Shape and flow fluctuations in ultracentral Pb + Pb collisions at the energies available at the CERN Large Hadron Collider

In ultracentral heavy-ion collisions, anisotropic hydrodynamic flow is generated by density fluctuations in the initial state rather than by geometric overlap effects. For a given centrality class, the initial fluctuation spectrum is sensitive to the method chosen for binning the events into centrality classes. We show that sorting events by total initial entropy or by total final multiplicity yields event classes with equivalent statistical fluctuation properties, in spite of viscous entropy production during the fireball evolution. With this initial entropy-based centrality definition we generate several classes of ultracentral Pb + Pb collisions at Cern Large Hadron Collider energies and evolve the events using viscous hydrodynamics with nonzero shear but vanishing bulk viscosity. Comparing the predicted anisotropic flow coefficients for charged hadrons with CMS data we find that both the Monte Carlo Glauber (MC-Glb) and Monte Carlo Kharzeev-Levin-Nardi (MC-KLN) models produce initial fluctuation spectra that are incompatible with the measured final anisotropic flow power spectrum, for any choice of the specific shear viscosity. In spite of this failure, we show that the hydrodynamic model can qualitatively explain, in terms of event-by-event fluctuations of the anisotropic flow coefficients and flow angles, the breaking of flow factorization for elliptic, triangular, and quadrangular flow measured by the CMS experiment. For elliptic flow, this factorization breaking is large in ultracentral collisions. We conclude that the bulk of the experimentally observed flow factorization breaking effects are qualitatively explained by hydrodynamic evolution of initial-state fluctuations, but that their quantitative description requires a better understanding of the initial fluctuation spectrum.

Figure 1

The root mean square (rms) initial eccentricity ${\epsilon }_n\left\{2\right\}$ for $n=2–5$, as a function of centrality, for different centrality selection criteria: using the number of participant nucleons $N_{\mathrm{part}}$ (solid black), the number of binary collisions $N_{\mathrm{coll}}$ (red dashed), impact parameter $b$ (green dotted), or initial total entropy $dS/dy$ (dotted blue). For this figure ${10}^6$ events were generated from the MC-Glauber model with minimum internucleon distance (“hard core radius”) $r_{\min }=0.9$ fm [22, 23], for Pb + Pb collisions at $\sqrt{s}=2.76A$ TeV, with binary collision to wounded nucleon ratio $\alpha =0.118$.

Figure 2

Correlation between the initial total entropy $dS/dy$ and final charged multiplicity $d{N}_{\mathrm{ch}}/dy$ for (ultra)central Pb + Pb collisions at the LHC (0–5 % centrality). The plot is based on 10  000 previously generated viscous hydrodynamic events with fluctuating MC-KLN initial conditions using a minimum internucleon distance $r_{\min }=0.4$ fm [24, 25] and evolved with $\eta /s=0.2$. The lines separate different centrality classes (as indicated), defined by ordering the events according to $d{N}_{\mathrm{ch}}/dy$ (horizontal lines) or $dS/dy$ (vertical lines). The normalized variance of the distribution is about 0.6% in both vertical and horizontal directions.

Figure 3

Initial entropy density profiles in the transverse plane from the MC-Glauber model, for one selected sampling of the nucleon positions inside the colliding nuclei. (a) does not contain $pp$ multiplicity fluctuations. (b) and (c) correspond to two different samplings of the entropy production associated with each nucleon-nucleon collision when $pp$ multiplicity fluctuations are included as described in the text.

Figure 4

The initial total entropy distribution (a) and the initial rms eccentricity ${\varepsilon }_n\left\{2\right\}$ as a function of harmonic order $n$ (b) from the MC-Glb and MC-KLN models for 0–0.2 % ultracentral Pb + Pb collisions at 2.76 $A$ TeV. The results shown with open symbols are calculated requiring a minimum internucleon distance (“hard core radius”) $r_{\min }=0.9$ fm [20, 22, 23]. The filled symbols in (b) are obtained by sampling nucleon configurations that incorporate a realistic repulsive two-body nucleon-nucleon correlation [31].

Figure 5

Charged hadron $p_T$-spectra from the MC-Glb and MC-KLN models, for different specific shear viscosities as indicated, for Pb + Pb collisions of 0–0.2 % centrality at $\sqrt{s}=2.76A$ TeV.

Figure 6

$p_T$-integrated $v_n\left\{2\right\}$ of charged hadrons in 2.76 $A$ TeV Pb + Pb collisions at 0–0.2 % centrality, from the MC-Glb (a,c) and the MC-KLN models (b,d) models. In (a,b) repulsive hard core correlations among nucleons are implemented by simply requiring a minimum internucleon distance $r_{\min }=0.9$ fm when sampling the nucleon positions, in (c,d) realistic two-nucleon correlations [31] are included in the sampling routine. The solid squares show the CMS measurements [12] for comparison. As in the CMS measurements, $v_n\left\{2\right\}$ is integrated over $p_T$ from 0.3 to 3 GeV.

Figure 7

MC-Glb model calculations, including realistic $NN$ correlations in the initial state, of the $p_T$-differential two-particle cumulant $v_n\left\{2\right\}$ ($n\in [1,6]$) of charged hadrons, compared with CMS measurements for 2.76 $A$ TeV Pb + Pb collisions at 0–0.2=,% centrality [12].

Figure 8

Similar to Fig. 7, but for MC-KLN initial conditions.

Figure 9

Flow factorization ratio $r_n(p_T^{\mathrm{trig}},p_T^{\mathrm{asso}})$ ($n=2,3,4$) from model calculations with MC-Glb (a–d) and MC-KLN (e–h) initial conditions, ultracentral 2.76 $A$ TeV Pb + Pb collisions at 0–0.2 % centrality. Four values of $\eta /s$ were explored in each case, as indicated. Theoretical results are compared with measurements by the CMS collaboration [12].

Figure 10

Similar to Fig. 9, but for $n=3$.

Figure 11

Similar to Fig. 9, but for $n=4$.