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Space Sciences
Cosmology
1922
Research

Friedmann Equation

(a˙a)2=8πG3ρ−kc2a2+Λc23\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}

Expansion rate of the universe depends on matter, curvature, and dark energy density.

By Alexander Friedmann

Space Sciences
Friedmann Equation
1922 · Alexander Friedmann
Human Reviewed
84%

Rabbit Hole Mode

Five doors into the universe behind this equation. Choose your path.

Story PortalWho discovered this?Discover how Alexander Friedmann and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — cmb analysis.Math PortalWhat does it derive from?Trace the mathematical lineage from hubble law.Future PortalWhere is it going?Quantum gravity cosmology
Why it matters: Mathematical framework for modern cosmology and the Big Bang model.

Discoverers: Alexander Friedmann (1922)

What does it mean?

Expansion rate of the universe depends on matter, curvature, and dark energy density.

Why should I care?

Mathematical framework for modern cosmology and the Big Bang model.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
aaScale factor—Universe expansion
ρρDensity—Energy density
kkCurvature—Spatial curvature
ΛΛCosmological constant—Dark energy

Worked Example

Matter-dominated era: a ∝ t^(2/3).

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Sources & further reading

Search WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Friedmann Equation

(a˙a)2=8πG3ρ−kc2a2+Λc23\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}

Real-world impact

Dark-energy cosmology

Expansion history measured by Planck, JWST, and supernova surveys.

Photo: Unsplash — deep-field galaxies

Expansion rate of the universe depends on matter, curvature, and dark energy density.

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