View source for Hubble's Law and Scale Factors

===Short Topical Videos===
* [https://www.youtube.com/watch?v=1V9wVmO0Tfg Hubble's Law (Khan Academy)]
* [http://study.com/academy/lesson/hubbles-law-hubbles-constant.html Hubble's Law & Hubble's Constant (Artem Cherpasov)]


===Reference Material===
* [https://en.wikipedia.org/wiki/Hubble%27s_law Hubble's Law (Wikipedia)]
* [https://en.wikipedia.org/wiki/Scale_factor_(cosmology) The Scale Factor (Wikipedia)]
* [http://timtrott.co.uk/scale-factor/ Cosmic Scale Factor (Tim Trott)]

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\begin{document}
\section*{ Hubble's Law (1929) }

Hubble's Law is an empirical law stating that, on large scales, recessional 
velocity is proportional to distance from observer.
$$\boxed{v=Hr}$$ 
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) $\approx3\cdot10^{18}cm
=3.26ly$, 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 
$H_0^{-1}\approx{h\over 10^{11}}yrs\approx 9.778h^{-1}Gyrs$.

\section*{ The Scale Factor, $a(t)$ }

$a(t)$ relates physical ($r$) and {\it comoving} ($x$) coordinates in an
expanding universe:
$$\begin{align}
r&=a(t)x\\
\dot r&=\dot ax+a\dot x=\underbrace{\aa}_{\equiv H}r+\underbrace{a\dot x}_{\equiv v_p}\\
\end{align}$$
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 $a(t_0)=1$.


\end{document}
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