Scale factor (cosmology)

The scale factor, cosmic scale factor or sometimes the Robertson-Walker scale factor[1] parameter of the Friedmann equations is a function of time which represents the relative expansion of the universe. It relates the proper distance (which can change over time, unlike the comoving distance which is constant) between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time $t$ to their distance at some reference time $t_0$. The formula for this is:

$d(t) = a(t)d_0,\,$

where $d(t)$ is the proper distance at epoch $t$, $d_0$ is the distance at the reference time $t_0$ and $a(t)$ is the scale factor.[2] Thus, by definition, $a(t_0) = 1$.

The scale factor is dimensionless, with $t$ counted from the birth of the universe and $t_0$ set to the present age of the universe: $13.798\pm0.037\,\mathrm{Gyr}$[3] giving the current value of $a$ as $a(t_0)$ or $1$.

The evolution of the scale factor is a dynamical question, determined by the equations of general relativity, which are presented in the case of a locally isotropic, locally homogeneous universe by the Friedmann equations.

The Hubble parameter is defined:

$H \equiv {\dot{a}(t) \over a(t)}$

where the dot represents a time derivative. From the previous equation $d(t) = d_0 a(t)$ one can see that $\dot{d}(t) = d_0 \dot{a}(t)$, and also that $d_0 = \frac{d(t)}{a(t)}$, so combining these gives $\dot{d}(t) = \frac{d(t) \dot{a}(t)}{a(t)}$, and substituting the above definition of the Hubble parameter gives $\dot{d}(t) = H d(t)$ which is just Hubble's law.

Current evidence suggests that the expansion rate of the universe is accelerating, which means that the second derivative of the scale factor $\ddot{a}(t)$ is positive, or equivalently that the first derivative $\dot{a}(t)$ is increasing over time.[4] This also implies that any given galaxy recedes from us with increasing speed over time, i.e. for that galaxy $\dot{d}(t)$ is increasing with time. In contrast, the Hubble parameter seems to be decreasing with time, meaning that if we were to look at some fixed distance d and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.[5]

According to the Friedmann–Lemaître–Robertson–Walker metric which is used to model the expanding universe, if at the present time we receive light from a distant object with a redshift of z, then the scale factor at the time the object originally emitted that light is $a(t) = \frac{1}{1 + z}$.[6][7]