Constraints on rapidity-dependent initial conditions from charged-particle pseudorapidity densities and two-particle correlations

We study the initial three-dimensional spatial configuration of the quark-gluon plasma (QGP) produced in relativistic heavy-ion collisions using centrality and pseudorapidity-dependent measurements of the medium's charged particle density and two-particle correlations. A cumulant-generating function is first used to parametrize the rapidity dependence of local entropy deposition and extend arbitrary boost-invariant initial conditions to nonzero beam rapidities. The model is then compared to $p+\text{Pb}$ and Pb + Pb charged-particle pseudorapidity densities and two-particle pseudorapidity correlations and systematically optimized using Bayesian parameter estimation to extract high-probability initial condition parameters. The optimized initial conditions are then compared to a number of experimental observables including the pseudorapidity-dependent anisotropic flows, event-plane decorrelations, and flow correlations. We find that the form of the initial local longitudinal entropy profile is well constrained by these experimental measurements.

Figure 1

Left: Unregulated skewness values lead to ill-behaved rapidity distributions. The distributions scale nonmonotonically with skewness parameter $\gamma$ and go negative at large rapidities. Right: Replacing $\gamma$ by Eq. (19) achieves desired monotonic scaling and suppresses negative regions.

Figure 2

Initial entropy density in sample Pb + Pb (upper) and $p+\text{Pb}$ (lower) events for cross sections of the $\eta =0$ and $x=0$ planes (left and right columns). Event is constructed using the relative skewness model in Table 1 with ${\mu }_0=1$, ${\sigma }_0=3$, and ${\gamma }_0=6$ along with midrapidity parameters from Ref. [10].

Figure 3

Left and middle: Centrality and pseudorapidity dependence of the charged particle pseudorapidity density $d{N}_{\text{PbPb}}/d\eta$ at $\sqrt{s_{NN}}=2.76$ TeV and $d{N}_{\text{pPb}}/d\eta$ at $\sqrt{s_{NN}}=5.02$ TeV. Bands represents model emulator calculations $d{N}_{\text{ch}}/d\eta$ from the Bayesian posterior. Symbols are experimental data from ALICE [51, 52] and ATLAS [53]. Rapidity cuts on model centrality selection are matched to experiment. Right: The root mean square of the Legendre expansion coefficient $a_1$ estimated from two-particle pseudorapidity correlations plotted as a function of collision centrality. Bands represents model emulator calculations $d{N}_{\text{ch}}/d\eta$ from the Bayesian posterior, while lines are results from full event-by-event viscous hybrid model simulations using selected parameters from the Bayesian posterior. Symbols with errors are experimental data from ATLAS [56].

Figure 4

Posterior distributions of the model parameters, listed in Table 2, for the relative-skewness (blue lower diagonal) and absolute-skewness (red upper diagonal) models. The diagonal panels are the marginal likelihood distributions of individual model parameters, while off-diagonal panels are joint distributions for pairs of model parameters.

Figure 5

Applying parameters sampled from the posterior probability distribution to Eq. (13) along with Eq. (19), this plot shows the resulting local entropy profile $ds/d\eta$ varying $T_A$ and $T_B$ from 0.2 to $2.6{\text{fm}}^{-2}$. The lines are the mean predictions and the bands denote $1\sigma$ model uncertainties.

Figure 6

Elliptic and triangular flow cumulants $v_2\left\{2\right\}$ and $v_3\left\{2\right\}$ as a function of centrality calculated from (3+1)-D hybrid model simulations using constant specific shear viscosity $\eta /s=0.17$ and 0.19 for relative- and absolute-skewness models respectively, zero bulk viscosity $\zeta /s=0$, and hydro-to-micro switching temperature $T_{\text{sw}}=154$ MeV. The initial condition parameters are selected from the Bayesian posterior.

Figure 7

Pseudorapidity dependence of anisotropic flow coefficients $v_2\left\{2\right\}$, $v_3\left\{2\right\}$, and $v_2\left\{4\right\}$ (blue circle, green triangle, and orange diamond symbols) calculated from the hybrid model with constant specific shear viscosities $\eta /s=0.25$ and 0.28 for relative- and absolute-skewness models respectively (solid and open symbols), compared to data from ALICE (smaller black symbols) with $p_T>0$ GeV (extrapolated) in different centrality bins. The bands for each theory calculation point indicate $1\sigma$ statistical error, while experimental bands and/or bars denote $1\sigma$ systematic and statistical errors respectively.

Figure 8

Left: The event-plane decorrelation for $n=2,3$ in different centrality bins with the reference particles from $3.0<|{\eta }^b|<4.0$. Right: The same quantities as the left panel but with the reference particles from $4.4<|{\eta }^b|<5.0$. Theory bands indicate $1\sigma$ statistical error, while experimental bands and/or bars denote $1\sigma$ systematic and statistical errors respectively.

Figure 9

Top row: calculated $\text{SC}(4,2)$ and $\text{SC}(3,2)$ as functions of centrality compared to ALICE measurements, with the same 3D hybrid model setup as used for Fig. 6. We conduct calculations in two kinematic ranges: $\left|\eta \right|<0.8$ is the pseudorapidity cut used by current the ALICE measurement and $2.5<\left|\eta \right|<3.5$ is our prediction for the symmetric cumulants away from midrapidity in the Pb + Pb system. Bottom row: $\text{SC}(m,n)$ normalized by $\langle v_m^2\rangle \langle v_n^2\rangle$ for two two kinematic ranges. The left column and right column show results using relative and absolute skewness respectively.