The redshift relations

The first is the definition of redshift and its relation to the scale factor of the Universe. This scale factor contains the whole of the dynamics of the expansion of the Universe. It comes from the Robertson-Walker metric that is needed to describe the Friedman cosmology that we are using troughout our calculations. The redshift of a galaxy is so-called because it was discovered by the pioneers of observational cosmology that the spectral lines of galaxies were all shifted towards the red end of the optical spectrum relative to their rest wavelengths. If is the emitted wavelength of some spectral feature and the wavelength with which it is observed, the redshift of the object is defined as

The usual interpretation of the redshift is as a velocity of recession and it is true that, provided the velocity of the galaxy is much less than the velocity of light, , . However, there is a much deeper meaning of redshift in cosmology. Using the Robertson-Walker metric, it can be shown that in general the redshift is directly related to the scale factor of the Universe through the relation

This relation is independent of the physics of the expansion. What this relation tells us is that the redshift measures the size of the Universe, i.e. the physical separation between comoving test particles, when the radiation was emitted relative to its present size. For example, at a redshift , the galaxies were all closer together by a factor of 4 relative to their present separation.

The second point is that, using the Robertson-Walker metric, it can be shown that Hubble's constant is just the present rate of expansion of the Universe, i.e.

The subscript 0 means the value of Hubble's constant at the present epoch. Note, however, that Hubble's constant may be defined at any epoch and in general changes with cosmic epoch. Thus, at any epoch .