6.4 Topology of the Universe:
Analysis of SDSS galaxies in terms of Minkowski functionals
All the observational results presented in the preceding
Sections 6.1, 6.2, and 6.3 were restricted to the
two-point statistics. As emphasized in Section 3, the clustering pattern
of galaxies has much richer content than the two-point statistics
can probe. Historically the primary goal of the topological
analysis of galaxy catalogues was to test Gaussianity of the
primordial density fluctuations. Although the major role for that
goal has been superseded by the CMB map analysis [41], the
proper characterization of the morphology of large-scale structure
beyond the two-point statistics is of fundamental importance in
cosmology. In order to illustrate a possibility to explore the
topology of the Universe by utilizing the new large surveys, we
summarize the results of the Minkowski Functionals (MF) analysis of
SDSS galaxy data [32
].
In an apparent-magnitude limited catalogue of
galaxies, the average number density of galaxies decreases with
distance because only increasingly bright galaxies are included in
the sample at larger distance. With the large redshift surveys it
is possible to avoid this systematic change in both density and
galaxy luminosity by constructing volume-limited samples of
galaxies, with cuts on both absolute-magnitude and redshift. This
is in particular useful for analyses such as MF and was carried out
in the analysis shown here.
Figure 31 shows the MFs as a
function of
defined from the volume
fraction [26]:
This is intended to map the threshold so that the volume fraction
on the high-density side of the isodensity surface is identical to
the volume in regions with density contrast
, for a Gaussian random field with
r.m.s. density fluctuations
. If the evolved density field
may approximately have a good one-to-one correspondence with the
initial random-Gaussian field, then this transformation removes the
effect of evolution of the PDF of the density field. Under this
assumption, the MFs as a function of volume fraction would be
sensitive only to the topology of the isodensity contours rather
than evolution with time of the density threshold assigned to a
contour. While the limitations of the approximation of monotonicity
in the relation between initial and evolved density fields are well
recognized [40], we plot the
result in this way for simplicity.
The good match between the observed MFs and the mock
predictions based on the LCDM model with the initial
random-Gaussianity, as illustrated in Figure 31, might be interpreted
to imply that the primordial Gaussianity is confirmed. A more
conservative interpretation is that, given the size of the
estimated uncertainties, these data do not provide evidence for
initial non-Gaussianity, i.e., the data are consistent with primordial Gaussianity.
Unfortunately, due to the statistical limitation of the current
SDSS data, it is not easy to put a more quantitative statement
concerning the initial Gaussianity. Moreover, in order to go
further and place more quantitative constraints on primordial
Gaussianity with upcoming data, one needs a more precise and
reliable theoretical model for the MFs, which properly describes
the nonlinear gravitational effect possibly as well as galaxy
biasing beyond the simple mapping on the basis of the volume
fraction. In fact, galaxy biasing is a major source of uncertainty
for relating the observed MFs to those obtained from the mock
samples for dark matter distributions. If LCDM is the correct
cosmological model, the good match of the MFs for mock samples from
the LCDM simulations to the observed SDSS MFs may indicate that
nonlinearity in the galaxy biasing is relatively small, at least
small enough that it does not significantly affect the MFs (the MFs
as a function of
remain unchanged for the linear
biasing).