Cosmological N-Body Simulation - Topology of Large scale Structure

Changbom Parkwith Juhan Kim

(Korea Institute for Advanced Study)amp J R Gott (Princeton) J Dubinski (CITA)

CCP 2006 8 29

History of Universe

Theme Theme Origin amp Formation Mechanism Origin amp Formation Mechanism of Cosmic Structuresof Cosmic Structures

1 Want to know1 Want to knowOriginOrigin ndash primordial density fluctuations from inflation ndash primordial density fluctuations from inflation

Formation MechanismFormation Mechanism ndash galaxies form at peaks in density field smoot ndash galaxies form at peaks in density field smoothed over galactic scalehed over galactic scale

2 Time is ripe2 Time is ripeLarge redshift surveys of galaxies Large redshift surveys of galaxies

High precision measurements ofHigh precision measurements of

1 Relations among 1 Relations among internal physical propertiesinternal physical properties

2 Relations between 2 Relations between internal properties and internal properties and

spatial amp temporal environmentsspatial amp temporal environments

SDSS2006

CfA1986

SDSS galaxies

h-1Mpc

(Park et al 2005 ApJ 633 11)

Effects of NL Gravitational Evolution Biasing amp Redshift Space Distortion on galaxy clustering amp properties

For PRECISION COMPARISONFor PRECISION COMPARISON between cosmological models with observationsbetween cosmological models with observations

Cosmological N-Body Simulation

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

History of Universe

Theme Theme Origin amp Formation Mechanism Origin amp Formation Mechanism of Cosmic Structuresof Cosmic Structures

1 Want to know1 Want to knowOriginOrigin ndash primordial density fluctuations from inflation ndash primordial density fluctuations from inflation

Formation MechanismFormation Mechanism ndash galaxies form at peaks in density field smoot ndash galaxies form at peaks in density field smoothed over galactic scalehed over galactic scale

2 Time is ripe2 Time is ripeLarge redshift surveys of galaxies Large redshift surveys of galaxies

High precision measurements ofHigh precision measurements of

1 Relations among 1 Relations among internal physical propertiesinternal physical properties

2 Relations between 2 Relations between internal properties and internal properties and

spatial amp temporal environmentsspatial amp temporal environments

SDSS2006

CfA1986

SDSS galaxies

h-1Mpc

(Park et al 2005 ApJ 633 11)

Effects of NL Gravitational Evolution Biasing amp Redshift Space Distortion on galaxy clustering amp properties

For PRECISION COMPARISONFor PRECISION COMPARISON between cosmological models with observationsbetween cosmological models with observations

Cosmological N-Body Simulation

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Theme Theme Origin amp Formation Mechanism Origin amp Formation Mechanism of Cosmic Structuresof Cosmic Structures

1 Want to know1 Want to knowOriginOrigin ndash primordial density fluctuations from inflation ndash primordial density fluctuations from inflation

Formation MechanismFormation Mechanism ndash galaxies form at peaks in density field smoot ndash galaxies form at peaks in density field smoothed over galactic scalehed over galactic scale

2 Time is ripe2 Time is ripeLarge redshift surveys of galaxies Large redshift surveys of galaxies

High precision measurements ofHigh precision measurements of

1 Relations among 1 Relations among internal physical propertiesinternal physical properties

2 Relations between 2 Relations between internal properties and internal properties and

spatial amp temporal environmentsspatial amp temporal environments

SDSS2006

CfA1986

SDSS galaxies

h-1Mpc

(Park et al 2005 ApJ 633 11)

Effects of NL Gravitational Evolution Biasing amp Redshift Space Distortion on galaxy clustering amp properties

For PRECISION COMPARISONFor PRECISION COMPARISON between cosmological models with observationsbetween cosmological models with observations

Cosmological N-Body Simulation

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

SDSS2006

CfA1986

SDSS galaxies

h-1Mpc

(Park et al 2005 ApJ 633 11)

Effects of NL Gravitational Evolution Biasing amp Redshift Space Distortion on galaxy clustering amp properties

For PRECISION COMPARISONFor PRECISION COMPARISON between cosmological models with observationsbetween cosmological models with observations

Cosmological N-Body Simulation

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

SDSS galaxies

h-1Mpc

(Park et al 2005 ApJ 633 11)

Effects of NL Gravitational Evolution Biasing amp Redshift Space Distortion on galaxy clustering amp properties

For PRECISION COMPARISONFor PRECISION COMPARISON between cosmological models with observationsbetween cosmological models with observations

Cosmological N-Body Simulation

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Effects of NL Gravitational Evolution Biasing amp Redshift Space Distortion on galaxy clustering amp properties

For PRECISION COMPARISONFor PRECISION COMPARISON between cosmological models with observationsbetween cosmological models with observations

Cosmological N-Body Simulation

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Cosmological N-Body Simulation

Requirement for galaxy formation studyRequirement for galaxy formation study

1 Several times larger than largest survey gtgt 1000 h1 Several times larger than largest survey gtgt 1000 h-1-1MpcMpc

for LSS formation + galaxy formation velocity field for LSS formation + galaxy formation velocity field

SDSS[2006] ~ 500 h SDSS[2006] ~ 500 h-1-1Mpc Hubble Depth S[2015] ~ 2000 hMpc Hubble Depth S[2015] ~ 2000 h-1-1MpcMpc

2 Should resolve objects with ltlt102 Should resolve objects with ltlt101111 h h-1-1MMsunsun (~ M (~ M+2)+2)

mean separation lt 02 h mean separation lt 02 h-1-1MpcMpc

currently 02~2000Mpccurrently 02~2000Mpc

Number of particles gt 5000Number of particles gt 500033 ~ 10000 ~ 1000033 will do will do

(100~1000 billion =10~100 current maximum)(100~1000 billion =10~100 current maximum)

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Cosmological N-Body Simulation

ProgressesProgresses

~ 104 CPUs

gt 1010 particles

Log N=02(Y-1970)+2

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

TreePM CodeTreePM Code11

About CodeAbout Code

1 Long range (rgt4 pixels 1 Long range (rgt4 pixels PMPM) + Short range() + Short range(PMPM++TreeTree) G-forces) G-forces

2 Tree generation in each slab amp in each cube of 42 Tree generation in each slab amp in each cube of 433 pixels pixels

3 Min of particles for tree generation ndash Direct P3 Min of particles for tree generation ndash Direct P22 if (cube) lt N if (cube) lt Ntreetree

4 Memory ~3 4 Memory ~3 xx [16] [16] xx words per particle words per particle

16 per particle index 16 per particle index22 position position33 velocity velocity33 acceleration acceleration33 mass mass11

softening length computational work measurement pointersoftening length computational work measurement pointer

factor ~3 for memory imbalance factor ~3 for memory imbalance

Buffer zone particles Buffer zone particles

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

TreePM Gravitational Force

PMPM

Tree + PMTree + PM

PMPM

ForceForce

GaussianSmoothed RG=09 pixels

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

TreePM CodeTreePM Code22

AdvantagesAdvantages

1 O(N log N) Tree operations for short range force ndash unlike P1 O(N log N) Tree operations for short range force ndash unlike P33MM

2 Periodic boundary condition solved by PM ndash unlike Tree2 Periodic boundary condition solved by PM ndash unlike Tree

3 No need to build a global tree ndash force correction only out to 4 pixels3 No need to build a global tree ndash force correction only out to 4 pixels

4 Local Trees 4 Local Trees

Parallelizable by domain decomposition (time)Parallelizable by domain decomposition (time)

amp disposable local trees keeping trees in 8amp disposable local trees keeping trees in 8xx88xxnnzz pixels (memory) pixels (memory)

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Parallelization

1 PM part 2 Tree part1 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 amptree force interactions amp Buffer zone particlesBuffer zone particles

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

TreePM CodeTreePM Code33

5 Accuracy ~ 05 5 Accuracy ~ 05 RMSRMS error in acceleration for error in acceleration for θθ=1=1

6 Performance6 Performance

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

CPU time per step

1024102433 particles particles

Regular backup amp Regular backup amp

Pre-halo finding Pre-halo finding

calculationcalculation

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Load balance

1024102433 particles particles

of particles of particles

in domain slabsin domain slabs

homogeneous homogeneous

distributiondistribution

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

ΛΛCDM SimulationsCDM Simulations (Ki

m amp Park 2004 7)

TreePM codeTreePM code GOTPM (Dubinski Kim Park 2003)

2048204833 mesh mesh (initial condition)

2048204833 CDM particles CDM particles

1024 amp 5632 h1024 amp 5632 h-1-1MpcMpc size boxes

50 amp 275 h50 amp 275 h-1-1kpckpc force resolutions

FOR PRECISION COMPARISON between cosmological models amp real universe

Using IBM SP3 at KISTI 128 CPUs 900 Gbytes

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Growth of Structures from initial Density Fluctuations

137b

118b

77b t=0

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Dark Halo Identification(Kimamp Park 2006

ΛΛCDMCDM 1024 h1024 h-1-1MpcMpc)

Physically Self-Bound Halos

Halo centers - local density peaks

Binding E wrt local halo centers

Tidal radii of subhalos wrt bigger halos

Halos with gt=53 particles (5x1011 M⊙)

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

PSB HalosVS

Others

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Topology studyTopology study

1 Gaussianity of the 1 Gaussianity of the linear (primordial) density fieldlinear (primordial) density field pr predicted by simple inflationary scenariosedicted by simple inflationary scenarios

2 Topology of galaxy distribution at NL scales sensitive 2 Topology of galaxy distribution at NL scales sensitive to to cosmological parameterscosmological parameters amp to amp to galaxy formation mechanismgalaxy formation mechanism

3 Direct Intuitive meaning3 Direct Intuitive meaning

Large ScalesLarge Scales Small ScalesSmall Scales

Primordial Gaussianity Galaxy FormationPrimordial Gaussianity Galaxy Formation

Cosmological ParametersCosmological Parameters

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

GenusGenus ndash A Measure of Topologyndash A Measure of Topology

DefinitionDefinition

G = of holes - of isolated regionsG = of holes - of isolated regions in iso-density contour surfacesin iso-density contour surfaces

= 14= 14ππ intintSS κ dA (Gauss-Bonnet Theorem) κ dA (Gauss-Bonnet Theorem)

[ex G(sphere)=-1 G(torus)=0 ][ex G(sphere)=-1 G(torus)=0 ]

2 holes ndash 1 body = +1

Gaussian FieldGaussian Field Genusunit volume g(ν) = A (1-νGenusunit volume g(ν) = A (1-ν22) exp(- ν) exp(- ν222)2) where ν=(ρ- ρwhere ν=(ρ- ρbb) ρ) ρbbσ amp σ amp

A=1(2π)A=1(2π)22 ltk ltk223gt3gt32 32

if P(k)~kif P(k)~knn A R A RGG33 =[8radic2π =[8radic2π22]]-1 -1 [(n+3)3][(n+3)3]32 32

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Non-Gaussian FieldNon-Gaussian Field (Toy models) (Toy models)

Clusters Bubbles HDM

(Weinberg Gott amp Melott 1987)(Weinberg Gott amp Melott 1987)

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Non-Gaussianity Genus-related statisticsNon-Gaussianity Genus-related statistics

1 Shift parameter 1 Shift parameter 2 Asymmetry parameters A2 Asymmetry parameters AC C AAVV

3 Amplitude drop R3 Amplitude drop RAAAAobsobsAAPSPS

ACAv

RA

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Biased Biased Formation Formation of Galaxiesof Galaxies

L-dependence L-dependence of 1 amp 2 point of 1 amp 2 point distribution distribution

but also but also topology topology

(Park et al 2005)

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

(Park Kim et al 2005)

Merger Merger Halo formation Halo formationvoid percolationvoid percolation

void splittingvoid splitting

LCDM1024LCDM1024

Matter field canrsquot

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Topology of LSS can be explained by Topology of LSS can be explained by GF modelsGF models

Direction of Direction of evolution evolution

~1 amp Little evoluti~1 amp Little evolution at low zon at low z

Mergers of hMergers of halosalos

AV lt 1

ltNltNsatsatgt = (MMgt = (MM11))αα for MgtM for MgtMminmin whe whe

re logMre logMminmin=1176 log M=1176 log M11=1=1

315 315 αα=113=113

HOD model for VL sHOD model for VL sample Mample Mrrlt-195lt-195

(Park et al 2005)Probably yes

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Comparison of topology SDSS vs Comparison of topology SDSS vs CDM CDM

SDSS amp 6 h-1Mpc scale Kim+Park(o) amp Springel(x)

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc

Future ofCosmological N-Body

Simulation1 Useful for cosmology amp galaxy formation study1 Useful for cosmology amp galaxy formation study

(until star formation can be properly simulated by radiati(until star formation can be properly simulated by radiative hydro-codes)ve hydro-codes)

2 Need to reach of particles gtgt 50002 Need to reach of particles gtgt 500033 ~ 10000 ~ 1000033

(10~100 current maximum)(10~100 current maximum)

Dynamic range for other studiesDynamic range for other studies Internal properties amp environment 1kpc ~ 100 Mpc Internal properties amp environment 1kpc ~ 100 Mpc

Galactic structure amp star formation 01pc ~ 100kpc Galactic structure amp star formation 01pc ~ 100kpc