Nonlinearities in the harmonic spectrum of heavy ion collisions with ideal and viscous hydrodynamics

Abstract

We determine the nonlinear hydrodynamic response to geometrical fluctuations in heavy ion collisions using ideal and viscous hydrodynamics. This response is characterized with a set of nonlinear response coefficients that determine, for example, the ${v}_{5}$ that is produced by an ${\ensuremath{\epsilon}}_{2}$ and an ${\ensuremath{\epsilon}}_{3}$. We analyze how viscosity damps both the linear and nonlinear 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 nonlinear 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 $\ensuremath{\sim}25%$ correction in midcentral 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 nonlinear response.