Hydrodynamics, flow, and flow fluctuations

Jean-Yves OllitraultIPhT-Saclay

Hirschegg 2010: Strongly Interacting Matter under Extreme ConditionsInternational Workshop XXXVIII on Gross Properties of Nuclei and Nuclear Excitations

January 19, 2010

In collaboration with Clément Gombeaud and Matt Luzum

Outline

• Elliptic flow and v4

• A simple, universal prediction from ideal hydrodynamics

• Comparison with data from RHIC

• v4 in ideal and viscous hydrodynamics

• Flow fluctuations, their effect on v4

• Conclusion

Gombeaud, JYO, arXiv:0907.4664, Phys. Rev. C81, 014901 (2010)

Luzum, Gombeaud, JYO, in preparation

Initial azimuthal distribution of particles

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

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 flowcreated by pressure gradients in the overlap area

φ

Anisotropic flow

Fourier series expansion of the azimuthal distribution:

Using the φ→-φ and φ→φ+π symmetries of overlap area:

dN/dφ=1+2v2cos(2φ)+2v4cos(4φ)+…

v2=<cos(2φ)> (<…> means average value) is elliptic flow

v4=<cos(4φ)> is a (much smaller) « higher harmonic »

higher harmonics v6, etc are 0 within experimental errors.

This talk is about v4

Azimuthal distribution without v4

The beauty is in the details!

A small effect: Average value 0.3%, maximum value 3%

Should we care?

A primer on hydrodynamics

• Ideal gas (weakly-coupled particles) in global thermal equilibrium. The phase-space distribution is (Boltzmann)

dN/d3pd3x = exp(-E/T) Isotropic!

• A fluid moving with velocity v is in (local) thermal equilibrium in its rest frame:

dN/d3pd3x = exp(-(E-p.v)/T) Not isotropic: Momenta parallel to v preferred

• At RHIC, the fluid velocity depends on φ:

typically v(φ)=v0+2ε cos(2φ)

The simplicity of v4

• Within the approximation that particle momentum p and fluid velocity v are parallel (valid for large momentum pt and low freeze-out temperature T)

dN/dφ=exp(2ε pt cos(2φ)/T)• Expanding to order ε, the cos(2φ) term is

v2=ε pt/T• Expanding to order ε2, the cos(4φ) term is

v4=½ (v2)2

Hydrodynamics has a universal prediction for v4/(v2)2 !Should be independent of equation of state, initial

conditions, centrality, particle momentum and rapidity, particle type

Borghini JYO nucl-th/0506045

PHENIX results for v4

PHENIX data for charged pions

Au-Au collisions at 100+100 GeV

20-60% most central

The ratio is independent of pT, 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?

v4 and coalescence

• The pt range where we have data for v4 is the range where quark coalescence is thought to be important

• Coalescence predicts, with n_q=2 or 3 constituent quarks

(dN/dφ)hadron(pt)=(dN/dφ)qn_q

(pt/n_q)• Our simple hydrodynamical picture

(dN/dφ)=exp(2ε pt cos(2φ)/T)

is stable under quark coalescence

• In particular, v4/(v2)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.

v4 in viscous hydrodynamics

Viscosity changes the fluid evolution and distorts the momentum distribution of particles emitted at freeze-out.

fviscous(p)=e-E/T(1+δf(p)) where

δf=Cχ(p) (pipj/p2-δij/3)∂iuj

and χ(p) depends on the microscopic interactions and C is a normalization fixed by matching with the fluid Tμν

Most calculations use χ(p)=p2 (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

Results from ideal and viscous hydro

M. Luzum, work in progress

Hydro parameters tuned to fit spectra and v2 at RHIC (Luzum and Romatschke)In particular, Tf=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 pt.

Hydro is unable to explain a ratio larger than 0.5. We need something more

More data : centrality dependence

Data > hydroSmall 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

Estimating experimental errors

Difference between STAR and PHENIX data compatible with non-flow error

How do we understand the discrepancy with hydrodynamics??

v2 and v4 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)

Eccentricity scaling

We understand elliptic flow as the consequence of the almond shape of the overlap area

It is therefore natural to expect that v2 scales like the eccentricity ε of the initial density profile, defined as :

(this is confirmed by numerical hydro calculations)

22

22

xy

xy

y

x

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 v2 scales like the eccentricity, eccentricity fluctuations translate into v2 (elliptic flow) fluctuations

We need fluctuations to understand v2 results(see next talk by Raimond Snellings)

Results using various methods(STAR)

After correcting for fluctuations and nonflowJYO Poskanzer Voloshin, PRC 2009

Eccentricity fluctuations in central collisions

Central collisions are azimuthally symmetric, except for fluctuations:

In the most central bin, v2 and v4 are all fluctuations!

Eccentricity fluctuations are to a good approximation Gaussian in the transverse plane (2-dimensional Gaussian distribution). This implies <ε4>=2 <ε2>2

The value of v4 for central collisions rather suggests <(v2)4>~3 <(v2)2>2. It is therefore unlikely that elliptic flow fluctuations are solely due to fluctuations in the initial eccentricity.

We need ideas.

Conclusions

• The fourth harmonic, v4, of the azimuthal distribution gives a further, independent indication that the matter produced at RHIC expands like a relativistic fluid

• v4 is mostly induced by v2 as a second order effect.

• v4 may help us constrain models based on viscous hydrodynamics, in particular viscous corrections at freeze-out : standard quadratic ansatz ruled out?

• v4 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.

Backup slides

More results from viscous hydro

Glauber initial conditions and smaller viscosity also reproduces the measured v2.