Description of PHOBOS data on v 2 ( h ) from s NN = 19.6 to 200 GeV

1. Description of PHOBOS data on v2(h) from sNN = 19.6 to 200 GeV A. Ster1,2, M. Csanád3, T. Csörgő2 1 KFKI-RMKI, 2 KFKI-MFA, 3 ELTE, Hungary • Buda-Lund hydrodynamical model • Buda-Lund ellipsoidal generalization • Buda-Lund fitting to PHOBOS recent data • Conclusion

2. Buda-Lund hydro model • 3D expansion with axial or ellipsoidal symmetry • Local thermal equilibrium • Analytic expressions for the observables (no numerical simulations, but formulas) • Reproduces known exact hydro solutions (nonrelativistic, Hubble, Bjorken limit) • Core-halo picture (long lived resonances)

3. An illustrative analogy • Core  Sun • HaloSolar wind • T0,RHIC  T0,SUN  16 million K • Tsurface,RHIC  Tsurface,SUN  6000 K • RG Geometrical size • t0 Radiation lifetime • <bt>  Radial flow of surface (~0) • DhLongitudinal expansion (~0) Fireball at RHICFireball Sun

4. Buda-Lund: spectra, HBT w/o puzzles J.Phys.G30: S1079-S1082, 2004; nucl-th/0403074

5. Ellipsoidal generalization • Axially symmetric case: RG, ut • Main axes of expanding ellipsoid: X, Y, Z • 3D expansion, 3 expansion rates: X, Y, Z • Introducing velocity space eccentricity • For Hubble type of expansions: X(t)Xt, ... • In this case: e  ev • In addition: DhZ

6. The ellipsoidal Buda-Lund model • The original model was developed for axial symmetry  central collisions • In the most general hydrodynamical form (‘Inspired by’ nonrelativistic solutions): • Assume special shapes: • Generalized Cooper-Frye prefactor: • Four-velocity distribution: • Temperature: • Fugacity: M.Csanád, T.Csörgő, B. Lörstad: Nucl.Phys.A742:80-94,2004; nucl-th/0310040

7. Observables from Buda-Lund hydro • Core-halo correction: • One-particle spectrum with core-halo correction: • Two-particle correlation function: • Flow coefficients:

8. The elliptic flow • One-particle spectrum: • The m-th Fourier component is the m-th flow • Depends on pseudorapidity and transverse momentum • Pseudorapidity dependence not understood in other hydro models (but ref: plenary talk of Hama/SPHERIO)

9. Hydro predicts universal scaling • Universal scaling variable • Universal scalig function: • Elliptic flow depends on every physical parameter (mass, colliding system, bombarding energy, pt, rapidity,centrality) only through universal scaling variable w • Do data collapse to the scaling curve of I1 / I0 ?

10. At large pseudorapidities • Under certain conditions, the even flows are: , where and • Here hs is the space-time rapidity of the saddlepoint • , and so • Rapidity grows  the asymmetry vanishes(saddlepoint goes to the z axis)  elliptic flow vanishes

11. Fits to PHOBOS data PHOBOS data from: Phys. Rev. Lett. 94, 122303 (2005)

12. Fit parameters • Used (non-essential) model parameters: • Fitted parameters: Increasing parameter values with energy

13. Error contours

14. Universal scaling • Data collapsing behavior to theoretically predicted scaling function The perfect fluid extends from very small to very large rapidities at RHIC

15. Conclusions • Buda-Lund model describes v2(h) data @RHIC • The vanishing elliptic flow at large rapidities is due to Hubble flow + finite longitudinal size • v2(h) data (2005) collapse to the theoretically PREDICTED (2003) scaling function of • The perfect fluid is present in AuAu in the whole h space

16. cs2 = 2/3 cs2 = 1/3 Sensitivity to the Equation of State • Different initial conditions, different equation of statebut exactly the same hadronic final state possible. (!!) • This is an exact, analytic result in hydro( !!).

17. Time dependence • Blastwave or Cracow model type of cooling vs Buda-Lund typeof cooling, cs2= 2/3, half freeze-out time (animated) http://csanad.web.elte.hu/phys/3danim/