and Cosmology

8. Cosmology III: The Cosmological Parameters
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the matter distribution. This effect, called cosmic shear,
is sketched in Fig. 8.18. In contrast to the case of a galaxy
cluster in which the tidal field is rather strong, the
large-scale distribution of matter causes a very much
weaker tidal field: a typical value for this shear is about
1% on angular scales of a few arcminutes, meaning that
the image of an intrinsically circular source attains an
axis ratio of 0.99:1.
The shear field results from the projection of the
three-dimensional tidal field along the line-of-sight.
Hence, we are able to obtain information about the
statistical properties of the density inhomogeneities in
the Universe, by a statistical analysis of the image
shapes of distant galaxies. For instance, the two-point
correlation function of the image ellipticities can be
measured. This is linked to the power spectrum P(k)
of the matter distribution. Thus, by comparing measurements
of cosmic shear with cosmological models
we obtain constraints on the cosmological parameters,
without the need to make any assumptions about the
Fig. 8.19. Early measurements of cosmic shear. Plotted is the
shear dispersion, measured from the ellipticities of faint and
small galaxy images on deep CCD exposures, as a function
of angular scale. Data from different teams are represented by
different symbols. For instance, MvWM+ resulted from a VLT
project, vWMR+ from a large survey (VIRMOS-Descartes) at
the CFHT. For this latter project, the images of about 450 000
galaxies have been analyzed; the corresponding error bars
from this survey are significantly smaller than those of the
earlier surveys. The curves indicate cosmic shear predictions
in different cosmological models, where the curves are labeled
by the cosmological parameters Ω m , Ω Λ , h, Γ and σ 8
relation between luminous matter (galaxies) and dark
matter.
Since the size of the effect is expected to be very
small, systematic effects like the anisotropy of the
point-spread function or distortions in the telescope optics
need to be understood very well, and they need
to be corrected for in the measurements. In principle,
the problems are the same as in the mass reconstruction
of galaxy clusters with the weak lensing effect
(Sect. 6.5.2), but they are substantially more difficult
to deal with since the measurable signal is considerably
smaller.
In March 2000, four research groups published,
quasi-simultaneously, the first measurements of cosmic
shear, and in the fall of 2000 another measurement was
obtained from VLT observations. Since then, several
teams worldwide have successfully performed measurements
of cosmic shear, for which a large number
of different telescopes have been used, including the
HST. The development of wide-field cameras and of
special software for data analysis are mainly responsible
for these achievements. Some of the early results are
compiled in Fig. 8.19.
By comparison of these measurement results with
theoretical models, constraints on cosmological parameters
are obtained; one example of this is presented
in Fig. 8.20. Currently, a major source of uncertainty
in this cosmological interpretation is our insufficient
knowledge of the redshift distribution of the faint galaxies
that are used for the measurements. In the coming
years this uncertainty will be greatly reduced, as extensive
redshift surveys of faint galaxies will be conducted
with the next generation of multi-object spectrographs
at 10-m class telescopes.
The most significant result that has been obtained
from cosmic shear so far is a derivation of a combination
of the matter density Ω m and the normalization σ 8
of the power spectrum of density fluctuations, which
can also be seen in Fig. 8.20. The near-degeneracy of
these two parameters has roughly the same functional
form as for the number density of galaxy clusters, since
with both methods we probe the matter distribution on
similar physical length-scales. For an assumed value
of Ω m = 0.3, σ 8 can thus be constrained. Although
the values obtained by different groups differ slightly,
they are compatible with σ 8 ≈ 0.8 within the range of
uncertainty.