Statistical physics for cosmic structures: Gravitational structure formation and the cosmological problem Francesco Sylos Labini In collaboration with T. Baertschiger, A. Gabrielli, L. Pietronero, B. Marcos Rome M. Joyce, Y. Baryshev, N. Vasilyev Paris Saint Petersburg “E. Fermi Center” & Institute for Complex Systems (ISC-CNR) Rome Italy

Summary The discrete gravitational N-body problem Primordial density fields and super-homogeneous distributions The observed distribution of galaxies Analogy with statistical physics systems Crucial observational tests for the standard theory of structure formation Role of dark matter

Early times density fields WMAP satellite 1st year, 2002 COBE DMR, 1992

150 Mpc/h (1990) 300 Mpc/h (2004) 5 Mpc/h Late times density fields

The problem of cosmological structure formation Initial conditions: Uniform distribution (small amplitude fluctuations) Final conditions: Stronlgy clustered, power-law correlations Dynamics: self-gravitating infinite system

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)

Substantially Poisson (finite correlation length) Super-Poisson (infinite correlation length) Sub-Poisson (ordered or super-homogeneous) Statistical properties of fluctuations in FRW models (II) Extremely fine-tuned distributions

Statistical properties of fluctuations in FRW models (III) For CDM models: For HDM models: (integral constraint…)

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)

Sampling fluctuations in FRW models (I)

Sampling fluctuations in FRW models (II)

Sampling fluctuations in FRW models (III) Normalization ? Origin the linear amplification? This is the only peculiar distinctive feature of HZ models in matter distribution

Conditional correlation properties (I)

Conditional correlation properties (II) Super-homogeneous Poisson-like Critical ?

Conditional correlation properties (III)

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/

Conditional correlation properties (VI)

Joyce M. & Sylos Labini F. Astrophys. J 554, L1 (2000) galaxies rotation curves Conditional correlation properties (VII)

Discrete gravitational N body problem (I) Classical interacting particles Collisionless Boltzmann Equation (Vlasov eq.)+ Poisson equation DM collisionless continuous medium Self-gravitating fluid equations in an expading universe Macro Particles N=10 10 (instead of N=10 80 ) as mass tracers Theoretical scheme of interpretation of numerical simulations Do the macro-particles correctly trace the evolution of the statistical and dynamical properties of theoretical models ? Statistical and dynamical effects of Discretization What Nbody really are

Problem: strong clustering up to scales ~ initial NN distance Self-gravitating continuous fluid interpretation: NO effects due to the granular nature of the particle distribution Convergence studies and stability No theory of discreteness Effects No theoretical knowledge of N dependence of the convergece Discrete gravitational N body problem (II)

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

Finite box + Periodic boundary conditions: infinite system (no center) Linear Perturbative Theory (I) Jeans swindle

Linear Perturbative Theory (II) Wigner crystal Bloch Theorem: diagonalization by plane waves

3X3 Real and symmetric matrix for each k Eigenmodes/ eigenvalues Wigner crystal: oscillations with plasma frequency Wigner crystal :unstable modes Gravity: Oscillating modes Wigner Crystal: oscillating modes Gravity: growing instabilities (even faster than fluid rate) Linear Perturbative Theory (III) Kohn sum rule Gravity: Fluid evolution (Lagrangian formalism)

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) 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 Linear Perturbative Theory (IV) What’s new

Poisson initial conditions (no correlations) Zero initial velocities Periodic Boundary conditions Growth of correlation in the gravitational problem (I) Two body collapse time No expansion

A simple test: NN force versus full gravity 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 Growth of correlation in the gravitational problem (II)

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 (III)

As for the Poisson and SL case: 1.Formation of first structures from discrete fluctuations 2. Propagation of correlation from small to large scales Growth of correlation in the gravitational problem (IV)

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 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 Growth of correlation in the gravitational problem (VI)

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 (VII)

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, B. Jancovici, M. Joyce, J.L. Lebowitz, L. Pietronero and F. Sylos Labini Generation of primordial cosmological perturbations from statistical mechanicalmodels, Phys.Rev. D67, (2003) T. Baertschiger, M. Joyce and F. Sylos Labini Power law and discreteness in cosmological N-body simulations, Astrophys.J.Lett 581, L63 (2002) 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 On the problem of initial conditions in cosmological N-body simulations Europhs.Lett. 57, 322 (2002) A. Gabrielli, M. Joyce and F. Sylos Labini The Glass-like universe: real space statistical properties of standard cosmological models, Phys.Rev.D, 65, (2002) T. Baertschiger and F. Sylos Labini Growth of correlations in gravitational N-body simulations Phys.Rev.D 69, , 2004 M. Joyce, B. Marcos, A. Gabrielli, T. Baertschiger, F. Sylos Labini Gravitational evolution of a perturbed lattice and its fluid limit Phys.Rev.Lett 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) A.Gabrielli, F. Sylos Labini, M. Joyce, L. Pietronero Statistical physics for cosmic structures Springer Verlag 2005