The cycloid / closed matter universe.)