Transcript Topology of large scale structure

Cosmological N-Body Simulation
- Topology of Large scale Structure
CCP 2006. 8. 29
Changbom Park
with Juhan Kim
(Korea Institute for Advanced Study)
& J. R. Gott (Princeton) , J. Dubinski (CITA)
History of Universe
Theme: Origin & Formation Mechanism
of Cosmic Structures
1. Want to know
Origin – primordial density fluctuations from inflation
Formation Mechanism – galaxies form at peaks in density field
smoothed over galactic scale?
2. Time is ripe
Large redshift surveys of galaxies
 High precision measurements of
1. Relations among internal physical properties
2. Relations between internal properties and
spatial & temporal environments
CfA1986
SDSS2006
h-1Mpc
SDSS galaxies
(Park et al. 2005, ApJ, 633, 11)
Cosmological N-Body Simulation
For PRECISION COMPARISON
between cosmological models with observations
Effects of NL Gravitational Evolution, Biasing,
& Redshift Space Distortion
on galaxy clustering & properties
Cosmological N-Body Simulation
Requirement for galaxy formation study
1. Several times larger than largest survey >> 1000 h-1Mpc
: for LSS formation + galaxy formation, velocity field
* SDSS[2006] ~ 500 h-1Mpc * Hubble Depth S.[2015] ~ 2000 h-1Mpc
2. Should resolve objects with <<1011 h-1Msun (~ M*+2)
: mean separation < 0.2 h-1Mpc
 currently 0.2~2000Mpc
Number of particles > 50003 ~ 100003 will do!
(100~1000 billion =10~100* current maximum)
Cosmological N-Body Simulation
Progresses

~ 104 CPUs

> 1010 particles
Log N=0.2(Y-1970)+2
TreePM Code1
About Code
1. Long range (r>4 pixels, PM) + Short range(PM+Tree) G-forces
2. Tree generation in each slab & in each cube of 43 pixels
3. Min. # of particles for tree generation – Direct P2 if #(cube) < Ntree
4. Memory : ~3 x [16] x words per particle
* 16 per particle: index2, position3, velocity3, acceleration3, mass1,
softening length, computational work measurement, pointer
* factor ~3 for memory imbalance
* Buffer zone particles
TreePM
Gravitational
Force
Tree + PM
PM
PM
Force
Gaussian
Smoothed
RG=0.9 pixels
TreePM Code2
Advantages
1. O(N log N) Tree operations for short range force – unlike P3M
2. Periodic boundary condition solved by PM – unlike Tree
3. No need to build a global tree – force correction only out to 4 pixels
4. Local Trees
 Parallelizable by domain decomposition (time)
& disposable local trees keeping trees in 8x8xnz pixels (memory)
Parallelization
1. PM part
: Domain slabs of equal thickness
2. Tree part
: Domain slabs of equal # of
tree force interactions &
Buffer zone particles
TreePM Code3
5. Accuracy : ~ 0.5% RMS error in acceleration for θ=1
6. Performance
CPU time
per step
10243 particles
Regular backup &
Pre-halo finding
calculation
Load
balance
10243 particles
# of particles
in domain slabs
/ homogeneous
distribution
ΛCDM Simulations
(Kim & Park 2004. 7)
TreePM code GOTPM (Dubinski, Kim, Park 2003)
20483 mesh (initial condition)
20483 CDM particles
1024 & 5632 h-1Mpc size boxes
50 & 275 h-1kpc force resolutions
* Using IBM SP3 at KISTI, 128 CPUs, 900 Gbytes,
FOR PRECISION COMPARISON
between cosmological models & real universe
Growth of Structures
from initial Density Fluctuations
13.7b
7.7b
11.8b
t=0
Dark Halo
Identification
(Kim& Park 2006:
ΛCDM 1024 h-1Mpc)
Physically SelfBound Halos
Halo centers
- local density peaks
Binding E wrt local
halo centers
Tidal radii of subhalos
wrt bigger halos
Halos with >=53
particles (5x1011 M⊙)
PSB Halos
VS
Others
Topology study
1. Gaussianity of the linear (primordial) density field
predicted by simple inflationary scenarios
2. Topology of galaxy distribution at NL scales sensitive
to cosmological parameters &
to galaxy formation mechanism
3. Direct Intuitive meaning
Large Scales
Primordial Gaussianity
Small Scales
Galaxy Formation
Cosmological Parameters
Genus – A Measure of Topology

Definition
G = # of holes - # of isolated regions
in iso-density contour surfaces
= 1/4π· ∫S κ dA (Gauss-Bonnet Theorem)
[ex. G(sphere)=-1, G(torus)=0,
]
: 2 holes – 1 body = +1

Gaussian Field
Genus/unit volume g(ν) = A (1-ν2) exp(- ν2/2)
where ν=(ρ- ρb)/ ρbσ &
A=1/(2π)2 <k2/3>3/2
if P(k)~kn, A RG3 =[8√2π2]-1 * [(n+3)/3]3/2

Non-Gaussian Field (Toy models)
Clusters
Bubbles
HDM
(Weinberg, Gott & Melott 1987)
Non-Gaussianity: Genus-related statistics
1. Shift parameter : 
2. Asymmetry parameters :AC, AV
3. Amplitude drop : RAAobs/APS
RA

Av
AC
Biased
Formation
of Galaxies
L-dependence
of 1 & 2 point
distribution,
but also
topology !
(Park et al. 2005)
Topology of LSS can be explained by GF models?
LCDM1024
Matter field can’t !
void splitting
Merger  Halo formation
void percolation
(Park, Kim et al. 2005)
Topology of LSS can be explained by GF models?
Probably yes!
(Park et al. 2005)
~1 & Little
evolution at low z
Direction of
evolution !
HOD model for VL :
sample Mr<-19.5
<Nsat> = (M/M1)α for M>Mmin
where logMmin=11.76, log
M1=13.15, α=1.13
Mergers of
halos
AV < 1 !
Comparison of topology: SDSS vs CDM
SDSS & 6 h-1Mpc scale; Kim+Park(o) & Springel(x)
Future of
Cosmological N-Body Simulation
1. Useful for cosmology & galaxy formation study
(until star formation can be properly simulated by
radiative hydro-codes)
2. Need to reach # of particles >> 50003 ~ 100003
(10~100 current maximum)
Dynamic range for other studies
* Internal properties & environment: 1kpc ~ 100 Mpc
* Galactic structure & star formation : 0.1pc ~ 100kpc