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

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===Reference Material=== | ===Reference Material=== | ||

* [https://en.wikipedia.org/wiki/Hubble%27s_law Hubble's Law (Wikipedia)] | * [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)] | ||

<latex> | <latex> | ||

− | \ | + | \documentstyle[11pt]{article} |

− | \ | + | |

+ | \def\:{\ddot } | ||

+ | \def\.{\dot } | ||

+ | \def\^{\hat } | ||

+ | \def\_{\bar } | ||

+ | \def\~{\tilde } | ||

+ | \def\hf{\frac12} | ||

+ | \def\imply{\Rightarrow} | ||

+ | \def\inv#1{{1\over #1}} | ||

+ | \def\ddt{{d\over dt}} | ||

+ | \def\aa{{\dot a \over a}} | ||

+ | \def\adda{{\ddot a \over a}} | ||

+ | \def\thnot{\theta_0} | ||

+ | \def\etot{\Omega_0} | ||

+ | \def\econs{\Omega_{0,\Lambda}} | ||

+ | \def\emat{\Omega_{0,M}} | ||

+ | \def\econs{\Omega_{0,\Lambda}} | ||

+ | \def\p{^\prime} | ||

+ | \def\iff{\Leftrightarrow} | ||

+ | \def\xv{{\vec x}} | ||

+ | \def\pv{{\vec p}} | ||

+ | \def\vv{{\vec v}} | ||

+ | \def\ppt{{\partial\over\partial t}} | ||

+ | \def\ddt{\frac{d}{dt}} | ||

+ | \def\epot{{8\pi \over 3}} | ||

+ | |||

+ | \usepackage{fullpage} | ||

\usepackage{amsmath} | \usepackage{amsmath} | ||

− | |||

− | |||

\usepackage{eufrak} | \usepackage{eufrak} | ||

\begin{document} | \begin{document} | ||

− | \section{Hubble's Law} | + | \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} | \end{document} | ||

− | < | + | </latex> |

## Latest revision as of 13:43, 17 January 2017

### Short Topical Videos[edit]

### Reference Material[edit]

## Hubble’s Law (1929)

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

where , the Hubble parameter, is not constant, but can vary slowly with time. By convention, is often expressed as , where 1 parsec (pc) , 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 , giving us that the current timescale for the expansion of the universe is .

## The Scale Factor,

relates physical () and *comoving* () coordinates in an expanding universe:

Thus, the two components of physical velocity are (the Hubble expansion parameter) and (the peculiar velocity, or motion relative to expansion) By convention, today and .