The fundamental observation underlying the Big Bang theory of cosmology is Edwin Hubbies 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, λrcc, to its wavelength at emission, Aemit, 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 = λrec/λemit = arec / aemit.
(a) Hubbies 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 = c-1HΔs, where c is the vacuum speed of light, H is the Hubble constant, and Δs 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 H0 = 72 (km/s)/Mpc, how far away is a galaxy whose light has the redshift z = 0.10? (The megaparsec (Mpc) is a unit of length equal to 3.26∙106 light years, for comparison, the Great Nebula in Andromeda is approximately 0.60 Mpc from Earth.)