Hubble Constant

In a simple ``Big Bang'' cosmology without inflation or deceleration, a pair of galaxies separated today by a distance $d$ have been moving apart for the entire history of the universe, a time $t_0$. Their relative velocity is then

$$v = \frac{d}{t_0}$$ (4)

The Hubble Law is the observed relationship

$$v = H d$$ (5)

where $v$ is the measured radial velocity, $d$ is the apparent distance of the galaxy, and $H$ is the Hubble Constant. We identify the Hubble Constant in the Big Bang model as one over the age of the universe -

$$H = \frac{1}{t_0}$$ (6)

Conventionally the Hubble Law is deduced by observing the redshift of spectral lines and interpreting that as a Doppler shift due to a velocity of recession. The connection is

$$\frac{\lambda_{obs}}{\lambda_e} = \sqrt{\frac{c+v}{c-v}}$$ (7)

where $\lambda_{obs}$ is the wavelength that we measure and $\lambda_e$ is the wavelength as it was emitted in the reference frame of galaxy. In interpreting observations this way we make the well-justified assumption that the laws of atomic and molecular physics were the same in the galaxy at the time the light was emitted as they are in our galaxy today. The redshift parameter $z$ is defined by

$$1 + z = \frac{\lambda_{obs}}{\lambda_e}$$ (8)

because in the limit of low $v$

$$z = \frac{\Delta\lambda}{\lambda} = \frac{v}{c}$$ (9)