Scale invariance of cosmic structure

Transcription

1 Scale invariance of cosmic structure José Gaite Instituto de Matemáticas y Física Fundamental, CSIC, Madrid (Spain) Scale invariance of cosmic structure p.1/25

2 PLAN OF THE TALK 1. Cold dark matter structure in the adhesion model 2. Renormalization group approach 3. Random fractals 4. Scaling of voids 5. Analysis of the SDSS catalogue 6. Multifractals: halos and voids 7. Conclusions. Scale invariance of cosmic structure p.2/25

3 CDM structure in the adhesion model Newtonian equations of motion in co-moving coordinates (peculiar velocity u = v Hr): du dt + Hu = g T g b g. No initial vorticity Zeldovich approx.: prolong the linear solution x(t,x 0 ) = x 0 + b(t)g(x 0 ). τ := b(t) free motion with velocity g(x 0 ) Singularities when lines cross caustics. Scale invariance of cosmic structure p.3/25

4 CDM structure in the adhesion model In caustics particles adhere to each other viscosity. Burgers equation (compressible turbulence): dũ dτ = ũ τ + ũ ũ = ν 2 ũ, ν 0, ũ = 0. Stabilization of caustics (pancakes, filaments, clusters). Random initial conditions self-similar distribution of caustics: cosmic web multifractal features (Vergassola et al 94, Aurell et al 97). Scale invariance of cosmic structure p.4/25

5 Simulations Burgers eq. with random initial conditions (Vergassola, Dubrulle, Frisch & Noullez) N-body cosmological simulation of Λ-CDM evolution (GIF2) Scale invariance of cosmic structure p.5/25

6 Renormalization group approach Stochastic adhesion model (KPZ): dφ k dt = F k + η k = νk 2 φ k ũ = φ q q (k q)φ q φ k q + η k RG has non-trivial IR fixed point in d = 1 but not in d = 3. Non-perturbative fractal distribution coarse-graining exact RG: L e V L[ ] = 1 2 dp L dl δ 2 δ 2e V L[ ], P L (k) = P(k)W L (k). Fokker-Planck eq. for KPZ (Langevin eq.): Π t = φ k [F k Π] D(k) 2 Π φ k φ k. Scale invariance of cosmic structure p.6/25

7 Renormalization group approach coarse graining of N -body simulation Evolution with L evolution with t Scale invariance of cosmic structure p.7/25

8 Fractals Self-similar fractals Cluster hierarchy Power-law mass L D, with D = 2 log 2/ log 3 = 1.26 Cantor set (2d) Scale invariance of cosmic structure p.8/25

9 Random fractals Number function N(r) = B r D Γ(r) = 1 dn(r) 4πr 2 dr = BD 4π rd 3 Transition to homogeneity at r 0 : Γ = (1 + ξ). Power-law 2-point correlator: ξ = ( r 0 r ) γ >> 1, γ = D 3. Random fractal with D = 0.8. Note clusters and voids Scale invariance of cosmic structure p.9/25

10 Fractal voids Void of size Λ in a fractal power law N(Λ > λ) λ D/3 Λ(R) R z, z = 3 D, Zipf s law Transition to homogeneity flattening of the Zipf law Log N r Log r N(r) for D = 1 fractal in two dimensions Zipf s law for voids with z = 2/D Scale invariance of cosmic structure p.10/25

11 Scaling of galaxy clusters and voids Γ(r) of SDSS VL sample. r 0 15 Mpc (Tikhonov). Slope γ D 2 The Zipf law for a 2dF VL sample (Tikhonov). Slope z D 2 Scale invariance of cosmic structure p.11/25

12 Scaling of galaxy clusters and voids Rank ordering of local voids (Karechentsev): Slope γ D 2. No transition to homogeneity Scale invariance of cosmic structure p.12/25

13 Analysis of the SDSS catalogue Correlation functions of galaxies from the SDSS catalogue according to their luminosity (SSDS Data Release 3, with 374,767 galaxies). Selection of 7 volumen limited samples (150,067 galaxies altogether) with slope fitting slope 1e changes s (h -1 Mpc) Γ * (s) ;-18 - γ= ;-19 - γ= ;-20 - γ= ;-21 - γ= ;-22 - γ= ;-23 - γ= ;-24 - γ = -1.4 Scale invariance of cosmic structure p.13/25

14 MULTIFRACTALS Scale invariance fractals Mass distributed irregularly + scale invariance multifractals Mass concentrations m[b(x,r)] r α( x) Multifractal spectrum f(α) is the function that gives the fractal dimension of the set of points with exponent α. Monofractal: constant α = f(α). Scale invariance of cosmic structure p.14/25

15 Example: multinomial MF Multinomial multifractals: the unit square is divided into (4) cells, the unit mass distributed among cells ({p i }), and the process it- 0 erated Random multinomial measure with distribution { 1 2, 1 4, 1 6, 1 12 }. Scale invariance of cosmic structure p.15/25

16 MASS CONCENTRATIONS: HALOS Mass concentrations of size l with singular power-law profile ρ(r) r β, β = 3 α > 0). Natural value for l is the lower cutoff to scaling. In N -body simulations, the larger of: (i) The linear size of the volume per particle. (ii) The gravitational softening length. We identify mass concentrations with equal-size halos Halo mass-function: N(m) l f(α), α = log m/ log l. Scale invariance of cosmic structure p.16/25

17 Multinomial bifractal A bifractal can be extracted: select {α 1,α 2 } {m 1,m 2 }. Multifractal models support halo populations with different levels of clustering. Note voids (may be non-empty) Two populations in a multinomial multifractal. Scale invariance of cosmic structure p.17/25

18 MASS DEPLETIONS: VOIDS Voids have regular power-law profile ρ(r) r β (β = 3 α). If α > 3 ρ(0) = Boundaries of voids: points 100 with α = 3 ρ(0) > 0 and finite. Not regular but fractal surfaces with D = f(3) > Fractal boundary of voids in multinomial MF: f(2) = Scale invariance of cosmic structure p.18/25

19 BIASING AND VOIDS Biasing: peculiar distribution of certain set of objects with respect to the total DM distribution. Bias from linear theory: δρ g ρ g = b δρ ρ ξ gg(r) = b 2 ξ(r), b > 1. Constant b bias in the nonlinear regime similar voids for every population false in MF. Voids not empty but harbor faint galaxies galaxy formation (Peebles, 2001). Distribution of dark matter inside voids (Gottlöber et al, 2003). Scale invariance of cosmic structure p.19/25

20 MULTIFRACTALITY IN N-BODY SIMULATIONS z = 0 positions in Virgo Λ-CDM GIF2 simulation: particles in a volume of (110 h 1 Mpc) 3 particle mass is h 1 M. Statistics by counts-in-cells: M q (l) = N(m)m q. m=1 Halos: l H = > (l H = 0.43 h 1 Mpc). Distribution of 5515 haloes (cutoff 1000) Scale invariance of cosmic structure p.20/25

21 Mass-function variation with l If MF scaling holds stable MF spectrum. For l > l H we expect a stable MF spectrum. How does it change for l < l H? log N log N log N log m log m log m Log-log plots of number of halos N(m) for l = (left), (middle), and (right). Power-law (Press-Schechter) for l < l H. Scale invariance of cosmic structure p.21/25

22 MF AS FRACTAL DISTRIBUTIONS OF HALOS log N log r log N log r GIF2 heavy haloes with 750 to 1000 particles (red); GIF2 light haloes with 100 to 150 particles (blue). Number function N[B(x,r)] r D : fractal dimensions D = 1.1 and D = 1.9. Transition to homogeneity starts at 14 h 1 Mpc. Scale invariance of cosmic structure p.22/25

23 Voids in the GIF2 simulation MF spectrum shows that f(3) 2.9 boundary is a fractal surface with large dimension Fractal boundary of voids in a GIF2 slice Scale invariance of cosmic structure p.23/25

24 SUMMARY and CONCLUSIONS Adhesion model self-similar cosmic web (multifractal) Renormalization group approach inconclusive Observations of galaxies scaling with variable γ Multifractals: most general scaling mass distributions, support halos (concentrations) and voids (depletions). Scale invariance of cosmic structure p.24/25

25 SUMMARY and CONCLUSIONS Adhesion model self-similar cosmic web (multifractal) MF spectrum halo mass function. Halos have nonlinear bias GIF2: fractal populations of haloes good scaling. Scale invariance of cosmic structure p.25/25