Cosmological N-Body Simulation - Topology of Large scale Structure Changbom Park with Juhan Kim (Korea Institute for Advanced Study) & J. R. Gott (Princeton), J. Dubinski (CITA) CCP

History of Universe

Theme: Origin & Formation Mechanism of Cosmic Structures 1. Want to know Origin – primordial density fluctuations from inflation Formation Mechanism – galaxies form at peaks in density field smoothed over galactic scale? 2. Time is ripe Large redshift surveys of galaxies  High precision measurements of 1. Relations among internal physical properties 1. Relations among internal physical properties 2. Relations between internal properties and 2. Relations between internal properties and spatial & temporal environments spatial & temporal environments

SDSS2006 CfA1986

SDSS galaxies h -1 Mpc (Park et al. 2005, ApJ, 633, 11)

  Effects of NL Gravitational Evolution, Biasing, & Redshift Space Distortion on galaxy clustering & properties For PRECISION COMPARISON between cosmological models with observations Cosmological N-Body Simulation

Requirement for galaxy formation study 1. Several times larger than largest survey >> 1000 h -1 Mpc : for LSS formation + galaxy formation, velocity field : for LSS formation + galaxy formation, velocity field * SDSS[2006] ~ 500 h -1 Mpc * Hubble Depth S.[2015] ~ 2000 h -1 Mpc * SDSS[2006] ~ 500 h -1 Mpc * Hubble Depth S.[2015] ~ 2000 h -1 Mpc 2. Should resolve objects with <<10 11 h -1 M sun (~ M * +2) : mean separation < 0.2 h -1 Mpc : mean separation < 0.2 h -1 Mpc  currently 0.2~2000Mpc  currently 0.2~2000Mpc Number of particles > ~ will do! Number of particles > ~ will do! (100~1000 billion =10~100* current maximum) (100~1000 billion =10~100* current maximum)

Cosmological N-Body Simulation Dynamic range for other studies * Internal properties & environment: 1kpc ~ 100 Mpc * Internal properties & environment: 1kpc ~ 100 Mpc * Galactic structure & star formation : 0.1pc ~ 100kpc * Galactic structure & star formation : 0.1pc ~ 100kpc

Cosmological N-Body Simulation Progresses  ~ 10 4 CPUs  > particles Log N=0.2(Y-1970)+2

TreePM Code 1 About Code 1. Long range (r>4 pixels, PM) + Short range(PM+Tree) G-forces 2. Tree generation in each slab & in each cube of 4 3 pixels 3. Min. # of particles for tree generation – Direct P 2 if #(cube) < N tree 4. Memory : ~3 x [16] x words per particle * 16 per particle: index 2, position 3, velocity 3, acceleration 3, mass 1, * 16 per particle: index 2, position 3, velocity 3, acceleration 3, mass 1, softening length, computational work measurement, pointer softening length, computational work measurement, pointer * factor ~3 for memory imbalance * factor ~3 for memory imbalance * Buffer zone particles * Buffer zone particles

TreePM Gravitational Force PM Tree + PM PMForce Gaussian Smoothed R G =0.9 pixels

TreePM Code 2 Advantages 1. O(N log N) Tree operations for short range force – unlike P 3 M 2. Periodic boundary condition solved by PM – unlike Tree 3. No need to build a global tree – force correction only out to 4 pixels 4. Local Trees  Parallelizable by domain decomposition (time)  Parallelizable by domain decomposition (time) & disposable local trees keeping trees in 8 x 8 x n z pixels (memory) & disposable local trees keeping trees in 8 x 8 x n z pixels (memory)

Parallelization 1. PM part 2. Tree part : Domain slabs of equal thickness : Domain slabs of equal # of : Domain slabs of equal thickness : Domain slabs of equal # of tree force interactions & tree force interactions & Buffer zone particles Buffer zone particles

TreePM Code 3 5. Accuracy : ~ 0.5% RMS error in acceleration for θ=1 6. Performance 6. Performance

CPU time per step particles Regular backup & Pre-halo finding calculation

Load balance particles # of particles in domain slabs / homogeneous distribution distribution

ΛCDM Simulations ΛCDM Simulations (Kim & Park ) TreePM code TreePM code GOTPM (Dubinski, Kim, Park 2003) mesh mesh (initial condition) CDM particles 1024 & 5632 h -1 Mpc 1024 & 5632 h -1 Mpc size boxes 50 & 275 h -1 kpc 50 & 275 h -1 kpc force resolutions FOR PRECISION COMPARISON between cosmological models & real universe * Using IBM SP3 at KISTI, 128 CPUs, 900 Gbytes,

N-body Simulations a) b) c) a)  IBM SP3 at KISTI, 128 CPUs, 900 Gbytes, b)  IBM SP3 at SNU, 16 CPUs, c)  QUEST at KIAS, 128 CPUs, 256 Gbytes,  b =0.046

Growth of Structures from initial Density Fluctuations 13.7b 11.8b 7.7b t=0

Dark Halo Identification (Kim& Park 2006: ΛCDM 1024 h -1 Mpc ΛCDM 1024 h -1 Mpc ) Physically Self- Bound Halos Halo centers - local density peaks Binding E wrt local halo centers Tidal radii of subhalos wrt bigger halos Halos with >=53 particles (5x10 11 M ⊙ )

PSB Halos VS Others

Topology study 1. Gaussianity of the linear (primordial) density field predicted by simple inflationary scenarios 2. Topology of galaxy distribution at NL scales sensitive to cosmological parameters & to galaxy formation mechanism 3. Direct Intuitive meaning Large Scales Small Scales Large Scales Small Scales Primordial Gaussianity Galaxy Formation Cosmological Parameters Cosmological Parameters

Genus – A Measure of Topology Definition Definition G = # of holes - # of isolated regions G = # of holes - # of isolated regions in iso-density contour surfaces in iso-density contour surfaces = 1/4π· ∫ S κ dA (Gauss-Bonnet Theorem) = 1/4π· ∫ S κ dA (Gauss-Bonnet Theorem) [ex. G(sphere)=-1, G(torus)=0, ] [ex. G(sphere)=-1, G(torus)=0, ] : 2 holes – 1 body = +1 Gaussian Field Gaussian Field Genus/unit volume g(ν) = A (1-ν 2 ) exp(- ν 2 /2) Genus/unit volume g(ν) = A (1-ν 2 ) exp(- ν 2 /2) where ν=(ρ- ρ b )/ ρ b σ & where ν=(ρ- ρ b )/ ρ b σ & A=1/(2π) 2 3/2 A=1/(2π) 2 3/2 if P(k)~k n, A R G 3 =[8√2π 2 ] -1 * [(n+3)/3] 3/2 if P(k)~k n, A R G 3 =[8√2π 2 ] -1 * [(n+3)/3] 3/2

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

Non-Gaussianity: Genus-related statistics 1. Shift parameter :  2. Asymmetry parameters :A C, A V 3. Amplitude drop : R A  A obs /A PS ACAC AvAv  RARA

Biased Formation of Galaxies L-dependence of 1 & 2 point distribution, but also topology ! (Park et al. 2005)

Topology of LSS can be explained by GF models? (Park, Kim et al. 2005) Merger  Halo formation void percolation void splitting LCDM1024 Matter field can’t !

Topology of LSS can be explained by GF models? Direction of evolution ! ~1 & Little evolution at low z Mergers of halos A V < 1 ! = (M/M 1 ) α for M>M min where logM min =11.76, log M 1 =13.15, α=1.13 = (M/M 1 ) α for M>M min where logM min =11.76, log M 1 =13.15, α=1.13 HOD model for VL : sample M r <-19.5 (Park et al. 2005) Probably yes!

Comparison of topology: SDSS vs  CDM SDSS & 6 h-1Mpc scale; Kim+Park(o) & Springel(x)

Future of Cosmological N-Body Simulation 1. Useful for cosmology & galaxy formation study (until star formation can be properly simulated by radiative hydro-codes) (until star formation can be properly simulated by radiative hydro-codes) 2. Need to reach # of particles >> ~ (10~100 current maximum) (10~100 current maximum) Dynamic range for other studies * Internal properties & environment: 1kpc ~ 100 Mpc * Internal properties & environment: 1kpc ~ 100 Mpc * Galactic structure & star formation : 0.1pc ~ 100kpc * Galactic structure & star formation : 0.1pc ~ 100kpc