Multipole solution of hydrodynamics and higher order harmonics

The time evolution of the medium created in heavy ion collisions can be described by hydrodynamical models. After expansion and cooling, the hadrons are created in a freeze-out. Their distribution describes the final state of this medium. In particular their azimuthal asymmetry, characterized by the elliptic flow coefficient $v_2$, is one of the most important observables in heavy ion physics. In recent years it has been revealed that, if measuring relative to higher order event planes ${\Psi }_n$, higher order flow coefficients $v_n$ for $n>2$ can be measured. This is due to initial state fluctuations, previously not described by analytic solutions of relativistic hydrodynamics. In this paper we show the first solutions that utilize higher order asymmetries and thus yield realistic $v_n$ flow coefficients. It is a clear consequence of this that different flow patterns may lead to the same observed flow coefficients. We also compare our results to PHENIX measurements and determine a possible parameter set corresponding to these data.

Figure 1

Heat map of $s$ values in the transverse plane for the $N=2,3,4$ solutions, respectively. If the temperature is a monotonic continuous function of $s$, then this is homomorphic with the actual temperature distribution of the solution.

Figure 2

Heat map of $s$ values in the transverse plane, with multiple superimposed symmetries. The more ${\epsilon }_N$ components are included, the more asymmetric the shape gets.

Figure 3

Curves for $v_2,v_3,v_4$ with only ${\epsilon }_2\ne 0$ are shown in the top panel (a), while for ${\epsilon }_3\ne 0$ only are shown in the bottom panel (b). Clearly there is no “interference” between odd and even harmonics.

Figure 4

The $v_2,v_3,v_4$ coefficients at different values of $b$ [panels (a)–(c)], $T_f$ [panels (d)–(f)], and $u_t$ [panels (g)–(i)]. The other, fixed parameters were taken from Table 1.

Figure 5

Fits to PHENIX 200 GeV Au+Au data [3] in five centrality bins. Fit parameters are summarized in Table 2. It is important to note that many set of parameters (and thus many different flow patterns) lead to the same observed $v_n$ values, and a large set of observables (or constraints on the initial conditions, as done for numerical calculations) are needed to determine model parameters.

Figure 6

Model parameters from data fit to PHENIX 200 GeV Au+Au data [3], as a function of centrality. The systematic error band comes from the correlation with $b$; see caption of Table 2. Note that if $b$ depends on centrality (which is a realistic scenario) then of course $u_t$ would also show a more pronounced centrality dependence. However, based on the available data, this ambiguity cannot be resolved.