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Expansion Rates and Radial Flow

Expansion Rates and Radial Flow

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Expansion Rates and Radial Flow

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

  2. 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 Expansion Rates at RHIC

  3. Timescales of Expansion Dynamics microscopic view vs macroscopic view um T t scattering ratenab ~ expansion ratem um dilution rate t s A macroscopic treatment requires that the scattering rate is larger than macroscopic rates Expansion Rates at RHIC

  4. Expansion Rate and Dilution Rate for a local dilution with n ~ t-a consider conserved charge n (net baryons, entropy, …) continuity equation in particular for v = 0 (i.e. a=1for 1-dim Bjorken,a=3 for 3-dim Hubble) In general: v = 0 and n has transverse gradients: n ~ n (R)t-a Expansion Rates at RHIC

  5. A simple Illustration boost invariantsource with ‘linear’ radial flow longitudinal + temporal transverse part limit r  0 Tomasik, Wiedemann nucl-th/0207074 Typical values at freezeout: tfo ~ 15 fm, x ~ 0.07 fm-1 Expansion Rates at RHIC

  6. Hydrdoynamic Evolution (b=0) Equations of Motion: m[(e +p)um un - pgmn] = 0 m[s um] = 0 + Equation of State: here a resonance gas EoS for Tcrit < 165 MeV with mixed phase and ideal gas EoS above + Initial Configuration: from an optical Glauber calculation t0 = 0.6 fm Expansion Rates at RHIC

  7. Evolution of expansion Parameter Region of decoupling Local expansion parameter: s(t) ~ t-a Hubble-like investigate its time evolution Bjorken-like initially: longitudinal 1-dim Bjorken transition to: fully 3-dim Hubble expansion Expansion Rates at RHIC

  8. Evolution of radial flow radial flow at fixed r as a function of time radial 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 vr =  r with  ~ 0.07 fm-1 Expansion Rates at RHIC

  9. Time evolution of expansion rate N p p p expansion rate m um times t expansion parameter Remember: For Expansion Rates at RHIC

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

  11. Particle spectra of central collisions Au+Au @ 200 A GeV 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 Parameters: 0 = 0.6 fm/c s0 = 110 fm-3 s0/n0 = 250 Tcrit=Tchem=165 MeV Tdec=100 MeV Expansion Rates at RHIC

  12. particle spectra, non-central collisionsAu+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 negative pions antiprotons Having the parameters fixed in central collisions, particle spectra at non-zero impact parameter are well reproduced up to b ~ 9 fm Expansion Rates at RHIC

  13. 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 pT ~ 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 Expansion Rates at RHIC

  14. 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 Rside ~ Rlong ? Thermal freeze-out is induced through the expansion rate, other observables (survival of deuterons, observability of r  p p) depend on the dilution rate. These detailed studies might help to resolve what really happens in the late stages of the fireball. Expansion Rates at RHIC

  15. Supplements Expansion Rates at RHIC

  16. (Strongly) Simplified image of a Heavy Ion Collision 10-15 fm 3-6 fm 1 fm 0 fm -1 fm longitudinal flow profilevz= z/t Expansion Rates at RHIC

  17. Relativistic Hydrodynamics PFK, J. Sollfrank, U. Heinz, Phys. Rev. C 62 (2000) 054909 Conservation of energy, momentum and baryon number With energy momentum tensor and 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 Tcrit= 0.165 MeV • bag constant B1/4 = 0.23 GeV • latent heat elat=1.15 GeV/fm3 Expansion Rates at RHIC

  18. 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 Expansion Rates at RHIC

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

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

  21. Transverse geometry of non-central collisions nWN(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. Expansion Rates at RHIC

  22. 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 pT the experimental results (as well as the parton cascade) saturate, indicating insufficient thermalization of the rapidly escaping particles to allow for a hydrodynamic description. Expansion Rates at RHIC

  23. Mass systematics is a flow effect (Blast wave parametrization for non-central collisions) Huovinen, PFK, Heinz, Ruuskanen, Voloshin, PLB 503 (01) 58 Huovinen, PFK, Heinz, NPA 698 (2002) 475 Radial rapidity-field with angular modulation: ( r, s) = 0(r) + a(r) cos 2s Freeze-out on azimuthally symmetric hypersurface of temperature Tdec: Withb = mT/T cosh randa = pT/T sinh r Collaps of the radial integration onto shell: Tdec= 140 MeV r= 0.58 ra = 0.09  Catches momentum and restmass dependence of elliptic flow Expansion Rates at RHIC

  24. Elliptic flow requires rapidthermalization 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(rT,t0+ t;pT) = f (rT-ep t,t0;pT) • Reduced spatial anisotropy •  as v2  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. -1 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! Expansion Rates at RHIC

  25. Sensitivity on the Equation of State PFK and U. Heinz, nucl-ex/0204061 Teaney, Lauret, Shuryak, nucl-th/0110037 The data favor an equation of state with a soft phase and a latent heat De between 0.8 and 1.6 GeV/fm3 Expansion Rates at RHIC

  26. 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 < h < 1.5 Larger rapidities hold pre-equilibrium information ( directed flow!) Expansion Rates at RHIC

  27. 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) Expansion Rates at RHIC

  28. 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 Expansion Rates at RHIC