# Difference between revisions of "Scale Factor"

## The Scale Factor, ${\displaystyle a(t)}$
${\displaystyle a(t)}$ relates physical (${\displaystyle r}$) and comoving (${\displaystyle x}$) coordinates in an expanding universe:
{\displaystyle {\begin{aligned}r&=a(t)x\\{\dot {r}}&={\dot {a}}x+a{\dot {x}}=\underbrace {{\dot {a}} \over a} _{\equiv H}r+\underbrace {a{\dot {x}}} _{\equiv v_{p}}\\\end{aligned}}\,\!}
Thus, the two components of physical velocity are ${\displaystyle H}$ (the Hubble expansion parameter) and ${\displaystyle v_{p}}$ (the peculiar velocity, or motion relative to expansion) By convention, ${\displaystyle t_{0}\equiv }$ today and ${\displaystyle a(t_{0})=1}$.