Hydrodynamic flow amplitude correlations in event-by-event fluctuating heavy-ion collisions

The effects of event-by-event fluctuations in the initial geometry of colliding nuclei are important in the analysis of final flow observables in relativistic heavy-ion collisions. We use hydrodynamic simulations to study the amplitude correlations between different orders of event-by-event fluctuating anisotropic flow harmonics. While the general trends seen in the experimental data are qualitatively reproduced by the model, deviations in detail, in particular for peripheral collisions, point to the need for more elaborate future calculations with a hybrid approach that describes the late hadronic stage of the evolution microscopically. It is demonstrated explicitly that the observed anticorrelation between $v_2$ and $v_3$ is the consequence of approximately linear hydrodynamic response to a similar anticorrelation of the corresponding initial eccentricities ${\epsilon }_2$ and ${\epsilon }_3$. For $n>3$, the hydrodynamic correlations between $v_{2,3}$ and $v_n$ deviate from the rescaled correlations among the corresponding initial eccentricities, demonstrating a nonlinear mode coupling effect in higher-order flows.

Figure 1

The correlations between $v_3$ and $v_2$ (a,b) and between $v_4$ and $v_3$ (c,d), as a function of event centrality for the 14 centrality bins described in the text, from viscous hydrodynamic simulations (a,c) and from experimental data [19] (b,d) for Pb-Pb collisions at the LHC. Solid triangles represent $v_m-v_n$ correlations in a given centrality bin; black triangles represent the values for centrality bins for which event-shape selected results are suppressed to avoid excessive clutter, while colored solid triangles represent the bin-averaged values for the centrality bins listed in the legend for which we show, with symbols of the same color, the $v_m-v_n$ correlations for event-shape selected events within the given centrality bin. In panels (a,b) event shapes are selected using $q_2$; for panels (c,d) we use $q_3$ for the event-shape selection. For the ATLAS data [19] (b,d) statistical and systematic uncertainties are shown as error bars and grey shaded boxes, respectively. Note that what is plotted is the rms value of $v_n$, i.e., $v_n\left\{2\right\}$, for charged hadrons (after resonance decays) integrated over the indicated $p_T$ range; this agrees with Ref. [19] where, however, the qualifier $\left\{2\right\}$ was dropped for clarity. The reader should also note the different vertical and horizontal ranges covered by the theoretical (a,c) and experimental (b,d) results.

Figure 2

The $v_n-v_2$ correlations for $n=3,4,5$, for six $q_n$ selections in seven centrality intervals of 5% width (only every other centrality bin is shown), using viscous hydrodynamics with MC-Glauber initial condition, $\eta /s=0.08$. Black circles show the hydrodynamic flow amplitude correlations; blue lines are the corresponding eccentricity correlations, with eccentricities ${\epsilon }_n$ rescaled according to Eq. (1). The red lines show fits according to Eqs. (2). Error bars indicate the statistical uncertainties arising from the finite number of events in each $q$ bin. For easier comparison with the ATLAS data [19] the charged hadron flows are integrated over the same $p_T$ range as in the experiment.

Figure 3

Similar to Fig. 2, but for $v_n-v_3$ correlations ($n=4,5$).

Figure 4

Centrality dependence of $v_4$ (a) and $v_5$ (b,c), together with their associated linear and nonlinear components defined in Eqs. (5) and (6), from the hydrodynamic simulations described in the text. Black solid circles show the full $v_n\left\{2\right\}$. Red dotted and blue dashed lines are the linear (L) and nonlinear (NL) parts as defined in Eqs. (5) and are identical in panels (b) and (c). Red circles (“Fit L”) and blues squares (“Fit NL”) show the linear and nonlinear components $v_n^{\mathrm{L},\mathrm{fit}}\left\{2\right\}$ and $v_n^{\mathrm{NL},\mathrm{fit}}\left\{2\right\}$, respectively, defined in Eqs. (6) based on the fit functions (2); they differ between panels (b) and (c). The shaded regions show the statistical uncertainties associated with the finite number (3000) of collision events per centrality bin.

Figure 5

$v_2, v_3, v_4$, and $v_5$ as function of collision centrality from ideal and viscous hydrodynamics with KLN initial conditions. Each centrality bin contains 1000 events. Solid black squares correspond to ideal hydrodynamics, while solid blue diamonds and solid red circles correspond to viscous hydrodynamics with $\eta /s=0.08$ and 0.2, respectively. The open blue diamonds and red circles are also from viscous hydrodynamics with $\eta /s=0.08$ and 0.2, respectively, but without including the viscous correction $\delta f$ at freeze-out.

Figure 6

Different types of $v_m-v_n$ correlation structures (“boomerang shapes”) from ideal and viscous hydrodynamics with KLN initial conditions. Each centrality bin contains 1000 events. See text for explanation.

Figure 7

The centrality dependence of the linear part of $v_5\left\{2\right\}$ obtained from the fit with Eqs. (2) and (6), as shown in Fig. 4, but now compared for two different fits using 6 (red circles) and 15 (blue diamonds) $q_2$ (a) or $q_3$ (b) bins when binning the 3000 events in each centrality class by their event shapes. The red lines show the linear parts as defined in Eqs. (5) for comparison. Note the expanded vertical scale compared to Fig. 4.

Figure 8

Centrality dependence of the fit parameters in Eqs. (2) for the fits shown in Figs. 2 and 3.