and Cosmology

8.7 Cosmological Parameters
Fig. 8.33. The power spectrum of density
fluctuations in the Universe, as determined
by different methods. Here, Δ 2 (k) ∝ k 3 P(k)
is plotted. Note that small length-scales
(or large k, respectively) are towards the
left in the plot. Going from large to small
scales, the results presented here are obtained
from CMB temperature fluctuations,
from the abundance of galaxy clusters, from
the large-scale distribution of galaxies, from
cosmic shear, and from the statistical properties
of the Lyα forest. One can see that the
power spectrum of a ΛCDM model is able
to describe all these data over many orders
of magnitude in scale
351
physical origin of this effect can be proven directly because
the integrated Sachs–Wolfe effect is produced at
relatively low redshifts (where the influence of a cosmological
constant is noticeable), as a result of the time
evolution of the gravitational potential. Therefore, it
should be directly correlated with the large-scale matter
overdensities which are observable in the distribution
of galaxies and clusters of galaxies, assuming a bias
model. For example, one can correlate the CMB temperature
map with luminous elliptical galaxies, as they are
observed photometrically in the Sloan Digital Sky Survey
over very large regions on the sky (see Sect. 9.1.2).
The significant correlation signal found in this analysis
yields very strong evidence for the temperature fluctuations
having originated from the Sachs–Wolfe effect
on large angular scales, hence providing a direct proof
of Ω Λ ̸= 0.
A big surprise in the WMAP results is the large value
of τ, which is derived in particular from the TE power
spectrum. This value for τ implies that the Universe was
reionized at a fairly high redshift of z ∼ 15.
The combination of CMB results with those from the
large-scale distribution of galaxies and the statistics of
the Lyα forest allows us to measure the power spectrum
at smaller length-scales, as shown in Fig. 8.33. This
combination therefore provides stronger constraints on
the cosmological parameters.
We can see from Table 8.1 that with this combination
the error margins of some parameters can indeed be reduced,
compared to considering the WMAP data alone;
in particular, this is the case for Ω m h 2 . It is important to
note that by combining the different data sets, the values
of the parameters change only within the range of their
error bars as determined from the CMB data, which
means that the different datasets are compatible with
each other (and with the flat ΛCDM model). With these
primary parameters known, further parameters may now
be derived; these are listed in Table 8.2.
The combined data yield, as a best value for the total
density of the Universe,
Ω m + Ω Λ = 1.02 ± 0.02 , (8.27)
in outstanding agreement with the prediction from the
inflationary model. Furthermore, the fraction of hot dark
matter can be constrained, for which the small-scale
observations (here from the Lyα forest) are of particular
importance, since HDM reduces the power on small