Relative flow fluctuations as a probe of initial state fluctuations

Elliptic flow, $v_2$, and triangular flow, $v_3$, are to a good approximation linearly proportional to the corresponding spatial anisotropies of the initial density profile, ${\varepsilon }_2$ and ${\varepsilon }_3$. Using event-by-event hydrodynamic simulations, we point out when deviations from this linear scaling are to be expected. When these deviations are negligible, relative $v_n$ fluctuations are equal to relative ${\varepsilon }_n$ fluctuations, and one can directly probe models of initial conditions using ratios of cumulants, for instance $v_n\left\{4\right\}/v_n\left\{2\right\}$. We argue that existing models of initial conditions tend to overestimate flow fluctuations in central Pb + Pb collisions, and to underestimate them in peripheral collisions. We make predictions for $v_3\left\{6\right\}$ in noncentral Pb + Pb collisions, and for $v_3\left\{4\right\}$ and $v_3\left\{6\right\}$ in high-multiplicity $p+\text{Pb}$ collisions.

Figure 1

Comparison between $v_n\left\{4\right\}/v_n\left\{2\right\}$ computed in hydrodynamics (full symbols) and ${\varepsilon }_n\left\{4\right\}/{\varepsilon }_n\left\{2\right\}$ computed from the corresponding initial energy density profiles (open symbols), for 2.76 TeV Pb + Pb collisions. Shaded bands: ATLAS data for $v_n\left\{4\right\}/v_n\left\{2\right\}$ [24]. Symbols are shifted horizontally for readability. (a) Elliptic flow ($n=2$). (b) Triangular flow ($n=3$).

Figure 2

Test of initial condition models using $v_2\left\{4\right\}/v_2\left\{2\right\}$ measured in Pb + Pb collisions at 2.76 TeV up to 20% centrality. Stars: CMS data [40]. Full circles: ALICE data [41]. Shaded band: ATLAS data [7]. Open symbols: Values of ${\varepsilon }_2\left\{4\right\}/{\varepsilon }_2\left\{2\right\}$ given by the $\mathrm{T}R\mathrm{ENTo}$ model with $p=-1$ (triangles), $p=0$ (circles), and $p=1$ (squares). Full diamonds: ${\varepsilon }_2\left\{4\right\}/{\varepsilon }_2\left\{2\right\}$ from the Monte Carlo rcBK model.

Figure 3

Test of initial condition models using $v_3\left\{4\right\}/v_3\left\{2\right\}$ measured in 2.76 TeV Pb + Pb collisions: (a) up to 20% centrality; (b) between 20% and 80% centrality. Stars: CMS data [47]. Full circles: ALICE data [41]. Shaded band: ATLAS data [24]. ALICE and CMS data are not shown in panel (b) for the sake of readability, but are compatible with ATLAS data. Remaining symbols correspond to values of ${\varepsilon }_3\left\{4\right\}/{\varepsilon }_3\left\{2\right\}$ from several models of initial conditions. Open symbols: $\mathrm{T}R\mathrm{ENTo}$, with $p=-1$ (triangles), $p=0$ (circles), $p=1$ (squares). Full crosses: IP-Glasma [18]. Full diamonds: Monte Carlo rcBK [16].

Figure 4

Predictions for $v_3\left\{6\right\}/v_3\left\{4\right\}$ in 2.76 TeV Pb + Pb collisions, from several models of initial conditions, in the $20–80\%$ centrality range. Empty symbols: Predictions of $\mathrm{T}R\mathrm{ENTo}$ with $p=1$ (squares), $p=0$ (circles), and $p=-1$ (triangles). Full diamonds: Prediction of Monte Carlo rcBK [16]. The red dashed line is the prediction of the power distribution [50].

Figure 5

$v_2\left\{4\right\}/v_2\left\{2\right\}$ (a) and $v_3\left\{4\right\}/v_3\left\{2\right\}$ (b) as functions of centrality percentile in 5.02 TeV $p+\text{Pb}$ collisions. Full circles: $\mathrm{T}R\mathrm{ENTo}$ parametrization with $\sigma =0.3$ fm. Triangles: $\mathrm{T}R\mathrm{ENTo}$ parametrization with $\sigma =0.4$ fm. Squares: CMS data [54]. The centrality binning of CMS data is taken from Table I of Ref. [60].

Figure 6

Eccentricity-driven predictions for $v_2\left\{6\right\}/v_2\left\{4\right\}$ and $v_2\left\{8\right\}/v_2\left\{6\right\}$ as function of $v_2\left\{4\right\}/v_2\left\{2\right\}$ in 5.02 TeV $p+\text{Pb}$ collisions. Full symbols: $\mathrm{T}R\mathrm{ENTo}$ parametrization with $\sigma =0.3$ fm (circles) and $\sigma =0.4$ fm (triangles). Empty symbols: CMS data [25]. The red dashed line represents the prediction of the power distribution [50].

Figure 7

Prediction for $v_3\left\{6\right\}/v_3\left\{4\right\}$ as function of $v_3\left\{4\right\}/v_3\left\{2\right\}$ in central 5.02 TeV $p+\text{Pb}$ collisions, from different $\mathrm{T}R\mathrm{ENTo}$ parametrizations. Circles: $\sigma =0.3$ fm. Triangles: $\sigma =0.4$ fm. The red dashed line represents the prediction of the power distribution [50].