The fundamental observation underlying the Big Bang theory of cosmology is
Edwin Hubble's 1929 discovery that the arrangement of galaxies throughout
space is expanding. Like the photons of the cosmic microwave background, the
light from distant galaxies is stretched to longer wavelengths by the
expansion of the universe. This is not a Doppler shift: Except for their local
motions around each other, the galaxies are essentially at rest in space; it
is space itself that expands. The ratio of the wavelength of light received at
Earth from a galaxy, \(\lambda_{\text {rec }}\), to its wavelength at emission,
\(\lambda_{\text {emit }}\), is equal to the ratio of the scale factor (radius
of curvature) \(a\) of the universe at reception to its value at emission. The
redshift, \(z\), of the light-which is what Hubble could measure \(-\) is defined
by \(1+z=\lambda_{\text {rec }} / \lambda_{\text {emit }}=a_{\text {rec }} /
a_{\text {emit }}\).
a) Hubble's Law states that the redshift, \(z\), of light from a galaxy is
proportional to the galaxy's distance from Earth (for reasonably nearby
galaxies): \(z \cong c^{-1} H \Delta s,\) where \(c\) is the vacuum speed of
light, \(H\) is the Hubble constant, and \(\Delta s\) is the distance of the
galaxy from Earth. Derive this law from the relationships described in the
problem statement, and determine the Hubble constant in terms of the scale-
factor function \(a(t)\).
b) If the Hubble constant currently has the value \(H_{0}=72(\mathrm{~km} /
\mathrm{s}) / \mathrm{Mpc},\) how far away is a galaxy whose light has the
redshift \(z=0.10 ?\) (The megaparsec \((\mathrm{Mpc})\) is a unit of length equal
to \(3.26 \cdot 10^{6}\) light-years. For comparison, the Great Nebula in
Andromeda is approximately 0.60 Mpc from Earth.)
