Anisotropic time evolution of sound modes in Bjorken expanding holographic plasma

TL;DR

The paper addresses how sound waves propagate in a strongly coupled, anisotropic plasma undergoing Bjorken expansion, challenging the use of a single equilibrium speed of sound in rapidly evolving media. By combining a self-consistent anisotropic hydrodynamic framework with a holographic N=4 SYM model, the authors compute time-dependent transverse and longitudinal sound speeds, attenuation, and relaxation times under a quasi-static approximation, and quantify the impact of time-derivative corrections. They find two distinct out-of-equilibrium speeds of sound, $c_\perp$ and $c_\parallel$, whose values range from near the conformal estimate to near the speed of light, with attenuation largely preserved near equilibrium and relaxation times evolving toward isotropic values, albeit with breakdown at early times. The results highlight the necessity of anisotropic transport coefficients in hydrodynamic analyses of heavy-ion collisions and provide a quantitative framework for incorporating non-equilibrium, direction-dependent sound properties into experimental interpretation and modeling.

Abstract

The speed of sound is a key parameter for characterizing equilibrium states. However, sound waves change their properties when propagating through rapidly evolving anisotropic media, such as the quark-gluon plasma created in heavy-ion collisions. This paper uses $\mathcal{N}=4$ Super-Yang-Mills theory to numerically study the time evolution of the speed and attenuation of sound modes along with the relaxation time in a plasma undergoing Bjorken expansion from various initial states in a quasi-static approximation. The longitudinal Bjorken expansion breaks the isotropy, resulting in two distinct sound speeds that range from just below the conformal value to the speed of light. An anisotropic hydrodynamic description is constructed and its applicability is discussed. Implications for the analysis of heavy ion data are considered.

Figures (13)

• Figure 1: Time evolution of the two speeds of sound. The left (right) figure shows the square of the transverse (longitudinal) speed of sound, $c_{s\perp}^2$ ($c_{s\parallel}^2$). Different colors represent different initial conditions (IC=5, 15, 20), see table \ref{['tab:IC']}. Dashed black lines indicate the conformal value of 1/3, and the solid black curves are the sound attractors computed from a thermodynamic definition of the speed of sound Cartwright:2022hlg. Solid curves in red, blue, and green show the speed of sound computed for the three initial conditions from that same thermodynamic definition of the speed of sound Cartwright:2022hlg. The triangles (dots) correspond to calculations with (without) time-derivative corrections. The red hollow circles display the corresponding results when using the time-dependent asymptotic metric \ref{['ASP:metric']}janikAsymptoticPerfectFluid2006Chesler:2009cy. The quasi-static approximation and fit procedure break down for the transverse (longitudinal) speed of sound including time-derivatives indicated by hollow triangles around $\tau T\lesssim 1.2$ ($\tau T\lesssim 1.4$). For an estimate of errors, see Figs. \ref{['figSET']} and \ref{['figSEL']}.
• Figure 2: Time evolution of the two sound attenuation coefficients. The top (bottom) figure shows the transverse (longitudinal) sound attenuation, $\Gamma_{\perp}$ ($\Gamma_{\parallel}$) normalized by a factor $\pi T$. Different colors represent different initial conditions (IC=5, 15, 20), see table \ref{['tab:IC']}. Dashed black lines indicate the isotropic equilibrium value of $\pi T \Gamma=1/3$. The triangles (dots) correspond to calculations with (without) time-derivative corrections. For an estimate of errors, see Figs. \ref{['figSET']} and \ref{['figSEL']}.
• Figure 3: Time evolution of the two relaxation time coefficients. The top (bottom) figure shows the transverse (longitudinal) relaxation time, $\tau{\perp}$ ($\tau{\parallel}$) normalized by a factor $2\pi T$. Different colors represent different initial conditions (IC=5, 15, 20), see table \ref{['tab:IC']}. Dashed black lines indicate the isotropic equilibrium value of $2\pi T \tau=2-\ln 2$. The triangles (dots) correspond to calculations with (without) time-derivative corrections. The quasi-static approximation and fit procedure break down for the relaxation times around $\tau T\lesssim 3$. The quasi-static approximation and fit procedure break down for the sound attenuations including time-derivatives (hollow triangles) around $\tau T\lesssim 1.3$. For an estimate of errors, see Figs. \ref{['figSET']} and \ref{['figSEL']}.
• Figure 4: The evolution of the constant terms $\mathrm{Re } C_{\perp,\parallel}$ from the real part of the dispersion relation under different initial conditions(IC). The left (right) plot corresponds to cases in the transverse (longitudinal) direction. Triangle and dot plots are for cases with and without time derivatives, respectively.
• Figure 5: The evolution of the constant terms $\mathrm{Im} C_{\perp,\parallel}$ from the imaginary part of the dispersion relation under different initial conditions IC. The left (right) plot corresponds to cases in the transverse (longitudinal) direction. Triangle and dot plots are for cases with and without time derivatives, respectively.