Why clusters? Large: WHY CLUSTERS? 2 dynamical time-scale comparable to the age of the universe! retains the cosmological initial condition Multi-band

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1 The universe traced by clusters Yasushi Suto Department of of Physics Research Center for the Early Universe The University of Tokyo, Japan July 5, Institut d'astrophysique de Paris Contents Why clusters? 2 2 An incomplete list: cosmology with clusters 3 3 SZ effect as a distance indicator 4 3. Sunyaev-Zel'dovich effect Determining the distance and peculiar velocity Reliabilities of the estimated H 0 and v pec Cluster abundances 9 4. Predicting cluster abundances Breaking the degeneracy Spatial correlation 7 6 Universal density profile Numerical results Observational confrontation Relation to nonlinear mass power spectrum Summary and conclusions 32

2 Why clusters? Large: WHY CLUSTERS? 2 dynamical time-scale comparable to the age of the universe! retains the cosmological initial condition Multi-band: current and future observations in various bands: optical survey: APM, 2dF, SDSS X-ray: Einstein, ROSAT, ASCA XMM-Newton, Chandra SZ in cm, mm & submm: BIMA, OVRO, SCUBA, PLANCK, LMSA Gravitational lensing: HST, Subaru, Simple: dark matter + gas + galaxies! theoretically well-defined and relatively easy to describe compared with galaxies Bright: important cosmological probe of high-z universe

3 2 AN INCOMPLETE LIST: COSMOLOGY WITH CLUSTERS3 2 An incomplete list: cosmology with clusters distance indicator: H 0, Ω 0, 0 Sunyaev & Zel'dovich (972), Silk & White (978), Inagaki, Suginohara + YS (996), Kobayashi, Sasaki + YS (996), Birkinshaw (999) peculiar velocity field: v pec Sunyaev & Zel'dovich (972), Rephaeli & Lahav (99), Holzapfel et al. (997), Yoshikawa, Itoh + YS (998) mass fluctuation amplitude:ff 8, Ω 0 Henry & Arnaud (99), Blanchard & Silk (99), White, Efstathiou & Frenk (993), Eke, Cole & Frenk (996), Viana & Liddle (996), Barbosa, Bartlett & Blanchard (996), Kitayama + YS (997) spatial clustering and its evolution: ο(r;z), b(r;z) Bahcall & Soneira (983), Klypin & Kopylov (983), Bahcall (988), Bahcall & West (992), Bahcall & Cen (993), Ueda, Itoh + YS (993), Watanabe, Matsubara + YS (994), Borgani et al. (999), Moscardini et al. (999), YS, Yamamoto, Kitayama & Jing (2000) baryon fraction and dark matter: Ω b, Ω 0 White & Frenk (99), Fabian (99), Makino + YS (993), White et al. (993) CMB anisotropy through the SZ effect: ffit=t Cole & Kaiser (988), Makino + YS (993), Komatsu & Kitayama (999) universal density profile/nonlinear clustering: ρ(r), P (k) Navarro, Frenk, & White (996, 997), Fukushige & Makino (997), Moore et al. (998), Jing + YS (2000), Seljak (2000), Ma & Fry (2000)

4 3 SZ EFFECT AS A DISTANCE INDICATOR 4 3 SZ effect as a distance indicator 3. Sunyaev-Zel'dovich effect Inverse Compton scattering of the CMB photon by the high temperature electron gas in clusters x 4 e x 0 x Iν thermalsz = y (e x coth ) A x hν kt ; y = Z cl (l)n e (l) ff kt CMB m e c 2 T dl Negative source at > :38mm (ν < 28GHz) Positive source in submm band Figure : Spectral feature of the thermal Sunyaev- Zel'dovich flux. Independent of sources nor z.

5 3 SZ EFFECT AS A DISTANCE INDICATOR Determining the distance and peculiar velocity? the angular diameter distance (H 0, Ω 0 & 0 ) SZ temperature decrement: T SZ / R cl T cl n e X-ray surface brightness: S X / R cl T =2 cl n 2 e ) R cl / ( T SZ ) 2 =S X The proportional constant in the above expression can be specified if one neglects clumping, and adopt a spherical isothermal fi-model with the observed X-ray temperature.! d A (z) = R cl = cl = cz=h 0 [ + O(z;Ω 0 ; 0 )] ) H 0, Ω 0 & 0 (e.g., Sunyaev & Zel'dovich 972; Silk & White 978; Birkinshaw, Hughes & Arnaud 99; Rephaeli 995; Kobayashi, Sasaki + YS 996; Hughes & Birkinshaw 998; Birkinshaw 999)? peculiar velocity of clusters Thermal and kinematic SZ fluxes have different frequency dependence ) cluster peculiar velocity (e.g., Sunyaev & Zel'dovich 980; Rephaeli & Lahav 99; Holzapfel et al. 997)

6 3 SZ EFFECT AS A DISTANCE INDICATOR 6? the Hubble diagram from the SZ effect reasonable but not accurate enough (yet): 0:4 < ο h < ο 0:8 000 A478 A228 d A (Mpc) A242 A263 CL006+6 A Coma h Ω 0 λ empty beam q 0 =0.0. redshift z Figure 2: The angular diameter distances from the SZ effect. Kobayashi, Sasaki + YS PASJ 48(996)L07

7 3 SZ EFFECT AS A DISTANCE INDICATOR Reliabilities of the estimated H 0 and v pec Departure fromthe isothermal fi-model (non-sphericity, substructure, non-isothermality, etc.) should produce different projection effects: + Examine the reliabilities of the estimates of H 0 and peculiar velocities using numerical simulations (Inagaki, Suginohara + YS 995; Yoshikawa, Itoh + YS 998) Number of clusters Number of clusters (a) z=.0 f H,2D = (b) z=0.05 f H,2D = f H,2D =r c,x,true /r c,x,est (c) z=.0 f H,3D = (d) z=0.05 f H,3D = f H,3D =r c,ne,true /r c,ne,est (e) z=.0 f v = (f) z=0.05 f v = f v =v r,est /v r,true Figure 3: Distribution of the estimated H 0 and v from 9 simulated clusters at z = 0:05 and z = :0 viewed from three different lineof-sight directions. Different patterns of the histogram correspond to different clusters. (Yoshikawa, Itoh + YS 998)

8 3 SZ EFFECT AS A DISTANCE INDICATOR 8? Inhomogeneous structure of RXJ 347 (z = 0:45) Cluster structure may not be so simple as you wish! (a) RX J / H22@2GHz (b) RX J / H22 vs ROSAT :43 :43 T(0)=.6mK Declination 45 Declination BEAM 47.55mJy central source subtracted 3:47: Right ascension 3:47: Right ascension (c) RX J / NOBA@50GHz (d) RX J / NOBA vs ROSAT 44:30 44:30 Declination :45:00 Declination :45: BEAM 46: :47:30 28 Right ascension 3.8mJy central source subtracted 46: :47:30 28 Right ascension mjy/beam Figure 4: The SZ maps of RX J (a) the 2 GHz map: (2:4 h 50 Mpc 2:4 h 50 Mpc). (b) the 2 GHz map after subtracting the central point source. (c) the 50 GHz map: (0:75 h 50 Mpc 0:75 h 50 Mpc). (d) the 50 GHz map after subtracting the central point source (assuming the flux of 3.8 mjy) overlaid with the X-ray contours. (Komatsu et al. 2000)

9 4 Cluster abundances 4 CLUSTER ABUNDANCES 9 X-ray Temperature function: Henry & Arnaud (99), White, Efstathiou, & Frenk (993), Kitayama + YS (996), Viana & Liddle (996), Eke, Cole, & Frenk (996) X-ray luminosity function and log N - log S: Evrard & Henry (99), Blanchard & Silk (99), Barbosa, Bartlett & Blanchard (996), Ebeling et al. (997), Rosati & Della Ceca (997), Oukbir, Bartlett, & Blanchard (996), Kitayama + YS (997) Mass function Bahcall & Cen (993), Ueda, Itoh + YS (993) Velocity function: Shimasaku (993), Ueda, Shimasaku, Suginohara + YS (994)? crucial assumption one-to-one correspondence between a virialized halo and an X-ray cluster. The former is defined through a spherical collapse model a la Press-Schechter, or selected from simulations with friend-of-friend or spherical overdensity algorithms. a reasonable working hypothesis, but needs justification if adopted for quantitative discussion

10 4 CLUSTER ABUNDANCES 0 4. Predicting cluster abundances model for halo mass function: n M (M;z) Press & Schechter (974), Sheth & Tormen (999), Sheth, Mo & Tormen (999), Jenkins et al. (2000) halo mass! gas temperature kt (z) = fl μm pgm 3r vir (M;z) gas temperature! luminosity L bol (z) = L 44 0 T (z) 6keV C A ff (fl = :2) ( + z) 0 44 h 2 erg sec (L 44 = 2:9, ff = 3:4, and = 0; David et al. 993) band correction: f(t;e a ;E b ) L band = L bol (z) f[t;e a ( + z);e b ( + z)] Observed band-limited X-ray flux S 0 [E a ;E b ] = L band[e a ( + z);e b ( + z)] 4ßd 2 L(z) Number of clusters per unit solid angle fi fi N(> S) = Z dz d 2 0 A (z)c fi fi fi dt fi fi fi fi fi fi fi fi fi dz Z S ds 0 ( + z) 3 n M (M;z) (dm=dt )(dt=dl band )(dl band =ds 0 )

11 4 CLUSTER ABUNDANCES? X-ray Log N - Log S in CDM models some models are consistent with observations. strong Ω 0 and ff 8 dependence. Figure 5: (a) ff 8 = :04, (b) Ω 0 = and Ω 0 = 0:45 models. (Kitayama + YS 997)

12 4 CLUSTER ABUNDANCES 2? Limits on Ω 0 and ff 8 in CDM models ff 8 ο 0:55Ω 0:55 0. Combined with COBE, (Ω 0 ; 0 ;ff 8 ) = (0:3; 0:7; :0), (0:45; 0:0; 0:85) Figure 6: n =, h = 0:7 with (a) 0 = Ω 0, and (b) 0 = 0. X-ray cluster Log N - Log S (solid) and XTF (dotted) are plotted as contours at ff(68%), 2ff(95%) and 3ff(99.7%) confidence levels. (Kitayama + YS 997)

13 4 CLUSTER ABUNDANCES 3? Systematic effects ( 0 = Ω 0 CDM models) Model uncertainties yield ±20% systematic errors in ff Figure 7: (a) L 44, (b) ff, (c), (d) fl, (e) s, and (f) h. Except for the parameters varied in each panel, our canonical set of parameters (L 44 = 2:9, ff = 3:4, = 0, fl = :2, h = 0:7). Dotted and dashed lines represent our best-fit for the canonical parameter set and the COBE 4 year results, respectively. (Kitayama + YS 997)

14 4 CLUSTER ABUNDANCES Breaking the degeneracy Many cosmological models are known to be more or less successful in reproducing the structure at redshift z ο 0 by construction. + This is because the models have still several degrees of freedom or cosmological parameters which can be appropriately adjusted to the observations at z ο 0 (Ω 0, ff 8, h, 0, b(r;z)). How to break the degeneracy among the viable models? + Wider: increase the statistics from wide-field surveys (SDSS, PLANCK,...) Deeper: observe clusters at higher redshifts Different bands: mm and submm bands in addition to the optical and X-ray bands. (e.g., Eke, Cole, & Frenk 996; Barbosa, Bartlett, Blanchard, & Oukbir 996; Fan, Bahcall, & Cen 997; Kitayama, Sasaki + YS 998)

15 4 CLUSTER ABUNDANCES 5? Breaking the degeneracy of number counts Figure 8: (a) soft X-ray ( kev), (b) hard X-ray (2-0 kev), and (c) submm (0.85 mm) bands. Upper: Different bands, Lower: different redshifts. Kitayama, Sasaki + YS (998)

16 4 CLUSTER ABUNDANCES 6? Predicted contours.6.4 hard X-ray COBE ROSAT 0.6 submm Figure 9: Shaded regions represent the ff significance contours derived in KS97 from the soft X-ray (0.5-2 kev) Log N - Log S. Dotted and solid lines indicate the predicted contours of the number of clusters in the hard X- ray (2-0 kev) band at S = 0 3 erg cm 2 s in the submm (0.85 mm) band at S ν = 0 2 mjy. Kitayama, Sasaki + YS (998)

17 5 Spatial correlation 5 SPATIAL CORRELATION 7 Strong spatial clustering of clusters relative to galaxies (Bahcall & Soneira 983; Klypin & Kopylov 983)! inspired the idea of biasing (Kaiser 984)! regularity in the clustering amplitude (Bahcall 988; Bahcall & West 992)? Two-point correlation functions of X-ray selected clusters (Borgani et al. 999; Moscardini et al. 2000; YS, Yamamoto, Kitayama, & Jing 2000) Two-point correlation functions of clusters on the lightcone brighter than the X-ray flux-limit S lim ο LC X cl (R; > S lim) = Z z min z max dz dv c dz n2 0 (z)οs cl (R; z(r); > S lim) Z z min z max dz dv c dz n2 0 (z) bias parameter b(z;m) analytic model (Mo & White 996) + N-body simulations (Jing 998) selection function halo mass function from the Press- Schechter theory + L X (M;z), T X (M;z), S X (M;z) redshift-space distortion Kaiser (987), Peacock & Dodds (996), Magira, Jing + YS (2000) cosmological light-cone effect Yamamoto + YS (999), Yamamoto, Nishioka + YS (999), Hamana, Colombi + YS (2000)

18 5 SPATIAL CORRELATION 8? Model predictions for X-ray cluster correlation functions with redshift-space distortion and light-cone effects z Figure 0: (YS, Yamamoto, Kitayama & Jing 2000) z z

19 5 SPATIAL CORRELATION 9? The redshift-distortion and light-cone effects on the two-point correlation functions in CDM SCDM SCDM h=0.5 h=0.5 SCDM SCDM h=0.5 h= Figure : Upper: 0 < z < 0:4, Lower: 0 < z < 2:0. Left: without selection functions, Lower: with selection functions. (Hamana, Colombi + YS 2000)

20 5 SPATIAL CORRELATION 20? Ω 0 -dependence of X-ray cluster correlation lengths Figure 2: The X-ray flux-limit S lim is 0 3 (solid lines), 0 4 (dotted) and 0 5 erg/s/cm 2 (dashed). For each S lim, we plot the case of 0 = Ω 0 in thick lines, and 0 = 0 in thin lines. Fluctuation amplitudes are normalized by the cluster abundance. (YS, Yamamoto, Kitayama & Jing 2000)

21 5 SPATIAL CORRELATION 2? Nonlinear and stochastic biasing of halos In reality, halo biasing may be very complicated Figure 3: 0 h Mfi < M < 0 3 h Mfi in LCDM model. Upper-left: z = 0; Upper-right: z = ; Lowerleft: z = 2; Lower-right: z = 3. Taruya + YS (2000)

22 6 UNIVERSAL DENSITY PROFILE 22 6 Universal density profile Original question: Are the observed flat rotation curves of galaxies (ρ / r 2 ) consistent with dark matter halo profile in the hierarchical clustering picture? Hoffman & Shaham (985) predicted ρ / r 3(3+n)=(4+n) in the outer region on the basis of the peak + secondary infall argument (n d ln P (k)=d ln k). Quinn, Salmon & Zurek (986) confirmed the Hoffman & Shaham profile for scale-free models using N-body simulations. Frenk, White, Davis & Efstathiou (988) reproduced the flat rotation curves from N-body simulations in CDM models. Suginohara + YS (992) showed that the profile is sensitive to the group-finding algorithm, and it is clear if CDM models account for the flat rotation curves. Navarro, Frenk & White (996, 997) claimed the universal density profile: ρ NFW (r) / (r=r s ) ( + r=r s ) 2 from higher resolution simulations. Fukushige & Makino (997) claimed that the inner profile is r :5ο 2 using GRAPE rather than r. Evans & Collett (997) found a steady-state, self-consistent cusped solution to the collisional Boltzmann equation corresponding to ρ / r 4=3. Syer& White (998) predicted ρ / r 3(3+n)=(5+n) analytically. Moore et al. (998) confirmed that the inner profile is steeper and the universal profile should be ρ / (r=r s ) :5 [+(r=r s ) :5 ] Jing + YS (2000) found the mass-dependence of the profile; the profile is not strictly universal.

23 6 UNIVERSAL DENSITY PROFILE Numerical results? Universal density profile of dark matter halo (Navarro, Frenk, & White 997, ApJ, 490, 493) Figure 4: Density profiles of one of the most and one of the least massive halos in each series. The arrows indicate the value of the gravitational softening.

24 6 UNIVERSAL DENSITY PROFILE 24? Higher resolution simulations using GRAPE (Fukushige & Makino 997, ApJL, 477, L9) Figure 5: Density and velocity dispersion profiles.

25 6 UNIVERSAL DENSITY PROFILE 25? The profiles of simulations are converged? (Moore et al. 998, ApJL, 499, L5) Figure 6: Dependence on the number of simulation particles and on the adopted gravitational softening length.

26 6 UNIVERSAL DENSITY PROFILE 26? Projected snapshots of simulated dark matter halos. (Jing + YS 2000, ApJL, 529, L69) Figure 7: Snapshots of the simulated halos at z = 0. Left, middle and right panels display the halos of galaxy, group and cluster masses, respectively (see Table ). The size of each panel corresponds to 2r vir of each halo.

27 6 UNIVERSAL DENSITY PROFILE 27? Profiles of simulated dark matter halos. (Jing + YS 2000, ApJL, 529, L69) Figure 8: Spherically-averaged radial density profiles of the simulated halos of galaxy (left), group (middle), and cluster (right) masses. The solid and dotted curves represent fits of fi =:5 andfi = respectively? Inner slope vs. halo mass Figure 9: Left panel: the concentration parameters for each halo for the fi = :5 form (filled circles) and for the NFW form (crosses). Right panel: Power-law index of the inner region (0:007 <r=r 200 < 0:02) as a function of the halo mass. The upper and lower dotted curves indicate the predictions of Hoffman & Shaham (985) and Syer & White (996), respectively.

28 6 UNIVERSAL DENSITY PROFILE Observational confrontation? Gas density profile from the universal dark halo The analytic solution for gas density (NFW mass profile + isothermal) turns out to be very well approximated by the conventional isothermal fi-model: ρ(r) / [ + (r=r c ) 2 ] 3fi=2 with fi = 0:9(8ßGμm p ffi c ρ c0 r 2 s=27kt) and r c = 0:22r s. Figure 20: Density profiles of gas (solid lines), the universal dark matter halo (dashed line), and The best-fit fi-models with fi = 0:9b and r c = 0:22r s (dotted lines). (Makino, Sasaki +YS, ApJ, 998, 497, 555)

29 6 UNIVERSAL DENSITY PROFILE 29? Predicted core radii from the universal dark halo. much smaller than the observed values for X-ray clusters 00 (a) (b).5 (c) Einstein Exosat Figure 2: Solid, dotted and dashed lines indicate SCDM (Ω 0 =, 0 = 0, = 0:5, ff 8 = :2), OCDM (Ω 0 = 0:3, 0 = 0, = 0:25, ff 8 = :0), and LCDM (Ω 0 = 0:3, 0 = 0:7, = 0:2, ff 8 = :0). (Makino, Sasaki +YS 998)

30 6 UNIVERSAL DENSITY PROFILE 30? c M vir, ffi c M vir relations from 63 ROSAT clusters. CDM + NFW predictions differ from X-ray data Figure 22: Ω M = 0:3 and Ω Λ = 0:7 are assumed. The MME and EF clusters are represented by the filled and open circles, respectively. (Wu & Xue, ApJL 2000, 529, L5)

31 6 UNIVERSAL DENSITY PROFILE Relation to nonlinear mass power spectrum? Recovering P DM (k) from universal profiles + halo bias Figure 23: Ma & Fry (2000); see also Seljak (2000)

32 7 SUMMARY AND CONCLUSIONS 32 7 Summary and conclusions SZ effect as a distance indicator: ffl reasonably useful, but not quite ffl more realistic (non-spherical, inhomogeneous :::) effects should be included (! talks by J.Bartlett, J.Carlstrom, Y.Rephaeli :::) Cluster abundances: ffl very successful constraints on ff 8 Ω 0 ffl will be significantly improved with multi-band and/or high-z observational data ffl Press-Schechter like objects = objects identified in simulations = observed optical/x-ray clusters are not guaranteed! (! talks by A.Blanchard, S.Borgani, M.Dickinson :::) Spatial correlation: ffl important clues to clustering on large scales ffl more reliable models for halo biasing, light-cone and distortion effects are essential (! talks by H.Böhringer, P.Schuecker, :::) Universal density profile: ffl surprising regularity in the density profile of virialized objects ffl another route to understanding nonlinear gravitational clustering of dark matter ffl inconsistent with observations?! beyond the universal density profile? or! beyond CDM (WDM, SIDM, :::)? (! talks by B.Moore, A.Evrard, :::)