1. ランダウ共鳴が与えるダークマ ターハロー密度構造への影響 ○ 扇谷 豪 森 正夫 ( 筑波大 )

2. Contents Introduction The Core-Cusp problem (GO & Mori; arXiv1206.5412) – Change of galactic potential – Analytical model – Results of simulations Summary & Discussion

3. Ishiyama et al. (2012)

4. What is the Core-Cusp Problem? Navarro et al.(1997) Mass-density profile of DM halo Observation vs Theory (CDM) log(ρ) [10 -3 M ☉ pc -3 ] -1 0 1 log(r) [kpc] van Eymeren et al. (2009) See also Burkert (1995) Constant : Core log(r) [kpc] log(ρ) [ 10 10 M ☉ kpc -3 ] Divergent : Cusp

5. Effects of Mass-Loss The central cusp becomes flatter when mass-loss occur in short timescale. But the central cusp still remains. Initial condition GO & Mori (2011)

6. Realistic Simulation Mashchenko et al. (2008) – Cosmoligical N-body+SPH simulation Supernova feedback etc. Blue: gas, Yellow: star – Gas Blown out (expansion) Fall back towards center (contraction) Repeat many times Gas Oscillation – Cusp-Core transition Physical mechanism??? Motivation: understand the mechanism for cusp flattening

7. Idealized Model 1) Gas heating by supernovae 3) Energy loss by radiative cooling 4) Contraction towards the center 5) Ignition of star formation again Repetition of these processes Gas Oscillation Change of potential ⇒ DM halo is affected gravitationally ⇒ Cusp to Core transition? 2) Gas expansion DM halo Gas

8. Linear analysis: Resonance model

9. Resonance between particles and density waves Some arithmetic calculations ( n = 1) CDM mass-density: Resonance condition for CDM halo Resonance occurs when the condition is satisfied ⇒ efficient energy transfer ⇒ system expands ⇒ Cusp-Core transition Gauss’s hyper-geometric function

10. Numerical Model Baryon (external potential): Hernquist potential ( Hernquist 1990 ) DM halo (N-body system): NFW model ( Navarro et al. 1997 ) Number of particles N 16M, 128M Softening parameter ε 0.004kpc 10kpc 2kpc Property of DM halo Oscillation period of the external force, T

11. Results1 -Density Profile- The cusp-core transition and the resultant core scale depends on the oscillation period of the external potential, T. Initial condition after 10 cycles

12. Initial condition Fourier spectrum of radial velocity Peaks appear when ω=2π/T. Each position of the peaks matches the core scale. Result2 -Fourier Spectrum of Velocity- Density profiles of DM halos after several oscillation periods r [kpc] Resonance & Core creation

13. Result3 -Overtones- ω=nΩ → Spectrum with peak (Resonance) r [kpc] 3 5 10 52 1 1 1

14. Summary -Core-Cusp problem- Study the dynamical response of a DM halo – The Core-Cusp problem – Oscillatory change of gravitational potential Analytical Model – Resonance between particles and density waves – Resonant condition: dynamical time oscillation period N-body simulations – Resonance plays a significant role to flatten central cusp Resonance → Efficient energy transfer → Cusp to Core transition – Core scale is well matched to our predictions

15. Discussion -Link to Observations- If DM halos resonate with their host galaxies, resonant condition may be satisfied in observed galaxies. SFHs of galaxies and density profiles of DM halos – ex. Holmberg II (Oh et al. 2011) McQuinn et al. (2010) Regarding as an oscillatory SFH, T ~ 100Myr

16. 以下、予備スライド

17. Structure Formation in the Universe Dark Matter (DM) – is the dominant element in mass – interacts only through gravity with others – assembles baryon (atoms) and DM – drives structure formation (DM halos, Galaxies etc.) Cold Dark Matter (CDM) cosmology – is the standard paradigm – matches observations in large scale – has serious problems in small scale Core-Cusp problem Ishiyama et al. (2011)

18. Resonance Model Resonance

19. Resonance between particles and density waves Some arithmetic calculations ( n = 1) Resonance Condition Resonance occurs when the condition is satisfied → efficient energy transfer → system expands → density change dramatically ⇒ Cusp to Core transition

20. Core Scale Density profile of CDM halo Mass profile Resonant condition Dynamical time ↑ Gauss’s hyper-geometric function

21. Resonant Solution When the condition is satisfied Using l'Hôpital's rule

23. Relation between T and γ γ NFW

24. Discussion –Energy Transfer Rate-

25. W/(dK/dt) r [kpc]

26. Number of Cycles W/(dK/dt)

27. Small amplitude Evolution

28. 以下、 GPU コードについて

29. N-body Simulations Aim: Whether we can resolve the Core-Cusp problem or not – Dynamical evolution of DM halos – Gravity from whole of the system is dominant Method: N-body simulations utilizing the Tree algorithm (Barnes & Hut 1986) – Sufficient accuracy – Reasonable computational cost – Manageable (no boundary, scale-free) – Commonly used in computational astrophysics

30. Tree Method (Barnes & Hut 1986) ｊ -particle (interact to i-particle) i-particle (compute gravity) (N: number of particles)

31. Tree-Construction & Tree-Traversal Tree-Construction – Separate cube (cell) contains all particles recursively – Link to child- and brother-cell Tree-Traversal – i-particles walk tree – Near cell -> child – Far cell -> brother Reduce computation ① ② ③ ④ ⑤ ① ② ③ ④ ⑤ ・・・・・・

32. Overview of Our Tree-Code Speed-up a Tree-code by using GPU cluster Serial: Nakasato (2012) – CPU: Tree-Construction – GPU: Tree-Traversal – Particle data: sorted by obeying space-filling curve Parallelization: MPI – Each MPI process Is host of a GPU Has data of (N/number of MPI processes) i-particles Constructs partial tree composed by its i-particles Communicates necessary tree data (Warren & Salmon 1993)

33. Warp Branch 32 GPU cores (Warp) work concurrently (SIMD) – Several kinds of processing in a warp (Warp branch) – -> All instructions should be executed (Large overhead!) Warp branches in tree-traversal – Occur whether each i-particle walks to child or brother – Become less frequent by consolidating routes of i- particles in tree

34. Vectorization & Grouping Vectorization – Consolidating routes of i-particles in a GPU core Grouping – Consolidating routes of i-particles among some cores Consolidating – Decrease frequency of Warp branch – Increase the amount of computation – Optimal value of vector length (V) and group member (G) Cell contains j-particles GPU core 1GPU core 2

35. Vectorization & Grouping Vectorization – Consolidating routes of i-particles in a GPU core Grouping – Consolidating routes of i-particles among some cores Consolidating – Decrease frequency of Warp branch – Increase the amount of computation – Optimal value of vector length (V) and group member (G) Cell contains j-particles

36. HA-PACS GPU cluster installed at CCS of U-Tsukuba in 2012 – Base cluster: 268 nodes – 4 Tesla M2090 cards / node

37. Improvement Computation time of Tree-Traversal – N=8M, single precision – Serial time [sec] Number of group members Vector length Compared with the case of without Vectorization and Grouping, Tree-Traversal becomes more than 3 times faster!! Optimal pair (V,G)=(4,4)

38. Strong Scaling (N=32M) Time [sec] Number of GPUs 1/Np

39. Weak Scaling (N=16M/GPU) Time [sec] Number of GPUs Log(Np)

40. Summary -Tree Code- Tree code on GPU clusters – Vectorization & Grouping To reduce the frequency of warp branch Consolidating the route of several i-particles – Kernel function (tree traversal) Becomes more than 3 times faster – Parallelization by using MPI Future: Tree-SPH code