Phase Space Structure of Dark Matter Halos

Transcription

1 Phase Space Structure of Dark Matter Halos Monica Valluri University of Michigan

2 Collaborators Ileana Vass (U. Florida, Gainesville) Andrey Kravtsov (KICP, U. Chicago) Stelios Kazantzidis (CCAPP, Ohio State U.) Victor Debattista (U. Central Lancashire) Tom Quinn (U. Washington) Ben Moore (U. Zürich)

3 Outline Evolution of phase space density in dark matter halos (Vass, Valluri, Kravtsov & Kazantzidis 2009 MNRAS, in press) (Vass, Kazantzidis,Valluri & Kravtsov 2009 ApJ, in press) Evolution of orbital structure of dark matter halos in response to the growth of baryonic structures (Valluri, Debattista, Quinn & Moore, MNRAS, in preparation)

4 Phase-Space density profiles Taylor & Navarro 2001 Q(r) ~ (r)/ 3 (r) ( (r) = f(r, v) d 3 v) CDM halos: have power-law Q(r) profiles over 2.5 decades in radius: Q = r/s 3 (r/r 200 ) - ~ 1.9±0.1 (Taylor & Navarro 2001, Dehnen & McLaughlin 2005) Power-law profiles also result from simple self-similar spherical infall models (Bertschinger 1985)

5 Evolution of Q in cosmological halos Q r - from z~5 to z~0 inside ~0.25r vir = 1.9 ± 0.1 Q is robust to mergers/infall (Hoffman et al. 2007)

6 Evolution of Collisionless Systems The true phase space distribution function is defined by (r) = f(x, v) d 3 v Fine grained phase space distribution function f(x, v) is described by collisionless Boltzmann equation (Liouville s Theorem): df dt = f t + vi f x i f v = 0 In the absence of collisions mass per unit volume of phase space f(x, v) is conserved But f (x,v) is never measured. Only possible to measure coarse grained density = <f(x,v)>

7 Evolution of the phase space density of dark matter How do power-law Q(r) profiles arise? How does Q(r) relate to the real phase-space distribution function f(x, v) of dark matter? (r) = f(r, v) d 3 v We address these questions using a series of N- body simulations

8 N-body simulations t Galaxies from hierarchically: major mergers accretion of sub halos or distributed material (quiescent infall) (Vass, Valluri, Kravtsov & Kazantzidis 2009 MNRAS, in press Arxiv: )) (Vass, Kazantzidis,Valluri & Kravtsov 2009 ApJ, in press Arxiv: )

9 Numerical computation of coarse grained phase-space density Binary Tree methods are accurate and metric-free FiEstAS (Ascasibar & Binney 2005) EnbiD (Sharma & Steinmetz 2006) 6-D density estimation problem. Need appropriate adaptive smoothing kernel

10 (Ir)Relevance of R vir Mass profile of a DM halo changes little from z=1 to z=0 but virial radius doubles static mass radius ~ 2r vir Also 40% of mass of progenitors in a merger major lies outside the formal virial radius (Kazantzidis et al. 2006, Valluri et al. 2007) Cuesta et al. 2008

11 Comparison of phase space density estimators in CDM halos z=0 Q(r) r - ( = -1.9±0.1) Mean (f) - sensitive to substructure Median (f) = F(r) Power law (r - ) => poor fit (Stadel et al. 2008, Maciejewski et al., 2009, Vass et al. 09a)

12 Evolution of z=0 all particles r < 2r vir are tracked back to z~9 F(r) not power-law at any redshift! F decreases: Central F by factor ~10 F(0.6 r vir ) by factor ~105 Mixing in phase space

13 Phase Mixing (Lynden-Bell 1967) Particle orbits to spread out because of spread in initial conditions The timescale for phase mixing depends on size of patch coarse grained density = <f(x,v)>

14 Violent Relaxation (Lynden-Bell 1967) Potential Energy A system out of equilibrium undergoes violent oscillations Particles exchange energy with rapidly changing background potential t [Gyr] T oscillation ~ T dynamical Relaxation time ~ orbital period

15 Evolution of Q and F within r vir (z) Q (r< r vir ) power-law F ~ decreases monotonically up to 0.6r vir (z) and increases thereafter

16 f(x,v) in CDM halos z=0 Sub halos have highest f values f(x,v) has range of 4-9 orders of magnitude at each radius (Arad et al. 2004)

17 Evolution of f(x,v) Large number of particles Small number of particles wiggles in F => subhalos streams from tidal shredding of subhalos At all z highest f values are in subhalos

18 Via Lactea II projected dark matter squared-density map Coarse-grained phase-space density f(x,v) Real space Density (r) J Diemand et al. Nature 454, (2008) doi: /nature07153

19 Evolution of N(f) Histograms of log(f) approximately lognormal f peak shifts to lower f f max today representative of the median f at z 9.

20 Evolution of f peak Evolution of f peak for 4 different galactic sized halos f peak (a) a 4.3±1.1

21 Physical implication of Q(r)? Q(r) = (r)/ 3 (r) 2 ~ measure of DM temperature Entropy 2/3 K gas = T / gas K DM = 2 / 2/3 dm Q 2/3 For Q r 1.8 K r 1.2 (Faltenbacher et al. 2007) (Hoffman et al. 2007) Power-law slope and entropy profile are very similar to that found for gas in the outer regions of clusters (also arises from spherical collapse/ spherical accretion) (Tozzi & Norman 07, Borgani et al 04, Voit et al. 05) Q(r) reflects average inverse entropy profile of DM not phase-space density!

22 Part1: Summary & Implications F(r) is not a simple power-law at any z! (not even monotonically decreasing) F(r) decreases by 1-5 orders of magnitude f(x,v) varies by 4-8 orders of magnitude at a given radius but is highest in subhalos (possibly also in DM streams) N(f) approx log-normal f peak (z) (1+z)( ) Measuring f at center of dsph galaxies (Strigari et al. 2007) and in tidal streams (Freese et al. 2004) will put tighter constraints primordial phase space density of DM/ nature of DM particle (e.g. Tremaine-Gunn 79) Q(r) is related to entropy and not a measure of phasespace density at all!

23 Part II: The shapes of DM halos Collisionless DM halos are triaxial: (Dubinski & Carlberg 91, Jing & Suto 02, Kasun & Evrard 05) <c/a> ~0.6 <b/a> ~0.8 Observationally preferred: oblate c/a > 0.8 b/a ~1 [polar rings (Sackett & Sparke 90), tidal streams (Johnston et al. 05), X-ray halos (Buote et al. 02), lensing (Kochenek 95, Oguri et al. 03)] prolate oblate b c a triaxial

24 Simulations with gas, star formation, cooling Kazantzidis et al prolate CDM simulations with gas cooling & SF spherical halos mergers of disk galaxies with gas spherical halos (caution: degree of shape change could be over-estimated due to over-cooling)

25 What causes the change in shape? Two possible options growth of central potential can deform orbits to make them rounder (Hernquist & Barnes 1987, Holley-Bockelmann et al. 2002) chaotic scattering by central deepened potential (Merritt & Quinlan 1998, Maccio et al. 2007) The goal of this study: distinguish between these possible mechanisms How does a baryonic disk change the phase space structure of DM particles locally?

26 Orbits in Triaxial Galaxies (de Zeeuw 1985, Statler 1987) Boxes Short axis Tubes (SAT) Inner long-axis Tubes (ILT) Outer long-axis Tubes (OLT) Triaxial figures are supported by 4 major orbit families. Box orbits and ILT dominate provide density along the major axis

27 Supermassive Black Holes Chaotic scattering by a central BH In a barred galaxy a BH with ~5% disk mass (Norman et al. 1996, Shen & Sellwood 2004) In a triaxial galaxy a BH with ~3% galaxy mass (Merritt & Quinlan 1998) Athanassoula, Lambert & Dehnen 2006

28 Chaotic Scattering => Mixing Strongly chaotic orbits diffuse ( T dyn ) Equipotential surface is rounder than the density distribution Diffusion of ensembles of 10,000 orbits near a chaotic orbit (Merritt & Valluri 1996)

29 Chaotic evolution => Irreversibility Chaotic evolution => exponential divergence Chaos => mixing Regular evolution => linear (adiabatic) response

30 Adiabatic growth of a baryonic (Debattista et al. 2008) component collisionless merger remnant After adiabatic growth of baryonic component Even a small central baryonic component ( %) makes halos rounder

31 Testing Reversibility Debattista et al Phase a : initial halo merger of two spherical NFW halos Halo B (prolate ; M ; 4 million particles) merger of two prolate merger remnants Halo A (triaxial; M ; 4 million particles) Phase b : after growth of baryonic component Disk galaxy (R d =3 kpc) Spherical galaxy (R s = 3kpc, R s = 1kpc) dense star cluster+smbh=cmc (R s = 0.1kpc) Phase c : baryonic component is evaporated Most models reverted to original shape

32 Orbits of DM particles in an equilibrium potential mostly regular Conserve integrals of motion Are quasi-periodic Characterized by fundamental frequencies of oscillation

33 Numerical Recovery of Fundamental Frequencies All oscillation frequencies are integer linear combinations three fundamental frequencies Frequencies can be used: To classify regular orbits into major orbit families To identify Resonant and Periodic orbits Map the phase space To identify chaotic orbits A k X(t) = A k e i kt k = k 1 + l 2 + m 3 Binney & Spergel 1984, Laskar 1990, 1996, Valluri & Merritt 1998 k

34 Application to N-body simulations Grainy N-body potential => noisy orbits Orbits in a spherical potential are regular Frequency drift: log( f) = log [ (t 1 )- (t 2 )]/ (t 1 ). 99.5% of orbits in spherical NFW halo have log( f) < -1.0 Defined as chaotic if Log( f) > -1. Analyze orbits in each halo potential in each phase of evolution

35 Baryons cause frequency changes Disk Point mass Almost perfect reversibility Irreversible scattering Extended baryonic distribution => frequencies increase Frequency change is nonlinear Point mass => scattering disk (elliptical) galaxy => frequency in phase a and phase c are correlated

36 Frequency change: distribution Disk Hard point mass Frequency change from phase a to b Frequency change from phase a to c Only a very hard point mass causes significant irreversibly scattering (i.e. black holes cause chaotic evolution) Extended distribution produce regular evolution

37 Orbit populations Phase a Phase b A+disk B+point

38 Orbital shapes s = y z x z Orbital shape parameter: s > 0 elongated along long axis s 0 for orbits with round shapes z x y

39 The same orbital type can change shape A+disk A+Sph 1.0 B+Sph 3.0 B+Sph 0.1 Pericenter radius Pericenter radius In all cases orbits with the smallest pericenter radii become round in phase b Short axis tubes always round Halo orbits respond to the baryonic component by becoming rounder within several R s

40 Frequency maps Orbits cluster around stable resonances Plot frequency ratios for ~10,000 different orbits at a single energy Gives a map of phase space (Like a Poincare surface of section) Resonant orbit condition: l x + m y + n z =0 (l, m, n) are small integers Unstable resonances depopulated Valluri & Merritt 1998

41 Frequency maps of N-body halos Halo A A+disk O,1,-1 ILT orbit family 1:-1,0 SAT orbit family 3,0,-2 fish family x z x z 2,0,-1 banana family Most bound particles Least bound particles Only 6000 orbits spanning full range of energies are plotted several global resonances appear following the growth of a disk

42 Resonant orbits In the presence of a baryonic component the resonances are stronger. Resonant orbits are two dimensional surfaces.

43 Sample of box-orbits in triaxial Halo A at different energies Y Z Y Z X Y X Y Y Z Z X Y X Y

44 Box Orbits =>Short Axis Tube orbits Baryonic disk and central bulge significantly alter phase space structure of DM particles

45 Part II: summary/implications Growth of disk (or elliptical galaxy) causes: REGULAR/ reversible change in shape Growth of central point mass causes CHAOTIC scattering of orbits Orbital shape changes are more significant than orbital type changes Centrally concentrated orbits tend to change shape Baryonic components enhance the fraction of DM particles trapped around resonances Future: How does this affect DAMA interaction cross sections for direct detection of DM particles? (upcoming with Savage & Freese)

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