Effects of viscosity on the mapping of initial to final state in heavy ion collisions

We investigate the correlation between various aspects of the initial geometry of heavy ion collisions at the BNL Relativistic Heavy Ion Collider energies and the final anisotropic flow, using v-USPhydro, a $2+1$ event-by-event viscous relativistic hydrodynamical model. We test the extent of which shear and bulk viscosity affect the prediction of the final flow harmonics, $v_n$, from the initial eccentricities, ${\varepsilon }_n$. We investigate in detail the flow harmonics $v_1$ through $v_5$ where we find that $v_1,v_4$, and $v_5$ are dependent on more complicated aspects of the initial geometry that are especially important for the description of peripheral collisions, including a nonlinear dependence on eccentricities as well as a dependence on shorter-scale features of the initial density. Furthermore, we compare our results to previous results from NeXSPheRIO, a $3+1$ relativistic ideal hydrodynamical model that has a nonzero initial flow contribution, and find that the combined contribution from $3+1$ dynamics and nonzero, fluctuating initial flow decreases the predictive ability of the initial eccentricities, in particular for very peripheral collisions, but also disproportionately in central collisions.

Figure 1

Measure of the quality $Q_2$ (6) of the elliptic flow estimator defined in Eq. (22); i.e., the event-by-event linear correlation coefficient (23) between the elliptic flow and initial eccentricity vectors over 150 events in each centrality bin. Computations were performed with ideal hydrodynamics (“ideal”), as well as including nonzero shear viscosity (“shear”), nonzero bulk viscosity (“bulk”), or both (“shearbulk”), using v-USPhydro. Previous results [24] from NeXSPheRIO are also shown (star). The points have been given an $x$ offset for readability.

Figure 2

Measure of the quality $Q_3$ (27) of initial triangularity as an estimator of triangular flow, Eq. (26), for ideal and several kinds of viscous hydrodynamics using v-USPhydro. NeXSPheRIO results are also shown (star) [24]. The points have been given an $x$ offset for readability.

Figure 3

Measure of the quality $Q_3$ (27) of initial triangularity as an estimator Eq. (26), for ideal and viscous hydrodynamics with and without the viscous correction to the equilibrium distribution function $\delta f$ using v-USPhydro. The points have been given an $x$ offset for readability.

Figure 4

Measure of the quality $Q_3$ (6) of improved estimators Eq. (29) in viscous hydrodynamics using v-USPhydro including both shear and bulk viscosity. The points have been given an $x$ offset for readability.

Figure 5

Measure of the quality $Q_4$ (6) of the two-term estimator Eq. (33), for ideal and several kinds of viscous hydrodynamics using v-USPhydro. NeXSPheRIO results are also shown (star). The points have been given an $x$ offset for readability.

Figure 6

Measure of the quality $Q_5$ (6) of the two-term estimator Eq. (35), for ideal and several kinds of viscous hydrodynamics using v-USPhydro. NeXSPheRIO results are also shown (star). The points have been given an $x$ offset for readability.

Figure 7

Measure of the quality $Q_4$ (6) for the two-term estimator Eq. (31) and improved estimator Eq. (38), for ideal and $\text{shear}+\text{bulk} \delta f$ viscous hydrodynamics using v-USPhydro. The points have been given an $x$ offset for readability.

Figure 8

Measure of the quality $Q_5$ (6) for the two-term estimator (35) and improved estimator (39), for ideal and $\text{shear}+\text{bulk} \delta f$ viscous hydrodynamics using v-USPhydro. The points have been given an $x$ offset for readability.

Figure 9

$Q_1$ (6) for various combinations of terms in estimator (41). The hydrodynamic calculation includes shear viscosity and the standard correction to the distribution function $\delta \mathrm{f}$. The points have been given an $x$ offset for readability.

Figure 10

Measure of the quality $Q_1$ (6) for $v_1$ computed by using the estimator given by Eq. (41), for ideal and several kinds of viscous hydrodynamics using v-USPhydro. The points have been given an $x$ offset for readability.

Figure 11

$Q_1$ (6) of estimator (41) for $p_T$-integrated directed flow ($p_T<5$ GeV), compared to $Q_1$ for the directed flow of hadrons with $p_T<1$ GeV. The hydrodynamic calculation includes shear viscosity and the standard correction to the distribution function $\delta \mathrm{f}$. The points have been given an $x$ offset for readability.