Relativity and gravitational theory Definition

Scale factor

Builds on Friedmann–Lemaître–Robertson–Walker metric

Definition

Let $(M, g)$ be a Friedmann–Lemaître–Robertson–Walker spacetime $M = I \times \Sigma$ with metric $g = -dt^2 + a(t)^2 \gamma_k$ , where $\gamma_k$ is a Riemannian metric of constant curvature $k$ on the three-manifold $\Sigma$ . The smooth positive function $a \colon I \to (0, \infty)$ is the scale factor. The function $H = \dot a / a$ on $I$ , where the dot is the derivative in $t$ , is the Hubble parameter.

Remarks

The pair $(a, \gamma_k)$ is fixed only up to $a \mapsto \lambda a$ and $\gamma_k \mapsto \lambda^{-2} \gamma_k$ for $\lambda > 0$ ; the Hubble parameter does not change. A common normalization is $a(t_0) = 1$ at the present time $t_0$ ; another, for $k \neq 0$ , is $k = \pm 1$ . Example: for $a(t) = t^{2/3}$ on $I = (0, \infty)$ , the Hubble parameter is $H = 2/(3t)$ . Non-example: a function $a$ that vanishes somewhere on $I$ is not a scale factor, because then $g$ is degenerate there. For two comoving points at comoving distance $\chi$ , the proper distance at time $t$ along the slice is $D(t) = a(t)\chi$ .

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Cosmological principle concept
Space form concept
Spacetime concept
Friedmann–Lemaître–Robertson–Walker metric example
Scale factor definition

Tag	`SHZHY79AWK`
URI	`m2://SHZHY79AWK`
Kind	definition
Writing	W0 stub
MSC	83F05
Defines	Scale factor
Uses	Friedmann–Lemaître–Robertson–Walker metric
Changed	3 November 2026

Scale factor

Definition

Remarks

Reading path

Further reading

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