Precision studies of $v_n$ fluctuations

The power spectrum of heavy-ion collisions is investigated by studying initial state fluctuations on top of a smooth hydrodynamic flow. In particular, the stability of the location of the first minimum of the power spectrum and the dependence of the hydrodynamic response $v_n/{\varepsilon }_n$ on $n$ and on $\eta /s$ are discussed. In our study we develop a new Green's function method for the analytic hydrodynamic flow by S. Gubser and make use of a fully nonlinear hydrodynamics code. We find that there will be no well-defined first minimum of the response for $n<10$, due to the fact that all minima in that region are found to be sensitive to the location of the initial perturbations. Also, we find that the often-proposed form of the hydrodynamical response, that $ln(v_n/{\varepsilon }_n)$ depend quadratically on $n$ and linearly on $\eta /s$, should not hold once many events have been averaged.

Figure 1

Fixed $p_T=1$ GeV differential power spectrum for $\eta =0$. Numerical error is smaller than the symbol size.

Figure 2

Left: A plot of the freeze-out surface for $T_0=630$ MeV. Right: The $\rho$ values corresponding to this freeze-out surface.

Figure 3

Inviscid response. Numerical error is smaller than the symbol size. The shaded region $m>3$ is cutoff dependent.

Figure 4

Response in nonlinear hydrodynamics. Here $\eta /s=0.08$. Errors are no larger than the symbol size.

Figure 5

Hydrodynamic response for different values of $\eta /s$. Note that this is a log plot and that the horizontal axis is $m^2$. Here $r_0=4$ fm. Errors are no larger than the symbol size.

Figure 6

Fit of the coefficients of $m^2$ in quadratic fits of $ln(v_m/e_m)$ [cf. (4.1)]. Each individual fit of $ln(v_m/e_m)$ includes the values $m=2$ to $m=6$. Errors are smaller than the symbol size.