234 The cosmic microwave background
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). Note that
Zaldarriaga and Seljak (1997) label these harmonics E and B, with a slightly
different normalization than defined here (see Kamionkowskiet al1996).
We now have three sets of multipole moments,a(Tlm),aG(lm),andaC(lm),which
fully describe the temperature/polarization map of the sky. These moments can be
combined quadratically into various power spectra analogous to the temperature
CTl. Statistical isotropy implies that
〈aT(lm∗)a(Tl′m′)〉=ClTδll′δmm′, 〈aG(lm∗)a(Gl′m′)〉=ClGδll′δmm′,
〈aC(lm∗)aC(l′m′)〉=ClCδll′δmm′, 〈a(Tlm∗)aG(l′m′)〉=ClTGδll′δmm′,
〈a(Tlm∗)a(Cl′m′)〉=ClTCδll′δmm′, 〈aG(lm∗)a(Cl′m′)〉=ClGCδll′δmm′, (7.26)where the angle brackets are an average over all realizations of the probability
distribution for the cosmological initial conditions. Simple statistical estimators
of the variousCls can be constructed from maps of the microwave background
temperature and polarization.
For fluctuations with Gaussian random distributions (as predicted by the
simplest inflation models), the statistical properties of a temperature/polarization
map are specified fully by these six sets of multipole moments. In addition,
the scalar spherical harmonicsY(lm)and the G tensor harmonicsY(Glm)abhave
parity(− 1 )l, but the C harmonicsY(Clm)abhave parity(− 1 )l+^1 .Ifthelarge-
scale perturbations in the early universe were invariant under parity inversion,
thenClTC = ClGC = 0. So generally, microwave background fluctuations
are characterized by the four power spectraCTl,CGl,ClC,andClTG.Theend
result of the numerical computations described in section 7.2.2 are these power
spectra. Polarization power spectraClGandClTGfor scalar perturbations in a
typical inflation-like cosmological model, generated with the CMBFAST code
(Seljak and Zaldarriaga 1996), are displayed in figure 7.2. The temperature
power spectrum in figure 7.1 and the polarization power spectra in figure 7.2
come from the same cosmological model. The physical source of the features in
the power spectra is discussed in the next section, followed by a discussion of
how cosmological parameters can be determined to high precision via detailed
measurements of the microwave background power spectra.
7.4 Acoustic oscillations
Before decoupling, the matter in the universe has significant pressure because it
is tightly coupled to radiation. This pressure counteracts any tendency for matter
to collapse gravitationally. Formally, the Jeans mass is greater than the mass
within a horizon volume for times earlier than decoupling. During this epoch,