Mapping the hydrodynamic response to the initial geometry in heavy-ion collisions F ERNANDO G. G ARDIM , Universidade de São Paulo based on arXiv:1111.6538 with Frédérique Grassi, Matt Luzum and Jean-Yves Ollitrualt August 17, 2012 F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Outline Motivation 1 The Almond Shape and Elliptic Flow Smooth & Realistic Initial Conditions Mapping the hydrodynamic response 2 How to map? The elliptic flow case; Generalization to higher harmonics Improving the predictor Conclusion 3 F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Motivation The azimuthal distribution of outgoing particles in a hydro event can be written as 2 π dN � = 1 + 2 v n cos [ n ( φ p − Ψ n )] N d φ p n � e in φ p � = v n e − in Ψ n or, equivalently: { . . . } = average in one event The largest source of uncertainty in hydro models is the initial conditions. Anisotropic flow v n and the event plane Ψ n are determined by initial conditions. We need to understand which properties of the initial state determine v n and Ψ n , so as to constrain models of initial conditions from data. F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
The almond shape and v 2 Average Initial Conditions - the smooth case With smooth initial conditions, the participant eccentricity ε 2 is proportional to z B the elliptic flow v 2 , ψ A Φ 2 And the participant plane Φ 2 is aligned with b b O x the event plane Ψ 2 . plano do evento ε 2 e i 2 Φ 2 = −{ r 2 e i 2 φ } . { r 2 } y {· · · } = average over initial density profile 7.1 < b < 10 fm φ But, in real collisions there are fluctuations, dN/d event-by-event. ψ x 2 In event-by-event hydrodynamics are these φ =0 O 120 240 b relations, v 2 ≈ k ε 2 and Ψ 2 ≈ Φ 2 , valid? plano do evento figures by R. Andrade F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Motivation Fluctuations: Event-by-event hydro NeXus: initial condition generator; NeXSPheRIO → SPheRIO: solves the equations of relativistic ideal hydrodynamics. Scatter plot of v 2 versus ε 2 Distribution of Ψ 2 − Φ 2 0.14 1 � 00 � 10 � 0.12 � � � Au � Au, s � 200GeV � 30 � 40 � � � � � Probability � � 2 �� 2 � � � 00 � 10 � 0.10 � � � � � � � � � fluctuations � � � � � � � � � 0.08 � � � � � � � � � � � � � 30 � 40 � � � � � � � 0.5 v 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � almond shape � � � � � � � 0.06 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� 0.04 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0.02 � � � � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � 0 � � � � � � � � � � � � � � � � � � � � � � � 0.00 � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � 0.0 0.2 0.4 0.6 0.8 0 Π � 4 Π � 2 Ε 2 � 2 �� 2 v 2 ≈ k ε 2 and Ψ 2 ≈ Φ 2 ? Reasonable, but not perfect. See also, F.G.G. et al 1110.5658, Petersen et al 1008.0625, Qiu & Heinz 1104.0650 F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Our goal Propose a simple quantitative measure of the correlation between ( v 2 , Ψ 2 ) and ( ε 2 , Φ 2 ) ; Generalization to higher harmonics; Find better scaling laws. F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
How To Characterize The Hydrodynamic Response Previously, the correlation of the flow with the initial geometry was studied through Distribution of Ψ 2 - Φ 2 Scatter plot v 2 versus ε 2 Our Proposal: A GLOBAL ANALYSIS v 2 e i 2 Ψ 2 =k ε 2 e i 2 Φ 2 + 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 �|E 2 |� ( �· · · � ≡ average over events). k = � ε 2 v 2 cos [ 2 (Ψ 2 − Φ 2 )] � / � ε 2 2 � �|E 2 |� = � v 2 2 �− k 2 � ε 2 2 � F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Results: Au+Au at the top RHIC energy Elliptic flow as a response to the almond-shaped overlap area 1 The quality response is given by: 0.9 � Quality � ε 2 k 2 � Quality = � 0.8 � v 2 2 � The closer Quality to 1, the 0.7 0 10 20 30 40 50 60 better the response. � centrality Central collisions: Mid-central collisions: All anisotropies due to fluctuations Elliptic flow is driven by the Quality 81 % almond shape: Quality 95 % ε 2 is a very good predictor of v 2 ! F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Results Generalization to higher harmonics Generalizing ε n ( Petersen et al 1008.0625 ). Natural estimators are: ε n e in Φ n = −{ r n e in φ } v n e in Ψ n =k ε n e in Φ n + E { r n } Teaney & Yan (1010.1876) showed ε n come from a cumulant expansion of the initial density energy 1 n=3 : 0.8 ε 3 is a very good predictor of v 3 . 0.6 v 3 from Ε 3 Quality v 4 from Ε 4 n=4,5 : 0.4 v 5 from Ε 5 Good quality for central collisions 0.2 Then decrease and even become 0 negative: Ψ n and Φ n are anticorrelated. � 0.2 0 10 20 30 40 50 60 Qiu & Heinz, 1104.0650 � centrality F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Finding better estimators The Almond Shape and v 4 With smooth IC, inspired by NeXus IC in the 30 − 40 % centrality bin ε n Φ n Ψ n n v n 2 .4 .069 0 0 4 0 .011 undf 0 odd 0 0 undf undf There is no ε 4 , so where does v 4 come from? v 4 is generated by ε 2 ! Ψ 4 is in the reaction plane, as Ψ 2 . Comparing with NeXSPheRIO (30-40 % ), � v 2 � ≈ . 066 and � v 4 � ≈ . 01 F.G.G. et al 1110.5658. F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Finding better estimators v 4 induced by ε 2 2 in event-by-event 1 0.8 Ε 4 0.6 2 A natural estimator is: Ε 2 Quality 0.4 ε 2 e i 2 Φ 2 � 2 + E 0.2 v 4 e i 4 Ψ 4 =k � 0 (preserves rotational symmetry) � 0.2 0 10 20 30 40 50 60 � centrality For mid-central collisions, where ε 2 is large, the non-linear term is important! This estimator is not as good as previous estimators of v 2 and v 3 . How to improve the estimator? F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
Finding a better estimator Combining both effects? Defining ε 2 e i 2 Φ 2 ) 2 + E v 4 e i 4 Ψ 4 = k ε 4 e i 4 Φ 4 + k ′ � And minimizing �|E| 2 � , with respect to k and k’. Then, the mean-square error is ε 2 e i 2 Φ 2 � 2 | 2 � 2 � − �| k ε 4 e i 4 Φ 4 + k ′ � �|E 2 |� = � v 2 This error is always smaller than with one parameter 2 Ε 4 and Ε 2 1 Ε 4 0.8 2 Ε 2 The combined estimator 0.6 results in an excellent Quality predictor for all centralities! 0.4 0.2 For v 5 , it is also possible to use both, 0 linear and non-linear, terms to obtain � 0.2 the best estimator: ε 5 and ε 2 ε 3 0 10 20 30 40 50 60 preserves rotational symmetry � centrality F ERNANDO G. G ARDIM , Universidade de São Paulo Mapping the hydrodynamic response to the initial geometry in heavy-ion collisio
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