Scale Factor

The scale factor a(t) is a dimensionless function of cosmic time that encodes the entire expansion history of a homogeneous, isotropic universe. In the FLRW metric every physical (proper) distance between two comoving objects is the product of a fixed comoving separation χ and the scale factor: d_proper(t) = a(t)·χ. By convention the scale factor is normalized to a = 1 at the present epoch, so a < 1 describes the smaller, denser past and a > 1 the larger future. A scale factor that doubles means every proper distance in the universe has doubled.

The time evolution of a(t) is not free: it is fixed by the Friedmann equations, which feed the matter, radiation, and dark-energy content of the cosmos into Einstein's field equations. Different compositions give qualitatively different histories — a universe that recollapses, one that coasts to a halt, or one that accelerates forever. The logarithmic rate of change of the scale factor, H = ȧ/a, is the Hubble parameter, so the present-day value H₀ is just today's expansion rate of a.

The scale factor is also the universe's clock and tape measure. Because wavelengths of light stretch in proportion to a, the cosmological redshift obeys 1 + z = a(now)/a(then): an observed redshift is a direct readout of how small the universe was when the light was emitted. Light from the cosmic microwave background, at z ≈ 1100, was released when a ≈ 1/1101 — when every cosmic distance was about 1100 times shorter than today.

Introduced implicitly by Alexander Friedmann in 1922 and by Georges Lemaître in 1927 as the time-dependent radius of curvature in their expanding-universe solutions, and given its modern coordinate-clean role by Howard Robertson and Arthur Walker in the 1930s.