= Scale factor
{disambiguate=cosmology}
{title2=$a(t)$}
{wiki}
= Scale factor
{synonym}
The <scale factor> $a(t)>0$ measures how spatial lengths in an <FLRW metric> change with <cosmic time> $t$. Choose $a$ to be a <dimensionless quantity> with $a(t_0)=1$ at a reference epoch, usually today. For two <comoving observers>, their fixed radial <comoving distance> $\chi$ and their <proper distance in cosmology> $d(t)$ satisfy
$$
d(t)=a(t)\chi,\qquad \chi=\text{constant}.
$$
The <proper distance in cosmology> is measured along a spatial slice at one <cosmic time>; it is not the distance that a travelling <photon> covers between emission and reception. Only ratios of the <scale factor> have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi/C$ leaves $d$ unchanged.
For example, take $\chi=100\,\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million <parsecs>. At $a=0.5,1,1.5$, the same pair of <comoving observers> has <proper distance in cosmology> $50,100,150\,\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal <comoving observers>; internal motions in <galaxies> need not follow this background scaling.
\Image[/expanding-comoving-grid.png]
{title=The same <comoving coordinates> at three values of the <scale factor>, and the <Hubble flow> recentered on two <comoving observers>}
{description=The top row keeps the same marked <comoving observers> and the same plotting scale: adjacent values of <proper distance in cosmology> are $0.5,1,1.5$ arbitrary length units. The bottom row shows equal local patches of an extended ideal grid at $a=1$; arrows represent the <Hubble flow> relative to each marked <comoving observer>. The drawn grid is a schematic, not a model of the arrangement of <galaxies>.}
The beauty of the <scale factor> is that one function describes the changing spatial ruler for every <comoving observer>. <Spatial homogeneity> gives no preferred centre of <cosmic expansion>: in a flat spatial patch, relative positions obey $\mathbf r-\mathbf r_O=a(\mathbf x-\mathbf x_O)$, so recentering on any <comoving observer> preserves <Hubble's law>. The evolution of the <scale factor> is determined by the <Friedmann equations>, while its ratio between emission and reception determines <cosmological redshift>.
Back to article page
