Scale Factor

Reference Material

• The Scale Factor (Wikipedia)
• Cosmic Scale Factor (Tim Trott)

The Scale Factor, $a(t)$

$a(t)$ relates physical ($r$) and comoving ($x$) coordinates in an expanding universe:

${\begin{aligned}r&=a(t)x\\{\dot {r}}&={\dot {a}}x+a{\dot {x}}=\underbrace {{\dot {a}} \over a} _{\equiv H}r+\underbrace {a{\dot {x}}} _{\equiv v_{p}}\\\end{aligned}}\,\!$

Thus, the two components of physical velocity are $H$ (the Hubble expansion parameter) and $v_{p}$ (the peculiar velocity, or motion relative to expansion) By convention, $t_{0}\equiv$ today and $a(t_{0})=1$.