26 May – 1 June 2008, Teipei-Hualian

Li-Zhi Fang University of Arizona

collaborators

supported by

US NSF Ast-0506734US NSF Ast-0507340

University of Arizona, Physics Li-Zhi Fang, Ji-Ren Liu Ping He Yi Lu

Brown University, Applied Mathematics Chi-Wang Shu Jing-Mei Qiu

Purple Mountain Observatory, Long-Long Feng, John Hopkins University Wei Zheng

High Order Accurate Weighted Essentially Non-Oscillatory (WENO) Algorithms with Applications to Cosmological Hydrodynamic Simulations

turbulence and large scale structure of the universe

structure formation of the universe, 2007

(1757)

(1923)

(1941)

quasar absorption spectrum: HI absorption

minihalo model of quasar’s Ly-alpha forests

• self-gravitating objects:

Bachall & Salpeter 1965 Black 1981

• pressure-confined clouds

Sargent et al. 1980 Ostriker 1988

• low mass object Bond, Szalay & Silk 1988

• miniholes Rees 1986 Murakami & Ikeuchi 1993

A

B

Yi-Hu Fang et al 1996

Lyman-alpha absorption clouds are un-clustered, not virialized (1996)

low mass object,pressure-confined clouds,self-gravitating objects:minihalo model are incorrect

log-normal model

QSO HP 1700+64

Zheng et al. 2004

quasar HE 2347-4342

b=

r2kTmc2

b(He) = b(H)=2

not thermal broadening

Zheng et al. 2004

quasar HS 1700+64Ly-alpha transmittance flux

clusters

Jackhedkar, Zhan, Fang 2000

±Fr (x) = F (x +r) ¡ F (x)

Jackhedkar, Zhan, Fang 2000

PDF of

sigmaLyman-alpha transmitted flux fluctuations are highly non-Gaussian (2000)

±F = F (x + r) ¡ F (x)

• non-thermal equilibrium

• not virialized objects

• not laminar flow

intermittency

intermittency is the alternation of phases of apparently periodic and chaotic dynamics. Consider a dynamical system. Let x be the observed variable. If x plotted as a function of time exhibits segments of relative constant values (laminar phase) interspersed by erratic bursts, the system dynamics is intermittent.

financial time series

Black Monday, October 1987

clusters

intermittent distribution random variables

random variable

PDF

»j can be 0 and 2 with probabilities 1=2

P =

8<

:

(2N ¡ 1)=2N »= 01=2N »= 2N

0 others

»= ¦ Nj =1»j = »1»2 ¢¢¢»N

h»i =0+0+¢¢¢+0+2N

2N =1

h»2i ¡ h»i2 =0+0+¢¢¢+22N

2N¡ 1=2N ¡ 1

ph»2i ¡ h»i2 À h»i

Probability Distribution Function (PDF)

P =

8<

:

(2N ¡ 1)=2N »= 01=2N »= 2N

0 othersP (») =1

¾p

2¼e¡ »2=2¾2

h»i =0+0+¢¢¢+0+2N

2N = 1

h»2i ¡ h»i2 =0+0+¢¢¢+22N

2N¡ 1= 2N ¡ 1

¾=p

h»2i ¡ h»i2 ' 2N =2

0

0.015

»= 8¾

long tail

N=6

Ut + f (U)X + g(U)Y +h(U)Z = F (t;U)

U = (½;½u;½v;½w;E )

f (U) = (½u;½u2 +P;½uv;½uw;u(E +P ))

g(U) = (½v;½uv;½v2 +P;½vw;v(E +P ))

h(U) = (½w;½uw;½vw;½w2 +P;w(E +P ))

E =P

° ¡ 1+

12½(u2 + v2 + w2)

F (t;U) =

µ0;¡

_aa

½V +½G;¡ 2_aa

E +½V ¢G ¡ ¤net

¶

G = ¡ r R © r 2©(x;t) = 4¼G[½tot(x;t) ¡ ½0(t)]=a

drDM

dt =1avDM ;

dvDM

dt =¡_aavDM +G

baryon fluid(Navier-Stokes)

gravity (Einstein)

dark matter (Newton)

a comsic factor

heating and cooling

peculiar velocity is irrotational, or potential.

structure formation: growth mode

the dynamical equation of baryon gas is

stochastic force driven Burgers' equation

or KPZ equation

v = ¡1a

r Á

Berera, Fang, PRL (1994), Jeans, ApJ (1999), Matarrese, Mohayaee, ApJ (2002)Feng, Pando, Fang, ApJ, (2003)

gravitational potential

@Á@t

¡1

2a2(r Á)2 +

ºa2

r 2Á = '

Jeans diffusion

Burgers’ turbulence

• Reynolds number

3/40 )/( arR c

Correlation length of random gravitational field Jeans smoothing length

Polyakov, PRE, (1995)Boldyrev, Linde, Polyakov, PRL, (2004)

scale free regime, hierarchical clustersfully developed turbulence

statistically quasi-equilibrium state

)()( xvrxvvr

prvS pr

pr

||

Kolmogorov (1941)

structure function

Gaussian field

S2pr = (2p¡ 1)!!(S2

r )p

S2pr =(S2

r )p = (2p¡ 1)!! r ¡ independent

Spr / r»(p)

»(p) = ®p

S2pr =(S2

r )p; r ¡ independent

self-similar field

Spr / r»(p)

S2pr =(S2

r )p / r»(2p)¡ p»(2); r ¡ dependent

»(p) is a nonlinear function of p:

intermittent field

models of hierarchical clusters

• beta model• linked-pair hierarchy• hierarchical Gaussian fluctuation (Press-Schechter

theory)• lognormal model• halo model

Soneira, Peebles 1977

beta modelradius R; R=̧ ; R=̧ 2;¢¢¢

N = 3

N(R=̧ )3 < R3

N¸¡ 3 = ¯ < 1

number of objects

test of beta model

p-dependence is linear. Not an intermittent field.

¹Spr / r»

»= ¡ (p¡ 1)(d¡ · ) < 0

N(R=̧ )3 < R3

N¸¡ 3 = ¯ < 1

N ?

linked-pair hierarchical clustering

Q_n are constant S. White 1979

Feng, Pando, Fang, 2001

d dimension

j / k

Q2n =1

2dj (n¡ 1)

¹S2nj

P n¡ 1j

testing linked-pair hierarchical clustering

Feng, Pando, Fang, 2001

hierarchical Gaussian fluctuation (Press-Schechter model)

Bond, Cole, Efsthathiou, Kaiser 1991

k-space

hierarchical models based on randomly additional process generally do notproduce intermittent field.

randomly multiplicational process

(central limitation theorem)

»= ¦ Nj =1»j = »1»2 ¢¢¢»j

ln»= ln»1 + ln»2 + ¢¢¢+ ln»N

»= »1 + »2 +¢¢+»N

hierarchical clusters

testing of hierarchical Gaussian fluctuation

Pando, Lipa, Greiner, Fang 1998

randomly multiplicational process

randomly additional process

halo modelmass fields are given by a superposition of the halos on various scales, and therefore, all non-Gaussian behaviors of the density field can be described by a universal density profile

The halo-halo correlation function

on scales larger than the halo size

is given by the two point correlation

function of the initially linear

Gaussian field.

log-Poisson hierarchical model

)()( xrxr

)1/()]/[ln(

!/)exp()(

)/(

)()(

21

21

12

21

2121

21

1212

rr

mmP

rrW

rWr

rr

mrrrr

mrr

rrrr

121

rrW

Liu, Fang, 2008, Lu, Fang, 2008

)]1/()1([

||

p

p

pr

pr

pp

rS p

Poisson random

m=1

m=2

m=N

r_1r_2

)1/()]/[ln(

!/)exp()(

)/(

)()(

21

21

12

21

2121

21

1212

rr

mmP

rrW

rWr

rr

mrrrr

mrr

rrrr

hierarchical clusteringrandomly multiplicative processscale invariance, self-similaritytranslational invarianceinfinite divisibility (the difference |r_1-r_2| can be finite or infinitesimal)

drr 21 lnln

tindependen- is )/(

spectrumpower ,||

)]1/()1([

1

22

222

rSS

rS

ppp

nr

nr

rr

pp

Gaussian field

S2nr = (2n ¡ 1)!!(S2

r )n

S2nr =(S2

r )n = (2n ¡ 1)!! r ¡ independent

¯ = 1 Gaussian ¯eld

¸ r1r2 ! 1 ; when ¯ ! 1

)1/()]/[ln(

!/)exp()(

)/(

)()(

21

21

12

21

2121

21

1212

rr

mmP

rrW

rWr

rr

mrrrr

mrr

rrrr

structures

rrFp

SSrF

rSp

r

pr

prp

ppr

r

)( ,

/)(

structures uppick

structuressinglar most by dominant is ,high for

structures

1

Fp(r) / r¡ ° (1¡ ¯ p )

log-Poisson hierarchical

. therefore,

tests of log-Poisson hierarchy

Intermittent exponent

beta-hierarchy

high order moment

scale-scale correlations

statistical properties

samplesmass density field of IGM, HI (simulation)

velocity field (simulation)

Lyman-alpha transmitted flux (simulation, observation)

scaling relations (simulation, observation)

He, Liu, Feng, Shu Fang, 2006Zhang, Liu, Feng, Fang, 2006Liu, Fang 2008Lu, Fang, 2008

)1(3/)]1(1[/ 3

3p

p CpC

He, Liu, Feng, Shu, Fang, PRL, (2006)

She-Leveque formula

intermittent exponent of mass density field

)]1/()1([

||

p

p

pr

p

r p

Liu, Fang, ApJ, 2008

beta- hierarchical

/123

/11

1

)](/)([)(/)(

........

)](/)([)(/)(

)(/)()(

rFrFrFrF

rFrFrFrF

rSrSrF

pp

ppp

p-invariance

)](/)(ln[)/1()(/)(ln

)](/)([)(/)(

231

/1231

rFrFrFrF

rFrFrFrF

pp

pp

beta-hierarchy

Liu, Fang, 2008

)](/)(ln[)/1()(/)(ln

)](/)([)(/)(

231

/1231

rFrFrFrF

rFrFrFrF

pp

pp

beta-hierarchy of Lyman-alpha transmitted flux

Lu, Fang 2008

scale-scale correlation

Lu & Fang 2008

lognormal vs. log-Poisson models

Lu, Fang 2008

evolution of IGM fields (in scale free range)

linear regime

P (k) / k2®

¯ = 1

»ln = ¡ ®p

»nl = 0

¯ < 1

»ln = ¡ ®p

»nl = ¡ °[p¡ (1¡ ¯p)=(1¡ ¯)]

nonlinear regime

Spr / r»p

»p = »ln +»nl

® power spectrum index

¯ non¡ Gaussianity

° singularity