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.

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

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