**1.1. Redshift**

Light emitted from a comoving galaxy located at
_{g} at time
*t*_{e} will reach an observer situated at
_{o}
0 at
a later time *t*_{o}, where

(5) |

Equation 5 provides the relation among
_{g},
*t*_{o},
and *t*_{e}. For a comoving galaxy,
_{g} is
unchanged so that differentiating eq. 5 leads to

(6) |

This result relates the evolution of the universe (*a*_{o}
/ *a*_{e}) as
the photon travels from emission to observation, to the change in its
frequency () or wavelength
(). As the universe
expands
(or contracts!), wavelengths expand (contract) and frequencies decrease
(increase). The redshift of a spectral line is defined by relating
the wavelength at emission (the "lab" or "rest-frame" wavelength
_{e}) to the
wavelength observed at a later time *t*_{o},
_{o}.

(7) |

Since the energies of photons are directly proportional to their
frequencies, as the universe expands photon energies redshift to
smaller values:
E_{}
= *h*
E_{}
(1 +
*z*)^{-1}. For **all** particles, massless or not,
de Broglie told us that wavelength and momentum are inversely related,
so that: p
^{-1}
p
(1 +
*z*)^{-1}. All momenta redshift; for non-relativistic
particles (e.g. galaxies) this implies that their "peculiar"
velocities redshift: v = p/M
(1 +
*z*)^{-1}.