Friedmann equations
/ FREED-mahn /
If you want to know how a thrown ball moves, you use Newton's laws. If you want to know how the entire universe expands, slows, or speeds up, you use the Friedmann equations. They are the master equations of cosmology: a pair of formulas, derived from Einstein's theory of gravity, that govern how the scale factor a(t) — the universe's size dial — changes through time, given what the universe is made of.
The idea behind them is intuitive even if the mathematics is not. A universe filled with matter and energy pulls on itself through gravity, which tends to slow expansion, just as throwing a ball upward is slowed by Earth's pull. The Friedmann equations balance the universe's expansion rate against the total density of everything inside it — matter, radiation, and dark energy — plus the curvature of space. From these, the equations predict whether expansion decelerates, holds steady, or accelerates. Each ingredient behaves differently as space grows: ordinary matter thins out, radiation thins out and reddens even faster, while dark energy's density stays constant, which is why dark energy eventually wins and drives acceleration. They also define the critical density — the exact density that makes space flat.
The Friedmann equations are the engine of the standard Lambda-CDM model. Feed in the measured amounts of matter, dark matter, and dark energy, and they output the universe's complete expansion history: how big it was at any redshift, when it switched from slowing to speeding up, and its age of 13.8 billion years. They assume the cosmological principle — that the universe is uniform on large scales — which is what lets a single number a(t) describe the whole cosmos. They are a triumph of theory, but they are only as good as our census of what the universe contains, which is why the still-mysterious dark sector keeps them honest.
Plug today's measured mix — about 5 percent ordinary matter, 27 percent dark matter, 68 percent dark energy — into the Friedmann equations, and they reproduce the observed expansion history exactly, including the moment about 5 billion years ago when expansion stopped slowing and began to accelerate.
Given the universe's contents, the Friedmann equations predict its whole expansion history.
The Friedmann equations assume a homogeneous, isotropic universe (the cosmological principle). Their predictions are only as reliable as our census of cosmic contents — most of which, the dark sector, remains poorly understood.