Statistical physics for cosmic structures:
Gravitational structure formation and the cosmological problem
Francesco Sylos Labini
“E. Fermi Center”
&
Institute for Complex Systems
(ISC-CNR)
Rome Italy
In collaboration with
T. Baertschiger, A. Gabrielli, L. Pietronero, B. Marcos
Rome
M. Joyce,
Paris
Y. Baryshev, N. Vasilyev
Saint Petersburg
Summary
Primordial density fields and super-homogeneous distributions
The observed distribution of galaxies
The discrete gravitational N-body problem
Analogy with statistical physics systems
Crucial observational tests for the standard theory of structure formation
Role of dark matter
Early times density fields
COBE DMR, 1992
WMAP satellite 1st year, 2002
Late times density fields
300 Mpc/h
(2004)
150 Mpc/h
(1990)
5 Mpc/h
The problem of cosmological structure formation
Initial conditions: Uniform distribution (small amplitude fluctuations)
Dynamics: self-gravitating infinite
system
Final conditions: Stronlgy clustered,
power-law correlations
Cosmological energy budget: the “standard model”
Non baryonic dark matter (e.g. CDM):
-never detected on Earth
-needed to make structures compatible with anisotropies
Dark Energy
-never detected on Earth
-needed to explain SN data
-troubles with the contribution of quantum vacuum energy…
What do we know about dark matter ?
Fundamental and observational constraints
Statistical properties of fluctuations in FRW models (I)


~
2
  (r1 )  (r2 )  0 (1   (r12 ))
 M ( R) 
 ( R) 
 M ( R)  2
 ~

1
2
 ( R) 
dr1   (r12 )dr2
2 
V ( R) V ( R ) V ( R )
2
2
 
1
lim R 
 (r )dr   0  0 x0

C
(
R
,
x
)
0
|| C ( R, x0 ) ||

P(k ) 
1
(2 )3
~  ikr 
  (r )e dr
lim R 

  (r )dr  P(0)  0
~
V ( R)
Statistical properties of fluctuations in FRW models (II)
Substantially Poisson (finite correlation length)
~
P(0)    (r )d 3r  const.  0
 M (r ) 2  M (r ) 
Super-Poisson (infinite correlation length)
~
P(0)    (r )d 3r  
 M (r ) 2  M (r )   1    2
Sub-Poisson (ordered or super-homogeneous)
Extremely fine-tuned
distributions
~
P(0)    (r )d 3r  0
 M (r ) 2  M (r )   0    1
Statistical properties of fluctuations in FRW models (III)
(integral constraint…)
P(k ) k 0  k  P(0)  0    (r )r 2 dr  0
1
P
(
k
)

k
for k  kc
For CDM models:
3/2
P
(
k
)

k
exp(

k
/
k
)
For HDM models:
c
 2 ( R  RH (t ))  const .
GM ( R)
M ( R )
 ( R) 
  ( R) 
  ( R) R 2   ( R)  const   2 ( R)  R  4
R
R
Statistical properties of fluctuations in FRW models (IV)
Anisotropies due to fluctuations
in the gravitational potential
(Sachs Wolfe)
Statistical properties of fluctuations in FRW models (V)
Angular correlation function vanishes at > 60 deg (COBE/WMAP teams)
Small quadrupole/octupole (COBE/WMAP teams)
Planar octupole, aligned with quadrulpole (de Olivera Costa et al., 2003)
Deficit of power in North ecliptic emisphere (Eriksen et al. 2003)
Quadrupole and octopole aligned and correlated with Ecliptic
(Schwartz et al. 2004)
Statistical properties of fluctuations in FRW models (VI)
 n(r )n(0) 
 (r ) 
1
2
n
 (r )  Ar 1.7
1 (r )  const   2 (r )
P(k )  FT  (r )
P(k )  Bk 1.3
P2 (k )  const.  P1 (k )
Sampling fluctuations in FRW models (I)
1


 ( x )   ( ( x )  )  
0

if  ( x )  

if  ( x )  
 (r )  f ( ,  c (r ))
 (r )   2 c (r ) we
P(0)  0
P(0)  const .
 2 ( R)  R 4
 ( R)  R
2
3
Sampling fluctuations in FRW models (II)
Pc (0)  const  P (0)  const
Pc (0)    P (0)  
 2 ( R)  R 4
P(0)  0
 const

P) (0const
)  const
Pc (0P)c (0)const
P
 (0
P (k )   Pc (k )k
2
Sampling fluctuations in FRW models (III)
 (r )  exp( 2 c (r ))  1 strong clustering
 (r )   2 c (r ) weak clustering
 (r )  r 4
P (k )   Pc (k )
2
Normalization ?
k
1.7

(
r
)

Ar
 (r )  Ar 1.7

1 ((rr )) 
 const
const 

 2 ((rr ))
1
1.3
1.3
2
P
P((k
k )) 
 Bk
Bk
P
P1 ((k
k )) 
 const
const.. 
P
P2 ((k
k ))
1
This is the only peculiar distinctive
feature of HZ models in matter distribution
2
Origin the linear amplification?
Conditional correlation properties (I)
Conditional correlation properties (II)
 n  n(r )  p  const
  (r )d 3r  0


 (r )  (r )d 3r  const.

3
   (r )d r  
 n( r )  p  r
 n  0
 (r ) ?
P(k)

Super-homogeneous
Poisson-like
Critical
Conditional correlation properties (III)
4 3
 N ( r )  P  Br ;V ( Rs ) 
Rs
3
 n(r )n(0) 
DB D 3
 n(r )  P 
r
n
4
N ( Rs ) 3B D 3
 n E 

Rs
V ( Rs ) 4
D
 n(r )n(0) 
D r 
 E (r ) 
 1   
2
n
3  Rs 
1 /( D 3)
D
 E (r0 )  1  r0   
6
Rs
D 3
1
 (r )  Ar 1.7
1 (r )  const   2 (r )
Conditional correlation properties (IV)
Sampling
Finite size
Conditional correlation properties (V)
Sylos Labini, F., Montuori M. & Pietronero L. Phys Rep, 293, 66 (1998)
Joyce M. & Sylos Labini F. Astrophys. J 554, L1 (2000)
Hogg et al. (SDSS Collaboration) astro-ph/0411197
Conditional correlation properties (VI)
 (r )  Ar
1 (r )  const   2 (r )
1.7
P(k )  Bk
1.3
P2 (k )  const.  P1 (k )
const  f ( Rs )
Conditional correlation properties (VII)
Joyce M. & Sylos Labini F.
Astrophys. J 554, L1 (2000)
 n( r )  p  r D 3

1 R
3
 j ( R)  p 
 n(r )  p d r   ( L) LdL

0
V ( R) 0
 j ( R )  p  R D 3
galaxies rotation curves
Discrete gravitational N body problem (I)
Theoretical scheme of interpretation of numerical simulations
Classical interacting particles
Collisionless Boltzmann Equation (Vlasov eq.)+ Poisson equation
Self-gravitating fluid equations in an expading universe
DM collisionless continuous medium
What Nbody really are
Macro Particles N=1010 (instead of N=1080) as mass tracers
Do the macro-particles correctly trace the evolution of the
statistical and dynamical properties of theoretical models ?
Statistical and dynamical effects of
Discretization
Discrete gravitational N body problem (II)
Self-gravitating continuous fluid interpretation:
NO effects due to the granular nature of the particle distribution
Problem: strong
clustering up to scales ~
initial NN distance
Convergence studies
and stability
No theory of discreteness
Effects
No theoretical knowledge
of N dependence of the
convergece
Discrete gravitational N body problem (III)
1) Static (generation of IC):
Effects of the pre-initial distribution on the
correlations imposed with displacements by the ZA.
2) Early time evolution (up to shell crossing):
Discreteness effects from spatial sampling
3) Growth of first correlations (two-body correlations):
Strong collisions between particles
4) Late times (many body correlations):
Self-similar evolution of the conditional density
Linear Perturbative Theory (I)
Finite box + Periodic boundary conditions: infinite system (no center)
Jeans swindle
Linear Perturbative Theory (II)
Wigner crystal
Bloch Theorem: diagonalization by plane waves
Linear Perturbative Theory (III)
3X3 Real and symmetric
matrix for each k
Eigenmodes/ eigenvalues
Kohn sum rule
Wigner crystal: oscillations with
plasma frequency
Gravity: Fluid evolution
(Lagrangian formalism)
Wigner crystal :unstable modes
Gravity: Oscillating modes
Wigner Crystal: oscillating modes
Gravity: growing instabilities
(even faster than fluid rate)
Linear Perturbative Theory (IV)
If initial perturbations contain modes such that
Same results as Lagrangian formalism of a pressureless self-gravitating fluid
Zeldovich approximation as asymptotic form of the discrete solutions
What’s new
Theoretical scheme for a perturbative treatment of the discrete N-body
Especially relevant because of the method to set up IC in cosmological N-body
Precise formalism up to shell crossing for calculating discreteness corrections
Oscillating modes
Breaking of isotropy
Divergence from fluid behavior
Extension of the perturbative treatment to higher orders
Body-centered cubic lattice is stable (only unstable mode for gravity)
Growth of correlation in the gravitational problem (I)
Poisson initial conditions (no correlations)
Zero initial velocities
Periodic Boundary
conditions
No expansion
~
 (r ) 
 (r )
n0
N
n0 
V
1 / 3
  0.55n0
 0  n0 m
1
 (r ) 
n0V (r ) 2
2
1

GmN / V
Two body collapse time
Growth of correlation in the gravitational problem (II)
Most of particles are mutually NN
Gravitational force is dominated by NN
Our hypothesis:
The full distribution can be treated for a time of the order of the dynamical
time as an ensamble of isolated two-body systems
A simple test:
NN force
versus
full gravity
 n( r , t )  p 
 (r , t )
r


2
4r 1    ( s, t )ds 
 0

Growth of correlation in the gravitational problem (III)
 n(r ) p  r     1.7
 n(r , t  t ) p  n(a  r , t ) p
Discrete fluctuations give rise to early
non-linear correlation (structures).
How discrete effects are “exported” at
larger scales and longer times ?
Growth of correlation in the gravitational problem (IV)
 n( r , t )  p 
 (r , t )
r


2
4r 1    ( s, t )ds 
 0

As for the Poisson and SL case:
1. Formation of first structures from
discrete fluctuations
 n(r , t  t ) p  n(a  r , t ) p
2. Propagation of correlation from small to
large scales
Growth of correlation in the gravitational problem (VI)
Our Conclusion:
Non-linear dynamics of cosmological NBS is
essentially the same as the
non-expanding Poisson case and it is
driven by the discrete fluctuations at the
smallest scales in the distribution:
-independent on IC
-Space expansion
Usual picture
Dynamical evolution of the small
fluctuations in the initial continuous
field (non-linear regime of the set
of fluid equations in an expanding
universe) dependence on:
- Initial conditions
- Space expansion
Growth of correlation in the gravitational problem (VII)
Discreteness of the gravitational field causes first correlations
(interaction of nearest neighbors and formation of small groups)
Each group begins to act as a single particle and the groups themselves become
correlated and more and more massive clusters are build up.
The clustering is rescaled to larger and larger distances whose limit is determined
by the time available for structure to form
(W.C. Saslaw “The distribution of the galaxies” 2002, CUP)
Coarse Grain approach:
Interplay between discrete effects and fluid linear evolution
Growth of correlation in the gravitational problem (VIII)
Summary
HZ tail: the only distinctive feature of FRW-IC in matter distribution
is the behavior of the large scales tail of the correlation function
Problem with large angle CMBR anisotropies
Homogeneity scale: not yet identified with galaxy distribution
Amplification (i.e.Bias): is due to a finite size effect and not to selection
different mechanism in simulations and galaxies
Structures in N-Body simulations: too small and maybe different in
nature from galaxy structures
Discrete gravitational clustering: results of N-body must be taken
with great care when interpreted as evolution of DM fluid
Discrete fluctuations play a central role for non-linear structures
Universality and independence on IC
Perturbative theory for the treatment of discrete systems
Conclusion: Plans for the Future….
Study of galaxy distribution in the SDSS survey
-- Homogeneity scale
-- Clustering of galaxies of different luminosity
Formation of non linear structures
-- Study of modified potentials (cut-off, softnening)
-- Non linear study of perturbed lattices
-- Coarse grain approach for the formation of structures
-- Statisical characterization of structures (Tsallis, Saslaw…)
Study of CMBR density fields
-- Real space analysis
-- Test for angular isotrotpy
Some references
A. Gabrielli, M. Joyce and F. Sylos Labini The Glass-like universe: real space statistical properties
of standard cosmological models, Phys.Rev.D, 65, 083523 (2002)
T. Baertschiger and F. Sylos Labini On the problem of initial conditions in cosmological N-body
simulations Europhs.Lett. 57, 322 (2002)
T. Baertschiger, M. Joyce and F. Sylos Labini Power law and discreteness in cosmological N-body
simulations, Astrophys.J.Lett 581, L63 (2002)
A.Gabrielli, B. Jancovici, M. Joyce, J.L. Lebowitz, L. Pietronero and F. Sylos Labini Generation of
primordial cosmological perturbations from statistical mechanicalmodels, Phys.Rev. D67,
043406 (2003)
R. Durrer, A. Gabrielli, M. Joyce and F. Sylos Labini ,``Bias and the power spectrum beyond the
turn-over'' Astrophys.J.Lett,585 , L1-L4 (2003)
F. Sylos Labini, T. Baertschiger and M. Joyce Universality of power-law correlation in
the gravitational clustering, Europhys.Lett. 66,171, (2004)
T. Baertschiger and F. Sylos Labini Growth of correlations in gravitational N-body
simulations Phys.Rev.D 69, 123001, 2004
A.Gabrielli, F. Sylos Labini, M. Joyce, L. Pietronero
Statistical physics for cosmic structures Springer Verlag 2005
M. Joyce, B. Marcos, A. Gabrielli, T. Baertschiger, F. Sylos Labini Gravitational evolution of a
perturbed lattice and its fluid limit Phys.Rev.Lett. 95 011304 2005