[Paper Review] Universal Flow in the Early Stages at RHIC

Scott Pratt|ArXiv.org|Mar 9, 2009

High-Energy Particle Collisions Research8 references3 citations

TL;DR

This paper demonstrates that transverse collective flow in the early stages of heavy-ion collisions at RHIC is universal across diverse models of pre-thermal matter—such as ideal hydrodynamics, free-streaming partons, and classical color fields—due to a universal initial growth rate independent of transverse pressure anisotropy. The key result is that flow initialization in hydrodynamic models can be reliably set at τ ≈ 1 fm/c with ~10% uncertainty, significantly reducing model dependence despite large uncertainties in early energy density and equation of state.

ABSTRACT

Pre-thermal flow plays an important role in the final state evolution of heavy ion collisions at RHIC. We show that flow has universal features for a wide range of models. This significantly reduces the uncertainty in initializing hydrodynamic models at RHIC.

Motivation & Objective

• To reduce uncertainty in initializing hydrodynamic models for heavy-ion collisions at RHIC by identifying universal features in early collective flow.
• To investigate whether collective flow evolution is insensitive to the microscopic nature of pre-thermal matter, such as partonic, hadronic, or classical field descriptions.
• To establish a robust initial condition for flow that is independent of transverse pressure anisotropy (κ), enabling consistent hydrodynamic simulations across models.
• To quantify how early flow generation affects final-state observables like elliptic anisotropy and two-particle correlations.
• To assess the impact of initial energy density and anisotropy uncertainties on final-state flow predictions.

Proposed method

• Derive the initial growth rate of transverse flow using energy-momentum conservation and boost-invariant longitudinal flow, leading to the universal flow formula αₓ = −∂ₓT₀₀/(2T₀₀).
• Assume a traceless stress-energy tensor Tαβ with Tₓₓ = κT₀₀ and T_zz = (1−2κ)T₀₀ to model diverse pre-thermal states, including ideal hydrodynamics (κ=1/3), free-streaming (κ=1/2), and longitudinal color fields (κ=1).
• Apply the condition of boost invariance (u_z = z/τ) at early times to derive the time evolution of T₀ₓ and T₀₀, showing that flow growth depends only on the spatial gradient of energy density.
• Use the Navier-Stokes and Israel-Stewart viscous hydrodynamics frameworks to test whether flow universality persists under viscous relaxation, with relaxation time τ_IS determined by viscosity and pressure fluctuations.
• Compare flow evolution across three models: ideal hydrodynamics, coherent classical fields, and incoherent classical fields, using identical initial energy density profiles.
• Match the stress-energy tensor at τ = 1 fm/c to hydrodynamic forms to assess whether flow and elliptic anisotropy (εₚ) converge across models despite differing Tᵢⱼ.

Experimental results

Research questions

• RQ1 Does the initial growth rate of transverse flow depend on the transverse stiffness κ of the stress-energy tensor in pre-thermal matter?
• RQ2 Can a universal initial flow condition be established at τ ≈ 1 fm/c that is independent of the underlying model of early-stage matter?
• RQ3 To what extent does the uncertainty in initial energy density and anisotropy affect final-state flow observables, given flow universality?
• RQ4 How does the elliptic anisotropy εₚ evolve during the pre-thermal stage, and can it be consistently initialized in hydrodynamic models?
• RQ5 Does matching to a viscous hydrodynamic form instead of an ideal form alter the universality of flow initialization?

Key findings

• The initial growth rate of transverse flow is universal and given by αₓ = −∂ₓT₀₀/(2T₀₀), independent of the transverse stiffness κ, due to the traceless nature of the stress-energy tensor and boost invariance.
• Flow evolution is nearly identical across models with vastly different Tₓₓ (from T₀₀/3 to T₀₀), confirming that flow is insensitive to early pressure anisotropy.
• Elliptic anisotropy εₚ is zero for incoherent classical fields (equivalent to free-streaming partons), but jumps to a positive value when the system is suddenly assumed to thermalize, demonstrating that conditions for elliptic flow can be generated even before εₚ is nonzero.
• Matching to a hydrodynamic form at τ = 1 fm/c causes all three models to converge to nearly identical flow and anisotropy evolution, indicating that flow initialization can be reliably set regardless of early-state model.
• The uncertainty in initial flow is reduced to ~10% due to universality, and since ~10% of final flow is generated in the first fm/c, initial flow uncertainty contributes only ~1% to final flow uncertainty.
• The initial energy density varies by a factor of 2 by τ = 1 fm/c depending on the model, but this does not significantly affect flow initialization due to flow universality.