232 The cosmic microwave background
only monopole and dipole directional dependences as explained earlier; therefore,
scattering can produce no net polarization and the radiation remains unpolarized.
As tight coupling begins to break down as recombination begins, a quadrupole
moment of the radiation field will begin to grow due to free-streaming of the
photons. Polarization is generated during the brief interval when a significant
quadrupole moment of the radiation has built up, but the scattering electrons have
not yet all recombined. Note that if the universe recombined instantaneously,
the net polarization of the microwave background would be zero. Due to
this competition between the quadrupole source building up and the density of
scatterers declining, the amplitude of polarization in the microwave background
is generically suppressed by an order of magnitude compared to the temperature
fluctuations.
Before polarization generation commences, the temperature fluctuations
have either a monopole dependence, corresponding to density perturbations, or
a dipole dependence, corresponding to velocity perturbations. A straightforward
solution to the photon free-streaming equation (in terms of spherical Bessel
functions) shows that for Fourier modes with wavelengths large compared to a
characteristic thickness of the last-scattering surface, the quadrupole contribution
through the last scattering surface is dominated by the velocity fluctuations
in the temperature, not the density fluctuations. This makes intuitive sense:
the dipole fluctuations can free stream directly into the quadrupole, but the
monopole fluctuations must stream through the dipole first. This conclusion
breaks down on small scales where either monopole or dipole can be the dominant
quadrupole source, but numerical computations show that on scales of interest
for microwave background fluctuations, the dipole temperature fluctuations are
always the dominant source of quadrupole fluctuations at the last scattering-
surface. Therefore, polarization fluctuations reflect mainly velocity perturbations
at last scattering, in contrast to temperature fluctuations which predominantly
reflect density perturbations.
7.3.3 Harmonic expansions and power spectra
Just as the temperature on the sky can be expanded into spherical harmonics,
facilitating the computation of the angular power spectrum, so can the
polarization. The situation is formally parallel, although in practice it is more
complicated: while the temperature is a scalar quantity, the polarization is
a second-rank tensor. We can define a polarization tensor with the correct
transformation properties, equation (7.15), as
Pab(nˆ)=1
2
(
Q(nˆ) −U(nˆ)sinθ
−U(nˆ)sinθ −Q(nˆ)sin^2 θ)
. (7.20)
The dependence on the Stokes parameters is the same as for the density matrix,
equation (7.17); the extra factors are convenient because the usual spherical
coordinate basis is orthogonal but not orthonormal. This tensor quantity must