Subleading harmonic flows in hydrodynamic simulations of heavy ion collisions

We perform a principal component analysis (PCA) of $v_3\left(p_T\right)$ in event-by-event hydrodynamic simulations of Pb+Pb collisions at the Large Hadron Collider (LHC). The PCA procedure identifies two dominant contributions to the two-particle correlation function, which together capture 99.9% of the squared variance. We find that the subleading flow (which is the largest source of flow factorization breaking in hydrodynamics) is predominantly a response to the radial excitations of a third-order eccentricity. We present a systematic study of the hydrodynamic response to these radial excitations in 2+1D viscous hydrodynamics. Finally, we construct a good geometrical predictor for the orientation angle and magnitude of the leading and subleading flows using two Fourier modes of the initial geometry.

Figure 1

Momentum dependence of flow components in central collisions. (a) Principal flow vectors, $V_3^{\left(a\right)}\left(p_T\right)$. (b) Principal flow vectors divided by the average multiplicity, $v_3^{\left(a\right)}\left(p_T\right)\equiv V_3^{\left(a\right)}\left(p_T\right)/\langle dN/d{p}_T\rangle$.

Figure 2

Centrality dependence of flow eigenvectors ${\psi }^a\left(p_T\right)$.

Figure 3

Centrality and viscosity dependence of scaled eigenvalues $\parallel v_3^{\left(a\right)}\parallel$. (The subleading flow has been magnified 5 times to bring to scale with leading flow.)

Figure 4

Average $\mathrm{geometry}\times r^3$ in the leading and subleading principal component planes in central collisions minus an averaged radially symmetric background, $r^3(\overline{S}(\mathit{x};{\xi }_a)-\langle S\left(\mathit{x}\right)|{\xi }_a|\rangle )$. Peak fluctuations are $\pm 10–20\%$ above the background.

Figure 5

Correlation between the principal components and the triangular geometry, $\langle S_3\left(r\right){\xi }_a^{*}\rangle$, for the leading and subleading flows in central collisions. The result has been multiplied by $r^4$ and normalized by ${\overline{S}}_{\mathrm{tot}}{R}_{\mathrm{rms}}^3$, so that the area under the leading curve is approximately ${\varepsilon }_{3,3}^{\mathrm{rms}}$.

Figure 6

Quality plot (or Pearson correlation coefficient) for ${\varepsilon }_3\left(k\right)$ as a single-term predictor for principal flows (central collisions).

Figure 7

Pearson correlation coefficient for 2 and 5 $k$-mode predictors [Eq. (26)], and a predictor based on the eccentricities ${\varepsilon }_{3,3}$ and ${\varepsilon }_{3,5}$. (a) Correlations for the leading flow (zero suppressed for clarity). (b) Correlations for the subleading flow.

Figure 8

Radial weight functions for the leading and subleading flow predictors in central collisions; see Eq. (34).

Figure 9

Comparison of the averaged geometry in the principal component plane, $\langle S_3\left(r\right){\xi }_a^{*}\rangle$, and two- and five-term predictor plane, $\langle S_3\left(r\right){\xi }_a^{*\text{pred}}\rangle$, in central collisions.

Figure 10

Comparison of event-by-event hydro (averaged response) and single-shot hydrodynamics (response to average geometry) in central collisions. The singe-shot hydrodynamic results are generated from the initial conditions in Fig. 4.

Figure 11

Hydrodynamic evolution of the subleading triangular flow for the averaged initial conditions shown in Fig. 4. The color contours indicate the radial momentum density per rapidity, $\tau T^{\tau r}$, while the arrows indicate the radial flow velocity.

Figure 12

Angle and magnitude correlations between the flow and and the two-term predictor in central collisions.