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 Although the major role for that goal has been superseded by the CMB map analysis [41], the proper characterization of the morphology of large-scale structure beyond the two-point statistics is of fundamental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 [32Jump To The Next Citation Point].

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 31View Image shows the MFs as a function of νf defined from the volume fraction [26]:

1 ∫ ∞ 2 f = √---- e−x ∕2dx. (178 ) 2π νf
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 δ = νfσ, for a Gaussian random field with r.m.s. density fluctuations σ. 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.
View Image

Figure 31: MFs as a function of νf for RG = 5h −1 Mpc for SDSS data. Averaged MFs of the mock samples are plotted for LCDM (solid lines) and SCDM (long dashed lines). Gaussian model predictions (see Equations (112View Equation, 113View Equation, 114View Equation, 115View Equation)) are also plotted with short dashed lines. The results favor the LCDM model with random–Gaussian initial conditions. (Figure taken from [32].)

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 31View Image, 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 νσ remain unchanged for the linear biasing).


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