Initial-state fluctuations from midperipheral to ultracentral collisions in an event-by-event transport approach

We have developed a relativistic kinetic transport approach that incorporates initial-state fluctuations allowing to study the buildup of elliptic flow $v_2$ and high-order harmonics $v_3$, $v_4$, and $v_5$ for a fluid at fixed $\eta /s\left(T\right)$. We study the effect of the $\eta /s$ ratio and its $T$ dependence on the buildup of the $v_n\left(p_T\right)$ for two different beam energies: RHIC for Au+Au at $\sqrt{s}=200\text{GeV}$ and LHC for $\text{Pb}+\text{Pb}$ at $\sqrt{s}=2.76\text{TeV}$. We find that for the two different beam energies considered the suppression of the $v_n\left(p_T\right)$ due to the viscosity of the medium have different contributions coming from the crossover or QGP phase. Our study reveals that only in ultracentral collisions $(0–0.2\%)$ the $v_n\left(p_T\right)$ have a stronger sensitivity to the $T$ dependence of $\eta /s$ in the QGP phase and this sensitivity increases with the order of the harmonic $n$. Moreover, the study of the correlations between the initial spatial anisotropies ${\epsilon }_n$ and the final flow coefficients $v_n$ shows that at LHC energies there is more correlation than at RHIC energies. The degree of correlation increases from peripheral to central collisions, but only in ultracentral collisions at LHC, we find that the linear correlation coefficient $C(n,n)\approx 1$ for $n=2,3,4$, and 5. This suggests that the final correlations in the $(v_n,v_m)$ space reflect the initial correlations in the $({\epsilon }_n,{\epsilon }_m)$ space.

Figure 1

Different parametrizations for $\eta /s$ as a function of the temperature. The orange area refers to the quasiparticle model predictions for $\eta /s$ [28]. The three different lines indicate different possible $T$ dependencies studied in this paper. Symbols are as in the legend. See the text for more details.

Figure 2

In the left column it is shown the initial transverse density ${\rho }_T(x,y)$ at midrapidity for two typical events for $\text{Au}+\text{Au}$ at $\sqrt{s_{NN}}=200\text{GeV}$ (top) and $\text{Pb}+\text{Pb}$ at $\sqrt{s_{NN}}=2.76\text{TeV}$ (bottom). In the right column is shown the corresponding initial temperature in transverse plane. These plots are for an impact parameter of $b=7.5\text{fm}$.

Figure 3

Initial spacial anisotropies ${\epsilon }_n$ as a function of the impact parameter. Different symbols are for different $n$. The solid lines refer to the Monte Carlo Glauber while the dashed ones to the optical Glauber model.

Figure 4

Results for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ for mid rapidity. Left: differential elliptic flow $v_2\left(p_T\right)$ at mid rapidity. The solid lines refer to the case with initial-state fluctuations the dashed lines are for the case without initial fluctuations. Thick lines are for $b=2.5\mathrm{fm}$ while thin lines for $b=7.5\mathrm{fm}$. Right: differential $v_4\left(p_T\right)$ at mid rapidity with the same legend as in the left panel.

Figure 5

${\epsilon }_n/{\epsilon }_n\left(t_0\right)$ as a function of the time for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ (dashed lines) and for $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ (solid lines). Different symbols refer to different harmonics $n$.

Figure 6

$\langle v_n\rangle /{\langle v_n\rangle }^{\max }$ as a function of time at midrapidity and for $(20–30)\%$ of centrality. Dashed lines are the results for $\text{Au}+\text{Au}$ at $\sqrt{s_{NN}}=200\text{GeV}$ while solid lines $\text{Pb}+\text{Pb}$ at $\sqrt{s_{NN}}=2.76\text{TeV}$. Different symbols correspond to different harmonics.

Figure 7

Left: Time evolution of $\langle v_n\rangle$ at midrapidity respectively for $(20–30)\%$ centrality collisions. Different symbols are for different harmonics. Right: Production rate $\frac{d\langle v_n\rangle }{dt}$ as a function of time at midrapidity and for the same centrality. These results are for $\text{Pb}+\text{Pb}$ collision at $\sqrt{s_{NN}}=2.76\text{TeV}$.

Figure 8

Top: Differential $v_2\left(p_T\right)$ (thick lines) and $v_4\left(p_T\right)$ (thin lines) green and blue lines respectively at midrapidity and for $(20–30)\%$ collision centrality. The comparison is between the two systems: $\text{Au}+\text{Au}$ at $\sqrt{s}=200\text{GeV}$ (left) and $\text{Pb}+\text{Pb}$ at $\sqrt{s}=2.76\text{TeV}$ (right). The dashed lines refer to the case with a constant $\eta /s={\left(4\pi \right)}^{-1}$ during all the evolution. The solid lines refer to the case with $\eta /s={\left(4\pi \right)}^{-1}$ at higher temperature and with an increasing $\eta /s$ ratio at lower temperature while the dot dashed lines refer to the case with $\eta /s\propto T$ at higher temperature and with an increasing $\eta /s$ ratio at lower temperature. Right: Differential $v_3\left(p_T\right)$ in red lines with the same legend as in the top panel.

Figure 9

Comparison between $v_n\left(p_T\right)$ for midperipheral (left) and central (right) collision. Different colors refer to different harmonics while solid lines correspond to $4\pi \eta /s=1$ in QGP phase and f.o. and dot dashed lines to $\eta /s\propto T$ in the QGP phase and f.o.

Figure 10

Ratio between $v_n\left(p_T\right)$ for two different parametrizations of $\eta /s$ as a function of the order of the harmonic $n$ and for $p_T=1.5\text{GeV}$ (open symbols) and $p_T=2\text{GeV}$ (full symbols). Solid lines refer to midperipheral collisions while dashed lines to ultracentral collisions.

Figure 11

${\epsilon }_n$ and $v_n$ for $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ and for three different centrality class. Top: ${\epsilon }_2$ and $v_2$ for $(10–20)\%$, $(20–30)\%$, and $(30–40)\%$ from left to right. Middle: ${\epsilon }_3$ and $v_3$ for the same centralities. Finally in the bottom panel ${\epsilon }_4$ and $v_4$. In these calculations we have fixed $\eta /s=1/\left(4\pi \right)$ for high temperature and the kinetic f.o. at lower temperature (see solid line in Fig. 1).

Figure 12

${\epsilon }_n$ and $v_n$ for $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$ and for $(20–30)\%$ and $(0–0.2)\%$ centrality cut, respectively right and left panel. In these calculations we have fixed $\eta /s=1/\left(4\pi \right)$ for high temperature and the kinetic f.o. at lower temperature (see solid line in Fig. 1).

Figure 13

Correlation coefficient $C(n,n)$ as a function of the impact parameter b. Different symbols refer to different harmonics $n$. In particular circles, triangles, squares and diamonds refer to $n=2,3,4$, and 5 respectively. The solid lines correspond to $\text{Au}+\text{Au}$ collisions at $\sqrt{s_{NN}}=200\text{GeV}$ while dashed lines to the system $\text{Pb}+\text{Pb}$ at $\sqrt{s_{NN}}=2.76\text{TeV}$.

Figure 14

(a)–(d) ${\sigma }_{v_n}/\langle v_n\rangle$ (full symbols) and ${\sigma }_{{\epsilon }_n}/\langle {\epsilon }_n\rangle$ (open symbols) as a function of $\langle N_{\mathrm{part}}\rangle$ respectively for $n=2,3,4$, and 5. The dashed lines indicate the value $\sqrt{4/\pi -1}$ expected for a 2D Gaussian distribution. These results are for $\text{Pb}+\text{Pb}$ collisions at $\sqrt{s_{NN}}=2.76\text{TeV}$.