# Difference between revisions of "Hubble's Law and Scale Factors"

## Hubble’s Law (1929)

Hubble’s Law is an empirical law stating that, on large scales, recessional velocity is proportional to distance from observer.

${v=Hr}\,\!$ where $H$ , the Hubble parameter, is not constant, but can vary slowly with time. By convention, $H$ is often expressed as $H=100\cdot h{km \over s\cdot Mpc}$ , where 1 parsec (pc) $\approx 3\cdot 10^{18}cm=3.26ly$ , is the distance at which 1 AU appears as 1 arcsec on the sky. The Hubble Space Telescope Key Project (Freedman et al. ApJ 553, 47, 2001) measured the present day value of Hubble Constant $H_{0}=72\pm 8{km \over s\cdot Mpc}$ , giving us that the current timescale for the expansion of the universe is $H_{0}^{-1}\approx {h \over 10^{11}}yrs\approx 9.778h^{-1}Gyrs$ .

## 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 $a(t_{0})=1$ .