The mass and anisotropy profiles of nearby galaxy clusters from the projected phase-space density
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1 The mass and anisotropy profiles of nearby galaxy clusters from the projected phase-space density Radek Wojtak Nicolaus Copernicus Astronomical Center collaboration: Ewa Łokas, Gary Mamon, Stefan Gottlöber, Anatoly Klypin, Yehuda Hoffman
2 Outline Motivation DM density profile/anisotropy profile Model of the phase-space density: brief description Tests on mock kinematic data Application to nearby galaxy clusters Summary/Perspective for future
![Motivation 4 3 2 v los [ 3 km/s] - -2-3 -4.5.5 2 2.](/docs-images/90/103107743/images/3-0.jpg "5 R [Mpc] no data binning f los (R, v los ) ρ DM (r), β(r)")
3 Motivation v los [ 3 km/s] R [Mpc] no data binning f los (R, v los ) ρ DM (r), β(r) breaking M β degeneracy
4 DM density profile 3 2 ρ r - r s =r v /c ρ(r) [ 4 M /Mpc 3 ] - ρ r -2 ρ r r/r v ρ NFW (r) (r/r s )(+r/r s ) 2
5 Anisotropy profile.5 β.3... r/r s β(r) = σ2 θ (r) σ 2 r(r)
6 Model of the phase-space density
7 f(r,v) spherical symmetry: f(r,v) = f(e, L) st component: ρ(r) f(e, L) 2nd component: β(r) f(e,l) β(r) = σ2 θ (r) σ 2 r(r) : radially-biased model : isotropic dispersion tensor : tangentially-biased model
8 History β(r) = β(r) = const. β(r) = r2 r 2 +r 2 a β(r) = r2 αr 2 a r 2 +r 2 a Eddington (96) Hénon (973) Osipkov (979), Merritt (985) Cuddeford (99) f(e, L) = f E (E)( + L 2 /(2L 2 )) β Cuddeford & Louis (995)
9 f L (L).5 r v Free parameters: β.3 r tran β β = lim r β(r) β = lim r β(r).. β. r/r s β(r tran ) β +β 2 f L (L) { L 2β for L L L 2β for L L, ansatz:f L (L) = ( + L2 2L 2 ) β +β L 2β
10 f E (E) f E (E) n gal (r) ρ DM (r) Ψ(r) n gal (r) ρ DM (r) ρ DM (r/r s )(+r/r s ) 2 n gal (r) = f E (E)f L (L)d 3 v n gal (r) = Ψ(r) f E (E)K(E, Ψ(r))dE n gal,i = j K i,jf E (Ψ j ) Wojtak et al. 28
11 The shape of β(r) profile.5 DF model α= α=2 α=.5.3 β. r /4 β(r)=.5r α /(r α +r /4 α ).. r/r s
12 Comparison with the simulation
13 f(e, L) from the simulation L/L s - r p >r v r p <r v <r a L max (E) r a <r v L/L s L max (E) E/V s N(E, L) = d2 M dedl E/V s f(e, L) = N(E,L) g(e,l) g(e, L) = δ( 2 v2 Ψ(r) E)δ( r v L)d 3 rd 3 v
14 Comparison: f(e, L) β r/r s f(e,l=const.) L=. L s L=. L s L=. L s E/E max (L) f(e,l=const.) L=. L s L=. L s L=. L s E/E max (L) f(e,l=const.) L=. L s L=. L s L=. L s E/E max (L)
15 Comparison: σ r (r) β σ r /V s r/r s r/r s σ r /V s.3 5 σ r /V s r/r s r/r s
16 Comparison: κ r = v 4 r /σ 4 r β κ r r/r s r/r s κ r 4. κ r r/r s r/r s
17 Projected phase-space density
18 Projected phase-space density f los (R, v los ) = 2πR zmax z max dz E> dv R dv φ f(r, v) V s (GM s /r s ) /2 r s v los /V s R/r s
19 Different ρ DM (r) and n gal (r) r s,dm /r s,gal = 5 r s,dm /r s,gal = / v los /V s v los /V s R/r s,gal R/r s,gal V s GM s,dm /r s,dm r s,gal Σ gal (R) r s,dm f los (R, v los ) = const.
20 f los (R, v los ) β v los /V s v los /V s R/r s R/r s -.5 β = β = β = β = 2
21 f los (R, v los ) β v los /V s v los /V s R/r s R/r s -. β =.5 β =.5 β = β =
22 Towards data analysis
23 Bayesian data analysis f los (R, v los ) = 2πR zmax z max dz E> dv R dv φ f(r, v) p post p prior L L = n i= f los(r i, v los,i {M s, r s,...}) v los [ 3 km/s] R [Mpc] (R i,v los,i )
24 Parameters n gal (r) ρ DM (r) ρ DM (r) /((r/r s )( + r/r s ) 2 ) Free parameters M s and r s M v and c ln( β ) and ln( β ) Fixed parameter L =.98L s r tran r s
25 Test on mock kinematic data
26 Tests on simulated data Motivation equilibrium finite size of the virial sphere infall zone substructures projection effects spherical symmetry Mock kinematic data relaxed haloes of the mass 4 5 M tracer: particles 3 particles per cluster
27 Mock kinematic data v los [ 3 km/s] R [Mpc]
28 Constraints on M v and c M v [ 4 M ] x y z c x y z 2 res cluster number res cluster number
29 c M v relation 2 c global formation history t form < t form < t form galaxies groups clusters M v [h - M ] local formation history
30 Recovering c M v relation 2 2 x z 6 6 c 4 c M v [ 4 M ] M v [ 4 M ] 2 y 6 c M v [ 4 M ]
31 Constraints on β and β 5rs (σ r /σ θ ) (σ r /σ θ ) 5rs cluster number cluster number x y z x y z β β 5rs
32 β and β : need for more data points (σ r /σ θ ) β (σ r /σ θ ) flat profiles n=3 n=3 n=9 β
33 β and β : hierarchical modelling cluster : p (r s,, M s ) cluster 2: p 2 (r s,2, M s,2 ) cluster N: p N (r s,n, M s,n ) p post (β, β, r s,, M s,,...) p prior L L = N n i f los (R j,i, v los,j,i {β, β, r s,i, M s,i }) i= j= p prior (r s,, M s,,...) = N p i (r s,i, M s,i ) i=
34 Constraints on β(r) from clusters (σ r /σ θ ) β (σ r /σ θ ) β(r)=const. β (σ r /σ θ ) β (σ r /σ θ ) β(r)=const. β σ r /σ θ β r/r s r/r s
35 Application to nearby galaxy clusters
36 Nearby (z <.) galaxy clusters Publically available database NASA/IPAC Extragalactic Database WINGS Cluster selection: symmetric X-ray image cool core regular velocity diagrams Final sample 4 clusters 2 redshifts per cluster within R < 2.5M pc
37 Constraints on c M v relation WMAP5 σ 8 =.796 c M v [h - M ]
38 Slope and normalization of c M α α σ 8 =.796 (WMAP5) σ 8 =.9 (WMAP) c(5 4 M )
39 Constraints on the anisotropy profile (σ r /σ θ ) β (σ r /σ θ ) flat profiles β. 4 clusters 577 redshifts β r/r s
40 Comparison with the simulations. β r/r s
41 n gal (r) ρ DM (r)? Σ [N(<r s )/r s 2 ] R/r s
42 n gal (r) ρ DM (r)? log L r s,dm /r s,gal
43 Summary model of distribution function tests on mock data application to nearby clusters constraints on c M v relation β, β(r v ) Perspective for future SDSS satellites ellipticals groups
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