“Ripples” on a relativistically expanding fluid

Recent studies have shown that fluctuations of various types play important roles in the evo lution of the fireball created in relativistic heavy-ion collisions and bear many phenomenological consequences for experimental observables. The bulk dynamics of the fireball is well described by relativistic hydrodynamic expansion and the fluctuations on top of such expanding background can be studied within the linearized hydrodynamic framework. In this paper we present complete and analytic sound-wave solutions on top of both Bjorken flow and Hubble flow backgrounds.

Figure 1

The longitudinal sound-wave amplitude $\delta =p_1/p_0$ (a) and the resulting pressure-pressure rapidity correlation (b) for an evolution time $\tau /{\tau }^{\prime }=1$ (blue), 2, (red), 3 (orange), 5 (magenta), 10 (purple), and 20 (black).

Figure 2

The longitudinal sound-wave amplitude $\delta =p_1/p_0$ (a) and the resulting pressure-pressure rapidity correlation (b) at a fixed freeze-out time ${\tau }_f=10$ fm/$c$, arising from fluctuations that occur at time ${\tau }^{\prime }=0.5$ (blue), 1 (red), 2 (magenta), and 5 (black) fm/$c$.

Figure 3

The longitudinal sound-wave amplitude $\delta =p_1/p_0$ (left) and the resulting pressure-pressure rapidity correlation (right) at a fixed freeze-out time ${\tau }_f=10$ fm/$c$, arising from fluctuations that occur at time ${\tau }^{\prime }=0.5$ (solid curves) and 5 (dashed curves) fm/$c$ with Gaussian fluctuation width $\sigma =0.1$ (blue), 0.2 (red), and 0.4 (black).

Figure 4

The sound-wave amplitude $\delta =p_1/p_0$ at fixed transverse distance $\rho$ and at evolution time $\tau /{\tau }^{\prime }=2$ (red), 3 (orange), 5 (magenta), and 10 (purple).

Figure 5

The sound-wave amplitude $\delta =p_1/p_0$ at different transverse distance $\rho$ [measured in unit of the transverse wave-front position ${\rho }_f=c_s(\tau -{\tau }^{\prime })$] at evolution time $\tau /{\tau }^{\prime }=2$ (red), 3 (orange), 5 (magenta), and 10 (purple).

Figure 6

The pressure-pressure rapidity correlation (after integration over transverse coordinates) at evolution time $\tau /{\tau }^{\prime }=2$ (red), 3 (orange), 5 (magenta), and 10 (purple).

Figure 7

The wave amplitude $\delta$ at evolution time $\tau /{\tau }^{\prime }=1$ (blue), 2 (red), 3 (orange), 5 (magenta), 10 (purple), and 20 (black).

Figure 8

The sound-wave amplitude $\delta =p_1/p_0$ (a)–(c) and the resulting pressure-pressure rapidity correlation (d)–(f) at fixed transverse distance $\rho$ and at evolution time ${\tau }_r/{\tau }_r^{\prime }=2$ (red), 3 (orange), 5 (magenta), 10 (purple), and 20 (black) fm/$c$. ${\rho }_f={\tau }_{r}sinh[c_{s}ln({\tau }_r/{\tau }_r^{\prime })]$ is the transverse wave front at the corresponding time.

Figure 9

The sound-wave amplitude $\delta =p_1/p_0$ at fixed transverse distance $\rho$ and at evolution time ${\tau }_r/{\tau }_r^{\prime }=2$ (red), 3 (orange), 5 (magenta), 10 (purple), and 20 (black) fm/$c$.

Figure 10

The pressure-pressure rapidity correlation at fixed transverse distance $\rho$ and at evolution time ${\tau }_r/{\tau }_r^{\prime }=2$ (red), 3 (orange), 5 (magenta), 10 (purple), and 20 (black) fm/$c$.

Figure 11

The early time evolution of 3D sound-wave amplitude $\delta =p_1/p_0$ from initial Gaussian perturbation at the center of Bjorken flow (a), Hubble flow (b) and static background (c) at evolution time $\tau /{\tau }^{\prime }=0$ (red), 0.05 (pink), 0.1 (orange), 0.15 (magenta), 0.2 (purple), 0.3 (blue), 0.4 (gray), 0.5 (brown), and 0.7 (black).