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The Friedmann equations: the rules of expansion
1922· Theory
ProvenTested many times; part of working physics.
Two equations, derived from general relativity, that link how fast the universe expands and accelerates to what it contains.
See it on the Science map →How it works
The first sets the expansion rate by the density of everything in space, minus a term for curvature. The second says matter and radiation brake the expansion, while anything with strong negative pressure, such as dark energy, speeds it up. Each ingredient thins differently as space grows: matter with volume, radiation faster because its waves also stretch, a cosmological constant not at all. So radiation ruled until about 50,000 years, then matter, and now dark energy.
Why it matters
They turn a list of ingredients into a full history: age, past temperature, how distances grew, and future fate. All of cosmology's headline numbers pass through them.
Think of it like
Like a ball thrown upward: its speed against Earth's pull decides whether it falls back or escapes. Dark energy is like a small rocket strapped to the ball that eventually wins.
How we know
With today's measured ingredients they give an age of about 13.8 billion years, just older than the oldest stars (about 12–13 billion), as it must be. The same history also predicts the afterglow and the hydrogen-helium mix correctly.
Still a mystery
They assume matter is spread evenly on large scales. Whether real-world lumpiness subtly changes the average expansion is debated, though most studies find the effect tiny.
Connections (4)
▸Alexander Friedmann derived The Friedmann equations: the rules of expansionProven
The relationship
Einstein's equations are ten tangled equations for spacetime's shape. Friedmann assumed the universe looks the same everywhere and in every direction on large scales. That collapsed them into two equations for one quantity, the scale factor: how big the universe is compared with today. Curvature became a choice among three shapes: closed like a sphere, flat, or open like a saddle.
How the link was found
Friedmann's 1922 paper treated closed, positively curved space; his 1924 paper added open space. The flat case was popularized by Einstein and de Sitter in 1932.
Why it's proven
The derivation is exact mathematics. Lemaître reached the same equations independently in 1927, and every modern cosmology calculation still solves them.
Open questions and mysteries
Is the universe uniform enough for the shortcut? Surveys find uniformity beyond about 300 million light-years, though a few claimed giant structures may strain it.
▸General relativity: gravity is curved spacetime yields The Friedmann equations: the rules of expansionProven
The relationship
General relativity links spacetime's curvature to the energy, momentum and pressure it contains. For a uniform universe, the curvature side reduces to the expansion rate and its acceleration, and the content side to density and pressure. One twist has no Newtonian counterpart: pressure itself gravitates. That is why negative pressure can make expansion accelerate.
How the link was found
Einstein's equations date from November 1915. In 1934 Edward Milne and William McCrea showed the first Friedmann equation also follows from Newton's gravity for an expanding ball; the pressure term needs Einstein.
Why it's proven
General relativity passes every precision test, from Mercury to gravitational waves. On cosmic scales, Euclid and DESI compare the expansion with how fast structures grow, a direct check.
Open questions and mysteries
Does general relativity hold unchanged over billions of light-years? Modified-gravity theories try to explain acceleration without dark energy; so far none beats the standard version.
▸The Friedmann equations: the rules of expansion predicted The universe is expandingProven
The relationship
Given today's expansion rate and contents, the equations predict the whole past: how much space has stretched since any light was emitted (its redshift), how distance relates to redshift, and the time since everything was densely packed. Run back to the first second, they give a temperature of about 10 billion degrees, which sets element formation.
How the link was found
Friedmann's 1922 solutions came before any expansion data; Hubble's 1929 measurements then fit them. Hubble's rate, however, gave an age of only about 2 billion years, younger than geologists' age for Earth.
Why it's proven
Walter Baade's 1952 recalibration of star distances fixed the age conflict. Today supernovae and galaxy surveys trace distance against redshift over more than 10 billion years, matching the predicted curve.
Open questions and mysteries
The equations need the contents as input; they cannot say what dark matter and dark energy are. And today's rate itself is disputed, the Hubble tension.
▸The Friedmann equations: the rules of expansion are the engine of ΛCDM: the standard model of cosmologyProven
The relationship
ΛCDM is the Friedmann equations with a specific recipe plugged in: radiation, ordinary matter, cold dark matter and a cosmological constant, in flat space. The equations then fix the background: the expansion rate and temperature at every moment. On that backdrop, extra equations track how small ripples grow, giving the afterglow's spots and later the cosmic web. Change the recipe and every prediction shifts together.
How the link was found
The ingredients were assembled from the 1980s; the full recipe, including Λ, became standard between 1998 and 2003. Computer codes such as CMBFAST (1996) and CAMB (1999) made the calculations routine.
Why it's proven
One solution of the equations fits the afterglow (from 380,000 years) and galaxy distances (up to the present) at once, a strong cross-check of the whole expansion history.
Open questions and mysteries
The Hubble tension and DESI's dark-energy hint could each mean the recipe is wrong, not the equations. Or the equations' smoothness assumption might matter more than thought.