and Cosmology

8. Cosmology III: The Cosmological Parameters
346
verification of the COBE measurements. In Fig. 8.30,
sky maps by COBE and by WMAP are displayed. The
dramatically improved angular resolution of the WMAP
map is obvious. In addition, it can clearly be seen that
both maps are very similar if one compares them at
a common angular resolution. This comparison can be
performed quantitatively by “blurring” the WMAP map
to the COBE resolution using a smoothing algorithm.
Since WMAP is not observing at exactly the same frequencies
as COBE, it is necessary to interpolate between
two frequencies in the WMAP maps to match the frequency
of the COBE map. The comparison then shows
that, when accounting for the noise, the two maps are
completely identical, with the exception of a single location
in the Galactic disk. This can be explained, e.g.,
by a deviation of the spectral behavior of this source
from the 2.73 K blackbody spectrum that was implicitly
assumed for the aforementioned interpolation between
two WMAP frequencies. The confirmation of the COBE
measurements is indeed highly impressive.
Fig. 8.30. Comparison of the CMB anisotropy measurements
by COBE (top) and WMAP (bottom), after subtraction of
the dipole originating from the motion of the Sun relative to
the CMB rest-frame. The enormously improved angular resolution
of WMAP is easily seen. Although these maps were
recorded at different frequencies, the similarity in the temperature
distribution is clearly visible and could be confirmed
quantitatively. From this, the COBE results have, for the first
time, been confirmed independently
frequency range, using five (instead of three) frequencies;
it has a much improved angular resolution (which
is frequency-dependent; about 20 ′ , compared to ∼ 7 ◦
for COBE), and in addition, WMAP is able to measure
the polarization of the CMB. Results from the
first year of observation with WMAP were published in
2003. These excellent results confirmed our cosmological
world model in such a way that we are now justified
in calling it the standard model of cosmology. The most
important results from WMAP will be discussed in the
following.
Comparison to COBE. Since WMAP is the first satellite
after COBE to map the full sky in the relevant
frequency range, its first year results allowed the first
Cosmic Variance. Before we continue discussing the
WMAP results we need to explain the concept of
cosmic variance. The angular fluctuation spectrum of
CMB anisotropies is quantified by the multipole coefficients
C l . For instance, C 1 describes the strength of
the dipole. The dipole has three components; these can
be described, for example, by an amplitude and two
angles which specify a direction on the sphere. Accordingly,
the quadrupole has five independent components,
and in general, C l is defined by (2l + 1) independent
components.
Cosmological models of the CMB anisotropies predict
the expectation value of the amplitude of the
individual components C l . In order to compare measurements
of the CMB with these models one needs
to understand that we will never measure the expectation
value, but instead we measure only the mean
value of the components contributing to the C l on our
microwave sky. Since the quadrupole has only five independent
components, the expected statistical deviation
of the average from the expectation value is C 2 / √ 5. In
general, the statistical deviation of the average of C l
from the expectation value is
ΔC l =
C l
√ 2l + 1
. (8.26)