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Topology of Large Scale Structures Introduction, Theory and Progress Report

KIAS Workshop on Cosmology and Structure Formation. Topology of Large Scale Structures Introduction, Theory and Progress Report. 2004. 10. 28-29 Changbom Park (Korea Institute for Advanced Study). 1. Genus – A Measure of Topology. Definition G = # of holes - # of isolated regions

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Topology of Large Scale Structures Introduction, Theory and Progress Report

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  1. KIAS Workshop on Cosmology and Structure Formation Topology of Large Scale StructuresIntroduction, Theory and Progress Report 2004. 10. 28-29 Changbom Park (Korea Institute for Advanced Study)

  2. 1. Genus– A Measure of Topology • Definition G = # of holes - # of isolated regions = 1/4π·∫S κ dA (Gauss-Bonnet Theorem) [ex. G(sphere)=-1, G(torus)=0 ] : 2 holes – 1 body = +1

  3. Why Topology? 1. Gaussianity of the primordial density field as predicted by many inflationary scenarios. 2. Topology of galaxy distribution at non-linear scales is sensitive to initial density power spectrum, cosmological parameters, and to galaxy formation mechanism. That is A. Initial density fluctuation  random phase Gaussian (L scale) B. Gravitational Instability  G formation  non-Gaussian (NL scale) C. Non-gravitational effects of G formation  non-Gaussian (NL scale) GENUS as a PRECISION MEASURE

  4. Genus- LS Galaxy Distribution • Genus of iso-density contour surfaces in smoothed galaxy density distribution as a function of density threshold level (Weinberg, Gott & Melott 1987)

  5. Genus- LS Galaxy Distribution • Gaussian Field Genus/unit volume g(ν) = A (1-ν2) e- ν2/2 where ν=(ρ- ρb)/ ρbσ & A=1/(2π)2 <k2/3>3/2 if P(k)~kn, A=[8√2π2 RG3]-1 *[(n+3)/3]3/2

  6. Clusters Bubbles HDM • Non-Gaussian Field (Toy models) (Weinberg, Gott & Melott 1987)

  7. Genus-Related Statistics • Amplitude dropRA RA = Aobs / APS • Shift parameterΔν By fitting Gobs(ν) over –1<ν<1 • Asymmetry parametersAC & AV A = ∫ Gobs(ν) d ν/∫ Gfit(ν) d ν where intervals are 1.2~2.2 (AC), -1.2~-2.2 (AV) UZC+SSRS2

  8. 2. Historyof Topology Study in Cosmology I. Early Works • 1986: Hamilton, Gott, Weinberg; Gott, Melott, Dickinson – smooth small-scale NL clustering to recover initial topology • 1987-8: GWM, WGM, MWG, Gott et al. – cosmological & toy models. RG>3rc to recover initial topology • 1989: Gott et al.– observed galaxies, dwarfs, clusters • 1991: Park, Gott– gravitational & biasing effects • 1992: Weinberg, Cole– PS, initial skewness, biasing effects • 1994: Matsubara – 2nd order perturbation in weakly NL regime • 1996: Matsubara – redshift space distortion in L regime • 1996: Matsubara, Suto – gravitational & z-space distortion • Etc….

  9. II. Recent Works • 2000: Colley et al.– Simulation of SDSS • 2001, 2003: Hikage, Taruya & Suto– dark halos (analytic & numerical) • 2003: Matsubara – 2nd orber perturbation theory • [ Minkowski functionals (Mecke, Buchert & Wagner 1994; Schmalzing & Buchert 1997 etc.)] III. 3D genus analysis of observational data 1989: Gott et al. - CfA 1 etc. 1992: Park, Gott, & da Costa - SSRS 1 1992: Moore et al. - IRAS QDOT 1994: Rhoads et al. - Abell Clusters 1994: Vogeley et al. - CfA 1+2 1997: Protogeros & Weinbergs - IRAS 1.2Jy 1998: Springel et al. - IRAS 1.2Jy 1998: Canavezes et al. - IRAS PSCz 2002: Hikage et al. - SDSS EDR 2003: Hikage et al. - SDSS LSS Sample 12 2004: Canavezes & Efstathious - 2dFRGS

  10. IV. 2D Genus (LSS) • 2D genus before SDSS • Suggested by Melott (1987) • Coles & Plionis (1991): Lick Galaxy Catalogue • Plionins, Valdarnini, & Coles (1992): Abell and ACO cluster catalogue • Park et al. (1992): CfA Slice • Colley (2000): Simulated SDSS • Park, Gott, & Choi (2001): HDF • Hoyle, Vogeley & Gott (2002): 2dFGRS • 2D genus with SDSS • Hoyle, Vogeley & Gott (2002): weak evidence for variation in the genus with galaxy type

  11. 3. Gravitational Evolution, Biasing, Redshift Space DistortionEffects on Topology ΛCDM Simulation(Kim & Park 2004) PMTree code(Dubinski, Kim, Park 2003) 20483 mesh (initial condition) 20483 (8.6G) CDM particles 1024 & 5632 h-1Mpc size boxes 50 & 275 h-1kpc force resolutions (Park et al. 1994) (Tegmark et al. 2004)

  12. Park, Kim & Gott (2004) Genus of matter distribution Amplitude drop RA= Asim / APS ShiftΔν # of Voids and Clusters AV & AC (at z = 0, 1, 2, 5, 8) 8 5 2 1 0

  13. Biasing  AV at small scales Redshift space distortion  small for AV

  14. 3.Analytic Model Matsubara (1994,2003) : perturbation theory Matsubara(1996): linear theory of z-effects

  15. Sloan Digital Sky Survey 1. Imaging of North Galactic Cap 2.5m APO telescope with a mosaic CCD camera u, g, r, i, z photometric bandpasses  selected for spectroscopy 2. Spectroscopy ~ 106 galaxies & 105 quasars with rms z-error ~ 30 km/s 3. Samples Main Galaxies: rPet < 17.77 ; Quasars Luminous Red Galaxies (LRG): z<0.4 & >0.4 samples Korean Scientist Group (KSG) KIAS: Changbom Park & SNU: Myeong Lee, Myungshin Im KNU: Myeong-Gu Park & SU : Hwankyung Sung

  16. As of Oct. 14, 2004

  17. SDSS LSS Sample 14 in equatorial coordinate (314K galaxies)

  18. SDSS LSS Sample 14 in survey coordinate (314K galaxies)

  19. SDSS galaxies in region 1

  20. SDSS galaxies in Region 1 (Park et al. 2004)

  21. Genus Analysis of SDSS LSS Sample 14 SDSS : Large volume & dense sampling of galaxies Best sample to test Gaussianity of primordial fluctuation (randomness of quantum fluctuation)  Not yet! Structure formation mechanism  OK ! G = # of holes - # of isolated regions If Gaussian, G(ν) = A(1- ν2)exp(- ν2/2)

  22. Volume-limited subsamples For scale dependence For luminosity dependence

  23. Best Subset -20.21< Mr <-21.59 160 < r < 314 h-1Mpc 0.054 < z < 0.107 31,580 galaxies d = 6.3 h-1Mpc

  24. Scale Dependence Within a Subset : Same place(structure) & luminosity

  25. LuminosityDependence Each Subset : Same place(structure) luminositysmoothing L1: -20.5 ~ -22.5 L3: -19.0 ~ -21.0 L2: -19.5 ~ -21.5 (25524, 154~235 h-1Mpc) 9 subsets with the same # of galaxies

  26. YOU CAN SEE Few bright galaxies in under dense regions !

  27. Luminosity bias: Park et al. (1994):nearly scale-independent biasing underdense regions lack bright galaxies

  28. Collective Physical Properties of Galaxy Subsets Galaxy Clustering Properties (z): Correlation Function, Power Spectrum, Count in Cell, Topology, etc Galaxy Biasing, Gaussianity of Initial Density Fluctuation, Ωm, σ8, b, etc. Morphology, Surface Brightness, Luminosity, Velocity Dispersion, Color, Spectral Type, SFR, etc Velocity Field Ωm, σ8, b, etc. Halo Mass Distribution, Luminosity Function (z), Color-Magnitude Relation, etc. Galaxy Formation Environment: Local Density Internal Physical Parameters of Galaxies Different Tracers of Structure Formation: Galaxy, Cluster, Group, Void, Quasar, etc.

  29. Conclusions 0. Wait for SDSS to finish for LS topology 1. Topology analysis does differentiate galaxy species Brighter : meat-ball topology, smaller voids Fainter :bubble topology, bigger voids 2. Topology changes below the characteristic magnitude Mr*=-20.4+5 log h 3. AV < 1 (few & big voids) consistently at all scales < 10 h-1 Mpc independantly of L  Not gravitational evolution effects. Existence of biasing !

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