** Next:** Integral Solutions
** Up:** Boltzmann Equation
** Previous:** Normal Modes

It is now straightforward to rewrite the Boltzmann equation
(19) as the evolution equations for the amplitudes
of the normal modes of the temperature and polarization
.The gravitational sources and scattering sources of these equations
follow from Eq. (20) and (21) by noting
that the spin harmonics are orthogonal,

| |
(26) |

The term is evaluated by use of the coupling relation
Eq. (25) for .It represents the fact that spatial gradients in the distribution become
orbital angular momentum as the radiation streams along its trajectory
.For example, a temperature variation on a distant surface surrounding the
observer appears as an anisotropy on the sky.
This process then simply reflects a projection relation that relates distant
sources to present day local anisotropies.
With these considerations, the temperature fluctuation evolves as

| |
(27) |

and the polarization as
The temperature fluctuation sources in Newtonian gauge are
| |
(28) |

and in synchronous gauge,
| |
(29) |

The source doesn't contain a curvature factor because we have
recursively defined the basis functions in terms of the lowest member,
which is in this case. In the above
| |
(30) |

and note that the photon density and velocities are related to the moments as
| |
(31) |

whereas the anisotropic stresses are given by
| |
(32) |

which relates them to the quadrupole moments () as
| |
(33) |

The evolution of the metric and matter sources are given in
Appendices A3--A5.

** Next:** Integral Solutions
** Up:** Boltzmann Equation
** Previous:** Normal Modes
*Wayne Hu*

*9/9/1997*