Initial conditions for hydrodynamics from weakly coupled pre-equilibrium evolution

We use effective kinetic theory, accurate at weak coupling, to simulate the preequilibrium evolution of transverse energy and flow perturbations in heavy-ion collisions. We provide a Green function which propagates the initial perturbations to the energymomentum tensor at a time when hydrodynamics becomes applicable. With this map, the complete pre-thermal evolution from saturated nuclei to hydrodynamics can be modelled in a perturbatively controlled way.


Introduction
Viscous relativistic hydrodynamic simulations of heavy ion collisions at the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider have shown tremendous success in describing simultaneously many of the soft hadronic observables, however initial conditions for hydrodynamics remain one of the largest uncertainties in hydrodynamic modeling of heavy ion collisions [1,2,3,4].In this work I use the effective kinetic theory of weakly coupled quasiparticles to study the equilibration and the onset of hydrodynamics in heavy ion collisions [5,6].In particular, I focus on the transverse perturbations, which are expected to initiate flow during the equilibration process [7].

Separation of scales
In the weak coupling limit, kinetic theory describes the evolution of the system from the microscopic formation time τ 0 ∼ Q −1 s to the onset of hydrodynamics at a much later time τ init ∼ τ equilibrium [8].For realistic values of coupling constant α s ∼ 0.3, the equilibration time is short and the causally connected region c(τ init − τ 0 ) ∼ 1 fm is much smaller than the transverse nuclear geometry R Pb ∼ 5 fm, but comparable to a single nucleon scale R p ∼ 1 fm (see Fig. 1 ) For this reason, the global nuclear geometry contributes a small gradient to a locally constant background, while event-by-event nucleon fluctuations are suppressed by 1/ N part , where N part is the number of participant nucleons.Therefore initial energy density can be expanded locally as e(x, τ 0 ) = e(τ 0 ) + δe(x, τ 0 ), where e(τ 0 ) = e(x, τ 0 ) |x−x 0 |≤c(τ init −τ 0 ) is the average energy density in the causal region and δe(x, τ 0 ) is a small perturbation.We will use effective kinetic theory described below .Kinetic theory describes the evolution from the microscopic formation time τ 0 to the equilibration time τ init , when hydrodynamics becomes applicable [8].By causality, for a given point in the transverse plane it is sufficient to analyze the pre-equilibrium evolution within the causal neighborhood of that point.

Effective kinetic theory
We use the effective kinetic theory of high temperature QCD at leading order in α s to model the pre-equilibrium evolution in heavy ion collisions [9].At early times gluons dominate over fermions in the plasma, so we solve the Boltzmann equation for boost invariant gluon distribution function f (τ, x, p) with leading order elastic 2 ↔ 2 and inelastic 1 ↔ 2 collision processes [5,6,9] We split the distribution function into translationally invariant background f (τ, p) and a linear perturbation with a wavenumber k ⊥ in the transverse plane δf k ⊥ (τ, p)e ik ⊥ •x .Then we solve the Boltzmann equation as a system of coupled differential equations with constant k ⊥ .We use a Color Glass Condensate (CGC) inspired initial background distribution function f (p), which possesses large initial pressure anisotropy, P T P L , and take the functional form of transverse perturbations to be |δf k ⊥ | ∼ p∂ p f (p) [5,6].In Fig. 2(a) we show the evolution of the background energy momentum tensor components relative to their asymptotic values in scaled time τ T /(η/s).We see that at sufficiently late times the longitudinal pressure P L in kinetic theory approaches the constitutive equation of viscous conformal hydrodynamics for Bjorken expansion [10] P Therefore at late times the system evolution can be smoothly passed from kinetic theory to hydrodynamics.In Fig. 2(b) we show that the linear perturbations of the energy momentum tensor δT µν can also be described with hydrodynamic constitutive equations similar to Eq. 4 at sufficiently late times and sufficiently small wavenumbers k ⊥ .

Linear response functions
The goal of the pre-equilibrium evolution is to construct the initialization conditions for hydrodynamics at τ init from a given initial state at τ 0 .Close to equilibrium the full energy momentum tensor T µν can be constructed via hydrodynamic constitutive equations from the local energy e + δe and momentum g densities, therefore we only need to know energy and momentum response functions to initial conditions.
In kinetic theory the linear energy response to the initial energy perturbations can be written as a convolution where E(r; τ init , τ 0 ) is the coordinate space Green function for energy perturbations.In Fig. 3(a), we show the radial profile of E(r; τ, τ 0 ) at τ T /(η/s) = 10 and compare it with the free streaming response.The coordinate space Green function has the meaning of a system response to initial δ-like perturbation (see Fig. 3(b)).In the absence of collisions, disturbances propagate at the speed of light and the free streaming response function shown in Fig. 3(a) is centered on the causality circle r = c(τ − τ 0 ).As seen in Fig. 3(a) collision processes modify the system response in kinetic theory and eventually it will become identical to the hydrodynamic response (not shown).Spatial Green functions are obtained by simulating the kinetic theory response to several values of wavenumber k ⊥ perturbations and then taking Fourier-Hankel transform to the coordinate space.Similarly, we find momentum response function to the initial energy gradients in the transverse plane [6].

Summary
We used effective kinetic theory to study equilibration and approach to hydrodynamics of linearized transverse energy perturbations around an initially anisotropic but boost invariant background.At the hydrodynamic initialization time all components of energy momentum tensor can be initialized from the local energy and momentum densities, which can be determined from the kinetic theory response to initial perturbations.Using the kinetic theory preequilibrium evolution in heavy ion collision models could reduce the dependence on hydro initialization time and better account for the pre-equilibrium flow.
Figure1.Kinetic theory describes the evolution from the microscopic formation time τ 0 to the equilibration time τ init , when hydrodynamics becomes applicable[8].By causality, for a given point in the transverse plane it is sufficient to analyze the pre-equilibrium evolution within the causal neighborhood of that point.

Figure 2 .
Figure 2. (a) Equilibration of background energy momentum tensor in kinetic theory relative to the asymptotic values.Dashed lines correspond to the asymptotic first and second order hydrodynamic constitutive equations.(b) Evolution of energy momentum tensor perturbation δT xx relative to the background energy density T 00 due to wavenumber k ⊥ energy perturbation in the transverse plane (in x-direction).