Fate of the initial state perturbations in heavy ion collisions. III. The second act of hydrodynamics

The hydrodynamical description of the Little Bang in heavy-ion collisions is surprisingly successful, mostly owing to the very small viscosity of the quark-gluon plasma. In this paper we systematically study the propagation of small perturbations, also treated hydrodynamically. We start with a number of known techniques allowing for the analytic calculation of the propagation of small perturbations on top of the expanding fireball. The simplest approximation is the geometric acoustics, which substitutes the wave equation by mechanical equations for the propagating phonons. Next we turn to the case in which variables can be separated, where one can obtain not only the eikonal phases but also the amplitudes of the perturbation. Finally, we focus on the so-called Gubser flow, a particular conformal analytic solution for the fireball expansion, on top of which one can derive closed equations for small perturbations. Perfect hydrodynamics allows all variables to be separated and all equations to be solved in terms of known special functions. We can thus collect the analytical expression for all the harmonics and reconstruct the complete Green's function of the problem. In the viscous case the equations still allow for variable separation, but one of the equations has to be solved numerically. Summing all the harmonics we show real-time perturbation evolution, observing the viscosity-induced changes in the spectra and the correlation functions. The calculated angular shape of the correlation function is remarkably similar to the shape emerging from the experimental data, for sufficiently large viscosity. We predict a minimum at m similar to 7 and maximum at m similar to 9 harmonics, which also have some experimental evidence for it. We conclude that local hot spots in the initial state are the only visible origin of the observed correlations.