Anisotropic time evolution of sound modes in Bjorken expanding holographic plasma

This paper numerically investigates the time evolution of sound modes in a Bjorken-expanding $\mathcal{N}=4$ Super-Yang-Mills plasma, revealing how longitudinal expansion-induced anisotropy splits the speed of sound into two distinct values and constructing an anisotropic hydrodynamic framework to interpret these findings for heavy-ion collision data.

The Gist

Imagine a giant, super-hot soup made of the smallest building blocks of the universe, created when heavy atoms smash together at nearly the speed of light. Physicists call this "quark-gluon plasma." For a long time, scientists assumed this soup was like a perfectly calm, uniform bowl of water, expanding evenly in all directions.

This paper argues that assumption is wrong. Instead of a calm bowl, the soup is more like a giant, expanding balloon being stretched rapidly in one direction. Because it's being stretched, the "sound" traveling through it behaves very differently depending on which way it's moving.

Here is a breakdown of what the paper discovered, using simple analogies:

1. The "Stretchy" Soup

When heavy ions collide, they create a plasma that expands incredibly fast along the direction of the collision (like a balloon stretching out). This stretching breaks the symmetry.

• The Old View: Scientists thought sound traveled at the same speed in every direction, like ripples in a still pond.
• The New View: Because the soup is being stretched, sound waves traveling sideways (across the stretch) behave differently than sound waves traveling lengthwise (along the stretch).

2. Two Different Speeds of Sound

The paper found that there aren't just one, but two different speeds of sound in this expanding plasma:

• The Sideways Speed: Sound traveling across the stretch moves at a speed that starts higher than expected and slowly settles down.
• The Lengthwise Speed: Sound traveling along the stretch starts slower and speeds up to catch up.

Think of it like running on a moving walkway at an airport. If you run with the walkway (lengthwise), you move differently than if you run across it (sideways). The paper shows that in this cosmic soup, the "walkway" (the expansion) is so strong that it creates two distinct rules for how sound moves.

3. The "Snapshot" Method

The plasma changes so fast that it's impossible to take a single, perfect photo of it while it's moving. To solve this, the researchers used a clever trick called the "Quasi-Static Approximation."

• The Analogy: Imagine trying to study a spinning fan. You can't see the blades clearly because they are moving too fast. So, you take a super-fast photo (a snapshot) where the fan looks like it's frozen in time. You measure the sound speed in that frozen moment, then take another snapshot a split second later, and so on.
• By stitching these snapshots together, they could map out how the speed of sound changes from the very first moment of the collision until the plasma cools down.

4. The "Thermometer" Problem

Scientists have been trying to measure the "stiffness" of this plasma (how hard it is to squeeze) by looking at the speed of sound. They used a formula that works perfectly for things in equilibrium (like a cup of coffee sitting still).

• The Paper's Claim: The paper shows that this standard formula is wrong for this expanding plasma, especially in the early stages. It's like trying to measure the temperature of a boiling pot using a thermometer meant for ice water; the reading will be misleading.
• The researchers found that the "thermodynamic" way of calculating sound speed often underestimated how fast sound was actually traveling in the sideways direction and overestimated it in the lengthwise direction. Their new method, which accounts for the rapid stretching, gives a much more accurate picture.

5. Why This Matters for Experiments

The paper suggests that when scientists analyze data from massive particle colliders (like the ones at CERN or RHIC), they need to stop treating the plasma as a uniform, calm fluid.

• The Takeaway: If you want to understand the "personality" of this cosmic soup, you have to acknowledge that it is anisotropic (different in different directions). Just as a stretched rubber band feels different if you pull it lengthwise versus widthwise, this plasma has different properties depending on the direction you look.

Summary

In short, this paper uses advanced computer simulations (based on a theory called "holography") to show that the hot plasma created in particle collisions is not a uniform, calm fluid. It is a rapidly stretching, anisotropic medium where sound travels at two different speeds depending on the direction. The authors argue that to understand these experiments correctly, we must stop using old, "equilibrium" formulas and start using new tools that account for this rapid stretching and the resulting differences in how sound moves.

Technical Summary

Technical Summary: Anisotropic Time Evolution of Sound Modes in Bjorken Expanding Holographic Plasma

Problem Statement
Experimental analyses of heavy-ion collisions (e.g., at LHC and RHIC) often assume the generated quark-gluon plasma (QGP) is effectively isotropic and in local thermodynamic equilibrium, replacing direction-dependent transport coefficients with isotropic time-averages. However, the plasma undergoes a boost-invariant Bjorken expansion, which inherently breaks isotropy between the beamline (longitudinal) and the transverse plane. Furthermore, the system is never in global thermodynamic equilibrium, and local equilibrium is not established until approximately $\tau \approx 1$ fm/c. This paper addresses the limitations of assuming isotropy and equilibrium by investigating how the speed of sound, attenuation, and relaxation time evolve in a strongly coupled, anisotropic plasma far from equilibrium. Specifically, it challenges the validity of extracting a single "speed of sound" via thermodynamic definitions ($c_s^2 = \partial P / \partial \epsilon$) in non-equilibrium regimes.

Methodology
The authors employ a holographic approach using $N=4$ Super-Yang-Mills (SYM) theory as a microscopic description of a strongly coupled plasma. The study numerically computes the time evolution of sound modes in a plasma undergoing Bjorken expansion from various non-equilibrium initial states.

1. Hydrodynamic Framework: An anisotropic hydrodynamic description is constructed for a conformal fluid with anisotropy along a fixed spacelike direction $n^\mu$. This framework distinguishes between longitudinal ($\parallel$) and transverse ($\perp$) directions relative to the expansion, introducing distinct pressures ($P_L, P_T$) and shear viscosities ($\eta_\parallel, \eta_\perp, \eta_1, \eta_2$).
2. Quasi-Static Approximation: To extract transport coefficients at specific moments, the authors apply a quasi-static approximation. They assume the time scale of sound perturbations is much shorter than the Bjorken expansion time scale. This allows the background metric (an inhomogeneous, anisotropic asymptotically Anti-de Sitter geometry) to be "frozen" on fixed time slices ($\tau = \tau_0$).
3. Holographic Computation:
   • Metric perturbations are Fourier-expanded on these fixed time slices.
   • The fluctuation equations (derived from covariant Einstein equations) are solved numerically using a pseudospectral method with ingoing boundary conditions at the apparent horizon and vanishing conditions at the AdS boundary.
   • The lowest quasinormal modes (hydrodynamic sound modes) are extracted as functions of momentum.
   • These modes are fitted to the derived anisotropic dispersion relations to extract the speed of sound ($c_{\perp, \parallel}$), attenuation coefficients ($\Gamma_{\perp, \parallel}$), and relaxation times ($\tau_{\perp, \parallel}$).
4. Corrections: The study compares calculations performed with and without time-derivative corrections of the background metric functions ($A, B, S$). Including these derivatives improves the accuracy of the quasi-static approximation, particularly at early times.

Key Contributions and Results

• Two Distinct Speeds of Sound: The longitudinal Bjorken expansion generates two distinct speeds of sound: a transverse speed ($c_\perp$) and a longitudinal speed ($c_\parallel$).
  • Transverse Mode ($c_\perp$): Initially exceeds the conformal equilibrium value ($1/\sqrt{3}$) and asymptotically converges to it from above.
  • Longitudinal Mode ($c_\parallel$): Initially drops below the conformal value and converges to it from below.
  • Range: Far from equilibrium, these values range from just below $1/\sqrt{3}$ up to the speed of light ($c_s \approx 1$).
• Discrepancy with Thermodynamic Definitions: The perturbative calculation of sound speeds (based on dispersion relations) deviates significantly from the thermodynamic definition ($c_s^2 = \partial P / \partial \epsilon$) used in previous works (e.g., Ref. [23]).
  • The thermodynamic definition systematically underestimates the magnitude of both anisotropic sound speeds out of equilibrium.
  • Deviations can be as large as a factor of two at early times ($\tau T \lesssim 1.4$).
  • Agreement between the perturbative and thermodynamic results is only achieved at later times ($\tau T \gtrsim 2.5$ fm/c for longitudinal, $\gtrsim 3.5$ fm/c for transverse).
• Attenuation and Relaxation:
  • Attenuation: Unlike the speeds of sound, the sound attenuation coefficients ($\Gamma_{\perp, \parallel}$) deviate only slightly (at most ~10%) from their isotropic equilibrium values, despite the large anisotropy.
  • Relaxation Time: The relaxation times evolve significantly and deviate from equilibrium values. Around $\tau T \approx 2.5$, the relaxation appears instantaneous (within the validity of the fit), before settling to the equilibrium value of $2 - \ln 2$.
• Validity of Approximations:
  • The quasi-static approximation breaks down at early times ($\tau T \lesssim 1.2$ for transverse, $\lesssim 1.4$ for longitudinal) when time-derivative corrections are omitted.
  • Including time-derivative corrections extends the reliability of the results to earlier times ($\tau T \approx 1.2$).
  • The breakdown is identified by the emergence of unphysical constant terms in the dispersion relation fits, reaching ~10% of the sound speed value.

Significance and Claims
The paper claims that the assumption of isotropy and the reliance on thermodynamic definitions for transport coefficients in non-equilibrium heavy-ion collision analysis can lead to drastically wrong conclusions.

• Relevance of Anisotropy: The results demonstrate that anisotropic hydrodynamic transport is highly relevant. The existence of two distinct sound speeds and the significant deviation of perturbative speeds from thermodynamic definitions imply that standard isotropic analyses may misinterpret the equation of state and transport properties of the QGP.
• Methodological Superiority: The authors argue that their perturbative calculation of sound waves in a time-dependent medium is more reliable than the thermodynamic definition ($c_s^2 = \partial P / \partial \epsilon$) in non-equilibrium regimes, as the latter is strictly valid only in global equilibrium.
• Implications for Experiment: The work motivates the inclusion of anisotropic transport coefficients (specifically $\eta_\perp, \eta_\parallel, \eta_1, \eta_2$) in hydrodynamic simulations (e.g., MUSIC, CLVisc) and Bayesian analyses of experimental data. It suggests that future experimental analyses should focus on dynamical, non-equilibrium observables and consider distinguishing between longitudinal and transverse momentum components to better constrain these anisotropic properties.

The authors explicitly state that their results provide a theoretical basis for re-evaluating experimental extractions of the speed of sound (such as the CMS analysis in Ref. [84]), noting that such analyses may be measuring ratios of pressure and energy density rather than the true speed of sound, particularly when local equilibrium is not fully established or when anisotropy is ignored.