# 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 ${\displaystyle t}$ to their distance at some reference time ${\displaystyle t_{0}}$. The formula for this is:

${\displaystyle d(t)=a(t)d_{0},\,}$

where ${\displaystyle d(t)}$ is the proper distance at epoch ${\displaystyle t}$, ${\displaystyle d_{0}}$ is the distance at the reference time ${\displaystyle t_{0}}$ and ${\displaystyle a(t)}$ is the scale factor.[2] Thus, by definition, ${\displaystyle a(t_{0})=1}$.

The scale factor is dimensionless, with ${\displaystyle t}$ counted from the birth of the universe and ${\displaystyle t_{0}}$ set to the present age of the universe: ${\displaystyle 13.798\pm 0.037\,\mathrm {Gyr} }$[3] giving the current value of ${\displaystyle a}$ as ${\displaystyle a(t_{0})}$ or ${\displaystyle 1}$.

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:

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

where the dot represents a time derivative. From the previous equation ${\displaystyle d(t)=d_{0}a(t)}$ one can see that ${\displaystyle {\dot {d}}(t)=d_{0}{\dot {a}}(t)}$, and also that ${\displaystyle d_{0}={\frac {d(t)}{a(t)}}}$, so combining these gives ${\displaystyle {\dot {d}}(t)={\frac {d(t){\dot {a}}(t)}{a(t)}}}$, and substituting the above definition of the Hubble parameter gives ${\displaystyle {\dot {d}}(t)=Hd(t)}$ 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 ${\displaystyle {\ddot {a}}(t)}$ is positive, or equivalently that the first derivative ${\displaystyle {\dot {a}}(t)}$ 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 ${\displaystyle {\dot {d}}(t)}$ 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 ${\displaystyle a(t)={\frac {1}{1+z}}}$.[6][7]