Phase Space Methods for the Analysis and Simulation of CDM Dynamics Oliver Hahn Laboratoire Lagrange, Observatoire de la Côte d’Azur, Nice, France Abel, Hahn, Kaehler (2012), MNRAS Kaehler, Hahn, Abel (2012), IEEE TVCG Hahn, Abel, Kaehler (2013), MNRAS Angulo, Hahn, Abel (2013), MNRAS Hahn, Angulo, Abel (2014), MNRAS subm. with Raul Angulo (CEFCA), Tom Abel (Stanford), Ralf Kaehler (SLAC) Hahn & Angulo (2015), MNRAS subm.
What is Dark Matter? microscopic continuum limit proton = 1GeV, WIMP 100GeV? -> 10 21 /g v thermal ≪ v bulk cold (or at most lukewarm) e.g. thermally produced at very early times, cooled since then σ DM ≪ σ em negligible cross-section collisionless weak-scale or even weaker …and also the dominant gravitating component (~80%) at first order, structure formation is well described by assuming all matter is dark matter
Dark Matter - properties on small scales P(k) CMB decr. particle mass CDM k dynamic range of simulation ICs hot warm cold y t i c o l e v 1+1D 1D position Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
1D behaviour under self-gravity cusp forms, shell-crossing, but no shock! velocity velocity time position Vanishing collision-term ⇒ not in hydro limit ⇒ velocity can be multi-valued ⇒ cannot stop at low order moments ⇒ have to discretize distribution function ⇒ singular caustics emerge Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Dark Matter - fluid flow Lagrangian description, evolution of fluid element Q ⇢ R 3 ! R 6 : q 7! ( x q ( t ) , v q ( t )) density constant density − 1 � � ∂ x i � � ρ = m DM � � ∂ q j � � For DM, motion of any point q depends only on gravity unlike hydro, no internal ( ˙ x q , ˙ v q ) = ( v q , − r φ ) temperature, entropy, pressure So the quest is to solve Poisson’s equation ∆ φ = 4 π G ρ Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
N-body vs. continuum approximation The N-body approximation: i 2 { 1 . . . N } 7! ( x i , v i ) ⇒ EoM are just Hamiltonian N-body eq. (method of characteristics) for small N, density field is poorly estimated, X ρ = m p δ D ( x − x i ) ⊗ W continuum structure is given up, but ‘easy’ to solve for forces hope that as N->very large numbers, approach collisionless continuum Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Lagrangian elements Define little piecewise maps: velocity Q i ⇢ R 3 ! R 6 : q 7! ( x q ( t ) , v q ( t )) a. position bi-linear density b. bi-quadratic − 1 � � ∂ x i � � ρ = m DM � � ∂ q j � � c. cost: tetrahedral truncation error in EoM! Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Describing the density field q 1 � (x,v) q 1 � (x,v) q 2 � (x,v) q 2 � (x,v) q 3 � (x,v) q 3 � (x,v) − 1 � � � det ∂ x i X X � � ρ = m p δ D ( x − x i ) ⊗ W ρ = m p � � ∂ q j � streams time particle locations Cosmic Structures… Cosmic Structures… Oliver Hahn Oliver Hahn ICTP ICTP , May 12, 2015 , May 12, 2015 IAU308 Tallinn, 06/23/2014 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ) Oliver Hahn
Three dimensions rendering points for particles. rendering tetrahedral phase space cells. Same simulation data! (Abel, Hahn, Kaehler 2012) Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Problem: How to measure the bulk velocity field? • Interpolate between neighbouring N-body particles • “neighbouring” in phase space, not configuration space • account for averaging over streams (“coarse-graining”) • Coarse-grained bulk velocity field: • result is discontinuous across caustics Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Derivatives of the bulk velocity field • Discontinuities make ordinary derivatives ill-defined without coarse-graining! • Away from discontinuities: Need to explicitly evaluate action of derivative on projected field: • Vorticity for std. gravity pure multi-stream phenomenon!! • At discontinuities: Derivatives are singular, but have finite measure. compressive singularities at caustics Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Properties of the cosmic velocity field II fluid mechanics! Dark matter Hahn et al. 2014a Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Spectral properties of the cosmic velocity field I CDM • Faster convergence (for WDM: convergence!) • Better small scale properties Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Problems of the N-body method: WDM Main Problem: two-body effects, directly related to force softening Scattering Clumping/ Fragmentation Wang&White 2007 Most obvious for non-CDM simulations! (e.g. Centrella&Melott 1983, Melott&Shandarin 1989, Wang&White 2007) Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Improving on N-body…. N-body Lagrangian phase-space element 1 x q = v q , ˙ and v q = � r x � | x q , ˙ with q 2 Q x i = ˙ and p i = � m r x φ | x i ˙ m a 2 p i continuum structure (diff w.r.t. q), approx by point-wise and Hamiltonian k X a αβγ q α 0 q β 1 q γ P k = { π ( q ) | π ( q ) = 2 } α , β , γ =0 need softening, -> EoM for polynomial coefficients no knowledge what it v ↵�� = � J � 1 f ↵�� , x ↵�� = v ↵�� , ˙ ˙ should be (empirical) explicit truncation error: 1 X f ↵�� q ↵ 0 q � 1 q � v = � J � 1 ∆ ˙ 2 ↵ , � , � = k +1 Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Using tets for simulations: 300eV toy WDM problem fixed mass resolution, varying force resolution: 20 std PM 15 force res. 3 /256 3 3 /512 3 3 /1024 3 PM 128 PM 128 PM 128 10 tetrahedra features become sharper monopole fragmentation appears h -1 Mpc 15 3 /256 3 3 /512 3 3 /1024 3 TCM 128 TCM 128 TCM 128 10 tetrahedra quadrupole sheet tesselation h -1 Mpc 15 based method cures artificial fragmentation 3 /256 3 3 /512 3 3 /1024 3 T4PM 128 T4PM 128 T4PM 128 10 10 15 10 15 10 15 20 h -1 Mpc h -1 Mpc h -1 Mpc Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
First determination of WDM halo mass function! Angulo, Hahn & Abel 2013 Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Limitations - diffusion/loss of energy cons. Mixing - (phase or chaotic) need increasingly larger number of elements to trace the sheet surface hi-res N-body tesselated cube orbiting in non-harmonic potential Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Need adaptive refinement adaptive refinement: a. element is flagged for b. positions and velocities are c. new elements are created refinement determined at mid-points using the mid-point values approximate element mass distribution by recursively deposited ‘mass carrier particles’ (these are not active, i.e. no degrees of freedom ) m=1 m=2 m=3 Hahn & Angulo 2015 Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
refinement + higher order! hi-res N-body tesselated cube orbiting adaptively refined tri-quadratic in non-harmonic potential phase-space element first alternative to N-body in highly non-linear regime! + able to track fine-grained phase space Hahn & Angulo 2015
Orbit test refinement no refinement Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
Self-gravitating tests 1D 32 3 particle plane wave, 32 3 particle plane wave, axis aligned oblique Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
let’s go cosmological no shotnoise!!! Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
+ able to track fine-grained phase space Hahn & Angulo 2015
Conclusions • Lagrangian elements can give new insights into existing simulations (density/velocity fields, multi-stream analysis,…) • Provide also self-consistent simulation technique. (functional when using high-order and adaptive refinement) • Solves two-body and fragmentation problems of N-body • First methodological test of N-body in deeply non-linear regime • Stay tuned for halo properties… Cosmic Structures… Oliver Hahn ICTP , May 12, 2015 GRAVASCO, Oct 14, 2013 Oliver Hahn (ETHZ)
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