and Cosmology

8.6 Angular Fluctuations of the Cosmic Microwave Background
In contrast to many other situations, in which the statistical
uncertainties can be reduced by analyzing a larger
sample, this is not possible in the case of the CMB: there
is only one microwave sky that we can observe. Hence,
we cannot compile a sample of microwave maps, but
instead depend on the one map of our sky. Observers
at another location in the Universe will see a different
CMB sky, and thus will measure different values C l ,
since their CMB sky corresponds to a different realization
of the random field which is specified by the
power spectrum P(k) of the density fluctuations. This
means that (8.26) is a fundamental limit to the statistical
accuracy, which cannot be overcome by any
improvements in instrumentation. This effect is called
cosmic variance. The precision of the WMAP measurements
is, for all l 350, better than the cosmic
variance (8.26). Therefore, the fluctuation spectrum for
l 350 measured by WMAP is “definite”, i.e., further
improvements of the accuracy in this angular range will
not provide additional cosmological information (however,
in future measurements one may test for potential
systematic effects).
The Fluctuation Spectrum. Since WMAP observes at
five different frequencies, the Galactic foreground radiation
can, in principle, be separated from the CMB
due to the different spectral behavior. Alternatively, external
datasets may be utilized for this, as described in
Sect. 8.6.4. This second method is preferred because,
by using multifrequency data in the foreground subtraction,
the noise properties of the resulting CMB map
would get very complicated. The sky regions in which
the foreground emission is particularly strong – mainly
in the Galactic disk – are disregarded in the determination
of C l . Furthermore, known point sources are also
masked in the map.
The resulting fluctuation spectrum is presented in
Fig. 8.31. In this figure, instead of plotting the individual
C l , the fluctuation amplitudes have been averaged
in l-bins. The solid curve indicates the expected fluctuation
spectrum in a ΛCDM-Universe whose parameters
are quantitatively discussed further below. The gray
region surrounding the model spectrum specifies the
width of the cosmic variance, according to (8.26) and
modified with respect to the applied binning.
The first conclusion is that the measured fluctuation
spectrum agrees with the model extraordinarily well.
Virtually no statistically significant deviations of the
data points from the model are found. Smaller deviations
which are visible are expected to occur as statistical
outliers. The agreement of the data with the model is in
fact spectacular: despite its enormous potential for new
discoveries, WMAP “only” confirmed what had already
been concluded from earlier measurements. Hence, the
results from WMAP confirmed the cosmological model
in an impressive way and, at the same time, considerably
improved the accuracy of the parameter values.
Fig. 8.31. As the central result from the first-year WMAP
measurements, the top panel shows the fluctuation spectrum of
the CMB temperature (TT), whereas the bottom panel displays
the power spectrum of the correlation between the temperature
distribution and polarization amplitude (TE). Besides the data
points from WMAP, which are plotted here in l-bins, the
results from two other CMB experiments (CBI and ACBAR)
are also plotted, at larger l. The curve in each panel shows the
best-fitting ΛCDM model, and the gray region surrounding
it indicates the cosmic variance. The large amplitude of the
point in the TE spectrum at small l indicates an unexpectedly
high polarization on large angular scales, which suggests an
early reionization of the Universe
347