Michael Issah Stony Brook University for the PHENIX Collaboration CIPANP 2006, Puerto Rico

1. What do the scaling characteristics of elliptic flow reveal about the properties of the matter at RHIC ? Michael Issah Stony Brook University for the PHENIX Collaboration CIPANP 2006, Puerto Rico

2. Outline • Motivation • Eccentricity scaling and thermalization • Equation of state and speed of sound estimation • Dependence of elliptic flow on kinetic energy and degrees of freedom • Summary CIPANP 2006, Puerto Rico

3. coordinate-space-anisotropy  momentum-space-anisotropy y py px x Initial/final conditions, dof, EOS Elliptic Flow Elliptic flow strength determined principally by initial conditions, initial eccentricity and EoS CIPANP 2006, Puerto Rico

4. PRL87, 052301 (2001) peripheral collisions Central collisions Are energy densities high enough? Extrapolation From ET Distributions • High energy densities are achieved, higher than required for phase transition to occur (~ 1 GeV/fm3) thermalization time (t0 ~ 0.2 – 1 fm/c) eBj~ 5 – 15 GeV/fm3 CIPANP 2006, Puerto Rico

5. PRC 72 (05) 014904 200 GeV Au+Au min-bias Hydro by Huovinen et al. hydro tuned to fit central spectra data. Elliptic flowat RHIC PRL 91, 2003 (PHENIX) • Large v2 created at RHIC • Indicates early thermalization • Elliptic flow data agrees well with hydrodynamic predictions • Can one exploit scaling properties of v2 to get insight into the degree of thermalization and test for hydrodynamic behavior ? CIPANP 2006, Puerto Rico

6. Eccentricity scaling as a test for the degree of thermalization • Large v2 indicative of high degree of thermalization of produced matter • Are there other observables showing that the matter is thermalized ? Eccentricity scaled v2 • Ideal hydrodynamics is scale invariant. If the matter behaves hydrodynamically and is thermalized, v2 should be independent of system size • Do we observe such independence in the data? • Data for different colliding systems (Au+Au, Cu+Cu) available to test this CIPANP 2006, Puerto Rico

7. Determination of eccentricity • Eccentricity usually obtained from a Glauber Model • One can also use experimental quantity sensitive to initial eccentricity, like the integrated v2 • Integrated v2 is proportional to the eccentricity STAR Collaboration CIPANP 2006, Puerto Rico

8. Eccentricity scaling Bhalerao, Blaizot, Borghini, Ollitrault , nucl-th/0508009 Eccentricity scaling observed in hydrodynamic model over a broad range of centralities R: measure of size of system CIPANP 2006, Puerto Rico

9. Eccentricity scaling and system size k ~ 3.1 (obtained from data ) PHENIX Preliminary v2 scales with eccentricity and across system size CIPANP 2006, Puerto Rico

10. Energy dependence of elliptic flow • Saturation of azimuthal anisotropy observed at RHIC energies • Suggests the matter has a soft EoS CIPANP 2006, Puerto Rico

11. Estimate of the speed of sound(I) • Energy dependence at RHIC energies seem to indicate a soft equation of state. How soft ? • Estimate of cs from elliptic flow measurementscan be made from eccentricity scaled v2 Bhalerao, Blaizot, Borghini, Ollitrault , nucl-th/0508009 CIPANP 2006, Puerto Rico

12. Estimation of the speed of sound (II) Assume simple EoS: relation between pressure and energy density v2/ε for <pT> ~ 0.45 GeV/c cs ~ 0.35 ±0.05 (cs2 ~ 0.12), soft EOS F. Karsch, hep-lat/0601013 CIPANP 2006, Puerto Rico

13. Scaling v2 with kinetic energy (I) Scaling breaks See H. Masui’s talk in this session Baryons scale together Mesons scale together PHENIX preliminary Kinetic energy of a particle in a relativistic fluid: = mT – m • Elliptic flow scales with KET up to KET ~1 GeV • Indicates hydrodynamic behavior • Possible hint of quark degrees of freedom become apparent at higher KET CIPANP 2006, Puerto Rico

14. Scaling v2 with kinetic energy (II) PHENIX preliminary Transverse kinetic energy scaling works for a large selection of particles CIPANP 2006, Puerto Rico

15. STAR preliminary 200 GeV Au+Au Test for partonic degrees of freedom PHENIX preliminary • Constituent quark scaling works better with KET as compared to pT • Hadron mass (hydro) scaling at low KET (KET < 1 GeV) is not perturbed by constituent quark scaling at higher KET ! CIPANP 2006, Puerto Rico

16. KET /n scaling and hydrodynamics Chiho Nonaka: 3-D Hydro PHENIX preliminary R. Fries, SQM2006 • Scaling works for a broad set of particles • Supports the observation that all particles emerge from a common flow field • Hydrodynamic calculation not incompatible with KET/n scaling CIPANP 2006, Puerto Rico

17. KET/n scaling across collision centralities KET/n scaling observed across centralities CIPANP 2006, Puerto Rico

18. Summary • Scaling properties of elliptic flow provide a way to infer some properties of the matter formed in heavy-ion collisions at RHIC • Eccentricity scaling shows that the medium is highly thermalized, with a soft EoS • Transverse kinetic energy scaling indicates perfect fluid hydrodynamic behavior • In addition, scaling with KET/n leads to comprehensive scaling of elliptic flow data, which prevails over a broad range of centralities • Taken together, these observations show that the matter is highly thermalized, has a soft equation of state and is made up of flowing constituent quarks CIPANP 2006, Puerto Rico