THE FRIEDMANN EQUATION

The expansion of the whole universe, read off its density. One line turns a snapshot of matter, radiation and curvature into the fate of everything: (ȧ/a)² = (8πG/3)ρ − kc²/a² + Λc²/3. The critical density splits flat from curved; Ω=ρ/ρc decides the geometry. Rendered, not quoted.

source Friedmann, A. — Über die Krümmung des Raumes, Z. Phys. 10, 377–386 (1922) · doi:10.1007/BF01332580 · ADS 1922ZPhy...10..377F

Blue Team · builds & defends
3

The Model

A homogeneous, isotropic universe (the cosmological principle). Its scale factor a(t) stretches every distance; Einstein's equations collapse to two ODEs.

H² = (ȧ/a)² = (8πG/3)ρ − kc²/a² + Λc²/3
ä/a = −(4πG/3)(ρ + 3p/c²) + Λc²/3

Gravity decelerates (the − sign on density); curvature k ∈ {−1,0,+1}; Λ pushes outward.

5

The Lineage

the-hubble-law → the-friedmann-equation. Hubble measured v = H₀d — a snapshot of one instant's expansion rate. Friedmann (1922, seven years earlier) had already written its engine: H itself is dynamical, driven by ρ against curvature.

The critical density ρc = 3H₀²/(8πG) is the exact seam between the two spheres — Hubble's number H₀ sets what density would make space flat.

7

The Witness

Live re-check of the density–curvature balance and the ρc/Ω geometry map. Confirms on load; flips red the instant the panel’s density term is tampered.

witness …
The Machine
4

Data In in ↓

Set the Hubble rate and the matter density, in units of the critical density (Ωm).

↓
0

The Panel lit

Live Friedmann engine. Every number below is computed from the constants and the two inputs — nothing is stored.

Scale factor a(t) integrated forward from today (a=1): closed recollapses, flat coasts, open expands free.

↓
8

Data Out out ↓

The proven result: at H₀≈70, the critical density is a few hydrogen atoms per cubic metre, and Ω alone fixes the sign of the curvature.

booting…
Red Team · attacks & breaks
1

The Adversary wall

“Ω is a free dial — pick any density and space stays flat.” Wall: flatness is k=0, which forces ρ=ρc exactly. Off that razor, curvature is non-zero by construction; the balance is not negotiable.
“Expansion needs a force pushing outward.” Wall: no — ȧ is an initial condition. Gravity only sets ä (deceleration). Nothing pushes unless Λ>0.
2

The Graveyard

Static, eternal universe (Einstein 1917, Λ tuned by hand).
→ That balance is unstable; Friedmann & Hubble showed a(t) actually evolves.

ρ ~ a−4 for matter.
→ Matter dilutes as a−3 (volume only); a−4 is radiation (volume + redshift).

Λ>0 decelerates.
→ A positive Λ gives ä>0 — it accelerates the expansion.

6

The Tamper

Planted void: flip the sign of the density term so gravity would push expansion instead of decelerating it. ρc goes negative, Ω inverts, the geometry map breaks — the Witness (7) catches it live.