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BETTI NUMBERS OF GAUSSIAN FIELDS
Park, Changbom,Pranav, Pratyush,Chingangbam, Pravabati,Van De Weygaert, Rien,Jones, Bernard,Vegter, Gert,Kim, Inkang,Hidding, Johan,Hellwing, Wojciech A. The Korean Astronomical Society 2013 Journal of The Korean Astronomical Society Vol.46 No.3
We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; ${\beta}_0$ is the number of connected regions, ${\beta}_1$ is the number of circular holes (i.e., complement of solid tori), and ${\beta}_2$ is the number of three-dimensional voids (i.e., complement of three-dimensional excursion regions). Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. ${\beta}_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). ${\beta}_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and ${\beta}_2$ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum n in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as n decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even though the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.
LARGE-SCALE STRUCTURE OF THE UNIVERSE AS A COSMIC STANDARD RULER
Park, Changbom,Kim, Young-Rae IOP Publishing 2010 ASTROPHYSICAL JOURNAL LETTERS - Vol.715 No.2
<P>We propose to use the large-scale structure (LSS) of the universe as a cosmic standard ruler. This is possible because the pattern of large-scale distribution of matter is scale-dependent and does not change in comoving space during the linear-regime evolution of structure. By examining the pattern of LSS in several redshift intervals it is possible to reconstruct the expansion history of the universe, and thus to measure the cosmological parameters governing the expansion of the universe. The features of the large-scale matter distribution that can be used as standard rulers include the topology of LSS and the overall shapes of the power spectrum and correlation function. The genus, being an intrinsic topology measure, is insensitive to systematic effects such as the nonlinear gravitational evolution, galaxy biasing, and redshift-space distortion, and thus is an ideal cosmic ruler when galaxies in redshift space are used to trace the initial matter distribution. The genus remains unchanged as far as the rank order of density is conserved, which is true for linear and weakly nonlinear gravitational evolution, monotonic galaxy biasing, and mild redshift-space distortions. The expansion history of the universe can be constrained by comparing the theoretically predicted genus corresponding to an adopted set of cosmological parameters with the observed genus measured by using the redshift-comoving distance relation of the same cosmological model.</P>
SIMULATION OF COSMIC MICROWAVE BACKGROUND POLARIZATION FIELDS FOR AMiBA EXPERIMENT
PARK CHAN-GYUNG,PARK CHANGBOM The Korean Astronomical Society 2002 Journal of The Korean Astronomical Society Vol.35 No.2
We have made a topological study of cosmic microwave background (CMB) polarization maps by simulating the AMiBA experiment results. A ACDM CMB sky is adopted to make mock interferometric observations designed for the AMiBA experiment. CMB polarization fields are reconstructed from the AMiBA mock visibility data using the maximum entropy method. We have also considered effects of Galactic foregrounds on the CMB polarization fields. The genus statistic is calculated from the simulated Q and U polarization maps, where Q and U are Stokes parameters. Our study shows that the Galactic foreground emission, even at low Galactic latitude, is expected to have small effects on the CMB polarization field. Increasing survey area and integration time is essential to detect non-Gaussian signals of cosmological origin through genus measurement.
THE CHALLENGE OF THE LARGEST STRUCTURES IN THE UNIVERSE TO COSMOLOGY
Park, Changbom,Choi, Yun-Young,Kim, Juhan,Gott III, J. Richard,Kim, Sungsoo S.,Kim, Kap-Sung IOP Publishing 2012 ASTROPHYSICAL JOURNAL LETTERS - Vol.759 No.1
<P>Large galaxy redshift surveys have long been used to constrain cosmological models and structure formation scenarios. In particular, the largest structures discovered observationally are thought to carry critical information on the amplitude of large-scale density fluctuations or homogeneity of the universe, and have often challenged the standard cosmological framework. The Sloan Great Wall (SGW) recently found in the Sloan Digital Sky Survey (SDSS) region casts doubt on the concordance cosmological model with a cosmological constant (i.e., the flat Lambda CDM model). Here we show that the existence of the SGW is perfectly consistent with the Lambda CDM model, a result that only our very large cosmological N-body simulation (the Horizon Run 2, HR2) could supply. In addition, we report on the discovery of a void complex in the SDSS much larger than the SGW, and show that such size of the largest void is also predicted in the Lambda CDM paradigm. Our results demonstrate that an initially homogeneous isotropic universe with primordial Gaussian random phase density fluctuations growing in accordance with the general relativity can explain the richness and size of the observed large-scale structures in the SDSS. Using the HR2 simulation we predict that a future galaxy redshift survey about four times deeper or with 3mag fainter limit than the SDSS should reveal a largest structure of bright galaxies about twice as big as the SGW.</P>