N-body Cosmology: The billion-body problem. Brian O Shea Michigan State University

Transcription

1 N-body Cosmology: The billion-body problem Brian O Shea (oshea@msu.edu) Michigan State University TIARA Winter School, Jan. 2011

2 This lecture Goals of cosmological N-body simulations Overview of an N-body simulation Requirements and assumptions Equations solved Numerical techniques for N-body simulations Generating cosmological initial conditions Some example applications Useful references and codes

3 Goals of cosmological N-body sims Model the growth of structure in the universe Can be treated as an experiment to verify theories of the origin, evolution of the universe More specific applications: galaxy cluster mass function, evolution of dark energy EOS, substructure in DM halos, base for semi-analytic models of galaxy formation/evolution If other physics is included: galaxy formation, galaxy cluster formation

4 Overview of a cosmological N- body simulation Generate initial conditions at high redshift using matter power spectrum and perturbation theory Evolve simulation forward in time: calculate forces acting on particles calculate max. allowable timestep advance all particles Done iteratively! Compare to observations!

5 Requirements and assumptions We calculate the evolution of matter in an expanding universe! For this, we need and assume: Comoving coordinates (and accompanying cosmological model) Periodic boundary conditions Gravity is the only force considered Dark matter can be treated in a statistical way as a collection of mobile, discrete particles (so we can use an N-body method rather than solving the collisionless Boltzmann equation directly) Other physical processes (particularly gas, stars and related physics) are not important

6 An aside on cosmology WMAP Year 7 map of the microwave sky

7 SDSS galaxy map All galaxies from SDSS survey with declination of o

8 WMAP 7 matter power spectrum Larson et al. 2011, ApJS, 192, 16

9 combined P(k) as a function of wave number (c/o Max Tegmark)

10 Komatsu et al. 2011, ApJS, 192, 18

11 Equations

12 Equations of motion But, you must include comoving coordinates:

13 Where a(t) is calculated assuming a homogeneous, isotropic universe and a Robertson-Walker metric, and reduces to the Friedmann equations: Often, calculate p assuming a perfect fluid:

14 In comoving coordinates, equations of motion and Poisson equation become:

15 Numerical techniques

16 Orbital integration It is not necessarily desirable to calculate a very high-order position update (eighth order Runge Kutte, for example): Gravitational potential for N-body systems very noisy! A particle is not really intended to be a single object: pairwise interactions undesirable Keeping many intermediate steps is memory-intensive and computationally costly Essentially all cosmology N-body codes use a leapfrog integrator:

17 Time steps in cosmological simulations Gravity causes particles to clump. The relevant timescale is dynamical time! Choice of time step is critical: Too small: lots of wasted CPU time Too large: excessive errors in integration! Not all particles need to take the same time step, either! In general, time steps in cosmological simulations governed by local forces and resolution.

18 A brief aside: hydrodynamics Gas is important in astrophysics, and we ignore it at our peril! BUT, large scale structure formation is dominated by gravity: can ignore hydrodynamics for certain problems! Can solve equations of hydrodynamics using either Eulerian (grid-based) or Lagrangian (particle-based) formalism: the choice of formalism depends on many considerations. Different types of N-body solvers favor either Eulerian or Lagrangian hydro solutions: I will attempt to specify.

19 N-body techniques

20 Direct summation Discussed previously by Douglas Heggie Sum forces by direct particle-to-particle gravitational force:

21 Direct summation Advantage: Exact forces calculated on each particle! Disadvantage: Takes N 2 summations to calculate force in one timestep Difficult to apply periodic boundary conditions Doable for N ~ 10 6, impossible for N ~ 10 9

22 The first N-body simulation 37 particles in 2D: Holmberg 1941, ApJ, 94, 385

23 Integration method: analog (light bulbs, photocells, and graduate students) 37 particles in 2D: Holmberg 1941, ApJ, 94, 385

24 Results: tidal deformation of disk nebulae, also interacting galaxies tend to merge Figure 4b in Holmberg 1941, ApJ, 94, 385

25 Tree method Space divided recursively into a hierarchy of cells Particles see neighbors directly: distant groups of particles are aggregated into a single apparent object with mass equivalent to the sum of the particle masses Original reference: Barnes & Hut 1986, Nature, 324,

26 Tree method Advantages O(N log N) operations Force errors can be easily understood and reduced Relatively easy to implement Spatially adaptive - good force resolution Disadvantages Large amount of memory required Periodic boundary conditions challenging

27 Particle-Mesh Uses Cartesian grid to calculate forces on particles. For each timestep: 1. Compute mass density on grid 2. Solve Poisson s equation for gravitational potential using gridded density 3. Calculate force on particles using gridded potential See Computer Simulations Using Particles by Hockney and Eastwood

28 Particle-Mesh Advantages Very fast: O(Np) + O(Ng log Ng) operations using FFT Periodic boundary conditions trivial Very simple to write and parallelize Disadvantages Poor approximation of gravitational force for nearby particles: resolution ~ 2Δx Some force anisotropy due to grid

29 APM (Adaptive Particle-Mesh) Improves particle-mesh algorithm by placing subgrids in places where particles are clustered: better force resolution! Potential on subgrids calculated using BCs from coarse grid, density field from fine grid, and a multigrid relaxation method This is the basis for most adaptive mesh refinement code gravity solvers: Enzo, FLASH, RAMSES, MLAPM Jessop et al. 1994, J. Comp. Phys., 115, Kravtsov et al. 1997, ApJS, 111, 73-94

30 P 3 M Combines particle-particle and particle-mesh methods At large distances, use particle-mesh to calculate gravitational potentials Up close, force computed by direct summation Advantages: better approximation of gravitational force for close particles than PM faster for given Np than PP Disadvantage: becomes slow when clustering occurs (cost of direct-summation dominates) See Efstathiou et al. 1985, ApJS, 57,

31 AP 3 M Improves the P3M method by using additional subgrids at regions where clustering occurs (like APM, but with additional particle-particle summation) Advantages: Good accuracy of pair forces on almost all pairs Very fast: O(Np log Np) Disadvantages: limited dynamic range in subgrids some ambiguity in subgrid placement (spurious forces) still eventually run into particle-particle expense See Couchman et al. 1991, ApJL, 368, L23-26

32 Example AP 3 M simulation. From Couchman et al. 1995, ApJ, 452, 797

33 TreePM Synthesizes tree algorithm and particle-mesh algorithm: At large distances, use PM to calculate smooth contribution to gravitational potential At small distances (< 2dx) use tree to calculate local neighbors contributions Very fast and efficient, high dynamic range, excellent scaling: current state-of-the-art for pure N-body Example code: Gadget-2 (Springel 2005, MNRAS, 364, 1105; Original method papers: Xu 1995 (ApJS, 98, 355), Bode & Ostriker 2003 (ApJS, 145, 1)

34 Evolution of simulation (and computer) size with time

35 2009: (68.7 billion) particles Teyssier et al. 2010, A&A, 497, : multiple simulations (Habib, Heitmann et al., in prep) 2011: particles, Norman, O Shea et al. (in prep.), From Springel et al. 2005, Nature, 435, 629

36 From top500.org

37 From top500.org

38 Initial conditions

39 Background ICs for cosmological simulations are generated at high redshift (z ~ ), when the universe is approximately linear (density variations of ~5-10%) Inputs: cosmological parameters and matter power spectrum Grav. potential calculated on a lattice using a Gaussian random field Particles are typically initialized on a lattice (or in a glass force-free initial condition) and given spatial, velocity perturbations using the Zeldovich approximation (or something similar)

40 Gaussian random field Given a matter power spectrum, P(k), and a lattice with (Nx, Ny, Nz) cells and known volume. Sample the power spectrum discretely at each grid location, for a given k: At each position, the k-space fluctuation is given random phase and (complex) fluctuation amplitude: We then perform a Fourier transform to get the relative matter density fluctuation at each spatial grid point in the simulation volume

41 Particles start out distributed uniformly on a grid, or in a forcefree ( glass ) configuration. Then, they are displaced using the Zel dovich approximation (Zel dovich 1970, A&A, 5, 84-89): q = unperturbed position D(a) = growth factor H = Hubble parameter f = logarithmic growth rate of D(a) = irrotational displacement field Note: the Zel dovich approximation has incorrect second- and higher-order growth rates: it s worth considering higher-order approximation methods if you are doing something particularly sensitive to this (Crocce et al. 2006, MNRAS, 373, 369)

42 Example ICs (dark matter) Log dark matter density z=40 FOV: 32 Mpc/h comoving Depth: 8 Mpc/h comoving simulation has 256^3 (16.8 million) particles c/o Bode, Cen, Xu

43 Example ICs (dark matter) Log dark matter density z=0 FOV: 32 Mpc/h comoving Depth: 8 Mpc/h comoving simulation has 256^3 (16.8 million) particles c/o Bode, Cen, Xu

44 Some example applications

45 Formation of filaments in CDM Movie c/o Andrey Kravtsov Box 40 Mpc across

46 Formation of a galaxy group Zoom-in of previous sim. c/o Andrey Kravtsov

47 Next slide: fly-through of the Millennium Simulation (Springel et al. 2005, Nature, 435, )

48

49 Springel et al. 2005, Nature, 435,

50 Springel et al. 2005, Nature, 435,

51 The Aquarius Project (Virgo collaboration) Formation of MW-halo mass galaxy

52 Growth of six different MW-sized halos Boylan-Kolchin et al. 2010, MNRAS, 406, 896 Formation of metalpoor globular clusters as f(z) (blue: halos that form stars) Griffen et al. 2010, MNRAS, 405, 375

53 Formation of a Milky Way-type galaxy

54 Galaxy formation with a supermassive black hole Springel, Hopkins et al.

55 Galaxy merger with small (1:8) companion (retrograde) Younger et al. 2007, ApJ, 670, 269

56 Useful references and codes

57 References, I Computer Simulations Using Particles, by Hockney & Eastwood (Taylor & Francis, 1989) by Aarseth (Cambridge University Press, 2003) Gravitational N-body Simulations: Tools and Algorithms, Simulations of Structure Formation in the Universe, Bertschinger 1998, Annual Reviews of Astronomy and Astrophyscis, 36, Numerical Simulations in Cosmology, I: Methods, Klypin, astro-ph/ (also see papers II and III in this series) A survey of all known N-body methods:

58 References, II Institute for Advanced Studies Computational Astrophysics Summer School lectures (particularly those by V. Springel): University of California HIPACC Astro-computing summer school (part. lectures by Klypin): SummerSchool_archive.html Cosmological N-body simulation: Techniques, scope and status, Bagla 2005, Current Science, 88, Galactic Dynamics, by Binney & Tremaine (Princeton University Press, 1988)

59 Codes you can download and use Enzo: (PM/APM) Gadget: (Tree/TreePM) FLASH: (APM) RAMSES: TPM: (APM) (TreePM) Hydra: (AP 3 M)