The last two sections have pulled a fast one. We began by discussing polarization as a two component tensor quantity, but then started discussing the production of polarization as if only its amplitude were relevant. A more complete formalism for describing the polarization field has been worked out and will be presented in this section (see Kamionkowski, Kosowsky, and Stebbins (1997) for a more extensive discussion). An equivalent formalism employing spin-weighted spherical harmonics has been used extensively by Zaldarriaga and Seljak (1997). Note that the normalizations employed by Seljak and Zaldarriaga are slightly different than those adopted here and by Kamionkowski, Kosowsky, and Stebbins (1997).

The microwave background temperature pattern on the sky
*T*() is
conventionally expanded in a
complete set of orthonormal basis functions, the spherical harmonics:

(25) |

where

(26) |

are the temperature multipole coefficients and *T*_{0} is the
mean CMB temperature.
Similarly, we can expand the polarization tensor for linear
polarization,

(27) |

(compare with Eq. 8; the extra factors are
convenient
because the usual spherical coordinate basis is orthogonal
but not orthonormal) in terms of
*tensor spherical harmonics*, a complete set of
orthonormal basis functions for symmetric trace-free 2 × 2 tensors
on the sky,

(28) |

where the expansion coefficients are given by

(29) | |

(30) |

which follow from the orthonormality properties

(31) | |

(32) |

These tensor spherical harmonics have been used
primarily in the literature of gravitational
radiation, where the metric perturbation can be
expanded in these tensors. Explicit forms can be
derived via various algebraic and group theoretic methods; see
Thorne (1980)
for a complete discussion.
A particularly elegant and useful derivation
of the tensor spherical harmonics (along with
the vector spherical harmonics as well) is provided by differential geometry
(Stebbins, 1996).
Given a scalar function on a manifold, the only related vector quantity
at a given point of the manifold is the covariant
derivative of the scalar function.
The tensor basis functions can be
derived by taking the scalar basis functions *Y*_{lm}
and applying to them two covariant derivative operators
on the manifold of the two-sphere (the sky):

(33) |

and

(34) |

where _{ab} is the
completely antisymmetric tensor,
the ``:'' denotes covariant differentiation on the 2-sphere, and

(35) |

is a normalization factor. Note that the somewhat more familiar vector spherical harmonics used to describe electromagnetic multipole radiation can likewise be derived as a single covariant derivative of the scalar spherical harmonics.

While the formalism of differential geometry may look imposing
at first glance, the expansion of the polarization
field has been cast into exactly the same form as for the familiar
temperature case, with only the extra complication of evaluating
covariant derivatives.
Explicit forms for the tensor harmonics are given in
Kamionkowski, Kosowsky, and Stebbins (1997).
Note that the underlying manifold, the two-sphere, is the
simplest non-trivial manifold, with a constant Ricci curvature
*R* = 2, so the differential geometry is easy.
One particularly useful property for doing calculations
is that the covariant derivatives are subject to integration
by parts:

(36) |

with no surface term if the integral is over the entire sky. Also, the scalar spherical harmonics are eigenvalues of the Laplacian operator:

(37) |

The existence of two sets of basis functions,
labeled here by ``G'' and
``C'', is due to the fact that the symmetric traceless
2 × 2 tensor describing linear polarization
is specified by two independent parameters.
In two dimensions, any symmetric traceless
tensor can be uniquely decomposed into a
part of the form
*A*_{: ab} - (1/2)*g*_{ab}*A*_{:
c}^{c} and another part of the form
*B*_{: ac}
^{c}_{b} +
*B*_{: bc}
^{c}_{a}
where *A* and *B* are two scalar
functions. This decomposition is quite similar to the decomposition of a
vector field into a part which is the gradient of a scalar field
and a part which is the curl of a vector field; hence we use the notation
G for ``gradient'' and C for ``curl''. In fact, this correspondence is
more than just cosmetic: if a linear polarization field is visualized
in the usual way with headless ``vectors'' representing the amplitude
and orientation of the polarization, then
the G harmonics describe the portion of
the polarization field which has no handedness associated with it,
while the C harmonics describe the other portion of the field which
does have a handedness (just as with the gradient and curl of
a vector field).

This geometric interpretation leads to an important physical
conclusion. Consider a universe containing only scalar
perturbations, and imagine a single Fourier mode of the
perturbations. The mode has only one direction
associated with it, defined by the Fourier vector **k**;
since the perturbation is scalar,
it must be rotationally symmetric around this axis.
(If it were not, the gradient of the perturbation
would define an independent physical direction, which
would violate the assumption of a scalar perturbation.)
Such a mode can have no physical handedness
associated with it, and as a result, the polarization
pattern it induces in the microwave background couples
only to the G harmonics. Another way of stating this
conclusion is that primordial density perturbations
produce *no* C-type polarization as long as
the perturbations evolve linearly.
This property is very useful for constraining
or measuring other physical effects, several of
which are considered below.

Finally, just as temperature fluctuations are commonly
characterized by their power spectrum *C*_{l}, polarization
fluctuations possess analogous power spectra.
We now have three sets of multipole moments,
*a*_{(lm)}^{T},
*a*_{(lm)}^{G}, and
*a*_{(lm)}^{C},
which fully describe the
temperature/polarization map of the sky.
Statistical isotropy implies that

(38) |

where the angle brackets are an average over all realizations of the
probability distribution for the cosmological initial conditions.
Simple statistical estimators of the various *C*_{l}'s can be
constructed from maps of the microwave background temperature
and polarization.

For Gaussian theories, the statistical properties of a
temperature/polarization map are specified fully by these six
sets of multipole moments.
In addition, the scalar spherical harmonics *Y*_{(lm)} and the G
tensor harmonics
*Y*_{(lm)ab}^{G} have parity (- 1)^{l}, but the C
harmonics *Y*_{(lm)ab}^{C} have parity
(- 1)^{l + 1}.
If the large-scale perturbations in the early universe were invariant
under parity inversion, then
*C*_{l}^{TC} = *C*_{l}^{GC} = 0.
The arguments in the previous paragraph about handedness
further imply that for *scalar* perturbations,
*C*_{l}^{C} = 0. A question of substantial theoretical and
experimental interest is what kinds of physics produce
measurable nonzero *C*_{l}^{C},
*C*_{l}^{TC}, and
*C*_{l}^{GC}. This question is addressed in the
following section.

The power spectra can be computed for a given cosmological model through well-known numerical techniques. A set of power spectra for scalar and tensor perturbations in a typical inflation-like cosmological model, generated with the CMBFAST code (Seljak and Zaldarriaga, 1996) are displayed in Fig. 2.