and Cosmology

8. Cosmology III: The Cosmological Parameters
324
us to draw conclusions about the statistical properties
of the dark matter distribution, e.g., its power spectrum.
At least on large scales, where structure evolution still
proceeds almost linearly today, this assumption seems
to be justified if an additional bias factor is allowed for.
Hence, it is obvious to also examine the large-scale distribution
of galaxy clusters, which should follow the
distribution of dark matter on linear scales as well,
although probably with a different bias factor.
The ROSAT All Sky Survey (see Sect. 6.3.5) allowed
the compilation of a homogeneous sample of
galaxy clusters with which the analysis of the largescale
distribution of clusters became possible for the
first time. Figure 8.12 shows that the power spectrum
of clusters has the same shape as that of galaxies, however
with a considerably larger normalization. The ratio
of the two power spectra displayed in this figure is
based on different bias factors for galaxies and clusters,
b clusters ≈ 2.6b g . For this reason the power spectrum
Fig. 8.12. The power spectrum of galaxies (open symbols) and
of galaxy clusters from the REFLEX survey (filled symbols).
The two power spectra have basically the same shape, but
they differ by a multiplicative factor. This factor specifies the
square of the ratio of the bias factors of optically selected
galaxies and of X-ray clusters, respectively. Particularly on
large scales, mapping the power spectrum from clusters is of
substantial importance
for clusters has an amplitude that is larger by a factor of
about (2.6) 2 than that for galaxies. Since clusters of galaxies
are much less abundant than galaxies, the density
maxima of the dark matter corresponding to the former
need to have a higher threshold than those of galaxies,
which will, in the biasing model illustrated in Fig. 8.3,
result in stronger correlations.
The analysis of the power spectrum by means of
clusters is interesting, particularly on large scales, yielding
an additional data point for the shape parameter
Γ = Ω m h. Together with the cluster abundance, their
correlation properties yield values of Ω m ≈ 0.34 and
σ 8 ≈ 0.71.
8.3 High-Redshift Supernovae
and the Cosmological Constant
8.3.1 Are SN Ia Standard Candles?
As mentioned in Sect. 2.3.2, Type Ia supernovae are supposed
to be the result of explosion processes of white
dwarfs which cross a critical mass threshold by accretion
of additional matter. This threshold should be
identical for all SNe Ia, making it at least plausible that
they all have the same luminosity. If this were the case,
they would be ideal for standard candles: owing to their
high luminosity, they can be detected and examined
even at large distances.
However, it turns out that SNe Ia are not really standard
candles, since their maximum luminosity varies
from object to object with a dispersion of about 0.4mag
in the blue band light. This is visible in the top panel
of Fig. 8.13. If SNe Ia were standard candles, the data
points would all be located on a straight line, as described
by the Hubble law. Clearly, deviations from the
Hubble law can be seen, which are significantly larger
than the photometric measurement errors.
It turns out that there is a strong correlation between
the luminosity and the shape of the light curve
of SNe Ia. Those of higher maximum luminosity show
a slower decline in the light curve, as measured from
its maximum. Furthermore, the observed flux is possibly
affected by extinction in the host galaxy, in addition
to the extinction in the Milky Way. With the resulting
reddening of the spectral distribution, this effect can be