5th Italian-Sino Workshop on Relativistic Astrophysics

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Intermittency of cosmic baryon fluid
Li-Zhi Fang
University of Arizona
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5th Italian-Sino Workshop on Relativistic
Astrophysics
26 May 1 June 2008, Teipei-Hualian
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collaborators
supported by
US NSF Ast-0506734 US NSF Ast-0507340
High Order Accurate Weighted Essentially
Non-Oscillatory (WENO) Algorithms with
Applications to Cosmological Hydrodynamic
Simulations
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
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turbulence and large scale structure
of the universe
(1757)
(1923)
(1941)
structure formation of the universe, 2007
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quasar absorption spectrum HI absorption
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minihalo model of quasars 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

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A
B
low mass object, pressure-confined
clouds, self-gravitating objects minihalo model
are incorrect
Lyman-alpha absorption clouds are
un-clustered, not virialized (1996)
Yi-Hu Fang et al 1996
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log-normal model
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QSO HP 170064
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quasar HE 2347-4342
Zheng et al. 2004
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not thermal broadening
Zheng et al. 2004
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quasar HS 170064Ly-alpha transmittance flux
clusters
Jackhedkar, Zhan, Fang 2000
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PDF of
sigma
Lyman-alpha transmitted flux fluctuations
are highly non-Gaussian (2000)
Jackhedkar, Zhan, Fang 2000
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• non-thermal equilibrium
• not virialized objects
• not laminar flow

intermittency
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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.
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financial time series
clusters
Black Monday, October 1987
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intermittent distribution
random variables
random variable
PDF
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Probability Distribution Function (PDF)
N6
0.015
0
long tail
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dark matter (Newton)
gravity (Einstein)
baryon fluid (Navier-Stokes)
heating and cooling
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structure formation growth mode
peculiar velocity is irrotational, or potential.
the dynamical equation of baryon gas is
stochastic force driven Burgers' equation or
KPZ equation
gravitational potential
Jeans diffusion
Berera, Fang, PRL (1994), Jeans, ApJ (1999),
Matarrese, Mohayaee, ApJ (2002) Feng, Pando,
Fang, ApJ, (2003)
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Burgers turbulence

• Reynolds number

Correlation length of random gravitational field
Jeans smoothing length
R gt 1 turbelence
Polyakov, PRE, (1995) Boldyrev, Linde, Polyakov,
PRL, (2004)
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scale free regime, hierarchical clusters fully
developed turbulence
Kolmogorov (1941)
statistically quasi-equilibrium state
structure function
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Gaussian field
self-similar field
intermittent field
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models of hierarchical clusters

• beta model
• linked-pair hierarchy
• hierarchical Gaussian fluctuation
  (Press-Schechter theory)
• lognormal model
• halo model

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beta model
radius
number of objects
Soneira, Peebles 1977
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test of beta model
p-dependence is linear. Not an intermittent
field.
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linked-pair hierarchical clustering
Q_n are constant
S. White 1979
Feng, Pando, Fang, 2001
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testing linked-pair hierarchical clustering
Feng, Pando, Fang, 2001
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hierarchical Gaussian fluctuation
(Press-Schechter model)
k-space
Bond, Cole, Efsthathiou, Kaiser 1991
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hierarchical clusters
hierarchical models based on randomly additional
process generally do not produce intermittent
field.
(central limitation theorem)
randomly multiplicational process
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testing of hierarchical Gaussian fluctuation
randomly multiplicational process
randomly additional process
Pando, Lipa, Greiner, Fang 1998
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halo model
mass 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.
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log-Poisson hierarchical model
Poisson random
Liu, Fang, 2008, Lu, Fang, 2008
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hierarchical clustering randomly multiplicative
process scale invariance, self-similarity translat
ional invariance infinite divisibility (the
difference r_1-r_2 can be finite or
infinitesimal)
mN
r_1
r_2
m1
m2
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Gaussian field
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structures
log-Poisson hierarchical
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tests of log-Poisson hierarchy
statistical properties

• Intermittent exponent
• beta-hierarchy
• high order moment
• scale-scale correlations

samples
mass density field of IGM, HI (simulation)
velocity field (simulation) Lyman-alpha
transmitted flux (simulation, observation) scaling
relations (simulation, observation)
He, Liu, Feng, Shu Fang, 2006 Zhang, Liu, Feng,
Fang, 2006 Liu, Fang 2008 Lu, Fang, 2008
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She-Leveque formula
He, Liu, Feng, Shu, Fang, PRL, (2006)
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intermittent exponent of mass density field
Liu, Fang, ApJ, 2008
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beta- hierarchical
p-invariance
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beta-hierarchy

Liu, Fang, 2008
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beta-hierarchy of Lyman-alpha transmitted flux
Lu, Fang 2008
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scale-scale correlation
Lu Fang 2008
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lognormal vs. log-Poisson models
Lu, Fang 2008
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evolution of IGM fields (in scale free range)
linear regime
nonlinear regime