1. Hydrodynamics, flow, and flow fluctuations Jean-Yves Ollitrault IPhT-Saclay Hirschegg 2010: Strongly Interacting Matter under Extreme Conditions International Workshop XXXVIII on Gross Properties of Nuclei and Nuclear Excitations January 19, 2010 In collaboration with Clément Gombeaud and Matt Luzum

2. Outline Elliptic flow and v 4 A simple, universal prediction from ideal hydrodynamics Comparison with data from RHIC v 4 in ideal and viscous hydrodynamics Flow fluctuations, their effect on v 4 Conclusion Gombeaud, JYO, arXiv:0907.4664, Phys. Rev. C81, 014901 (2010) Luzum, Gombeaud, JYO, in preparation

3. Initial azimuthal distribution of particles Random parton-parton collisions occurring on scales << nuclear radius. No preferred direction in the production process. Isotropic azimuthal distribution

4. Final azimuthal distribution of particles Bar length = number of particles in the direction = Azimuthal (φ) distribution plotted in polar coordinates (for pions with transverse momentum ~ 2 GeV/c) Strong elliptic flow created by pressure gradients in the overlap area φ

5. Anisotropic flow Fourier series expansion of the azimuthal distribution: Using the φ→-φ and φ→φ+π symmetries of overlap area: dN/dφ=1+2v 2 cos(2φ)+2v 4 cos(4φ)+… v 2 = ( means average value) is elliptic flow v 4 = is a (much smaller) « higher harmonic » higher harmonics v 6, etc are 0 within experimental errors. This talk is about v 4

6. Azimuthal distribution without v 4 The beauty is in the details! A small effect: Average value 0.3%, maximum value 3% Should we care?

7. A primer on hydrodynamics Ideal gas (weakly-coupled particles) in global thermal equilibrium. The phase-space distribution is (Boltzmann) dN/d 3 pd 3 x = exp(-E/T) Isotropic! A fluid moving with velocity v is in (local) thermal equilibrium in its rest frame: dN/d 3 pd 3 x = exp(-(E-p.v)/T) Not isotropic: Momenta parallel to v preferred At RHIC, the fluid velocity depends on φ: typically v(φ)=v 0 +2ε cos(2φ)

8. The simplicity of v 4 Within the approximation that particle momentum p and fluid velocity v are parallel (valid for large momentum p t and low freeze-out temperature T) dN/dφ=exp(2ε p t cos(2φ)/T) Expanding to order ε, the cos(2φ) term is v 2 =ε p t /T Expanding to order ε 2, the cos(4φ) term is v 4 =½ (v 2 ) 2 Hydrodynamics has a universal prediction for v 4 /(v 2 ) 2 ! Should be independent of equation of state, initial conditions, centrality, particle momentum and rapidity, particle type Borghini JYO nucl-th/0506045

9. PHENIX results for v 4 PHENIX data for charged pions Au-Au collisions at 100+100 GeV 20-60% most central The ratio is independent of p T, as predicted by hydro. The ratio is also independent of particle species within errors. But… the value is significantly larger than 0.5. Can detailed (ideal or viscous) hydro calculations explain this?

10. v 4 and coalescence The p t range where we have data for v 4 is the range where quark coalescence is thought to be important Coalescence predicts, with n_q=2 or 3 constituent quarks (dN/dφ) hadron (p t )=(dN/dφ) q n_q (p t /n_q) Our simple hydrodynamical picture (dN/dφ)=exp(2ε p t cos(2φ)/T) is stable under quark coalescence In particular, v 4 /(v 2 ) 2 is closer to ½ for hadrons than for parent quarks Discrepancy data/hydro is worse for the parent quarks: coalescence does not help here.

11. v 4 in viscous hydrodynamics Viscosity changes the fluid evolution and distorts the momentum distribution of particles emitted at freeze-out. f viscous (p)=e -E/T (1+δf(p)) where δf=Cχ(p) (p i p j /p 2 -δ ij /3)∂ i u j and χ(p) depends on the microscopic interactions and C is a normalization fixed by matching with the fluid T μν Most calculations use χ(p)=p 2 (quadratic ansatz) but it has been recently pointed out that other choices are possible such as χ(p)=p (linear ansatz) Dusling Moore Teaney arXiv:0909.0754

12. Results from ideal and viscous hydro M. Luzum, work in progress Hydro parameters tuned to fit spectra and v 2 at RHIC (Luzum and Romatschke) In particular, T f =140 MeV Ideal hydro predicts a flat ratio as expected Viscous hydro with a linear ansatz is also OK Viscous hydro with (usual) quadratic ansatz fails badly at large p t. Hydro is unable to explain a ratio larger than 0.5. We need something more

13. More data : centrality dependence Data > hydro Small discrepancy between STAR and PHENIX data Au-Au collision per nucleon 100+100 GeV STAR: Yuting Bai, PhD thesis Utrecht PHENIX: Roy Lacey, private communication

14. Estimating experimental errors Difference between STAR and PHENIX data compatible with non-flow error How do we understand the discrepancy with hydrodynamics?? v 2 and v 4 are not measured directly but inferred from azimuthal correlations (more later on this). There are many sources of correlations (jets, resonance decays,…): this is the « nonflow » error which we can estimate (order of magnitude only)

15. Eccentricity scaling We understand elliptic flow as the consequence of the almond shape of the overlap area It is therefore natural to expect that v 2 scales like the eccentricity ε of the initial density profile, defined as : (this is confirmed by numerical hydro calculations) yy xx

16. Eccentricity fluctuations Depending on where the participant nucleons are located within the nucleus at the time of the collision, the actual shape of the overlap area may vary: the orientation and eccentricity of the ellipse defined by participants fluctuates. Assuming that v 2 scales like the eccentricity, eccentricity fluctuations translate into v 2 (elliptic flow) fluctuations

17. We need fluctuations to understand v 2 results (see next talk by Raimond Snellings) Results using various methods (STAR) After correcting for fluctuations and nonflow JYO Poskanzer Voloshin, PRC 2009

20. Eccentricity fluctuations in central collisions Central collisions are azimuthally symmetric, except for fluctuations: In the most central bin, v 2 and v 4 are all fluctuations! Eccentricity fluctuations are to a good approximation Gaussian in the transverse plane (2-dimensional Gaussian distribution). This implies =2 2 The value of v 4 for central collisions rather suggests ~3 2. It is therefore unlikely that elliptic flow fluctuations are solely due to fluctuations in the initial eccentricity. We need ideas.

21. Conclusions The fourth harmonic, v 4, of the azimuthal distribution gives a further, independent indication that the matter produced at RHIC expands like a relativistic fluid v 4 is mostly induced by v 2 as a second order effect. v 4 may help us constrain models based on viscous hydrodynamics, in particular viscous corrections at freeze-out : standard quadratic ansatz ruled out? v 4 is a sensitive probe of elliptic flow fluctuations. The standard model of eccentricity fluctuations fails for central collisions. We need a better understanding of fluctuations.

22. Backup slides

23. More results from viscous hydro Glauber initial conditions and smaller viscosity also reproduces the measured v 2.