Non linearities in the harmonic spectrum of heavy ion collisions with ideal and viscous hydrodynamics (1206.1905v2)

Abstract: We determine the non-linear hydrodynamic response to geometrical fluctuations in heavy ion collisions using ideal and viscous hydrodynamics. This response is characterized with a set of non-linear response coefficients that determine, for example, the $v_5$ that is produced by an $\epsilon_2$ and an $\epsilon_3$. We analyze how viscosity damps both the linear and non-linear response coefficients, and provide an analytical estimate that qualitatively explains most of the trends observed in more complete simulations. Subsequently, we use these nonlinear response coefficients to determine the linear and non-linear contributions to $v_1$, $v_4$ and $v_5$. For viscous hydrodynamics the nonlinear contribution is dominant for $v_4$, $v_5$ and higher harmonics. For $v_1$, the nonlinear response constitutes an important $\sim 25%$ correction in mid-central collisions. The nonlinear response is also analyzed as a function of transverse momentum for $v_1$, $v_4$ and $v_5$. Finally, recent measurements of correlations between event-planes of different harmonic orders are discussed in the context of non-linear response.

Overview of Non-linear Response in Heavy Ion Collisions Using Hydrodynamics

The research conducted by D. Teaney and L. Yan systematically characterizes the non-linear hydrodynamic response to geometrical fluctuations in heavy ion collisions, using both ideal and viscous hydrodynamics frameworks. This paper dives into the dynamic complexities of the Quark Gluon Plasma (QGP) formed in such collisions and inspects the momentum response using non-linear coefficients. Through rigorous simulations, it contributes a detailed analysis on how viscosity affects linear and non-linear response coefficients, providing an analytical approximation that aligns with more extensive simulations.

The paper investigates the contribution of non-linear interactions to various harmonic flows such as $v_1$, $v_4$, and $v_5$. A remarkable 25% non-linear correction to $v_1$ is found in mid-central collisions, while the non-linear contributions are dominant for $v_4$, $v_5$, and higher harmonics in viscous hydrodynamics. This highlights the essential role of non-linear interactions, especially in peripheral collisions.

Key Findings and Methodological Approach

1. Non-linear Response Coefficients: By introducing and defining a set of non-linear response coefficients, the paper models the hydrodynamic behavior and examines the effects of non-linearities in heavy ion collision scenarios. These coefficients allow the analysis of various flow harmonics and their interplay with geometrical deformations, such as $\epsilon_2$ and $\epsilon_3$.
2. Analytical and Numerical Methods: The researchers develop an analytical estimate alongside comprehensive simulations (ideal and viscous hydrodynamic models) to investigate how viscosity influences both linear and non-linear response phenomena. This approach helps elucidate the relationship between geometrical fluctuations and the resulting flow harmonics.
3. Implications of Viscosity: The investigation uncovers how viscous effects disproportionately impact linear harmonics compared to non-linear ones. Viscous damping exhibits a pattern influenced by harmonic order and spatial moments, particularly affecting $v_4$ and $v_5$ more than $v_1$. This differentiation indicates that non-linear contributions become more prominent as viscosity rises.
4. Transverse Momentum Analysis: The paper thoroughly examines the transverse momentum dependence of the non-linear response for $v_1$, $v_4$, and $v_5$. Results demonstrate notable scaling behavior and the quadratic nature of higher momentum responses in both ideal and viscous hydrodynamics.

Practical and Theoretical Implications

The use of non-linear response coefficients adds an informative layer to the characterization of heavy ion collisions, particularly in estimating the QGP transport properties. The paper's findings suggest measurable differences in harmonic flow responses due to non-linear interactions, providing key insights into the distribution and correlation of harmonic orders.

Specifically, understanding the geometric and non-linear contributions to flow patterns aids in constraining the shear viscosity to entropy density ratio ($\eta/s$) of the QGP. This could advance the calibration of hydrodynamic models against experimental observations, enabling more precise predictions and interpretations.

Speculations on Future Developments

Future work could enhance the methodological framework to include higher-order non-linear interactions and explore alternative initial state models like Color Glass Condensate. As more experimental data becomes available, particularly from upcoming detector capabilities, additional pressure gradients and non-linear effects can be mapped and integrated into hydrodynamic simulations to refine QGP characterization.

Overall, this paper contributes a substantial theoretical foundation, offering a nuanced exploration of the non-linear phenomena in heavy ion collisions and inviting further studies to elucidate the complex dynamics of QGP transport properties under varied conditions.