Relativistic viscous hydrodynamics for heavy-ion collisions with ECHO-QGP

Abstract: We present ECHO-QGP, a numerical code for $(3+1)$-dimensional relativistic viscous hydrodynamics designed for the modeling of the space-time evolution of the matter created in high energy nuclear collisions. The code has been built on top of the \emph{Eulerian Conservative High-Order} astrophysical code for general relativistic magneto-hydrodynamics [\emph{Del Zanna et al., Astron. Astrophys. 473, 11, 2007}] and here it has been upgraded to handle the physics of the Quark-Gluon Plasma. ECHO-QGP features second-order treatment of causal relativistic viscosity effects in both Minkowskian or Bjorken coordinates; partial or complete chemical equilibrium of hadronic species before kinetic freeze-out; initial conditions based on the optical Glauber model, including a Monte-Carlo routine for event-by-event fluctuating initial conditions; a freeze-out procedure based on the Cooper-Frye prescription. The code is extensively validated against several test problems and results always appear accurate, as guaranteed by the combination of the conservative (shock-capturing) approach and the high-order methods employed. ECHO-QGP can be extended to include evolution of the electromagnetic fields coupled to the plasma.

• The paper develops ECHO-QGP, a simulation code that adapts shock-capturing techniques for modeling relativistic viscous hydrodynamics in heavy-ion collisions.
• It employs a (3+1)-dimensional framework using Minkowski and Bjorken coordinates and integrates second-order viscous corrections following the Israel-Stewart formalism.
• The code is validated against analytical solutions like Gubser flow, accurately simulating QGP temperature evolution and spatial anisotropy conversion.

Overview of Relativistic Viscous Hydrodynamics for Heavy-Ion Collisions with ECHO-QGP

The paper details the development and capabilities of ECHO-QGP, a numerical code designed for modeling the relativistic viscous hydrodynamics of the quark-gluon plasma (QGP) produced in high-energy nuclear collisions. The approach models the space-time evolution through an adaptation of the Eulerian Conservative High-Order (ECHO) framework, originally developed for general relativistic magnetohydrodynamics. The significance of the ECHO-QGP code lies in its ability to address the challenges posed by the complex dynamics of QGP, relying on high-order numerical accuracy and the shock-capturing approach inherited from astrophysical applications.

Key Features

1. Dimensional Flexibility and Coordinate Systems: ECHO-QGP facilitates the simulation of QGP in the highly energetic environments typical of collider experiments, employing a (3+1)-dimensional framework. It supports both Minkowski and Bjorken coordinates, accommodating expansions in different space-time directions, a crucial feature for simulating realistic collision events.
2. Viscosity and Relativistic Extensions: The code incorporates second-order causal treatments of viscosity, addressing the inherent anisotropic stress and energy-momentum tensor components, consistent with Israel-Stewart formalisms. Such considerations are critical for capturing the non-ideal fluid behavior observed in relativistic experiments.
3. Initial Conditions and Freeze-out: The initial conditions in ECHO-QGP can be tailored using parameters derived from the Glauber model, including Monte Carlo routines for event-by-event initialization. This setup reflects the fluctuating conditions typical of QGP formation in nuclear collisions. The code includes advancements such as a robust freeze-out procedure based on the Cooper-Frye approximation, facilitating the transition from fluid dynamics to particle spectra.
4. Validation and Numerical Testing: The code undergoes extensive validation against established analytical solutions and other numerical results across a suite of test problems. This validation exhibits its efficacy in handling shocks, expansion dynamics, and viscous corrections. Test cases include situations with exact solutions, such as Gubser flow, which ECHO-QGP successfully replicates.

Numerical Results and Observations

ECHO-QGP demonstrates its robustness and accuracy through various simulations, notably capturing the anisotropies in momentum spectra and spatial configurations that arise during collision events. The code's results are in alignment with expectations from analytical models and experimental data, particularly noteworthy in its handling of both shock propagation and viscous behavior.

The simulations revealed:

• Temperature Evolution: ECHO-QGP can accurately predict the cooling dynamics of QGP, essential for evaluating the critical points in the QGP transition process.
• Eccentricity Dynamics: The conversion of spatial anisotropies to momentum anisotropies is effectively simulated, reflecting on the success of hydrodynamic models to capture elliptic flow in non-central collisions.

Implications and Future Directions

ECHO-QGP represents a significant computational tool for the study of QGP, poised to deepen the understanding of its transient properties and behavior under extreme conditions. Its ability to incorporate electromagnetic effects, following further developments, could unveil new facets of QGP dynamics and advance the study of phenomena like the Chiral Magnetic Effect.

Future enhancements aimed at integrating finite baryon density will extend its applications to additional environments, such as those expected in upcoming collision experiments at the Facility for Antiproton and Ion Research (FAIR). Moreover, coupling with electromagnetic fields offers new research horizons in the context of relativistic heavy-ion collisions, bridging gaps between theoretical approaches and experimental results.

In conclusion, ECHO-QGP sets a foundational stage for advanced studies in the relativistic hydrodynamics domain, providing insights into the evolution of the QGP in collider experiments with a promising trajectory towards more comprehensive models in future research endeavors.