# Hubble's Law and Scale Factors

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## Hubble’s Law (1929)

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

${\displaystyle {v=Hr}\,\!}$

where ${\displaystyle H}$, the Hubble parameter, is not constant, but can vary slowly with time. By convention, ${\displaystyle H}$ is often expressed as ${\displaystyle H=100\cdot h{km \over s\cdot Mpc}}$, where 1 parsec (pc) ${\displaystyle \approx 3\cdot 10^{18}cm=3.26ly}$, 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 ${\displaystyle H_{0}=72\pm 8{km \over s\cdot Mpc}}$, giving us that the current timescale for the expansion of the universe is ${\displaystyle H_{0}^{-1}\approx {h \over 10^{11}}yrs\approx 9.778h^{-1}Gyrs}$.

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