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

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 v₅ that is produced by an ε₂ and an ε₃. 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 v₁, v ₄, and v₅. For viscous hydrodynamics the nonlinear contribution is dominant for v₄, v₅, and higher harmonics. For v₁, the nonlinear response constitutes an important ∼25% correction in midcentral collisions. The nonlinear response is also analyzed as a function of transverse momentum for v₁, v₄, and v ₅. Finally, recent measurements of correlations between event planes of different harmonic orders are discussed in the context of nonlinear response.