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

We investigate how the initial geometry of a heavy-ion collision is transformed into final flow observables by solving event-by-event ideal hydrodynamics with realistic fluctuating initial conditions. We study quantitatively to what extent anisotropic flow (vₙ) is determined by the initial eccentricity εₙ for a set of realistic simulations, and we discuss which definition of εₙ gives the best estimator of vₙ. We find that the common practice of using an r² weight in the definition of εₙ in general results in a poorer predictor of vₙ than when using rⁿ weight, for $n>2$. We similarly study the importance of additional properties of the initial state. For example, we show that in order to correctly predict v₄ and v₅ for noncentral collisions, one must take into account nonlinear terms proportional to ε₂² and ε₂ε₃, respectively. We find that it makes no difference whether one calculates the eccentricities over a range of rapidity or in a single slice at $z=0$, nor is it important whether one uses an energy or entropy density weight. This knowledge will be important for making a more direct link between experimental observables and hydrodynamic initial conditions, the latter being poorly constrained at present.