Friedmann's Three Fates

Plate 71 speaks for itself: “One equation, three possible endings.”

The Physics

Friedmann's equation ties the expansion rate to density, curvature and the cosmological constant. Depending on their balance, a universe recollapses, expands forever, or accelerates.

H² = 8πGρ/3 − k c²/a² + Λ c²/3

• H: Hubble parameter, s⁻¹
• G: gravitational constant, m³ kg⁻¹ s⁻²
• ρ: density, kg m⁻³
• k: curvature constant, m⁻² (comoving)
• c: speed of light, m s⁻¹
• a: scale factor, dimensionless
• Λ: cosmological constant, m⁻²

Planck 2018: H₀ = 67.36 ± 0.54 km/s/Mpc. Friedmann, Z. Phys. 10, 377 (1922).

Honest caveat: With Λ the 'three fates' are not tied to curvature alone; a closed universe with enough Λ can still expand forever.

Sources

• A. Friedmann, Z. Phys. 10, 377 (1922).
• Planck Collaboration, Astron. Astrophys. 641, A6 (2020).

On the poster

Plate 071, “Friedmann's Three Fates”: the pixel-art plate with its slogan “One equation, three possible endings.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.