Pr

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The problem of galaxy clustering
Francesco Sylos Labini Laboratoire de Physique
Théorique Université Paris XI Orsay, France
sylos_at_th.u-psud.fr http//qcd.th.u-psud.fr/page_p
erso/Sylos-Labini
in collaboration with ....
Luciano Pietronero (Dept.Phys., Univ.Roma 1,
Italy) Andrea Gabrielli (E.Fermi Center, Roma 1,
Italy) Michael Joyce (LPNHE Paris VI)
STATISTICAL PHYSICS AND COMPLEXITY FOR COSMIC
STRUCTURES SPRINGER VERLAG, 2003
Pavia 09.09.03
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Summary
Observations of galaxy structures and correlation
properties
Clustering and fluctuations in standard
cosmological models Statistical properties of
homogeneous and isotropic density fields. Nature
of correlations in standard cosmological models
super-homogeneous distributions.
Characterization of intrinsically irregular
distributions clustering and fluctuations.
Properties of asymptotically empty distributions
The cosmological problem of structure formation
and the normalization of late-time
(galaxies) to early times (CMB) fluctuations
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SSRS2
Random
CfA2
5 Mpc/h
SDSS-DR1
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Bahcall Soneira Astrophys.J., 270, 4 (1983)
Benoist et al., Astrophys. J. 472, 452 (1996)
Norberg et al. MNRAS 328, 64 (2001)
Zehavi et al. (Sdss Collaboration) Astrophys.J.
(2002)
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Statistically homogeneous and isotropic mass
distributions
Ergodicity in a single realization of the
Sationary Stochastic Process, the existence of a
positive average density implies that
any point in space
Homogeneity scale
Scale beyond which mass fluctuations can be
considered small
Note that there are SSP where the ensemble
average density is zero
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Reduced correlation function and mass variance
Normalized mass variance
Homogeneity
Always verified for if the average density is
positive. Very different from
Not a trivial condition ! (Kolb Turner,
Padhmanabam,...)
Power Spectrum
Window function
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Classification of correlated distributions
(statistically isotropic and homogeneous)

• Gabrielli, M. Joyce and F. Sylos Labini
  Phys.Rev.D, 65, 083523 (2002)

Substantially Poisson (finite correlation length)
Super-Poisson (infinite correlation length)
Sub-Poisson (ordered or super-homogeneous)
Super-homogeneous systems are ordered compact
distributions with small amplitude correlated
perturbations (e.g. glasses, phonons in
lattices, etc. )
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Harrison-Zeldovich correlationsorigin
Fundamental characteristic length scale
distance light can travel from t0 until any
given time. It growths linearly with time
HZ criterion
Mass variance at the horizon scale be constant
fluctuations have the same amplitude as a
function of the only scale of the model (in this
sense scale invariant)
surface fluctuations
Note that
(Poisson volume fluctuations)
will always break down in the past or future as
the amplitude of perturbations become arbitrarily
large (not compatible with FRW metric !)
Scale-invariance in statistical physics
Power laws (infinite corr. length)
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Harrison-Zeldovich Super-homogeneity

• Gabrielli, M. Joyce and F. Sylos Labini
  Phys.Rev.D, 65, 083523 (2002)

This is a global condition on the systems
fluctuations
For CDM models
For HDM models
The H-Z Spectrum corresponds to a SH
distribution example one-component plasma
A.Gabrielli, B. Jancovici, M. Joyce, J.L.
Lebowitz, L. Pietronero and F. Sylos Labini
Phys.Rev.D 67, 043406-043513 (2003)
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Biasing a Gaussian random field (comparison
models/observations of galaxies)
Basic idea Kaiser ApJ 284,L9 (1984) Underlying
Gaussian field dark matter (i.e. CDM)
Biased field (visible) peaks galaxies,
clusters ...
Correlation function of the Gaussian field
(i.e CDM)
Regions above a certain threshold (selection)
threshold density (set of equal weigh points)
Correlation function of the biased field
Only on very small scales non-linearities are
developed on large scales the field is
Gaussian. Peaks correspond to different objects
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Change of nature of correlations a simple example
Consider a perfect cubic
lattice Surface fluctuations
Volume fluctuations
Lets consider the distribution obtained by
keeping or rejecting each point with probability
p (or 1-p)
The resulting distribution is described by a
binomial probaibility density with a variance
This selection introduces a new term of noise
which dominates with respect to the original
noise of the system.
The biasing scheme is a deterministic process on
a disordered system the effect of this sampling
is of the same kind of that given by random
sampling
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Power spectrum of the threshold density
R. Durrer, A. Gabrielli, M. Joyce F. Sylos Labini
Astrophys.J.Lett 585, L1 (2003) A. Gabrielli F.
Sylos Labini and R. Durrer Astrophys.J.Lett
531,L1 (2000)
Substantially poisson
Critical
Super-homogeneous
In the super-homogeneous case the threshold
field changes its statistical nature
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amplitude rescaling
Different from observations !?
The comparison between CMB and structures
fails Fundamental problem for standard models
of galaxy formation (e.g.N-body
simulations) How to normalize early and late
fluctuations ? How to interpret the linear
amplification of x(r) ?
change of the nature of fluctuations for SH
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Analytical Fluctuaction Characterized
by -background -density, -position -size
-intensity -profile
Non-Analytical Fluctuations Strongly
irregular structure -no backgound density -no
specific size or intensity. Simple stochastic
fractal regularity in the scale transformation
(scale-invariance)
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Standard Statistical analysis Assumes
that one point properties are well defined
(implict assumption of small fluctuations) Ave
rage number of points in a randomly chosen
ball Fluctuations characterized by excess of
points (with respect to the mean value) around
an occupied point.
(2)
More General statistical method
Tests whether one point properties are
well-defined if fluctuations are too wild it is
not possible to define an average density
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Fractal Geometry Basic statistical tools
Conditional average density
Homogeneous Dd -average density
constant. -amplitude of fluctuations
-homogeneity scale -correlation length.
Fractal Dltd -esitimation of the
average density in a finite sample is sample size
dependent. -infinite sample average density
zero. -asymptotically empty. -scaling
properties of fluctuations.
p
conditional properties
p
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Extendend classification of structures
Statistically isotropic and homogeneous
Super-homogeneous Poisson-like Critical
Fractals isotropic but not homogeneous
not defined in the ensemble sense, but in a
finite sample its estimator can be measured, but
....
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For a fractal.....
Let us suppose to have a fractal in a
spherical sample of radius Rs
The amplitude depends on the sample size The
exponent is not well defined The linear
amplification of the amplitude of the correlation
function is a natural cosequence of a
finite-size effect
Hence it is necessary to test whether ltngt gt0 in
any given finite sample
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Absolute magnitude versus distance diagram
Deeper the sample Brighter the galaxies
Volume limited samples
3-d samples
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Cfa2SSRS2 redshift surveys
Joyce M. Sylos Labini F. Astrophys. J 554, L1
(2000)
4 15 Mpc/h
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Sylos Labini, F., Montuori M. Pietronero L.
Phys Rep, 293, 66 (1998)
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The case of cluster catalogs
10 25 Mpc/h
Resolution of the mistmatch galaxy-cluster
Sylos Labini, F., Montuori M. Pietronero L.
Phys Rep, 293, 66 (1998)
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The problem of cosmological structure formation
Initial Conditions
Cosmic microwave background fluctuations small
amplitude, continuous Gaussian field with
super-homogeneous correlations (plus non-trivial
physics at small angular scales - oscillations)
WMAP
02.03
Final distribution
Large scale galaxy structures large
fluctuations, power-law correlations (fractal)
discrete set of points
0.1 1 10 50 100 1000 Mpc
now................. ....new surveys (SDSS
2003-2006)
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The problem of time scales
Galaxy peculiar velocity
Age of the universe
Maximum distance travelled by an object in a
universe life time
How to form fractal structures with size at least
one order of magnitude larger ?
Structures form up to 3 Mpc/h
There is factor TEN (at least of difference) !
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Questions about Nbody simulations
How to discretize a continuous field ? Create a
particle distribution with the same statistical
properties as the continuous field, in a certain
range of scales. Importance of discretization
in the dynamical evolution introduces a new
kind of fluctuations at all and, in particular,
at small scales. Are these fluctuations
important ? What does drive non-linear structure
formation ? Small-scale large-amplitude particle
fluctuations Large-scale small-amplitude
continuous fluctuations
T. Baertschiger, M. Joyce and F. Sylos Labini
Astrophys.J.Lett 581, L63 (2002) F. Sylos Labini,
T. Baertschiger and M. Joyce astro-ph/0207029
T. Baertschiger and F. Sylos Labini
Europhs.Lett. 57, 322 (2002)
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CfA2 150Mpc/h
SDSS 600 Mpc/h
5 Mpc/h