N-body Simulations. Initial conditions: What kind of Dark Matter? How much Dark Matter? Initial density fluctuations P(k) GRAVITY

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1 N-body Simulations

2 N-body Simulations

3 N-body Simulations Initial conditions: What kind of Dark Matter? How much Dark Matter? Initial density fluctuations P(k) GRAVITY Final distribution of dark matter.

4 N-body Simulations Start with a grid of particles representing dark matter. Give them initial smooth density fluctuations. Compute the force of gravity on every particle from every other particle in a series of time steps. A very computationally expensive technique! Only made possible recently with fast computers.

5 N-body Simulations: Initial Conditions Cosmological model h,ω m,ω b,ω ν,σ 8,n s P( k) Phases for modes Gaussian random phase δ(r) Non-Gaussian? Evolve to starting redshift z init = Zel dovich approximation 2LPT

6 N-body Simulations: Initial Conditions Assign initial positions and velocities using Zel dovich approximation x = q + D t ( ) ψ q ( ) q : initial position Ψ : dispacement field D : growth function δ : initial density field v = a dd dt ψ q ( ) ψ = δ q D t ( ) ( )

7 N-body Simulations: Force calculations Direct particle-particle (N 2 ) Particle-Mesh (PM) (N g logn g ) Particle-particle particle-mesh (P 3 M) (N 2 / N g logn g ) Tree (NlogN) Tree-PM (NlogN / N g logn g ) Adaptive mesh refinement (AMR) Adaptive refinement tree (ART) Moving mesh (AREPO)

8 N-body Simulations: Direct N-body

9 N-body Simulations: Particle-Mesh 2 Φ = 4πGρ ˆΦ = 4πG ˆρ k 2

10 N-body Simulations: Particle-Particle, Particle-Mesh

11 N-body Simulations: Tree Code

12 N-body Simulations: AREPO

13 N-body Simulations: Code comparisons Heitmann et al. (2005)

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16 e.g., dark matter models We can constrain cosmological models. SDSS observations

17 We can constrain cosmological models. van Dalen & Schaefer (1992)

18 We can constrain cosmological models. Fraction of present age: 1/10 1/2 1 Flat universe with dark energy Ω m = 0.3 Ω Λ = 0.7 Flat universe with high DM Ω m = 1 Different initial P(k) Ω m = 1 Open universe with low DM Ω m = 0.3

19 We can constrain cosmological models.

20 We can constrain cosmological models. Jenkins et al. (1998)

21 Virgo Collaboration

22 What is a dark matter halo? Dark Matter collapses under its own self-gravity into virialized regions, or halos. Halos are typically defined as regions with density of ~200 times the mean density, but there are several halo-finding algorithms. Halos come in different sizes, masses, and shapes. High mass halo (the kind that would host a galaxy cluster) Intermediate mass halo (the kind that would host a galaxy group) Very low mass halo (the kind that would host no galaxy at all) Low mass halo (the kind that would host a single galaxy)

23 What is a dark matter halo? Friends-of-Friends (FoF) linking length b Spherical Overdensity (SO) choice of center, density threshold Δ vir Density Maxima (DENMAX, BDM) choice of center, density threshold Δ vir, criteria for unbinding Other (e.g., Voronoi tesselation)

24 What is a dark matter halo? Bound dense regions within a larger halo are referred to as subhalos, substructure, or satellite halos. Subhalos have a density higher than ~200 times the mean. A halo can host a single galaxy, or a cluster of galaxies. Within a cluster, individual galaxies would sit inside subhalos. Subhalo / Satellite halo Halo Host/Parent/Central halo Kravtsov et al.

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26 Halo properties: Abundance (mass function) We now know the z=0 mass function to ~5% for reasonable choices of cosmological parameters. (For one N-body code and one halo-finder) There may be larger uncertainties for higher redshifts or more exotic cosmological models. M Warren et al. (2006)

27 Halo properties: Abundance (mass function) Press-Schechter (1974) theory Halos collapse from regions in the primordial density field that exceed a threshold density. One halo forming does not influence the likelihood of other halos forming nearby. The halo mass function thus depends on: The distribution of initial densities The linear growth of fluctuations The density threshold for collapse P(k) D(z) δ crit

28 Halo properties: Abundance (mass function)

29 Halo properties: Abundance (mass function)

30 Halo properties: Abundance (mass function)

31 Halo properties: Abundance (mass function)

32 Halo properties: Abundance (mass function)

33 Halo properties: Abundance (mass function)

34 Halo properties: Abundance (mass function) Consider a spherical region of mass M. This region corresponds to a scale: R = 3M 4πρ The density field smoothed on this scale has a variance of: 1 3 σ R 2 = P( k) W R ( k) 2 d 3 k σ ( M ) The probability of the density having a value between δ and δ+dδ is: P( δ M )dδ = 1 2πσ M ( ) exp δ 2 2σ ( M ) 2 dδ

35 Halo properties: Abundance (mass function) The fractional volume in this smoothed density field with δ>δ c is: ( ) = P δ M F > M δ c ( )dδ The fractional volume corresponding to masses in the range M to M+dM is: df ( > M ) dm dm

36 Halo properties: Abundance (mass function) The volume of a region that will make a single halo of mass M is: M ρ The number of halos of mass in the range M to M+dM is: fraction of volume V tot volume of 1 halo = ρ M df ( > M ) dm dm V tot The number density of halos of mass in the range M to M+dM is: dn dm dm = ρ M df ( > M ) dm dm

37 Halo properties: Abundance (mass function) dn dm = 2 π ρ M 2 δ c σ d lnσ d ln M exp δ 2 c 2σ 2 Spherical collapse model Power spectrum Linear theory growth rate σ ( M, z) = σ ( M,z = 0) D( z) There are numerous improvements to the Press-Schechter mass function

38 Halo properties: Abundance (mass function) Cooray & Sheth (2002)

39 Halo properties: Abundance (mass function)

40 Halo properties: Abundance (mass function)

41 Halo properties: Abundance (mass function)

42 Halo properties: Abundance (mass function)

43 Halo properties: Abundance (mass function) LasDamas

44 Halo properties: Abundance (mass function) Zel dovich 2LPT z>>0 z in

45 Halo properties: Abundance (mass function) LasDamas

46 Halo properties: Abundance (mass function) LasDamas

47 Halo properties: Clustering (halo bias) LasDamas

48 Halo properties: Clustering (halo bias) LasDamas

49 Halo properties: Clustering (halo bias) Good to ~3% for standard cosmology. b h = ( ) ( ) P hh k < 0.1 P mm k < 0.1 Seljak & Warren (2005)

50 Halo properties: Clustering (halo bias) Good to ~10% across different cosmologies. Seljak & Warren (2005)

51 Halo properties: Structure (density profile) r

52 Halo properties: Structure (density profile) Navarro, Frenk & White (NFW) ρ( r) = ρ s ( 1+ r r s ) 2 ( r r s ) logr Navarro et al. (2004)

53 Halo properties: Structure (density profile) Navarro, Frenk & White (NFW) ρ( r) = ρ s ( 1 + r r ) 2 ( s r r ) s c R vir r s M vir = 4 3 π R 3 virδ vir ρ M vir = R vir 0 4πr 2 ρ( r)dr

54 Halo properties: Structure (density profile) NFW, varying mass

55 Halo properties: Structure (density profile) NFW, varying concentration

56 Halo properties: Structure (density profile) concentration mass relation c R vir r s c c c * ( ( 1+ z) M M *) 0.13 M vir Bullock et al. (2001)

57 Halo properties: Structure (density profile) Density profile slope vs. r Einasto profile ( ) α 1 ρ = ρ 2 e 2 α e r r 2 Navarro et al. (2010)

58 Halo properties: History (merger tree) Assembly History High mass halos have accreted more of their mass recently relative to low mass halos. time M a Wechsler et al. (2002)

59 Halo properties: History (merger tree) Halo concentrations are determined by their accretion history. Wechsler et al. (2002)

60 Halo properties: Assembly Bias Wechsler et al. (2006)

61 Halo properties: Assembly Bias fast spinning slow spinning Gao & White (2007)

62 Halo properties: Assembly Bias early assembly high concentration late assembly low concentration Gao & White (2007)

63 Halo properties: Assembly Bias high substructure low substructure Gao & White (2007)