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Mapping the hydrodynamic response to the initial geometry in - - PowerPoint PPT Presentation

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Mapping the hydrodynamic response to the initial geometry in heavy-ion collisions F ERNANDO G. G ARDIM , Universidade de So Paulo based on arXiv:1111.6538 with Frdrique Grassi, Matt Luzum and Jean-Yves Ollitrualt August 17, 2012 F ERNANDO

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Mapping the hydrodynamic response to the initial geometry in heavy-ion collisions

FERNANDO G. GARDIM, Universidade de São Paulo

based on arXiv:1111.6538 with Frédérique Grassi, Matt Luzum and Jean-Yves Ollitrualt August 17, 2012

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Outline

1

Motivation

The Almond Shape and Elliptic Flow

Smooth & Realistic Initial Conditions

2

Mapping the hydrodynamic response

How to map? The elliptic flow case; Generalization to higher harmonics Improving the predictor

3

Conclusion

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Motivation

The azimuthal distribution of outgoing particles in a hydro event can be written as 2π N dN dφp = 1 + 2

  • n

vn cos[n(φp − Ψn)]

  • r, equivalently:
  • einφp

= vne−inΨn

{. . .} =average in one event

The largest source of uncertainty in hydro models is the initial conditions. Anisotropic flow vn and the event plane Ψn are determined by initial conditions. We need to understand which properties of the initial state determine vn and Ψn, so as to constrain models of initial conditions from data.

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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The almond shape and v2 Average Initial Conditions - the smooth case

With smooth initial conditions, the participant eccentricity ε2 is proportional to the elliptic flow v2, And the participant plane Φ2 is aligned with the event plane Ψ2. ε2ei2Φ2 = −{r 2ei2φ}

{r 2}

.

{· · · } = average over initial density profile

z

Φ

b

x

O A B

b

ψ

2

plano do evento

y dN/d ψ

2

φ

120 240

7.1 < b < 10 fm

O

plano do evento b

φ=0 x

But, in real collisions there are fluctuations, event-by-event. In event-by-event hydrodynamics are these relations, v2 ≈ kε2 and Ψ2 ≈ Φ2, valid?

figures by R. Andrade

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Motivation Fluctuations: Event-by-event hydro

NeXSPheRIO→

NeXus: initial condition generator; SPheRIO: solves the equations of relativistic ideal hydrodynamics.

Scatter plot of v2 versus ε2

  • 0.0

0.2 0.4 0.6 0.8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Ε2 v2

3040 0010

Distribution of Ψ2 − Φ2

  • Π4

Π2 0.5 1 22 Probability22

  • 30 40

almond shape

  • 00 10

fluctuations AuAu, s 200GeV

v2 ≈ kε2 and Ψ2 ≈ Φ2? Reasonable, but not perfect.

See also, F.G.G. et al 1110.5658, Petersen et al 1008.0625, Qiu & Heinz 1104.0650

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Our goal

Propose a simple quantitative measure of the correlation between (v2, Ψ2) and (ε2, Φ2); Generalization to higher harmonics; Find better scaling laws.

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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How To Characterize The Hydrodynamic Response

Previously, the correlation of the flow with the initial geometry was studied through Distribution of Ψ2-Φ2 Scatter plot v2 versus ε2 Our Proposal: A GLOBAL ANALYSIS

v2ei2Ψ2=kε2ei2Φ2 + E

k: It is the same for all events (in each centrality class). E: event-by-event error. The best linear fit is achieved minimizing the mean-square error |E2| (· · · ≡average over events). k= ε2v2 cos[2(Ψ2 − Φ2)]/ε2

2

|E2| = v 2

2 −k2ε2 2

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Results: Au+Au at the top RHIC energy Elliptic flow as a response to the almond-shaped overlap area

The quality response is given by: Quality = k

  • ε2

2

  • v 2

2

The closer Quality to 1, the better the response.

10 20 30 40 50 60 0.7 0.8 0.9 1

centrality Quality

Central collisions: All anisotropies due to fluctuations Quality 81% Mid-central collisions: Elliptic flow is driven by the almond shape: Quality 95%

ε2 is a very good predictor of v2!

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Results Generalization to higher harmonics

Natural estimators are: vneinΨn=kεneinΦn + E Generalizing εn (Petersen et al 1008.0625). εneinΦn = −{r neinφ}

{r n} Teaney&Yan (1010.1876) showed εn come from a cumulant expansion of the initial density energy

10 20 30 40 50 60 0.2 0.2 0.4 0.6 0.8 1 centrality Quality

v5 from Ε5 v4 from Ε4 v3 from Ε3

n=3: ε3 is a very good predictor of v3. n=4,5: Good quality for central collisions Then decrease and even become negative: Ψn and Φn are anticorrelated.

Qiu&Heinz, 1104.0650

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Finding better estimators The Almond Shape and v4

With smooth IC, inspired by NeXus IC in the 30 − 40% centrality bin n εn vn Φn Ψn 2 .4 .069 4 .011 undf

  • dd

undf undf

There is no ε4, so where does v4 come from?

v4 is generated by ε2! Ψ4 is in the reaction plane, as Ψ2.

Comparing with NeXSPheRIO (30-40%), v2 ≈ .066 and v4 ≈ .01 F.G.G. et al 1110.5658.

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Finding better estimators v4 induced by ε2

2 in event-by-event A natural estimator is:

v4ei4Ψ4=k

  • ε2ei2Φ22 + E

(preserves rotational symmetry) 10 20 30 40 50 60 0.2 0.2 0.4 0.6 0.8 1 centrality Quality Ε2

2

Ε4

For mid-central collisions, where ε2 is large, the non-linear term is important! This estimator is not as good as previous estimators of v2 and v3.

How to improve the estimator?

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Finding a better estimator

Combining both effects? Defining v4ei4Ψ4 = kε4ei4Φ4 + k′ ε2ei2Φ2)2 + E And minimizing |E|2, with respect to k and k’. Then, the mean-square error is |E2| = v 2

2 − |kε4ei4Φ4 + k′

ε2ei2Φ22 |2

This error is always smaller than with one parameter 10 20 30 40 50 60 0.2 0.2 0.4 0.6 0.8 1 centrality Quality Ε2

2

Ε4 Ε4and Ε2

2

The combined estimator results in an excellent predictor for all centralities!

For v5, it is also possible to use both, linear and non-linear, terms to obtain the best estimator: ε5 and ε2ε3

preserves rotational symmetry

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio

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Conclusions

We have defined a quantitative measure of the quality of estimators of vn from initial conditions event-by-event hydro; v2 can be understood as a response to the almond-shaped

  • verlap area ε2, even for central collisions;

The triangularity ε3 is a very good predictor to v3; Non-linear terms are necessary to predict v4 (and v5) from initial energy density, for all centralities. (See TeaneyYan 1206.1905) These results provide an improved understanding of the hydro response to the initial state in realistic heavy-ion collisions.

FERNANDO G. GARDIM, Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio