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The explicit form of the
Boltzmann equations for the temperature and polarization
follows directly from the Clebsch-Gordan relation of
Eqn. (58).
For the temperature (*s*=0),

| |
(52) |

The term in the square brackets is the free streaming effect
that couples the -modes and tells us that in the absence of
scattering power is transferred down the hierarchy when . This transferral merely represents geometrical
projection of fluctuations on the scale corresponding to *k* at
distance which subtends an angle given by .
The main effect of scattering
comes through the term and implies
an exponential suppression of anisotropies with optical depth in
the absence of sources. The source accounts for
the gravitational and residual scattering effects,
| |
(53) |

The presence of represents the fact that
an isotropic temperature fluctuation is not destroyed by
scattering. The Doppler effect enters the dipole equation through the baryon velocity
*v*_{B}^{(m)} term. Finally the anisotropic
nature of Compton scattering is expressed through
| |
(54) |

and involves the quadrupole moments of the temperature and
*E*-polarization
distribution only.
The polarization evolution follows a similar pattern for
, from Eqn. (58) with
,^{}

| |
(55) |

Notice that the source of polarization *P*^{(m)}
enters only in the *E*-mode quadrupole
due to the opposite parity of and *B*_{2}. However, as discussed in §IIB, free
streaming or projection couples the two parities except for
the *m*=0 scalars. Thus by geometry
regardless of the source.
It is unnecessary to solve separately for the *m*=-|*m*| relations
since they satisfy the same equations and solutions with
and all other quantities
equal.
To complete these equations, we need to express the evolution of
the metric sources . It is to this subject we
now turn.

** Next:** Scalar Einstein Equations
** Up:** Evolution Equations
** Previous:** Free Streaming
*Wayne Hu*

*9/9/1997*