^{*}

^{†}

^{‡}

^{§}

^{∥}

a.k@cern.ch

a.mazeliauskas@thphys.uni-heidelberg.de

jeanfrancois.paquet@duke.edu

sschlichting@physik.uni-bielefeld.de

derek.teaney@stonybrook.edu

^{3}.

We develop a macroscopic description of the space-time evolution of the energy-momentum tensor during the pre-equilibrium stage of a high-energy heavy-ion collision. Based on a weak coupling effective kinetic description of the microscopic equilibration process (à la “bottom-up”), we calculate the nonequilibrium evolution of the local background energy-momentum tensor as well as the nonequilibrium linear response to transverse energy and momentum perturbations for realistic boost-invariant initial conditions for heavy-ion collisions. We demonstrate how this framework can be used on an event-by-event basis to propagate the energy-momentum tensor from far-from-equilibrium initial-state models to the time

High-energy heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) probe nuclear matter at extreme densities and temperatures. One of the primary goals of heavy-ion research is to study the properties of the new phase of deconfined matter created in such collisions: the quark-gluon plasma (QGP).

Sophisticated multistage models, in which the QGP evolution is described by viscous relativistic hydrodynamics, have been remarkably successful in describing the soft hadronic observables measured in heavy-ion collisions

Considerable progress has been made to increase the predictive power of hydrodynamic simulations and to fully understand the assumptions built into such models. These advances include an evolving understanding of the conditions necessary for hydrodynamics to be applicable

The initial conditions of hydrodynamic models remain one of the major sources of uncertainty in phenomenological studies of heavy-ion collisions. We will provide a practical way to propagate the energy-momentum tensor in a far-from-equilibrium initial state to a time when viscous hydrodynamics becomes applicable. Our goals are to have consistent overlapping descriptions of the early time dynamics and to limit the dependence of the hydrodynamic model on

One approach to the initial conditions is simply to parameterize the initial energy density and its fluctuations, sidestepping the thermalization process with additional parameters. Glauber-based models are commonly used for this purpose and provide the energy density at

At weak coupling, significant progress has been made in constructing a complete picture of the early time dynamics before

Although the output of classical field simulations has been used to initialize hydrodynamic codes

Elaborating on the ideas formulated in Ref.

In order to obtain a smooth transition from kinetic description to realistic viscous hydrodynamic evolution with typical shear viscosity over entropy ratio

The present paper provides a detailed exposition of the formalism; a compact summary of the key results along with a discussion of some of their phenomenological consequences in high energy heavy-ion collisions is given in the companion Letter

Although the initial state shortly after the collision of two heavy nuclei is presumably very complicated, many of the microscopic details wash out over the first

The transverse energy density distribution for boost-invariant IP-Glasma initial conditions at the start of the kinetic theory pre-equilibrium evolution at

We first note that, by virtue of causality, all contributions to the energy-momentum tensor at a given space-time point ^{1}

We will consider a boost-invariant form of the energy-momentum tensor throughout this work and bold spatial vectors

By applying the propagation formula in Eq.

Equation

While the details of the calculation are described in the following sections, we point out an important feature of the dynamics that greatly simplifies the practical use of Eq.

In this work, the microscopic description of equilibration is provided by QCD effective kinetic theory ^{2}

Here

We determine the microscopic input to Eq.

Before we analyze the evolution of the background energy-momentum tensor in detail, we briefly review how the phase space distribution of gluons evolves in the “bottom-up” equilibration scenario

Different phases of weak coupling equilibration (à la “bottom-up”) for a Bjorken expanding plasma. Different panels show the gluon-momentum distribution function

Based on the above picture of the underlying microscopic dynamics, we will now discuss how the boost-invariant homogeneous background evolves toward viscous hydrodynamics. As explained in Sec.

Our results for the evolution of the diagonal elements of the background energy-momentum tensor,^{3}

Off-diagonal elements of the background energy-momentum tensor vanish by the symmetries of the underlying distribution. Hence, off-diagonal contributions of

Equilibration of the different components of the background energy-momentum tensor. After normalizing the vertical axis by the asymptotic values, Eq.

Such a scaling is guaranteed to work at late times where kinetic theory matches viscous hydrodynamics. Indeed, in second-order conformal hydrodynamics the energy density in a Bjorken expansion has the following asymptotic form (see Ref. ^{4}

In plots, we include an additional factor of

Returning to Fig. ^{5}

We note that at very weak coupling this near-equilibrium scaling ansatz will fail. Indeed, in the bottom-up scenario, the hydrodynamization time scales parametrically as

Exploiting the observed scaling property of the kinetic evolution, we will express the entire nonequilibrium energy-momentum tensor evolution in terms of a universal function of the scaled time

The universal scaling function, Eq.

Based on the results in Fig.

where ^{6}

In Eq.

Hydrodynamic simulations usually adjust the energy density

One additional consequence of the pre-equilibrium evolution is a rather rapid entropy production associated with the increase of the gluon number density per rapidity. With the full gluon distribution at our disposal, we can immediately calculate the Boltzmann entropy

The entropy and gluon number density per rapidity as a function of scaled time

Continuing the discussion of Sec.

By normalizing the perturbations to the background energy density ^{7}

Note that vectors

Similarly to the discussion in

The independent components of macroscopic response functions

We follow the methodology of Ref.

Even though the effective kinetic description of the pre-equilibrium dynamics requires the knowledge of the phase-space distribution

Based on a weak-coupling picture of the initial state, where the properties of the background distribution

Similarly, considering that gradients of

Even though at the level of the linearized Boltzmann equation, the actual magnitude of the perturbations is irrelevant in computing the response functions (as long as it remains sufficiently small to justify the linearized approximation at the relevant momentum scale), we find it convenient to choose an appropriate normalization. Defining the moments of the distribution function

the corresponding energy and momentum perturbations associated with

Given the above explicit form of the initial perturbations, one can then determine the response of the energy-momentum tensor at any later time by numerically solving the kinetic equations, Eqs.

We now present our numerical results for the nonequilibrium response functions calculated in effective kinetic theory. Even though generally the response functions

Indeed, we find that in the relevant range of parameters the postulated scaling property holds and the response functions can be compactly expressed in terms of a universal function of the scaling variables such that, for example,
^{8}

This hydrodynamic regime is discussed in more detail in Appendix

(a) The universal scaling function

Because of the scaling of the response functions with

Finally, the coordinate space response functions used in the propagation formula Eq.

At late times when the system starts behaving hydrodynamically, the different components of response function decomposition Eqs.

Comparison of the response functions for

Similar constitutive relations can be derived for momentum response components and are given in Eq.

Based on the general formalism and results of linearized effective kinetic theory presented in the previous sections, we will now describe a practical implementation of the pre-equilibrium evolution for hydrodynamic modeling of heavy-ion collisions—KøMPøST. Starting from a given profile of the energy-momentum tensor ^{9}

Equilibration is not necessarily achieved everywhere at the same time

Decompose

Determine asymptotic temperature

Convolute initial perturbations

Decompose

The typical KøMPøST propagation (with default settings) of

We first split the initial energy-momentum tensor ^{10}

Note that changing the smearing width

^{11}

Note that in this implementation the diagonal components of the background energy-momentum tensor are entirely given by

Once the decomposition in background and perturbations is determined at each point

Next we propagate the initial energy and momentum perturbations, ^{12}

We note that in practice the high

Once the full energy-momentum tensor is obtained at hydrodynamic initialization time ^{13}

In this paper, we use mostly plus metric convention

^{14}

Note that the linearized kinetic theory evolution does not guarantees the existence of a local fluid rest frame for arbitrary inputs; for sick cases, the procedure fails to find a meaningful rest frame. Such instances appear for the cases where the initial gradients are particularly steep (e.g., edges or peaks of the medium). Problems in extracting the flow velocity

Once the hydrodynamic fields are initialized on a constant

At the end of hydrodynamic evolution, the hadronic observables are computed from a constant temperature freeze-out surface with

Below, we document the regulator procedure for the limited instances when the energy-momentum tensor

Physically in heavy-ion collisions the low-density regions at the edges of the fireball can be never meaningfully described as a hydrodynamic medium. Hence, there is no need to assume that ^{15}

^{16}

Typically

The remaining instabilities arise when nonlinearities become important. Examining when the

Profile of the energy density

Technically the event is processed in two passes. In the first pass, no regulator is used, and the scaling variable

Effect of the regulator as quantified by relative change in integrated energy density, Eq.

We will now illustrate the applicability of our framework to perform event-by-event simulations of the pre-equilibrium dynamics of high-energy heavy-ion collisions. Since the kinetic theory equilibration scenario described in the previous sections provides a smooth crossover from the early stage of heavy-ion collisions to the viscous hydrodynamics regime, we will demonstrate with the example of two initial-state models how initial conditions for hydrodynamic simulations can be obtained within our framework.

We first consider the Monte Carlo–Glauber (MC-Glauber) model

In the second part of this section, we will also consider the IP-Glasma model

While the microscopic IP-Glasma model provides an initialization for the entire energy-momentum tensor of the collision in 2+1D, the MC-Glauber model is typically used as an ansatz only for the transverse energy density,^{17}

It is also common to use Glauber model as an ansatz for the entropy density, which is then related to the energy density through the equation of state. In this work, the Glauber model is used as an ansatz for the energy density directly.

without specifying the other components of the energy-momentum tensor. In the language of this paper, this means that IP-Glasma initial conditions provide both energy and momentum perturbations,^{18}

We note that IP-Glasma also provides higher order fluctuations, e.g., of the different

We note that the hydrodynamic initialization time

We start our discussion with the Monte Carlo Glauber initial conditions, which provides an ansatz for the energy density distribution ^{19}

We used the publicly available

^{20}

We checked explicitly that the sensitivity of our results to this choice is relatively small, as long as the initial energy density is rescaled by an appropriate factor, which can be deduced from the relation

We used a single MC-Glauber event with

Transverse density of

The energy momentum tensor in Eq.

Since realistic fluctuating initial conditions are used, hydrodynamic fields have a complicated profile in the transverse

In Fig.

(top row) The transverse averages [as defined in Eq.

In Fig.

In order to follow the evolution of the azimuthal anisotropy, we computed the integrated transverse stress tensor

Our results for momentum ellipticity are shown in Fig.

The effective kinetic theory approach used in this work assumes that the system is conformal, i.e., the pressure is

Deviation of the QCD equation of state

Based on our analysis in the previous section, the average energy, transverse velocity and momentum anisotropy were found to have a weak dependence on the value of

In Fig.

Single event profiles along the

One can further probe the approach of kinetic theory toward a hydrodynamic evolution by comparing the out-of-equilibrium shear-stress tensor

(a) Comparison of the out-of-equilibrium shear stress tensor [cf. Eq.

Based on the successful matching of the early time pre-equilibrium stage to the subsequent hydrodynamic regime discussed in the previous sections, we now investigate the impact of a consistent description of the early time dynamics on the final-state hadronic observables computed after the freeze-out of the hydrodynamic evolution. We focus on the multiplicity ^{21}

We reiterate, as noted in Sec.

Starting with the same Glauber initial conditions at ^{22}

Our results for free-streaming evolution are obtained by replacing the kinetic evolution of the background energy

^{23}

Specifically, the initial energy density profile is rescaled

The thermal freeze-out pion (a) multiplicity

In Fig.

In Fig.

Besides the different

Besides being applicable to general initial condition ansatzs, our framework of kinetic pre-equilibrium evolution can also be applied to microscopically motivated initial states. In this section, we use the IP-Glasma model, where the evolution at early times of heavy-ion collisions is described in terms of classical Yang-Mills (CYM) dynamics

Since classical-statistical field theory and effective kinetic theory have an overlapping range of validity, the combination of the two allows for a consistent weak coupling description of early time dynamics

In what follows, we use the original version of the IP-Glasma model, which is effectively 2+1D with boost-invariant fields in the longitudinal direction. Even though a qualitative matching between classical-statistical field theory and effective kinetic theory has been demonstrated in previous works

In order to illustrate the smooth matching at early times, we analyze the evolution of transversely averaged longitudinal

Transverse “entropy” density

In Fig.

Time evolution of the longitudinal and transverse stress tensor components,

In Fig.

Having investigated the different stages of the evolution, we will evaluate the transition between kinetic theory and hydrodynamics in greater detail by monitoring the stress tensor, and by checking that hadronic observables are independent of the crossover time

As in Sec.

(Top row) Transverse average of

We note that both IP-Glasma and the kinetic theory are conformal, which means that the breaking of conformality discussed in Sec.

We also vary the transition time between IP-Glasma and the kinetic theory

Transverse average of

Next, we scrutinize the matching between KøMPøST with IP-Glasma initial conditions and hydrodynamic evolution by looking at the transverse energy and velocity profiles along the

Transverse profile of

To test the convergence of viscous components of energy momentum tensor to hydrodynamic expectations, in Fig.

(a) Comparison of

After checking the smooth matching between individual stages of the evolution, we now test the effect of the hydrodynamic initialization time

The thermal freeze-out pion (a) multiplicity

So far we have demonstrated the practical performance of our framework to describe early time dynamics of high-energy collisions. We will now investigate in more detail theoretical relations and practical comparison to other approaches previously discussed in the literature

In viscous hydrodynamics, small-scale fluctuations dampen rapidly, and many final-state observables are only sensitive to long-wavelength perturbations

In order to study the low-^{24}

Note that even if the initial energy-momentum tensor features fluctuations smaller on length scales smaller than

which is a generalization of the previous result for the transverse preflow derived in Refs.

We extract the first- and second-order coefficients in

Linear and quadratic long wavelength response coefficients to initial energy perturbations (a) and initial momentum perturbations (b) from nonequilibrium kinetic theory evolution. Dashed lines show comparison to viscous hydrodynamic asymptotic derived in Appendix

Before we compare the long-wavelength results to the full treatment in KøMPøST, it is important to point out that the above separation into long- and short-wavelength modes introduces an artificial regulator dependence not present the full treatment. Specifically for the long-wavelength filter

We now turn to the comparison between the full kinetic theory and the low-

Comparison of hydrodynamic fields obtained in the long-wavelength (top) and free-streaming (bottom) limits, with the results of full kinetic theory pre-equilibrium (KøMPøST). Different columns show profiles of energy density

As is visible from Fig.

In Fig.

Similar features can be observed in Fig.

It is constructive to compare in the same fashion the kinetic theory response with a simple model of the pre-equilibrium evolution prescription based on free streaming

Finally, in Fig.

To summarize, the low-

In this paper, we developed a linear response framework to describe the nonequilibrium evolution of the energy-momentum tensor during the early stages of high-energy heavy-ion collisions. Based on microscopic input from effective kinetic theory simulations, we presented a practical implementation of the “bottom-up” thermalization scenario with realistic fluctuating heavy-ion initial conditions and demonstrated a consistent matching between the pre-equilibrium and the hydrodynamic evolution on an event-by-event basis. Our linear kinetic pre-equilibrium propagator KøMPøST

We found that for typical QGP parameters the leading-order kinetic theory predicts a hydrodynamization time around

We applied our formalism to two widely used initial-state descriptions: a phenomenological MC-Glauber ansatz for transverse energy density deposition in Sec.

Finally, we studied the kinetic response in the “low-wavelength” (hydrodynamic) and “fast-expansion” (free-streaming) limits to establish connections with the existing literature

Our publicly available kinetic propagator package KøMPøST

More importantly, the formalism derived in this work to propagate linear perturbations on top of a smoother background can be used with response functions computed in limits other than weakly coupled effective QCD kinetic theory. By using this framework to compare systematically the macroscopic description of equilibration from a weakly coupled regime and a strongly coupled one, one can hope to better constrain the real dynamics of the medium produced in heavy-ion collisions.

The authors would like to thank Björn Schenke for insightful discussions and for his help adapting the hydrodynamics code

In this section, we summarize the procedure of extracting the transport coefficients in kinetic theory for different values of the coupling constant

First, the transport coefficients are obtained from the relaxation of pressure anisotropy. For Bjorken expansion and in second-order conformal hydrodynamics one can show that it takes the following form (see Ref.

Evolution of pressure anisotropy, Eq.

As discussed in Sec.

Finally, for completeness we list the asymptotic expansions for temperature, energy, and entropy at second order in terms of scaling variable

Below, we provide explicit parametrizations of the evolution of the background components of the energy momentum tensor. Based on our discussion in Sec.

The remaining fit parameters are

The transverse and longitudinal pressure is determined by functions

A comparison of the parametrization with the simulation data is compiled in Fig.

The universal scaling function fit, Eq.

Similar to the discussion in Sec.

Different components of the out-of-equilibrium Green functions are determined from the kinetic evolution of the perturbations defined in Eqs.

If initial amplitude of perturbations relative to the background is chosen such that

Specifically, for

Finally, the coordinate space response functions are obtained by Fourier transform according to Eqs. ^{25}

Note that Ref.

Independent tensor components of kinetic theory coordinate space response functions to initial-energy (scalar) perturbations, Eq.

Independent tensor components of kinetic theory coordinate space response functions to initial momentum (vector) perturbations, Eq.

Because the momentum space response functions

Simple analytic results for the response functions can be obtained in the free-streaming limit, for which

The independent structures of response functions for energy perturbations in Fourier space, Eq.

In coordinate space

We also note for later comparison the long-wavelength limit of the free streaming response functions for initial energy

Below we detail the derivation of the hydrodynamic limit of the response functions for energy and momentum perturbations. We employ comoving coordinates denotes as

Hydrodynamic constitutive relations take the form
^{26}

We neglect the term associated with vorticity

where

Background:

Background:

Specifically, for a conformal system, the speed of sound

By closer inspection of the relations, one immediately observes that the perturbation of the flow velocity can be inferred as

Linearized perturbations:

By comparing the hydrodynamic constitutive relation with the explicit expressions in terms of the nonequilibrium Green's functions, we find the following.

Energy perturbations:

Energy perturbations:

In the limit of

Evolution equations for linearized energy perturbations then take the form

Similarly, for momentum perturbations, the relevant equation of motion takes the form

Energy perturbations:

Energy perturbations:

Momentum perturbations:

Energy perturbations:

Energy perturbations:

Momentum perturbations:

To specify the initial conditions, we use the late time expansion of free streaming given by Eqs.

Energy perturbations:

Momentum perturbations:

Note that for initial energy perturbations the leading Taylor expansion terms in energy and momentum response agree between free streaming and hydrodynamic response. However, at higher orders in

Once the response to conserved quantities is calculated, the shear response, i.e.,

One important feature of the hydrodynamic Green's functions is the fact that the solutions can be expressed entirely in terms of the two scaling variables

Below, we provide the details of the derivation of the long-wavelength limit of kinetic theory response, including the response in coordinate space as well as some additional discussion on the regulator dependence.

Starting point of our discussion is Eq.

Explicitly, the background

Expressing the Green's functions on the right-hand side of Eq.

Energy perturbations:

Energy perturbations:

Energy perturbations:

which is the result given in the main text. We note that corresponding expressions for long-wavelength response of the shear-stress (

We will now discuss in more detail how the long-wavelength response depends on the choice of the regulator ^{27}

Note that in the following discussion, we will neglect for simplicity the (weak) dependence of the response functions on the background energy scale