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Travel-Nature

Published on August 27, 2007

Author: Richie

Source: authorstream.com

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Cosmological N-Body SimulationHow to build your own Universe in 5 easy steps!:  Cosmological N-Body Simulation How to build your own Universe in 5 easy steps! December 13, 2004 Paul Edmon James Klassen 5-Step Program:Towards your Apotheosis! Or maybe just insanity.:  5-Step Program: Towards your Apotheosis! Or maybe just insanity. See the POWER! The Equations and Conditions The Code Interpreting what you have just done Apply to the Universe and feel the POWER! Virgo Simulation:The Granddaddy of them all!:  Virgo Simulation: The Granddaddy of them all! 5 days of CPU time 512 Processors (7 years of computing time!) 200 GB of data Particle Mass: 1.1x1015 Solar Masses 1 Billion Particles 3000 h-1Mpc sides Coma Cluster would be resolved by 500 particles Virgo Simulation Results:  Virgo Simulation Results For CDM 3000 clusters of the size of Coma 1 million clusters of 32 particles or more Virgo Simulation Results:  Virgo Simulation Results Box size 70.5 (h-1Mpc)3 Adaptively Smoothed Periodic Boundary Conditions Jenkins et al, 1998 Astrophysical Journal,499,20-40 Einstein-deSitter Simulation:The Obligatory Simulation:  Einstein-deSitter Simulation: The Obligatory Simulation 16.7 Million Particles 100 Mpc Periodic Box 40 hours with 1015 Steps Evolved from z=32 Rotations at 8=0.8 and 1.3 Michael S. Warren 1995, Los Alamos National Laboratory Methods in the madness:What gives us any reason to think we can do this anyway?:  Methods in the madness: What gives us any reason to think we can do this anyway? Basic Assumptions Considering Initial Conditions Formulas and formalisms Choosing Variables Basic Assumptions:Not all assumptions are bad, just most.:  Basic Assumptions: Not all assumptions are bad, just most. Expanding U All calculations carried out in co-moving coordinates Only gravity as significant force Limits smallest scale that is valid in the model as it ignores pressure effects It is a model of cold dark matter only (no interactions, no thermal velocities) Work in Newtonian limit – particles are non-relativistic Can model U as few massive particles each representing about 1015 solar masses Treat particles as average density over a finite volume (not point particles) Periodic Boundary Conditions Assumes U is on large scales basically the same everywhere limits largest scale that can be seen in the model – or – a large fixed size sphere is simulated Assumes contributions from scales larger than the box can be neglected Initial conditions:What starts it all.:  Initial conditions: What starts it all. Inflation says: need near homogeneous initial particle distribution Zel’dovich power spectrum Initial perturbations about homogeneous small Linear regime  dominated EdS/Matter dominated Need to be in growing mode (eventual collapse) From Virial thm -andgt; linearized overdensity andgt; 1.7 Methods for generating IC’s:  Methods for generating IC’s Generate an initial gravitational potential to ensure growing mode Realize that potential by: Place particles on a cubic grid Random distribution of particles Particles placed on a lattice and the randomly perturbed Run N-body simulation with repulsive potential Initial Conditions: Cubic grid:  Initial Conditions: Cubic grid Particles are placed on a (uniform) cubic grid Particle mass is varied to match potential Assign velocities to particles to put system in growing mode Pros: Creates a homogeneous distribution Easy to implement Cons: Distribution is not random Interferes with power spectrum Initial Conditions: Random Distribution:  Initial Conditions: Random Distribution Start with a uniform distribution of identical particles Calculate particle velocities to put system in growing mode (from initial potential) Displace the particles from uniform positions by amount corresponding to assigned velocity Check displacements andlt;andlt; inter-particle separation Otherwise will violate initial potential Pros: Produces both a homogeneous and random distribution Cons: Requires an uniform distribution as input Initial Conditions: Generating Uniform distribution:  Initial Conditions: Generating Uniform distribution Use a cubic lattice Uniform by definition Regularity creates peaks in power spectrum Randomly displace particles from cubic lattice Also uniform Removes problems with regularity above N-body simulation with repulsive potential Creates uniform distribution Reuses existing N-body code Takes awhile to run Formulas:Hey, these look familiar.:  Formulas: Hey, these look familiar. Equation of motion Freedman Equation (expansion), flat U Gravitational Effects Too complicated – requires following all particles Use solutions of Vlasov equation Variables:  Variables Numerical considerations Want variables that change slowly in time Requires less iterations Less errors accumulated from numerical integration (Less than if used variables that change quickly in time) Choose f=const over f’=a for early steps Use ‘a’ as time parameter since displacements should scale as ‘a’ and not ‘t’ early in the simulation. N-Body Codes:The source of the power!:  N-Body Codes: The source of the power! Particle-Particle (PP) Particle Mesh (PM) Adaptive-Refinement-Tree (ART) Hybrids Particle-Particle/Particle Mesh (P3M) Adaptive TreePM (ATreePM) Particle-Particle (PP):  Particle-Particle (PP) Sums Forces by direct Particle to Particle Gravitational Force Takes N2 Summations to calculate force in one timestep Inefficient for large numbers of particles Difficult to apply periodic boundary conditions Particle Mesh (PM):  Particle Mesh (PM) Phase-space is divided up into a mesh of equal sizes Mesh is then treated as the particles and then forces are applied Mass can change in each Mesh space at each timestep Resolution limited to mesh size Particle Mesh (PM):  Particle Mesh (PM) In Fourier space Poisson’s equation is a simple algebraic equation so Fast Fourier Transforms (FFT) can be used to get to real space Since it is a Fourier method, Boundary Conditions are obtained for free Particle Mesh (PM):  Particle Mesh (PM) Mesh softens force at small scale which leads to an underestimation of the force at large scales Very Fast and Efficient Code and uses a large number of particles Adaptive-Refinement-Tree:  Adaptive-Refinement-Tree Divides space in 1/8 size boxes Then looks inside the box and sees if there are any particles in the box It will continue to divide each box into smaller boxes until each box contains a particle Klypin 2000, astro-ph/0005502 Adaptive-Refinement-Tree:  Adaptive-Refinement-Tree Forces are summed such that: Close forces use the small 1 particle boxes as particles Forces from far away use the bigger multiparticle boxes as large pseudoparticles Klypin 2000, astro-ph/0005502 Adaptive-Refinement-Tree:  Adaptive-Refinement-Tree Tree only recalculated when there is an overdensity of particles (i.e. greater than 2-5) Uses NlogN Summations Force Resolution is down to 2 grid spaces away Adaptive-Refinement-Tree:  Adaptive-Refinement-Tree Each level of the Tree moves at its own time step so particles crossing the boundary between levels needs to have its position and momentum interpolated Periodic Boundary Conditions are hard to implement Hybrid Codes:In the works as a Sci-fi Channel Original Movie!:  Hybrid Codes: In the works as a Sci-fi Channel Original Movie! Particle-Particle/Particle Mesh (P3M) Uses PP for short range force and PM for long range force Underestimates Forces at large scale Adaptive TreePM (ATreePM) Uses a Tree hierarchy and PM for force softening Interpreting results:Now that you’ve generated 200GB of data, what are you supposed to do with it?:  Interpreting results: Now that you’ve generated 200GB of data, what are you supposed to do with it? Goal: compare simulated Universe with the real Universe Problem 1: Simulation models only dark matter We can directly observe only baryonic matter Solution: Assume baryons trace dark matter Problem 2: Halo Identification Comparing results of simulations with observed large scale structure of the U Halo Identification:Note you are not the Master Chief.:  Halo Identification: Note you are not the Master Chief. Identify condensed objects in simulated U Non-trivial (in dense regions) Filaments make two objects look like one object Close objects often result in double counting Large galaxy with a smaller satellite Problem since satellite is within the virial radius of the galaxy Stripping of particles from one halo by another In dense regions it is likely much of the mass of a halo is stripped off by another halo in passing. This however, is not evidence the first halo was destroyed. The core is likely still overdense enough to form a galaxy. Slide28:  A large Galaxy: http://www.onysd.wednet.edu/~g98s57/cars/ford/67galaxy500.JPG And a smaller Satellite: http://www.allpar.com/model/images/belvgtx67.jpg It doesn’t look like a problem to me… Slide29:  http://www.mpa-garching.mpg.de/Virgo/mosaic_lcdm_new.gif Here’s the problem… Methods of Identifying Objects:FOF, MRE, Master Chief? No we’re not in the military.:  Methods of Identifying Objects: FOF, MRE, Master Chief? No we’re not in the military. Friends of Friends (FOF) DENMAX Overdensity 200 Friends of Friends:  Friends of Friends Looks for objects close together to define halo All objects linked in a chain of objects within a sphere of radius (bd/2) of each other d is average inter-particle distance b is a free parameter Results are very dependant on b Can conglomerate two close objects Tends to follow filaments in largely over-dense regions As such, it is somewhat depreciated DENMAX:  DENMAX Looks for density maximums Associates particles with the maximums Treats density field as an attractive potential Simulates particles moving under that potential Assigns particles to density maximum they 'fall' into Can loose galaxies if sum to positive energy Example: two galaxies passing at high velocities Not a common enough situation to be a big problem Real problem: computationally complex Overdensity 200:  Overdensity 200 Uses virial density and radius to define halos Locate density maximums Assign particles to halo if they fall within a sphere where the over-density is greater than 177. We showed this in class that regions with an over-density greater than 200 will collapse eventually. Most like method used in observations Easy to compare results to real universe Simple to implement Limitations:What happens as you head towards infinity or zero.:  Limitations: What happens as you head towards infinity or zero. Unreliable at scales shorter than mean particle separation in high density (very non-linear) regions with low numbers of particles Additional tuning required to model small scales Addition of short range effects Smooth Particle Hydrodynamics State of the art can simulate The Intergalactic Medium The Intracluster Medium The formation of first stars Cosmology:Not to be confused with Cosmetology!:  Cosmology: Not to be confused with Cosmetology! Work backwards Assume cosmological parameters (initial conditions) Run simulation Compare distributions of condensed objects statistically with observations of our U Important tests of Cosmological theories Allows theories to be tested without the need for as many simplifications used in class Allows simplifications used in class to be verified Links theory with observation Theories/Results:With great power comes great responsibility...:  Theories/Results: With great power comes great responsibility... Yet another reason EdS is out Zel'dovich spectrum is good U is about flat, Wm=0.3 WL=0.7 Gravitational Clustering only removes dependence of final power spectrum on initial conditions in very massive halos Non-linear small scales can influence large scales if the large scale has small fluctuations Power transfer between K-modes Non-linear fluctuations at the large scale removes dependence on smaller scales References:…to credit those guys who wrote about this.:  References: …to credit those guys who wrote about this. Evrard, 'Simulating Large-Scale Structure' 1998, astro-ph/9812377 Klypin, 'Numerical Simulations in Cosmology I: Methods' 2000, astro-ph/0005502 Splinter et. al., 'Dark Matter, Discreteness and Collision Error in Cosmological N-Body Simulations' 1997, astro-ph/9711105 Bagla, 'Cosmological N-Body Simulation: Techniques, Scope, and Status' 2004, astro-ph/0411043 www.lanl.gov movie: Michael S. Warren 1995, Los Alamos National Laboratory VIRGO collaboration : Jenkins et al, 1998 Astrophysical Journal,499,20-40 http://www.mpa-garching.mpg.de/Virgo/mosaic_lcdm_new.gif

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