伊藤 誠

With the help of a graph-theoretical approach, we have quantified two-dimensional galaxy distributions in observation and Cold Dark Matter (CDM) simulations. In our analysis, we adopted the Lyon-Meudon Extragalactic Database as a typical two-dimensional observation. To estimate adjacency matrices, we constructed constellation graphs from galaxy distributions, and calculated the distribution functions of the eigenvalues of these matrices. Using the mean absolute deviations, we compared the two-dimensional galaxy distributions in observations with CDM simulations in a statistical way. From our analysis we found that the CDM model with a density parameter of less than one is preferable to reproduce the galaxy distributions in the observations.

We use a graph theory for quantifying galaxy distributions in the cold dark matter (CDM) universe. Cosmological N-body simulations with CDM spectra are performed, and a constellation graph is constructed from these simulations. We apply graph theory to these constellation graphs and calculate the distribution functions of the eigenvalues of the adjacency matrices. In addition to a three-dimensional analysis, a two-dimensional analysis, an analysis in a slicelike geometry, and a redshift space analysis are also carried out. From our analyses, we find that the kurtosis and the average deviation of the distribution function of the eigenvalues are useful statistical measures for quantifying the galaxy distributions. We also find that the graph-theoretical approach possesses a discriminative ability with regard to the two-dimensional galaxy distributions. The slicelike geometry, which covers a rather narrow region of the sky, is not sufficient for the graph theoretical analysis. However, we find that the discriminative ability of the graph theory is recovered in redshift space.

We propose a graph-theoretical approach for quantifying the galaxy distributions in the universe. In order to examine the validity of this approach, we construct graphs based on our cosmological N-body simulations with scale-free power-lay spectra, and apply graph theory to them. constellation graph from our simulations, which is one of the most basic graphs. distribution function of the edge-length, order, degree, and the eigenvalue of the adjacency matrix of the constellation graph. From our analysis we find that the eigenvalue of the adjacency matrix is a good statistical measure for quantifying the galaxy distributions in a clear manner. For supplementary purposes we also construct a separated minimal spanning tree and a group graph, and examine the usefulness of the graph-theoretical approach.

Previous analyses have revealed that thermodynamic distribution function can represent the galaxy distributions in various observational and numerical data very well. In their analyses, the value of b in the thermodynamic function was used as a fitting parameter. The thermodynamic function is determined by b and the mean number density ($) over bar. Therefore, the statistical properties which are derived from the distribution function, for example the k-th order moments, can be represented by b and ($) over bar n. We compared the second-order moment, skewness and kurtosis of numerical data of power-law models with those that are predicted by fitted values of b. We found that the fitted values of b cannot give the second-order moment, skewness and kurtosis correctly, though agreements between the thermodynamic and experimental. distribution functions are fairly good. Small deviations between them cause large deviations in the moments. However, these deviations can give clues concering the initial density fluctuations, because they strongly depend on the initial power-law indices. We also confirm that the galaxy distributions in our data are homogeneous and isotropic, even in evolved stages, by using White's relation. It is necessary to confirm it, because the thermodynamic function and the general properties of the distribution function are derived under the condition that the distributions are homogeneous and isotropic.

Following recent observational studies of counts-in-cells statistic, we consider the evolution of the third and fourth moments of distribution functions of the density contrast 6. We define a normalized skewness as S = [delta3]/[delta2]2 and a normalized kurtosis as K = ([delta4] - 3 [delta2]2)/[delta2]3, which are expected from previous theoretical and phenomenological arguments to be '' constants '' in the mildly nonlinear regime. We extended the study of S and K into the highly nonlinear regime 1 less than or similar to delta(rms) = [delta2]1/2 less than or similar to 30 using N-body simulations and taking into account the effect of Poisson shot noise. We find that the simple scaling relation which holds in the mildly nonlinear regime breaks down in the highly nonlinear regime. Both S and K exhibit variation which strongly depends on the primordial power spectrum of density fluctuations, but only weakly on the epoch. For a low-density (OMEGA = 0.2; lambda = 0.8) cold dark matter model at the ''present epoch,'' the normalized skewness and kurtosis in cubic cells are well approximated by S(F) almost-equal-to 3 delta(rms)0.4 and K(F) almost-equal-to 15 delta(rms)0.9 (after removing the shot-noise contribution), over the range 1 less than or similar to delta(rms) less than or similar to 27. For a simulation with initial white-noise spectrum the moments vary more slowly, S(F) almost-equal-to 2 delta(rms)0.2 and K(F) almost-equal-to 8 delta(rms)0.3. Our present findings are in conflict with simple hierarchical clustering models with a power-law correlation function which predict that S(F) and K(F) are independent of the scale. Furthermore, we find that in redshift space the moments, as well as S and K, are smaller than in real space for delta(rms) > 1. Consequently S and S(F) appear to be '' constants '' in redshift space analysis which might explain several observational indications in favor of hierarchical clustering models. We also discuss sparse sampling and the effect of filtering scale and shape. Counts-in-cells of IRAS and optical galaxies suggest S approximately 2 for delta(rms) less than or similar to 1, while from the correlation functions of optical galaxies S approximately 4 for delta(rms) greater than or similar to 1. These values are in accord with that expected from gravitational clustering from Gaussian initial conditions and may carry important information on the shape of the power spectrum.