1. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC1 Expansion Rates and Radial Flow Transverse Dynamics at RHIC Brookhaven National Laboratory Friday, March 7, 2003 Department of Physics and Astronomy State University of New York Stony Brook, NY 11794 with support from the Alexander von Humboldt Foundation Peter F. Kolb

2. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC2 Outline Introduction –expansion rate - expansion parameter A simple example Evolution of thermodynamic fields in hydro –central collisions –non-central collisions Experimental observables –particle spectra –elliptic flow Summary

3. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC3 Timescales of Expansion Dynamics microscopic viewmacroscopic viewvs scattering rate ab ~ expansion rate   u  dilution rate   s A macroscopic treatment requires that the scattering rate is larger than macroscopic rates uu T t

4. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC4 Expansion Rate and Dilution Rate in particular for v = 0 for a local dilution with n ~   -  consider conserved charge n (net baryons, entropy, …) continuity equation (i.e. =1 for 1-dim Bjorken, =3 for 3-dim Hubble) In general: v = 0 and n has transverse gradients: n ~ n (R)   - 

5. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC5 A simple Illustration boost invariant source with ‘linear’ radial flow longitudinal + temporal transverse part limit r  0 Typical values at freezeout:  fo ~ 15 fm,  ~ 0.07 fm -1 Tomasik, Wiedemann nucl-th/0207074

6. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC6 Hydrdoynamic Evolution (b=0) Equations of Motion: + Equation of State: + Initial Configuration:   [(e +p)u  u - pg  ] = 0   [s u  ] = 0 here a resonance gas EoS for T crit < 165 MeV with mixed phase and ideal gas EoS above from an optical Glauber calculation  0 = 0.6 fm

7. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC7 Evolution of expansion Parameter Local expansion parameter: s(  ~   -  investigate its time evolution initially: longitudinal 1-dim Bjorken transition to: fully 3-dim Hubble expansion Bjorken-like Hubble-like Region of decoupling

8. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC8 Evolution of radial flow radial flow at fixed r as a function of timeradial flow at fixed time as a function of r + mixed phase obstructs the generation of transverse flow + the transverse flow profile rapidly adopts a linear behavior v r =  r with  ~ 0.07 fm -1

9. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC9 Time evolution of expansion rate expansion rate   u  times  expansion parameter Remember: For N  

10. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC10 Time evolution of non-central collisions (b=7 fm) PFK, J. Sollfrank and U. Heinz, PRC 62 (2000) 054909 spatial excentricity momentum anisotropy evolution of the energy density initial energy density distribution

11. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC11 Particle spectra of central collisions Au+Au @ 200 A GeV Parameters:  0 = 0.6 fm/c s 0 = 110 fm -3 s 0 /n 0 = 250 T crit =T chem =165 MeV T dec =100 MeV Data: PHENIX: n-ex/0209027; STAR: n-ex/0210034; PHOBOS: n-ex/0210037; BRAHMS: n-ex/0212001 Hydro-calcultations including chemical potentials: PFK and R. Rapp, hep-ph/0210222 Particle spectra: hydro vs. data

12. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC12 particle spectra, non-central collisions Au+Au @130 A GeV PHENIX collab., PRL 88 (2002) 242301; STAR collab., PRL 87 (2001) 262302, nucl-ex/0111004 Hydro calc.: U.Heinz, PFK, NPA 702(02)269 Having the parameters fixed in central collisions, particle spectra at non-zero impact parameter are well reproduced up to b ~ 9 fm negative pionsantiprotons

13. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC13 Elliptic flow at RHIC (130): Heinz, PFK, NPA 702(02)269; Huovinen et al. PLB 503(01)58; Teaney et al. PRL 68(01)4783; Hirano, PRC 65(01)011901 Mass, momentum and centrality dependence are well described up to p T ~ 2 GeV and b ~ 7 fm Over 99 % of the emitted particles follow hydro systematics Data: STAR collab., J. Phys. G 28 (2002) 20; PRL 87 (2001) 182301

14. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC14 Summary and Outlook From thermalization to freeze-out the transverse pressure gradients gradually lead to a transition from initial 1-dimensional expansion to a fully 3- dimensional expansion at freeze-out. Locally the system looks isotropic! Is this related to the observed R side ~ R long ? Thermal freeze-out is induced through the expansion rate, other observables (survival of deuterons, observability of ) depend on the dilution rate. These detailed studies might help to resolve what really happens in the late stages of the fireball.

15. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC15 Supplements

16. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC16 (Strongly) Simplified image of a Heavy Ion Collision -1 fm 1 fm 3-6 fm 10-15 fm 0 fm longitudinal flow profile v z = z/t

17. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC17 Relativistic Hydrodynamics Conservation of energy, momentum and baryon number With energy momentum tensorand conserved current Equation of state: - EOS I : ultrarelativistic, ideal gas, p = e/3 - EOS H: interacting resonance gas, p ~ 0.15 e - EOS Q: Maxwell construction of those two: critical temperature T crit = 0.165 MeV bag constant B 1/4 = 0.23 GeV latent heat e lat =1.15 GeV/fm 3 PFK, J. Sollfrank, U. Heinz, Phys. Rev. C 62 (2000) 054909

18. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC18 Initialization of the Fields Central collisions: density of wounded nucleons: density of binary collisions: nuclear thickness function: Non-central collisions: wounded nucleons: binary collisions: PFK, Heinz, Huovinen, Eskola, Tuominen, Nucl. Phys. A 696 (2000) 197

19. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC19 Systematic studies of non-central collisions offer:- varying energy content and maximum energy - different system sizes - broken azimuthal symmetry  additional observables! central temperature and energy density number of participants and binary collisions spatial anisotropy

20. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC20 Scattering rate meets expansion rate Macroscopic timescale: Local exponent of dilution s ~  -  Expansion rate: Microscopic timescale: Scattering rate For a hydrodynamic description Scattering rate > expansion rate Hung and Shuryak PRC 57 (1998) 1891 R=0 Bjorken-like Hubble-like

21. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC21 Transverse geometry of non-central collisions n WN (x,y) ; b = 7 fm Anisotropic distribution of matter in the overlap region leads to anisotropies in the observed final particle spectra (elliptic flow). Strong rescattering is a prerequisite for large signals. Self quenching effect, generated by early pressure, insensitive to later stages.

22. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC22 Elliptic flow requires strong rescattering PFK et al., PLB 500 (2001) 232; D. Molnar and M. Gyulassy, NPA 698 (2002) 379 Cross-sections and/or gluon densities of at least 80 times the perturbative estimates are required to deliver sufficient anisotropies. At larger p T the experimental results (as well as the parton cascade) saturate, indicating insufficient thermalization of the rapidly escaping particles to allow for a hydrodynamic description.

23. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC23 Mass systematics is a flow effect Huovinen, PFK, Heinz, Ruuskanen, Voloshin, PLB 503 (01) 58 Huovinen, PFK, Heinz, NPA 698 (2002) 475 (Blast wave parametrization for non-central collisions) Radial rapidity-field with angular modulation:  ( r,  s ) =  0 (r) +  a (r) cos 2  s Freeze-out on azimuthally symmetric hypersurface of temperature T dec : With  = m T /T cosh  and  = p T /T sinh  Collaps of the radial integration onto shell:  Catches momentum and restmass dependence of elliptic flow T dec = 140 MeV    a = 0.09

24. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC24 Elliptic flow requires rapid thermalization PFK, J. Sollfrank and U. Heinz, PRC 62 (2000) 054909 Free flow for an interval  t changes the initial distribution function f(r,t;p). For massless particles in the transverse plane ( z = 0 ): f(r T,t 0 +  t;p T ) = f (r T -e p  t,t 0 ;p T )  Reduced spatial anisotropy  as v 2   x, the elliptic flow is reduced accordingly. With typical dimensions of non-central collisions, one obtains a reduction of 30 % for  t = 2 fm/c. Collisions of deformed nuclei (e.g. U+U) deliver anisotropic initial conditions even in central collisions. We expect the full signal already at smaller beam energies!

25. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC25 Sensitivity on the Equation of State Teaney, Lauret, Shuryak, nucl-th/0110037PFK and U. Heinz, nucl-ex/0204061 The data favor an equation of state with a soft phase and a latent heat  e between 0.8 and 1.6 GeV/fm 3

26. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC26 Elliptic flow at finite rapidity T. Hirano and K. Tsuda, nucl-th/020868 Boost invariance and thermodynamic concepts seem to be justified over a pseudo-rapidity interval from -1.5 <  Larger rapidities hold pre-equilibrium information (  directed flow!)

27. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC27 Open and untouched issues: Although the momentum space observables are well described by hydrodynamics, the geometry of the freeze-out hypersurface seems to be much different than inferred from HBT-observables (‘HBT-puzzle’, Heinz, PFK, NPA 702 (02)269). Viscous effects? Positive x-t correlations? Further insight can be obtained through - Anisotropic HBT- observations (Heinz, PFK, PLB 542 (02)216) -  reconstruction, a direct probe of freeze-out conditions (PFK, Prakash, nucl-th/0301003(PRC)) - deuterium distribution and deuterium elliptic flow to probe the surface of the very last rescattering (PFK, Shuryak, work in progress)

28. Peter Kolb, BNL, Mar 7, 2003Expansion Rates at RHIC28 Collaborators and Contributors Ulrich Heinz and Josef Sollfrank Pasi Huovinen, Vesa Ruuskanen Kimmo Tuominen, Kari Eskola Sergei Voloshin, Henning Heiselberg Ralf Rapp, Prakash, Edward Shuryak