Probing cosmic structure formation in the wavelet representation

Li-Zhi FangUniversity of Arizona

IPAM, November 10, 2004IPAM, November 10, 2004

outline

• introduction

• phenomenological hierarchical clustering models and scale-scale correlation

• intermittency and small scale problem of LCDM model

• quasi-local evolution

• summary

cosmic random fields

• density distribution• velocity field • temperature • entropy • …

),(

),(

),(

),(

tS

tT

t

t

x

x

xv

x

density distribution of hydrogen gas in the LCDM model ),( tx

Feng,Shu, Zhang (2004)

Information of structures: position, scaledark matter baryon gas

representations• x-representation f(x,t)

• k(p)-representation f(k,t), f(p,t),

• Wigner distribution function (WDF), Wigner representation W(x,k,t), W(x,p,t),

phase-space behavior (position and scale) coherent structure phase-sensitive effects

WDF is density matrix, not a linear transform of f(x,t).

2 kx

j=1

j=2

j=3

j=4

mode (j, l)

)12...0( jl

space-scale decompositionby discrete wavelet mode

)()(~),(

)(),()(~

0

12

1

0

xttxf

dxxtxft

jlj l

jl

L

jljl

j

WFCs (wavelet functioncoefficients) of mode (j, l) represents fluctuations on a length scale L/2j

jl~

dynamical models of cosmic structure formation cosmological parameters, initial perturbations

simulation sample

observational samples

predictions

N-body simulationhydrodynamic simulationpseudo-hydro simulationlognormal model

1-D, 2-D, 3-Dbackground radiationIGM, galaxies

wavelet

wavelet

wavelet

hierarchical clustering

scale

j

j+1

j+2

small halos

massive halo

block model

Cole, Kaiser (1988)

2j

branching model

Pando, Lipa, Greiner, Fang (1998)

The first and second orders statistical properties of the block model and branching model are the same if

Pando, Lipa, Greiner, Fang (1998)

scale-scale correlation

1, ppjC

j

j

jC 22

422,2

)1(

)61(

block model

12

0

12

0' ',,

12

0 2,1,1

,1 ~~

~~2j j

j

l l

plj

plj

l

plj

plj

jpp

jC

branching model

Pando, Lipa, Greiner, Fang (1998)

Pando, Lipa, Greiner, Fang (1998)

Lu, Wolfe, Turnshek, sample of Ly-alpha absorption

branching model

Feng, Fang 2000

lognormal model

Feng, Deng, Fang 2000

APM-BGC galaxy sample

mock samples

problem of the LCDM model small scale powers

reducing the power on small scales relative to the `standard' LCDM model is favored by the analysis of WMAP plus galaxy and Lyman-alpha forest data. It can also solve the halo concentration and substructure problems.

Warm dark matter (WDM) models leads to exponential damping of linear power spectrum, and results in better agreement with observations

intermittencyThe lognormal model of IGM (baryon

gas)

(x,t)Jeans is a Gaussian random field derived from the density contrast

DM of dark matter filtered on the Jeans scale.

]2

),([

0

2

),(

JeansJeans t

IGM et

xx

Bi, Davidsen 1997

lognormal model at small scales

Feng, Fang 2000

lognormal field is intermittent

)()]()([

)]()([

][)(

2

2

2

2 nL

r

xrx

xrx

S

Sn

n

nr

nr

)1()( nnn

structure function

intermittent exponent

A field is intermittent if

Lognormal Field:

0)( n

structure function with DWT

][)(

ln1

)1()(

|~|2

1

2

2

2

212

0,

2

nj

nj

n

lljj

nj

S

S

jn

Sj

structure functions of Ly-alpha samples

Samples: 28 Keck HIRES QSO spectra. redshift z from 2.19 – 4.11

km/s24

s/km24.013

11

j

jk

Pando, Feng, Fang 2002

Structure Functions of Ly-alpha transmission

Pando, Feng,Jamkhedkar,Fang, 2002

LCDM

WDM

mw =1000ev

mw =800ev

mw =600ev

mw =300ev

pseudo hydro simulation

Feng, Pando, Fang 2003

simulation box 6 Mpc/h 12 Mpc/h

Fit Keck Data to

Pando, Feng, Jamkhedkar, Fang, 2002

j=11, 12 h-1 kpcj=10, 24 h-1 kpc

)1(log nnSF

Fit LCDM Data to

j=11, 12 h-1 kpcj=10, 24 h-1 kpc

)1(log nnSF

Pando, Feng, Jamkhedkar, Fang, 2002

Fit WDM 300 eV Data to

j=11, 12 h-1 kpcj=10, 24 h-1 kpc

)1(log nnSF

Pando, Feng, Jamkhedkar, Fang, 2002

quasi-local evolution of cosmic clustering

in the wavelet representation

Gaussian field in the DWT representation

''''~~

lljjljjl

j

l

2 kx

strongweak

quasi-locality (second order)

0l if ,0)(

~~

~~)(

',

2/12''

2/12

''',

l

l

jj

ljjl

ljjljj

different physical position

quasi-locality (higher order)

1)0(

~~

~~)(

,',

',',

',',,',

lC

lC

qpjj

qlj

plj

qlj

pljqp

jj

quasi-locality of mass field inZeld’ovich approximation

(weak linear regime)

If the initial perturbations are Gaussian, the cosmic gravitation clustering in weakly nonlinear regime given by the Zeldovich approximation is quasi-localized in the DWT representation.

Pando, Feng, Fang 2001

Mpch 2/189 1jPando, Feng, Fang 2001

second order quasi-locality : Ly-alpha data

Pando, Feng, Fang 2001Mpch 2/189 1j

second order quasi-locality : Ly-alpha data

Mpch 2/189 1j

Pando, Feng, Fang 2001

Ly-alpha data: second order

Pando, Feng, Fang 2001Mpch 2/189 1j

4th order locality: Ly-alpha data

halo model(highly nonlinear regime)

),()()( iii

iii

i mum xxxxx

mrm

rrrmru

sc

s

c

,

)(),(

Feng,Shu, Zhang (2004)

halo model

Xu, Fang, Wu 2000

quasi-locality of mass field in halo model

If the initial perturbations is Gaussian (no correlation between modes at different physical positions), the evolved field will also spatially weakly correlated. The nonlinear evolution of gravitational clustering has memory of the initial spatial correlations.

Feng, Fang 2004

Box size

(Mpc/h)

mp

(M_sun/h)

Lmin

(L_sun/h2)

Realizations

LCDM 100 6.5 x 108 2 x 108 4

LCDM 300 1.8 x 1010 3 x 109 4

N-body simulation samples

Jing, Suto (2000), Jing (2001)

Feng, Fang 2004

N-body simulation sample (Jing, Suto 2002)

One-point distribution of WFCs

Mpch 2/50 -1j

Feng, Fang 2004

two-point correlation functions: N-body simulation sample (Jing, Suto 2002)

Feng, Fang 2004

Scale-scale correlation: N-body simulation sample (Jing, Suto 2002)

Mpch 2/50 -1j

Feng, Fang 2004

DWT mode-mode correlations: N-body simulation sample (Jing, Suto 2002)

Feng, Fang 2004

Feng, Fang 2004

galaxy sample: 2MASS XSC (Jarrett et al. 2000)987,125 galaxies

Guo, Chu, Fang 2004

angular correlation functions: 2MASS data

Guo, Chu, Fang 2004

one-point distribution of WFCs: 2MASS data

Guo, Chu, Fang 2004degreeangular 2/28 j

mode-mode correlations of WFCs: 2MASS data

Guo, Chu, Fang 2004degreeangular 2/28 j

4th order correlations of WFCs: 2MASS data

degreeangular 2/28 j

Guo, Chu, Fang 2004

scale-scale angular correlation functions: 2MASS data

non-local scale-scale angular correlation functions: 2MASS data

Guo, Chu, Fang 2004

Initial perturbations probably are Gaussian on scales > 0.1 Mpc/h

other topics

• scaling of velocity field

• halos and the difference between the Fourier and DWT power spectrum

• redshift distortion and DWT power spectrum of non-diagonal modes

• the statistical decoupling of baryon gas from dark matter

summary• cosmic gravitational clustering essentially is

hierarchical and self-similar. A phase space description of clustering dynamics is necessary.

• The DWT with the features of similarity, admissibility, invertibility, orthogonal and locality in phase space provides an effective representation to describe the nonlinear evolution of the cosmic fields