9.4 Correlation function and multipoles 365energy density itself and the Sachs–Wolfe effect, and the second term is related
to the velocities of the baryon–radiation plasma at recombination. The latter term
is therefore often referred to in the literature as the Doppler contribution to the
fluctuations. The characteristic peaks in the temperature anisotropy power spectrum
(described below) are sometimes called “Doppler peaks,” but we will see that the
Doppler term is not the dominant cause of these peaks.
9.4 Correlation function and multipoles
A sky map of the cosmic microwave background temperature fluctuations can be
fully characterized in terms of an infinite sequence of correlation functions. If the
spectrum of fluctuations is Gaussian, as predicted by inflation and as current data
suggest, then only the even order correlation functions are nonzero and all of them
can be directly expressed through the two-point correlation function (also known
as the temperature autocorrelation function):
C(θ)≡〈
δT
T 0
(l 1 )δT
T 0
(l 2 )〉
, (9.33)
where the brackets〈〉denote averaging over all directionsl 1 andl 2 ,satisfying
the conditionl 1 ·l 2 =cos(θ). The squared temperature difference between two
directions separated by angleθ,averaged over the sky, is related toC(θ)by
〈(
δT
T 0
(θ)) 2 〉
≡
〈(
T(l 1 )−T(l 2 )
T 0) 2 〉
= 2 (C( 0 )−C(θ)). (9.34)The temperature autocorrelation function is a detailed fingerprint that can be used
first to discriminate among cosmological models and then, once the model is fixed,
to determine the values of its fundamental parameters. The three-point function,
also known as thebispectrum, is a sensitive test for a non-Gaussian contribution to
the fluctuation spectrum since it is precisely zero in the Gaussian limit.
The cosmic microwave background is also polarized. An expanded set ofn-
point correlation functions can be constructed which quantifies the correlation of
the polarization over long distances and the correlation between polarization and
temperature fluctuations. For the purpose of this primer, however, we focus on com-
puting the temperature autocorrelation function, since this has proven to be the most
useful to date. The generalization to other correlation functions is straightforward.
The universe is homogeneous and isotropic on large scales. Consequently,
averaging over all directions on the sky from a single vantage point (e.g., the
Earth) should be close to the average of the results obtained by other observers in
many points in space for given directions. The latter average corresponds to the
cosmic meanand is determined by correlation functions of the random field of