###### Abstract

Relativistic hydrodynamics has been extensively applied to high energy heavy-ion collisions. We review hydrodynamic calculations for Au+Au collisions at RHIC energies and provide a comprehensive comparison between the model and experimental data. The model provides a very good description of all measured momentum distributions in central and semiperipheral Au+Au collisions, including the momentum anisotropies (elliptic flow) and systematic dependencies on the hadron rest masses up to transverse momenta of about 1.5–2 GeV/. This provides impressive evidence that the bulk of the fireball matter shows efficient thermalization and behaves hydrodynamically. At higher the hydrodynamic model begins to gradually break down, following an interesting pattern which we discuss. The elliptic flow anisotropy is shown to develop early in the collision and to provide important information about the early expansion stage, pointing to the formation of a highly equilibrated quark-gluon plasma at energy densities well above the deconfinement threshold. Two-particle momentum correlations provide information about the spatial structure of the fireball (size, deformation, flow) at the end of the collision. Hydrodynamic calculations of the two-particle correlation functions do not describe the data very well. Possible origins of the discrepancies are discussed but not fully resolved, and further measurements to help clarify this situation are suggested.

## Hydrodynamic description of ultrarelativistic heavy-ion collisions

###### Contents

### 1 Introduction

The idea of exploiting the laws of ideal hydrodynamics to describe the expansion of the strongly interacting matter that is formed in high energy hadronic collisions was first formulated by Landau in 1953.[1] Because of their conceptual beauty and simplicity, models based on hydrodynamic principles have been applied to calculate a large number of observables for various colliding systems and over a broad range of beam energies. However, it is by no means clear that the highly excited, but still small systems produced in those violent collisions satisfy the criteria justifying a dynamical treatment in terms of a macroscopic theory which follows idealized laws (see Section 2.1). Only recently, with first data[2, 3] from the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory came striking evidence for a strong collective expansion which is, for the first time, in quantitative agreement with hydrodynamic predictions,[4] both in central and non-central collisions (in this case Au+Au collisions with center of mass (c.m.) energies of 130 and 200 GeV per nucleon-pair).

The validity of ideal hydrodynamics requires local relaxation times towards thermal equilibrium which are much shorter than any macroscopic dynamical time scale. The significance and importance of rapid thermalization of the created fireball matter cannot be over-stressed: Only if the system is close to local thermal equilibrium, its thermodynamic properties, such as its pressure, entropy density and temperature, are well defined. And only under these conditions can we pursue to study the equation of state of strongly interacting matter at high temperatures. This is particularly interesting in the light of the expected phase transition of strongly interacting matter which, at a critical energy density of about 1 GeV/fm, undergoes a transition from a hadron resonance gas to a hot and dense plasma of color deconfined quarks and gluons. Lattice QCD calculations indicate[5, 6] that this transition takes place rather rapidly at a critical temperature somewhere between 155 and 175 MeV.

In this article we review and discuss data and calculations which provide strong evidence that the created fireball matter reaches temperatures above and which indicate a thermalization time below 1 fm/. Due to the already existing extensive literature on relativistic hydrodynamics, in particular in the context of nuclear collisions, we will be rather brief on its theoretical foundations, referring instead to the available excellent introductory material[7, 8, 9] and comprehensive reviews.[10, 11, 12]

In Section 2 we briefly present the formulation of the hydrodynamic framework. Starting with the microscopic prerequisites, we perform the transition to macroscopic thermodynamic fields and formulate the hydrodynamic equations of motion. These are equivalent to the local conservation of energy and momentum of a relativistic fluid, together with continuity equations for conserved charges in the system. The set of equations is closed by providing a nuclear equation of state whose parameterization is also presented in that section. This is followed by a discussion of the initial conditions at thermalization which are used to start off the hydrodynamic part of the collision fireball evolution. Towards the end of the expansion, local thermalization again breaks down because the matter becomes dilute and the mean free paths become large. Hence the hydrodynamic evolution has to be cut off by hand by implementing a “freeze-out criterion” which is discussed at the end of Section 2.

Phenomenological aspects of the expansion are studied in Section 3. First we elaborate on central collisions and their characteristics – their cooling behavior, their dilution, their expansion rates, etc. We then proceed to discuss the special features and possibilities offered by non-central collisions, due to the breaking of azimuthal symmetry. We will see how the initial spatial anisotropy of the nuclear reaction zone is rapidly degraded and transferred to momentum space. This results in a strong momentum anisotropy which is easily observable in the measured momentum spectra of the finally emitted hadrons.

Experimental observables reflecting the hydrodynamic fireball history are the subject of Section 4. These include both momentum and coordinate space observables. Momentum space features are discussed in Section 4.1. We begin by analyzing the angle-averaged transverse momentum spectra of a variety of different hadron species for evidence of azimuthally symmetric radial flow. The measured centrality dependences of the charged multiplicity near midrapidity, of the mean transverse energy per particle, and of the shapes and mean transverse momenta of identified hadron spectra are compared with hydrodynamic calculations. We then discuss azimuthal momentum anisotropies by decomposing the same spectra into a Fourier series with respect to the azimuthal emission angle relative to the reaction plane. We investigate in particular the mass and centrality dependence of the second Fourier coefficient , the differential elliptic flow. Combining the experimental observations with a simple and quite general theoretical arguments, we will build a compelling case for the necessity of rapid thermalization and strong rescattering at early times. We will show that this provides a very strong argument for the creation of a well-developed quark-gluon plasma at RHIC, with a significant lifetime of about 5-7 fm/ and an initial energy density which exceeds the critical value for color deconfinement by at least an order of magnitude.

In Section 4.2 we describe how two-particle momentum correlations can be used to explore the spatial geometry of the collision fireball at the time of hadron emission. This so-called Hanbury Brown-Twiss (HBT) intensity interferometry supplements the momentum space information extracted from the single particle spectra with information about the size and shape of the fireball in coordinate space. Whereas the momentum anisotropies are fixed early in the collision, spatial anisotropies continue to evolve until the very end of the collision, due to the early established anisotropic flow. By combining information on the momentum and spatial anisotropies one can hope to constrain the total time between nuclear impact and hadronic freeze-out rather independently from detailed theoretical arguments. We will show that, contrary to the momentum spectra, the experimentally measured HBT size parameters are not very well described by the hydrodynamical model. Since, contrary to the momentum anisotropies, the HBT correlations are only fixed at the point of hadronic decoupling, one might suspect that they are particularly sensitive to the drastic and somewhat unrealistic sharp freeze-out criterion employed in the hydrodynamic simulations. However, this so-called HBT-puzzle[13, 14] is shared by most other available dynamical models, including those which start hydrodynamically but then describe hadron freeze-out kinetically,[15, 16, 17, 18, 19, 20] and still awaits its resolution.

In our concluding Section 5 we give a summary and also highlight the need for future studies of uranium-uranium collisions. Since uranium nuclei in their ground state are significantly deformed, the long axis being almost 30% larger than the short one, they offer a significantly deformed initial geometry even for central collisions with complete nuclear overlap.[4, 21, 22] The resulting deformed fireballs are much larger and significantly denser than those from equivalent semicentral gold-gold collisions, providing better conditions for local thermalization and the validity of hydrodynamic concepts even at lower beam energies where data from Au+Au and Pb+Pb collisions indicate a gradual breakdown of ideal hydrodynamics.[23] Central U+U collisions may thus provide a chance to explore the hydrodynamic behavior of elliptic flow down to lower collision energies and confirm the hydrodynamic prediction[4, 24] of a non-monotonic structure in its excitation function which can be directly related to the quark-hadron phase transition and its softening effects on the nuclear equation of state in the transition region (see also Section 4.1).

### 2 Formulation of hydrodynamics

#### 2.1 Hydrodynamic prerequisites

For macroscopic systems with a large number (say, of the order of Avogadro’s number) of microscopic constituents, thermodynamics takes advantage of the fact that fluctuations in the system are small, microscopic dynamics drives such systems rapidly to a state of maximum disorder, and the system’s global behavior can then be expressed in terms of a few macroscopic thermodynamic fields. Thermalization happens locally and on microscopic time scales which are many orders of magnitude smaller than the macroscopic time scales related to the reaction of the system to small inhomogeneities of the density, pressure, temperature, etc. Under such conditions, the system can be described as an ideal fluid which reacts instantaneously to any changes of the local macroscopic fields, by readjusting the slope of its particles’ momentum distribution (i.e. its temperature) locally on an infinitesimally short time scale. The resulting equations of motion for the macroscopic thermodynamic fields are the equations of ideal (i.e. non-viscous) hydrodynamics, i.e. the Euler equations and their relativistic generalizations.[25] They describe how macroscopic pressure gradients generate collective flow of the matter, subject to the constraints of local conservation of energy, momentum, and conserved charges.

The systems produced in the collision of two large nuclei are much smaller: In central Au+Au or Pb+Pb collisions at RHIC energies (i.e. up to 200 GeV per nucleon pair in the center of mass system), about 400 nucleons collide with each other, producing several thousand secondary particles. Recent experiments with trapped cold fermionic atoms with tunable interaction strength have shown that systems involving a few hundred thousand particles behave hydrodynamically if the local re-equilibration rates are sufficiently large.[26, 27] Similar experiments involving much smaller numbers of atoms are under way. In fact, one can argue that the number of particles is not an essential parameter for the validity of the hydrodynamic approach, and that hydrodynamics does not even rely on the applicability of a particle description for the expanding system at all. The only requirement for its validity are sufficiently large momentum transfer rates on the microscopic level such that relaxation to a local thermal equilibrium configuration happens fast on macroscopic time scales. Local thermal equilibrium can also be formulated for quantum field theoretical systems which are too hot and dense to allow for a particle description because large scattering rates never let any of the particles go on-shell.

If the fireballs formed in heavy ion reactions were not expanding, the typical macroscopic time scales would be given by the spatial dimensions of the reaction zone divided by the speed of sound , i.e. of the order of 10 fm/. Collective expansion reduces this estimate, and the geometric criterion must be replaced by a dynamical one involving the local expansion rate (“Hubble constant”), where is the local flow 4-velocity.[28, 29, 30, 31] Typical values for are of the order of only one to several fm/.[32] The hydrodynamic description of heavy-ion collisions thus relies on local relaxation times below 1 fm/ which, until recently, was thought to be very difficult to achieve in heavy-ion collisions, causing widespread skepticism towards the hydrodynamic approach. The new RHIC data have helped to overcome this skepticism, leaving us with the problem to theoretically explain the microscopic mechanisms behind the observed fast thermalization rates.

From these considerations it is clear that in heavy-ion collisions a hydrodynamic description can only be valid during a finite interval between thermalization and freeze-out. Hydrodynamics can never be expected to describe the earliest stage of the collision, just after nuclear impact, during which some of the initial coherent motion along the beam direction is redirected into the transverse directions and randomized. The results of this process enter the hydrodynamic description through initial conditions for the starting time of the hydrodynamic stage and for the relevant macroscopic density distributions at that time. The hydrodynamic evolution is ended by implementing a freeze-out condition which describes the breakdown of local equilibrium due to decreasing local thermalization rates. These initial and final conditions are crucial components of the hydrodynamic model which must be considered carefully if one wants to obtain phenomenologically relevant results.

#### 2.2 Hydrodynamic equations of motion

The energy momentum tensor of a thermalized fluid cell in its local rest frame is given by[25] where labels the position of the fluid cell and and are its energy density and pressure. If in a global reference frame this fluid cell moves with four-velocity (where with and ), a corresponding boost of yields the fluid’s energy momentum tensor in the global frame:

(1) |

Note that this form depends on local thermal equilibrium at each point of the fluid in its local rest frame, i.e. it corresponds to an ideal fluid where dissipative effects can be neglected. The local conservation of energy and momentum can be expressed by

(2) |

If the fluid carries conserved charges , with charge densities in the local rest frame and corresponding charge current densities in the global reference frame, local charge conservation is expressed by

(3) |

Examples for such conserved charges are the net baryon number, electric charge, and net strangeness of the collision fireball.

If local relaxation rates are not fast enough to ensure almost instantaneous local thermalization, the expressions for the energy momentum tensor and charge current densities must be generalized by including dissipative terms proportional to the transport coefficients for diffusion, heat conduction, bulk and shear viscosity.[7, 8, 25] The solution of the correspondingly modified equations is very challenging.[33] We will later discuss some first order viscous corrections in connection with experimental observables.

#### 2.3 The nuclear equation of state

The set (2,3) of differential equations involves undetermined fields: the 3 independent components of the flow velocity, the energy density, the pressure, and the conserved charge densities. To close this set of equations we must provide a nuclear equation of state which relates the local thermodynamic quantities. We consider only systems with zero net strangeness and do not take into account any constraints from charge conservation which are known to have only minor effects.[34] This leaves the net baryon number density as the only conserved charge density to be evolved dynamically.

The equation of state for dense systems of strongly interacting particles can either be modeled or extracted from lattice QCD calculations. We use a combination of these two possibilities: In the low temperature regime, we follow Hagedorn[35] and describe nuclear matter as a noninteracting gas of hadronic resonances, summing over all experimentally identified[36] resonance states.[37, 38] As the temperature is increased, a larger and larger fraction of the available energy goes into the excitation of more and heavier resonances. This results in a relatively soft equation of state (“EOS H”) with a smallish speed of sound: .[24]

As the available volume is filled up with resonances, the system approaches a phase transition in which the hadrons overlap and the microscopic degrees of freedom change from hadrons to deconfined quarks and gluons. Due to the large number of internal quark and gluon degrees of freedom (color, spin, and flavor) and their small or vanishing masses, this transition is accompanied by a rather sudden increase of the entropy density at a critical temperature . Above the transition, the system is modeled as a noninteracting gas of massless , , quarks and gluons, subject to an external bag pressure .[39] The corresponding equation of state which is more than twice that of the hadron resonance gas. In the following we refer to this equation of state as “EOS I”. is quite stiff and yields a squared sound velocity

We match the two equations of state by a Maxwell construction, adjusting the bag constant MeV such that for a system with zero net baryon density the transition temperature coincides with lattice QCD results.[5, 6] We choose[38] MeV and call the resulting combined equation of state “EOS Q”. It is plotted as at vanishing net baryon density and strangeness in Figure 1, together with the hadron resonance gas EOS H and the ideal gas of massless partons, EOS I. The Maxwell construction inevitably leads to a strong first order transition,[37] with a large latent heat GeV/fm (between upper and lower critical values for the energy density of GeV/fm and GeV/fm).[24] This contradicts lattice QCD results[5, 6] which suggest a smoother transition (either very weakly first order or a smooth cross-over). However, the total increase of the entropy density across the transition as observed in the lattice data[5, 6] is well reproduced by the model, and it is unlikely that the artificial sharpening of the transition by the Maxwell construction leads to significant dynamical effects. This is in particular true since the numerical algorithm used to solve the hydrodynamic equations tends to soften shock discontinuities such as those which might be generated by a strong first order phase transition. We will return to the possible influence of details of the EOS on certain observables in Section 4 when we discuss experimental data.

#### 2.4 Initialization

As discussed in Section 2.1, the initial thermalization stage in a heavy-ion collision lies outside the domain of applicability of the hydrodynamic approach and must be replaced by initial conditions for the hydrodynamic evolution. Different authors have explored a variety of routes to arrive at reasonable such initial conditions. For example, one can try to treat the two colliding nuclei as two interpenetrating cold fluids feeding a third hot fluid in the reaction center (“three-fluid dynamics”[11]). This requires modelling the source and loss terms describing the exchange of energy, momentum and baryon number among the fluids. Alternatively, one can model the early stage kinetically, using a transport model such as the parton cascades VNI[40], VNI/BMS[41], MPC[42], AMPT[43] or one of several other available transport codes to estimate the initial energy and entropy distributions in the collision region[44] before switching to a hydrodynamic evolution.

However, the microscopic effects which generate the initial entropy are still poorly understood, and it is quite likely that, due to the high density and collision rates, transport codes which simulate the solution of a Boltzmann equation using on-shell particles are not really valid during the early thermalization stage. In our own calculations, we therefore simply parameterize the initial transverse entropy or energy density profile geometrically, using an optical Glauber model calculation[45] to estimate the density of wounded nucleons and binary nucleon-nucleon or parton-parton collisions in the plane transverse to the beam and superimposing a “soft” component (scaling with the number of wounded nucleons) and a “hard” component (scaling with the number of binary collisions) in such a way[46, 47] that the experimentally observed rapidity density of charged hadrons at the end of the collision[48, 49] and its dependence on the collision centrality[50, 51] are reproduced.[46]

For the Glauber calculation we describe the density distributions of the colliding nuclei (with mass numbers and ) by Woods-Saxon profiles,

(4) |

with the nuclear radius fm.[52] The nuclear thickness function is given by the optical path-length through the nucleus along the beam direction: fm and the surface diffuseness

(5) |

The coordinates parametrize the transverse plane, with pointing in the direction of the impact parameter (such that span the reaction plane) and perpendicular to the reaction plane. For a non-central collision with impact parameter , the density of binary nucleon-nucleon collisions at a point in the transverse plane is proportional to the product of the two nuclear thickness functions, transversally displaced by :

(6) |

is the total inelastic nucleon-nucleon cross section; it enters here only as a multiplicative factor which is later absorbed in the proportionality constant between and the “hard” component of the initial entropy deposition.[46] Integration over the transverse plane (the -plane) yields the total number of binary collisions,

(7) |

Its impact parameter dependence, as well as that of the maximum density of binary collisions in the center of the reaction zone, , are shown as the dashed lines in Fig. 2.

The “soft” part of the initial entropy deposition is assumed to scale with the density of “wounded nucleons”,[53] defined as those nucleons in the projectile and target which participate in the particle production process by suffering at least one collision with a nucleon from the other nucleus. The Glauber model gives the following transverse density distribution of wounded nucleons:[53]

(8) |

Here the value of the total inelastic nucleon-nucleon cross section plays a more important role since it influences the shape of the transverse density distribution , and its dependence[36] on the collision energy must be taken into account. The total number of wounded nucleons is obtained by integrating Eq. (2.4) over the transverse plane. Its impact parameter dependence, as well as that of the maximum density of wounded nucleons in the center of the reaction zone, , are shown as the solid lines in Fig. 2.

Our hydrodynamic calculations were done with initial conditions which ascribed 75% of the initial entropy production to “soft” processes scaling with and 25% to “hard” processes scaling with . This was found[46] to give a reasonable description of the measured[50, 51] centrality dependence of the produced charged particle rapidity density per participating (“wounded”) nucleon.

Figure 2 shows that exploring the centrality dependence of heavy-ion collisions provides access to rich physics: With increasing impact parameter both the number of participating nucleons and the volume of the created fireball decrease. Except for effects related to the deformation of the reaction zone in non-central collisions, increasing the impact parameter is thus equivalent to colliding smaller nuclei, eventually reaching the limit of collisions in the most peripheral nuclear collisions. Furthermore, at fixed beam energy, the initially deposited entropy and energy densities decrease with increasing impact parameter. To a limited extent, heavy-ion collisions at fixed beam energy but varying impact parameter are therefore equivalent to central heavy-ion collisions at varying beam energy, i.e. one can map sections of the “excitation function” of physical observables without changing the collision energy, but only the collision centrality.

In one respect, however, non-central collisions of large nuclei such as Au+Au are fundamentally different from central collisions between lighter nuclei: A finite impact parameter breaks the azimuthal symmetry inherent in all central collisions. In a strongly interacting fireball, the initial geometric anisotropy of the reaction zone gets transferred onto the final momentum spectra and thus becomes experimentally accessible. As we will see, this provides a window into the very early collision stages which is completely closed in central collisions between spherical nuclei. (The same information is, however, accessible, with even better statistics due to the larger overlap volume and number of produced particles, in completely central collisions between deformed nuclei, such as W+W or U+U.)

The left panel of Fig. 3 shows the distribution of binary collisions in the transverse plane for Au+Au collisions at impact parameter fm. Shown are lines of constant density at 5, 15, 25, …% of the maximum value. The dashed lines indicate the Woods-Saxon circumferences of the two colliding nuclei, displaced by from the origin. The obvious geometric deformation of the overlap region can be quantified by the spatial eccentricity

(9) |

where the averages are taken with respect to the underlying density ( or or a combination thereof, depending on the exact parametrization used). The centrality dependence of is displayed in the right panel of Fig. 3.

#### 2.5 Decoupling and freeze-out

As already mentioned in Section 2.1, the hydrodynamic description begins to break down again once the transverse expansion becomes so rapid and the matter density so dilute that local thermal equilibrium can no longer be maintained. Detailed studies[32, 54] comparing local mean free paths with the overall size of the expanding fireball and the local Hubble radius (inverse expansion rate) have shown that bulk freeze-out happens dynamically, i.e. it is driven by the expansion of the fireball and not primarily by its finite size. This is similar to the decoupling of the primordial nuclear abundances and the cosmic microwave background in the early universe which was also entirely controlled by the cosmic expansion rate. Nonetheless, some part of the initially produced matter never becomes part of the hydrodynamic fluid, but decouples right away even though no transverse flow has developed yet. Figure 2 shows that already at initialization the density distribution has dilute tails where the mean free path is never short enough to justify a hydrodynamic treatment. These tails should be considered as immediately frozen out, i.e. they describe particles which carry their momentum information directly and without further modification to the detector. Their decoupling is obviously not a result of (transverse) dynamics, but a geometric effect. However, for both geometric and dynamical freeze-out the local scattering rate (density times cross section) is the controlling factor, with the density showing the largest variations across the fireball, and it was found[32, 54] that the hypersurface along which the local mean free path begins to exceed the local Hubble radius or the global fireball size can be characterized in good approximation as a surface of constant temperature. Note that for almost baryon-antibaryon symmetric systems such as the ones generated near midrapidity at RHIC, the entropy density, energy density, particle density and temperature profiles are directly related and all have similar shapes. A surface of constant temperature is therefore, in excellent approximation, also a surface of constant energy and particle density.

A traditional way of describing the breakdown of hydrodynamics and particle freeze-out is the Cooper-Frye prescription[55] which postulates a sudden transition from perfect local thermal equilibrium to free streaming of all particles in a given fluid cell once the kinetic freeze-out criterion (obtained, for example, in the way just described, by comparing local scattering and expansion rates etc.) is satisfied in that cell. In the Cooper-Frye formalism, one first lets hydrodynamics run up to large times, then determines the space-time hypersurface along which the hydrodynamic fluid cells first pass the freeze-out criterion, and computes the final spectrum of particles of type from the formula

(10) |

where is the outward normal vector on the freeze-out surface such that is the local flux of particles with momentum through this surface. For the phase-space distribution in this formula one takes the local equilibrium distribution just before decoupling,

(11) |

boosted with the local flow velocity to the global reference frame by the substitution . and are the chemical potential of particle species and the local temperature along , respectively. The temperature and chemical potentials on are computed from the hydrodynamic output, i.e. the energy density , net baryon density and pressure , with the help of the equation of state.[38]

This formalism is used to calculate the momentum distribution of all directly emitted hadrons of all masses. Unstable resonances are then allowed to decay (we include all strong decays, but consider weakly decaying particles as stable), taking the appropriate branching ratio of different decay-channels into account.[36] The contribution of the decay products is added to the thermal momentum spectra of the directly emitted stable hadrons to give the total measured particle spectra.[56]

Initial particle production at high is controlled by hard QCD processes which produce transverse momentum spectra which strongly deviate from an exponential thermal or hydrodynamic shape. Since the momentum transfer per collision is limited, such particles require a larger number of rescatterings than soft particles for reaching thermal equilibrium. They have a much higher chance of escaping from the fireball before being thermalized than soft hadrons. This is not captured by the Cooper-Frye formula which freezes out all particles in a given fluid cell together, irrespective of their momenta. A modified freeze-out criterium which takes the momentum dependence of the escape probability into account has recently been advocated.[57, 58, 59] We will discuss phenomena at large transverse momenta in the last part of Section 4.1.

The Cooper-Frye formula has another shortcoming which materializes if the freeze-out normal vector is spacelike (as it happens in certain regions of our hydrodynamic freeze-out surfaces), in which case the Cooper-Frye integral also counts (with negative flux) particles entering the thermalized fluid from outside. However, simple attempts to cut these contributions by hand[60] generate problems with energy-momentum conservation, and only recently a possible resolution of this problem has been found.[61]

Clearly, any Cooper-Frye like prescription implementing a sudden transition from local equilibrium (infinite scattering cross section) to free-streaming (zero cross section) is ultimately unrealistic and should be replaced by a more realistic prescription. A preferred procedure would be the transition from hydrodynamics to kinetic transport theory just before hydrodynamics begins to break down,[15, 16, 17, 18, 19, 20] thus allowing for a gradual decoupling process which is fully consistent with the underlying microscopic physics. Such a realistic treatment of the freeze-out process is clearly much more involved than the Cooper-Frye formalism, and so far it has not led to strong qualitative differences for the emitted hadron spectra, even though in detail some phenomenological advantages of the hybrid (hydro+transport) approach can be identified.[19] Also, the crucial question whether for rapidly expanding heavy-ion fireballs there is really an overlap window where both the macroscopic hydrodynamic and the microsopic transport approach using on-shell particles work simultaneously has not been finally settled. Most of the results presented in this review have therefore been obtained using the simple Cooper-Frye freeze-out algorithm.

#### 2.6 Longitudinal boost invariance

Most of the observables to be discussed below have been collected near mid-rapidity. This region is of particular interest as one expects there the energy- and particle density to be the largest, giving the clearest signals of the anticipated phase-transition, and many components of the large heavy-ion experiments have therefore been optimized for the detection of midrapidity particles. Furthermore, rapidity distributions are more difficult to analyze theoretically than transverse momentum distributions since they are strongly affected by “memories of the pre-collision state”: Whereas all transverse momenta are generated by the collision itself, a largely unknown fraction of the beam-component of the momenta of the produced hadrons is due to the initial longitudinal motion of the colliding nuclei. In hydrodynamics one finds that final rapidity distributions are very sensitive to the initialization along the beam direction, and that hydrodynamic evolution is not very efficient in changing the initial distributions.[64] Collective transverse effects are thus a cleaner signature of the reaction dynamics than longitudinal momentum distributions, and the best way to isolate oneself longitudinally from remnants of the initially colliding nuclei is by going as far away as possible from the projectile and target rapidities, i.e. by studying midrapidity hadron production.

Near midrapidity one is far from the kinematic limits imposed by the finite collision energy, and the microscopic processes responsible for particle production, scattering, thermalization and expansion should therefore be locally the same everywhere and invariant under limited boosts along the beam direction.[62] In a hydrodynamic description this implies a boost-invariant longitudinal flow velocity[62] whose form is independent of the transverse expansion of the fluid while the latter is identical for all transverse planes in their respective longitudinal rest frames. Under these assumptions the analytically solvable longitudinal evolution decouples from the transverse evolution[63, 70] which greatly reduces the numerical task of solving the hydrodynamic equations of motion. Limitations of Bjorken’s solution and boost invariance will be discussed in Section 4.1.4, but most of the review reports results obtained under the assumption of longitudinal boost invariance.

Bjorken showed[62] that the boost invariant longitudinal flow field has the scaling (Hubble) form and that the hydrodynamic equations preserve this form in proper time if the initial conditions for the thermodynamic variables do not depend on space-time rapidity . With this profile the flow 4-velocity can be parametrized as , where the transverse flow rapidity does not depend on and is related to the radial flow velocity at midrapidity by . It is then sufficient to solve the hydrodynamic equations for at , and the transverse velocity at other longitudinal positions is given by[63]

(12) |

Solving the hydrodynamic equations in the transverse plane with longitudinally boost-invariant boundary conditions becomes easiest after a coordinate transformation from to longitudinal proper time and space-time rapidity :

(13) |

In these coordinates, the equations of motions become[4]

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) |

where the lower case comma indicates a partial derivative with respect to the coordinate following it. One sees that the evolution in -direction is now trivial, and that only 4 coupled equations remain to be solved.

### 3 Phenomenology of the transverse expansion

In this section we study the transverse fireball expansion at midrapidity as it follows from the hydrodynamic equations of motion (Sections 2.2 and 2.6) with the equation of state described in Section 2.3 and the initial conditions from Section 2.4.

In the first part of this section we study central collisions (). These are used to tune the initial conditions of the calculations, by requiring the calculation to reproduce the measured rapidity density of charged hadrons at midrapidity and the shape of the pion and proton spectra in central collisions. For Au+Au collisions with a center of mass energy of 200 GeV per nucleon pair, we find for the initial equilibration time (i.e. for the beginning of the hydrodynamic stage) fm/ and an initial entropy density in the center of the fireball of fm.[65] Freeze-out occurs when the energy density drops below GeV/fm. How these parameters are fixed will be described in some detail in Section 4.1. In the present section we will simply use them to illustrate some of the characteristic features of the transverse hydrodynamic expansion.

In the second part we address non-central collisions and discuss the special opportunities provided by the breaking of azimuthal symmetry in this case. We discuss how the initial spatial deformation transforms rapidly into a momentum space anisotropy which ultimately manifests itself through a dependence of the emitted hadron spectra and their momentum correlations on the azimuthal emission angle relative to the reaction plane.

#### 3.1 Radial expansion in central collisions

As seen from the terms on the right hand side in Eqs. (14)-(16), the driving force for the hydrodynamic expansion are the transverse pressure gradients which accelerate the fireball matter radially outward, building up collective transverse flow. As a result, the initial one-dimensional boost-invariant expansion along the beam direction gradually becomes fully three-dimensional. For the adiabatic (ideal fluid) expansion discussed here, this implies that the entropy and other conserved charges spread out over a volume which initially grows linearly with time, but as time evolves increases faster and ultimately as . Accordingly, the entropy and baryon densities follow inverse power laws with a “local expansion coefficient” which changes smoothly from 1 to 3.

This is seen in the left panel of Figure 4 which shows a double logarithmic plot of the entropy density at three points in the fireball (at the origin as well as 3 and 5 fm away from it) as a function of time. At early and late times the solid lines representing the numerical solution are seen to follow simple and scaling laws, indicated by dashed lines. For an ideal gas of massless particles (such as our model for the quark-gluon plasma above the hadronization phase transition) , and a scaling of the entropy density translates into a scaling of the temperature. The right panel of Figure 4 shows that the numerical solution follows this simple scaling rather accurately almost down to the phase transition temperature . At this point the selected fluid cell enters the mixed phase and the temperature remains constant until the continued expansion dilutes the energy density below the lower critical value GeV/fm. This is an artefact of the Maxwell construction employed in Section 2.3; for a more realistic rapid but smooth crossover between the QGP and hadron gas phases the horizontal plateau in Figure 4 would be replaced by a similar one with smoothed edges and a small but finite slope. As the fluid cell exits the phase transition on the hadronic side, its temperature is seen to drop very steeply; this is caused not only by the now much more rapid three-dimensional expansion, but also by the different temperature dependence of in the hadronic phase, generated by the exponential dependence of the phase-space occupancy on the hadron rest masses. The horizontal dashed line indicates the freeze-out temperature of about 100 MeV (see Section 4.1); one sees that in central collisions freeze-out occurs about 15 fm/ after equilibration.

Figure 5 compares the time-dependence of the local expansion coefficient

One sees that in the numerical solutions both quantities increase with time from a Bjorken-like behavior initially to a Hubble-like behavior at later times. Weak structures in the time evolution at the beginning and end of the mixed phase are probably due to our artificially sharp phase transition and should disappear for a realistic equation of state. Note that both the local expansion coefficient and the normalized local expansion rate exceed the limiting global value of 3 at large times. This does not violate causality, but is due to the existence of density gradients and their time evolution.[66]

At late times the expansion rate is larger for points at the edge of the fireball than in the center, again due to the stronger density and pressure gradients near the edge. In contrast, the local dilution rate shows the opposite dependence on the radial distance, being smaller at large radial distance than in the center. This reflects the transport of matter from the center to the edge, due to radial flow and density gradients.[66] The relation between and can be established by using entropy conservation, , to write . Assuming longitudinal boost-invariance and a temporal power law 66] for the local entropy dilution rate one finds the relation[

(19) |

The last term involving the radial flow and radial entropy density gradient is positive, especially at large radii, explaining the different ordering of the three curves in the left and right panels of Figure 5.

To further illustrate the transverse dynamics we show in Figure 6 the radial velocity as a function of time and radial distance. The left panel shows the time evolution of at fixed radii of 3 and 5 fm. After a steep initial rise of , the radial velocity at fixed position is seen to decrease with time while the matter there changes from QGP into a hadron gas. Inside the mixed phase the pressure is constant (i.e. the pressure gradient vanishes) and the matter is not further accelerated. As a result, the system expands without acceleration, with rapidly flowing matter moving to larger radii while more slowly moving matter from the interior arrives at the fixed radius . Only after the mixed phase has completely passed through the radius does the radial expansion accelerate again, caused by the reappearance of pressure gradients.

The right panel of Figure 6 shows the radial velocity profile at selected times. Initially whereas the pressure profile features strong radial gradients (especially near the surface), except for a moderately thin layer between the QGP and hadron gas phases where the matter is in the phase transition and the corresponding softness of the equation of state does not allow for pressure gradients. Accordingly, the initial acceleration happens mostly in the outer part of the QGP core and in the hadron gas shell, with no acceleration in the mixed phase layer which only gets squashed by the accelerating matter pushing out from the interior. This is clearly seen in the solid line in Figure 6 which represents the radial velocity profile at fm/. As time proceeds the structure caused by the weak acceleration in the mixed phase gets washed out and the velocity profile becomes more uniform. It very rapidly approaches a nearly linear shape with an almost time-independent limiting slope of fm.

The near constancy of this slope implies that one also obtains an almost linear transverse velocity profile along a hypersurface of fixed temperature or energy density rather than fixed time. Figure 7 compares the radial flow rapidity profile for Pb+Pb or Au+Au collisions at SPS and RHIC energies for three different equations of state,[19] with LH8 corresponding most closely to EOS Q shown in Figure 1. Figure 7 provides welcome support for the phenomenologically very successful blast-wave parametrization[67, 68] which is usually employed with a linear transverse velocity or rapidity profile for reasons of simplicity. (Note that for the range of velocities covered in the figure the difference between rapidity and velocity can be neglected.)

As discussed in Sections 2.1 and 2.5, the freeze-out of particle species is mostly controlled by the competition between the macroscopic expansion time scale[32, 31] and the microscopic scattering time scale ; here the sum goes over all particle species (with density ) in the fireball and are the corresponding thermally averaged and velocity weighted scattering cross sections. The scattering rates drop steeply with temperature,[32, 68, 69] enabling us to idealize freeze-out as a relatively sudden process which happens along a freeze-out surface of approximately constant decoupling temperature . The most important contributions to the local scattering rate arise from +meson (due to their large abundance) and +(anti)baryon collisions (due to their large resonant cross sections). At RHIC the total baryon density is somewhat lower than at the SPS, due to the smaller baryon chemical potential (note that this does not reduce the contribution from baryon-antibaryon pairs!), and one thus expects that, at the same temperature, the mean scattering time should be slightly longer at RHIC than at the SPS. The magnitude of this effect should be small, however, and its sign could even be reversed if at RHIC the pion phase-space is significantly oversaturated.[59]

On the other hand, the expansion time scale does change significantly between SPS and RHIC: For boost-invariant longitudinal flow and a linear transverse flow rapidity profile (as suggested by Figs. 6 and 7) the expansion rate is easily calculated as[66]

(20) |

where the approximate expression[59] holds in the region . Equation (20) gives ; reading off fm from Figures 6 and 7 at RHIC energies, this linear function reproduces well the almost linear behavior seen in the right panel of Fig. 5. On the other hand, Figure 7 shows at SPS energies a transverse flow rapidity slope that is only about 2/3 of the value at RHIC. At freeze-out ( fm/[4, 19]) the expansion rate at RHIC is thus about 25% larger than at the SPS fm for Au+Au at GeV vs. fm for Pb+Pb at GeV. The corresponding “Hubble times” at freeze-out are fm/ and fm/. Combining this with the already mentioned rather similar microscopic scattering time scales at both energies one is led to the conclusion that at RHIC freeze-out should happen at a somewhat higher decoupling temperature than at the SPS.

#### 3.2 Anisotropic flow in non-central collisions

In Section 2.4 we have already addressed some of the great opportunities offered by non-central collisions. The most important ones are related to the broken azimuthal symmetry, introduced through the spatial deformation of the nuclear overlap zone at non-zero impact parameter (see Figure 3). If the system evolves hydrodynamically, driven by its internal pressure gradients, it will expand more strongly in its short direction (i.e. into the direction of the impact parameter) than perpendicular to the reaction plane where the pressure gradient is smaller.[70] This is shown in Figure 8 where contours of constant energy density are plotted at times 2, 4, 6 and 8 fm/ after thermalization. The figure illustrates qualitatively that, as the system evolves, it becomes less and less deformed. In addition, some interesting fine structure develops at later times: After about 6 fm/ the energy density distribution along the -axis becomes non-monotonous, forming two fragments of a shell that enclose a little ’nut’ in the center.[71] Unfortunmately, when plotting a cross section of the profiles shown in Figure 8 one realizes that this effect is rather subtle, and it was also found to be fragile, showing a strong sensitivity to details of the initial density profile.[4]

A more quantitative characterization of the contour plots in Figure 8 and their evolution with time is provided by defining the spatial eccentricity

(21) |

where the brackets indicate an average over the transverse plane with the local energy density as weight function, and the momentum anisotropy

(22) |

Note that with these sign conventions, the spatial eccentricity is positive for out-of-plane elongation (as is the case initially) whereas the momentum anisotropy is positive if the preferred flow direction is into the reaction plane.

Figure 9 shows the time evolution of the spatial and momentum anisotropies for Au+Au collisions at impact parameter fm, for RHIC initial conditions with a realistic equation of state (EOS Q, solid lines) and for a much higher initial energy density (initial temperature at the fireball center = 2 GeV) with a massless ideal gas equation of state (EOS I, dashed lines).[73] The initial spatial asymmetry at this impact parameter is , and obviously since the fluid is initially at rest in the transverse plane. The spatial eccentricity is seen to disappear before the fireball matter freezes out, in particular for the case with the very high initial temperature (dashed lines) where the source is seen to switch orientation after about 6 fm/ and becomes in-plane-elongated at late times.[74] One also sees that the momentum anisotropy saturates at about the same time when the spatial eccentricity vanishes. All of the momentum anisotropy is built up during the first 6 fm/.

Near a phase transition (in particular a first order transition) the equation of state becomes very soft, and this inhibits the generation of transverse flow. This also affects the generation of transverse flow anisotropies as seen from the solid curves in Figure 9: The rapid initial rise of suddenly stops as a significant fraction of the fireball matter enters the mixed phase. It then even decreases somewhat as the system expands radially without further acceleration, thereby becoming more isotropic in both coordinate and momentum space. Only after the phase transition is complete and pressure gradients reappear, the system reacts to the remaining spatial eccentricity by a slight further increase of the momentum anisotropy. The softness of the equation of state near the phase transition thus focusses the generation of anisotropic flow to even earlier times, when the system is still entirely partonic and has not even begun to hadronize. At RHIC energies this means that almost all of the finally observed elliptic flow is created during the first 3-4 fm/ of the collision and reflects the hard QGP equation of state of an ideal gas of massless particles (4] Microscopic kinetic studies of the evolution of elliptic flow lead to similar estimates for this time scale.[75, 76, 77, 78] ).[

We close this Section with a beautiful example of elliptic flow from outside the field of heavy-ion physics where the hydrodynamically predicted spatial expansion pattern shown in Figure 8 has for the first time been directly observed experimentally:[26] Figure 10 shows absorption images of an ensemble of about 200,000 Li atoms which were captured and cooled to ultralow temperatures in a CO laser trap and then suddenly released by turning off the laser. The trap is highly anisotropic, creating a pencil-like initial spatial distribution with an aspect ratio of about 29 between the length and diameter of the pencil. The interaction strength among the fermionic atoms can be tuned with an external magnetic field by exploiting a Feshbach resonance. The pictures shown in Figure 10 correspond to the case of very strong interactions. The right panels in Figure 10 show that the fermion gas expands in the initially short (“transverse”) direction much more rapidly than along the axis of the pencil. As argued in the paper,[26] the measured expansion rates in either direction are consistent with hydrodynamic calculations.[27] At late times the gas evolves into a pancake oriented perpendicular to the pencil axis. The aspect ratio passes through 1 (i.e. ) about 600 s after release and continues to follow the hydrodynamic predictions to about 800 s after release. At later times it continues to grow, but more slowly than predicted by hydrodynamics, perhaps indicating a gradual breakdown of local thermal equilibrium due to increasing dilution. The authors of the paper[26] argue that, although the scattering among the fermions is very strong by design, it does not seem to be enough to ensure rapid local thermalization, and that sufficiently fast healing of deviations from local equilibrium caused by the collective expansion might require the fermions to be in a superfluid state.

### 4 Experimental observables

Unfortunately, the small size and short lifetime prohibits a similar direct observation of the spatial evolution of the fireball in heavy-ion collisions. Only the momenta of the emitted particles are directly experimentally accessible, and spatial information must be extracted somewhat indirectly using momentum correlations. In the present Section we discuss measurements from Au+Au collisions at RHIC and compare them with hydrodynamic calculations. Most of the available published data stem from the first RHIC run at GeV, but a few selected preliminary data from the 200 GeV run in the second year will also be studied. Occasionally comparison will be made with SPS data from Pb+Pb collisions at GeV.

This Section is subdivided into two major parts: In Section 4.1 we discuss single-particle momentum spectra, first averaged over the azimuthal emission angle and later analyzed for their anisotropies around the reaction plane. These data provide a complete characterization of the momentum-space structure of the fireball at freeze-out. In particular the analysis of momentum anisotropies yields a strong argument for rapid thermalization in heavy-ion collisions and for the creation of a quark-gluon plasma.

In the second part, Section 4.2, we discuss the extraction of coordinate-space information about the fireball at freeze-out from Bose-Einstein interferometry which exploits quantum statistical two-particle momentum correlations between pairs of identical bosons. The general framework of this method is discussed in the accompanying article by Tomášik and Wiedemann.[79] Here we will discuss specific aspects of Bose-Einstein correlations from hydrodynamic calculations, in particular their dependence on the azimuthal emission angle relative to the reaction plane and its implications for the degree of spatial deformation of the fireball at freeze-out.

#### 4.1 Momentum space observables

The primary observables in heavy-ion collisions are the triple-differential momentum distributions of identified hadrons as a function of collision centrality (impact parameter ):

(23) |

We have expanded the dependence on the azimuthal emission angle relative to the reaction plane into a Fourier series.[80] Due to reflection symmetry with respect to the reaction plane, only cosine terms appear in the expansion. As explained before, we restrict our attention to midrapidity,

As already mentioned, the parameters of the hydrodynamic model are fixed by reproducing the measured centrality dependence of the total charged multiplicity as well as the shape of the pion and proton spectra in central collisions at midrapidity (see below). The shapes of other hadron spectra, their centrality dependence and the dependence of the elliptic flow coefficient on , centrality and hadron species are then all parameter free predictions of the model.[81] The same holds for all two-particle momentum correlations.[13, 14, 74] These predictions will be compared with experiment and used to test the hydrodynamic approach and to extract physical information from its successes and failures.

##### 4.1.1 Single particle spectra

The free parameters of the hydrodynamic model are the starting (thermalization) time , the entropy and net baryon density in the center of the reaction zone at this time, and the freeze-out energy density . The corresponding quantities at other fireball points at are then determined by the Glauber profiles discussed in Sec. 2.4 (see discussion below Fig. 2). The ratio of net baryon to entropy density is fixed by the measured proton/pion ratio. Since the measured chemical composition of the final state at RHIC was found[82] to accurately reflect a hadron resonance gas in chemical equilibrium at the hadronization phase transition, we require the hydrodynamic model to reproduce this ratio on a hypersurface of temperature . By entropy conservation, the final total charged multiplicity fixes the initial product .[24, 62, 70] The value of controls how much transverse flow can be generated until freeze-out. Since the thermal motion and radial flow affect light and heavy particles differently at low ,[67, 83] a simultaneous fit of the final pion and proton spectra separates the radial flow from the thermal component. The final flow strength then fixes whereas the freeze-out temperature determines the energy density at decoupling.

The top left panel of Fig. 11 shows the hydrodynamic fit[14] to the transverse momentum spectra of positive pions and antiprotons, as measured by the PHENIX and STAR collaborations in central () Au+Au collisions at GeV.[84, 85, 86] The fit yields an initial central entropy density fm at an equilibration time fm. This corresponds to an initial temperature of MeV and an initial energy density GeV/fm in the fireball center. (Note that these parameters satisfy the “uncertainty relation” .) Freeze-out was implemented on a hypersurface of constant energy density with GeV/fm.

SPS | RHIC 1 | RHIC 2 | |

(GeV) | 17 | 130 | 200 |

(fm) | 43 | 95 | 110 |

(MeV) | 257 | 340 | 360 |

(fm/) | 0.8 | 0.6 | 0.6 |

Table 1. Initial conditions for SPS and RHIC energies used to fit the particle spectra from central Pb+Pb or Au+Au collisions. and refer to the maximum values at in the fireball center.

The fit in the top left panel of Fig. 11 was performed with a chemical equilibrium equation of state. Use of such an equation of state implicitly assumes that even below the hadronization temperature chemical equilibrium among the different hadron species can be maintained all the way down to kinetic freeze-out. With such an equation of state the decoupling energy GeV/fm translates into a kinetic freeze-out temperature of MeV. The data, on the other hand, show[82] that the hadron abundances freeze out at , i.e. already when hadrons first coalesce from the expanding quark-gluon soup the inelastic processes which could transform different hadron species into each other are too slow to keep up with the expansion. The measured ratio thus does not agree with the one computed from the chemical equilibrium equation of state at the kinetic freeze-out temperature MeV, and the latter must be rescaled by hand if one wants to reproduce not only the shape, but also the correct normalization of the measured spectra in Fig. 11.

A better procedure would be to use a chemical non-equilibrium equation of state for the hadronic phase[87, 88, 89] in which for temperatures below the chemical potentials for each hadronic species are readjusted in such a way that their total abundances (after decay of unstable resonances) are kept constant at the observed values. This approach has recently been applied[65] to newer RHIC data at GeV and will be discussed below.

For a single hadron species, the shape of the transverse momentum spectrum allows combinations of temperature and radial flow which are strongly anticorrelated.[67] By using two hadron species with significantly different masses this anticorrelation is strongly reduced albeit not completely eliminated. Consequently, the above procedure still leaves open a small range of possible variations for the extracted initial and final parameters. Within this range, we selected a value for which is, if anything, on the large side; some of the hadron spectra would be fit slightly better with even smaller values for or a non-zero transverse flow velocity already at (see below). Table 1 summarizes the initial conditions applied in our hydrodynamic studies at SPS,[24, 90] RHIC1[13, 23, 46, 81] and RHIC2[65] energies.

Once the parameters have been fixed in central collisions, spectra at other centralities and for different hadron species can be predicted without introducing additional parameters. The remaining three panels of Fig. 11 show the transverse momentum spectra of pions, kaons and antiprotons in five different centrality bins as observed by the PHENIX[84, 91] and STAR[85, 86] collaborations. For all centrality classes, except the most peripheral one, the hydrodynamic predictions (solid lines) agree pretty well with the data. The kaon spectra are reproduced almost perfectly, but for pions the model consistently underpredicts the data at low . This has now been understood to be largely an artifact of having employed in these calculations a chemical equilibrium equation of state all the way down to kinetic freeze-out. More recent calculations[65] with a chemical non-equilibrium equation of state, to be compared to GeV data below, show that, as the system cools below the chemical freeze-out point , a significant positive pion chemical potential builds up, emphasizing the concave curvature of the spectrum from Bose effects and increasing the feeddown corrections from heavier resonances at low . The inclusion of non-equilibrium baryon chemical potentials to avoid baryon-antibaryon annihilation further amplifies the resonance feeddown for pions.

Significant discrepancies are also seen at large impact parameters and large transverse momenta

For the calculations shown in Fig. 11 the same value was used for all impact parameters. Phenomenological studies[92] using a hydrodynamically motivated parametrization[67] to describe pion and antiproton spectra from 200 GeV Au+Au collisions in a large number of centrality bins indicate somewhat earlier freeze-out, at higher temperature and with less transverse flow, in the most peripheral collisions (see Fig. 12). In the hydrodynamic model this can be accommodated by allowing the freeze-out energy density to increase with impact parameter. A consistent determination of from the kinetic decoupling criterion is expected to automatically yield such a behavior. Such a calculation would use the fitted value for extracted from central collision data to determine the unknown proportionality constant between the local expansion and scattering time scales at decoupling (see discussion in Secs. 2.5 and 3.1), and then calculate for other impact parameters by using the kinetic freeze-out criterion with the same constant extracted from central collisions. So far this has not been done, though.

Without transverse flow, thermal spectra exhibit -scaling[35], i.e. after appropriate rescaling of the yields all spectra collapse onto a single curve. Transverse collective flow breaks this scaling at low (i.e. for non-relativistic transverse particle velocities) by an amount which increases with the particle rest mass .[68, 83, 93] When plotting the spectra against instead of , any breaking of -scaling is at least partially masked by a kinematic effect at low which unfortunately again increases with the rest mass . To visualize the effects of transverse flow on the spectral shape thus requires plotting the spectrum logarithmically as a function of or . Such plots can be found in recent experimental publications,[94, 95, 96] and although the viewer’s eye is often misled by superimposed straight exponential lines , a second glance shows a clear tendency of the heavier hadron spectra to curve and to begin to develop a shoulder at low transverse kinetic energy , as expected from transverse flow.

One such example is shown in Fig. 13 where preliminary spectra of hyperons[97] are compared with hydrodynamic predictions. For this comparison the original calculations for 130 GeV Au+Au collisions[81] were repeated with RHIC2 initial conditions and a chemical non-equilibrium equation of state in the hadronic phase.[65]

The solid lines are based on default parameters (see Table 1) without any initial transverse flow at . (The dashed lines will be discussed further below.) Following a suggestion that hyperons, being heavy and not having any known strong coupling resonances with pions, should not be able to participate in any increase of the radial flow during the hadronic phase and thus decouple early,[98] we show two solid lines, the steeper one corresponding to decoupling at GeV/fm, i.e. directly after hadronization at , whereas the flatter one assumes decoupling together with pions and other hadrons at GeV/fm. The data clearly favor the flatter curve, suggesting intense rescattering of the ’s in the hadronic phase. The microscopic mechanism for this rescattering is still unclear. However, without hadronic rescattering the hydrodynamic model, in spite of its perfect local thermalization during the early expansion stages, is unable to generate enough transverse flow to flatten the spectra as much as required by the data. Partonic hydrodynamic flow alone can not explain the spectrum.

We close this subsection with a comparison of pion, kaon and hadron spectra from 200 GeV Au+Au collisions (RHIC2) with recent hydrodynamic calculations which correctly implement chemical decoupling at . Figure 14 shows a compilation of preliminary spectra from the four RHIC collaborations.[99, 100, 101, 102] For better visibility, the , and spectra are separated artificially by scaling factors of 10 and 100, respectively. The lines reflect hydrodynamic results.

In these calculations the particle numbers of all stable hadron species were conserved throughout the hadronic resonance gas phase of the evolution, by introducing appropriate chemical potentials.[87, 88, 89] It turns out that such a chemical non-equilibrium equation of state has almost the same relation between the pressure and energy density as the equilibrium one, and that the hydrodynamical evolution remains almost unaltered.[87, 89] However, the relation between the decoupling energy density and the freeze-out temperature changes significantly, since the non-equilibrium equation of state prohibits the annihilation of baryon-antibaryon pairs as the temperature drops. Consequently, at any given temperature below the non-equilibrium equation of state contains more heavy baryons and antibaryons than the equilibrium one and thus has a higher energy density.

The same energy density GeV/fm then translates into a significantly lower freeze-out temperature MeV.[87, 89, 65] The corresponding results are given as thick solid (red) lines in Fig. 14. The thin solid (blue) lines in the Figure, shown for comparison, were calculated by assuming kinetic freeze-out already at hadronization, MeV. The data clearly favor the additional radial boost resulting from the continued buildup of radial flow in the hadronic phase. Still, even at GeV/fm, the spectra are still steeper than the data and the previous calculations with a chemical equilibrium equation of state shown in Fig. 11, reflecting the combination of the same flow pattern with a lower freeze-out temperature. Somewhat unexpectedly, the authors of the study[65] were unable to significantly improve the situation by reducing even further: The effects of a larger radial flow at lower were almost completely compensated by the accompanying lower freeze-out temperature, leading to only modest improvements for kaons and protons and almost none for the pions.

This motivated the authors[65] to introduce a small but non-vanishing transverse “seed” velocity already at the beginning of the hydrodynamic stage. The dashed lines in Fig. 14 (and also earlier in Fig. 13) show hydroynamic calculations with an initial transverse flow velocity profile given by fm. This initial transverse kick is seen to significantly improve the agreement with the pion, kaon and antiproton data up to GeV/ for pions and kaons and up to GeV/ for (anti)protons.[65] It can be motivated by invoking some collective (although not ideal hydrodynamic) transverse motion of the fireball already during the initial thermalization stage, although the magnitude of the parameter requires further study. with

In Figure 14 the kaon and antiproton spectra were divided by factors of 10 and 100, respectively, for better visibility. If this is not done one notices that the antiproton spectrum crosses the pion spectrum at around GeV/,[91, 99] i.e. at larger antiprotons are more abundant than pions! This became known as the “ anomaly” and has attracted significant attention.[103] Here the word “anomaly” arises from a comparison of this ratio in central Au+Au with collisions and with string fragmentation models which both give ratios much below 1. However, in Au+Au collisions string fragmentation is expected to explain hadron production only at rather large ,[104] and in the hydrodynamic picture which is successful at GeV/ there is actually nothing anomalous about a ratio that exceeds 1.

To see this let us first look at a thermal system without flow. The corresponding transverse mass spectra are in good approximation simple exponentials in whose ratios at fixed are simply given by the ratios of their spin-isospin degeneracies and fugacities. For sufficiently large , and the same holds true for the ratio of the -spectra at fixed . It still holds true in the presence of transverse flow which, at sufficiently large , simply flattens all -slopes by a common blueshift factor[67, 83, 93] . Since antiprotons have a 2-fold spin degeneracy and pions have none, we see that the asymptotic hydrodynamic ratio is above unity if the chemical potentials are sufficiently small.

164 MeV | 29 MeV | -29 MeV | 0 | 0 |

100 MeV | 379 MeV | 344 MeV | 81 MeV | 167 MeV |

164 MeV | 0.7 | 2.4 | 1.7 | 1.0 |

100 MeV | 0.7 | 40 | 28 | 2.4 |

Table 2. Upper part: Chemical potentials of protons, antiprotons, pions and kaons at the chemical (164 MeV) and kinetic (100 MeV) freeze-out temperatures for 200 GeV Au+Au collisions.[65] Lower part: Asymptotic particle ratios for these hadrons at fixed large , for two assumed hydrodynamic freeze-out temperatures of 164 and 100 MeV, respectively (see text).

In Au+Au collisions at GeV the baryon chemical potential at chemical freeze-out is small[82] ( MeV) and the pion chemical potential vanishes. Correspondingly, the asymptotic and ratios are both close to 2 (see Table 2).

As the system cools below the chemical freeze-out temperature, however, pions, kaons and both protons and antiprotons all develop significant positive chemical potentials which are necessary to keep their total abundances (after unstable resonance decays) fixed at their chemical freeze-out values[88] (second row in Table 2). As a consequence, the asymptotic and ratios increase dramatically, to 40 and 28, respectively, and even the asymptotic ratio increases from 1 to 2.4.

We see that cooling at constant particle numbers strongly depletes the pions at high in favor of high- baryons and kaons. Correspondingly, the ratios of the hydrodynamic spectra, shown in Fig. 15, rise far above unity at large . We should stress that the increase with of these ratios at small is generic for thermalized spectra and independent of whether or not there is radial flow. It is a simple kinematic consequence of replotting two approximately parallel exponentials (more exactly: K-functions) in as functions of and taking the ratio. Due to the larger rest mass the -spectrum of the heavier particle is flattened more strongly at low than that of the lighter particle, yielding for their ratio a rising function of . The additional flattening from radial flow, which again affects the heavier particles more strongly than the light ones, further increases this tendency.

It is worth pointing out that a ratio well above 2 and and a ratio well above 1, as hydrodynamically predicted for GeV/ (see Fig. 15) would, when taken together with the measured global thermal yields, provide a unique proof for chemical and kinetic decoupling happening at different temperatures. Unfortunately, as we will see in more detail later, hydrodynamics begins to seriously break down exactly in this interesting domain, and the experimentally observed ratios[99] never appear to grow much beyond unity before decreasing again at even higher , eventually perhaps approaching the small asymptotic values expected from jet fragmentation.[104]

##### 4.1.2 Mean transverse momentum and transverse energy

The good agreement of the hydrodynamic calculations with the experimental transverse momentum spectra is reflected in a similarly good description of the measured average transverse momenta.[72] Figure 16 shows a comparison of for identified pions, kaons, protons and antiprotons measured by PHENIX in 200 GeV Au+Au collisions[99, 105] with the hydrodynamic results.[65] The bands reflect the theoretical variation resulting from possible initial transverse flow already at the beginning of the hydrodynamic expansion stage, as discussed at the end of the previous subsection. The figure shows some discrepancies between hydrodynamics and the data for peripheral collisions (small ) which are strongest for the kaons whose spectra are flatter at large impact parameters than predicted by the model.

Figure 17 shows the total transverse energy per emitted charged hadron as a function of collision centrality. The data are from Pb+Pb collisions at the SPS[106] and Au+Au collisions at two RHIC energies.[107, 108] Although both the charged particle multiplicity and total transverse energy vary strongly with the number of participating nucleons (and

one or the other are therefore often used to determine the collision centrality), the transverse energy per particle is essentially independent of the centrality. It also depends only weakly on the collision energy.

The superimposed band in Figure 17 reflects hydrodynamic calculations for Au+Au collisions at GeV with and without initial transverse flow, as before. The slight rise of the theoretical curves with increasing can be attributed to the larger average transverse flow developing in more central collisions, resulting from the higher initial energy density and the somewhat longer duration of the expansion until freeze-out.[46] Successful reproduction of the data requires a correct treatment of the chemical composition at freeze-out (by using a chemical non-equilibrium hadron equation of state below ). If one instead assumes chemical equilibrium of the hadron resonance gas down to kinetic freeze-out, hydrodynamics overpredicts the transverse energy per particle by about 15-20%.[46]

##### 4.1.3 Momentum anisotropies as early fireball signatures

In non-central nuclear collisions, or if the colliding nuclei are deformed, the nuclear overlap region is initially spatially deformed (see Fig. 3). Interactions among the constituents of the matter formed in that zone transfer this spatial deformation onto momentum space. Even if the fireball matter does not interact strongly enough to reach and maintain almost instantaneous local equilibrium, and a hydrodynamic description therefore fails, any kind of re-interaction among the fireball constituents will still be sensitive to the anisotropic density gradients in the reaction zone and thus redirect the momentum flow preferably into the direction of the strongest density gradients (i.e. in the “short” direction).[75, 76, 78, 109] The result is a momentum-space anisotropy, with more momentum flowing into the reaction plane than out of it.

Such a “momentum-space reflection” of the initial spatial deformation is a unique signature for re-interactions in the fireball and, when observed, proves that the fireball matter has undergone significant nontrivial dynamics between creation and freeze-out. Without rescattering, the only other mechanism with the ability to map a spatial deformation onto momentum space is the quantum mechanical uncertainty relation. For matter confined to smaller spatial dimensions in than in direction it predicts for the corresponding widths of the momentum distribution. However, any momentum anisotropy resulting from this mechanism is restricted to momenta /(size of the overlap zone) which for a typical fireball radius of a few fm translates into a fraction of 200 MeV/. This is the likely mechanism for the momentum anisotropy observed[110] in calculations of the classical dynamical evolution of a postulated deformed “color glass condensate” created initially in the collision. Unlike the experimental data, this momentum anisotropy is concentrated around relatively low .[110]

Whatever the detailed mechanism responsible for the observed momentum anisotropy, the induced faster motion into the reaction plane than perpendicular to it (“elliptic flow”) rapidly degrades the initial spatial deformation of the matter distribution and thus eliminates the driving force for any further increase of the anisotropic flow. Elliptic flow is therefore “self-quenching”,[75, 76] and any flow anisotropy measured in the final state must have been generated early when the collision fireball was still spatially deformed (see Fig. 9). If elliptic flow does not develop early, it never develops at all (see also Sec. 4.1.4). It thus reflects the pressure and stiffness of the equation of state during the earliest collision stages,[75, 76, 77, 4, 24] but (in contrast to many other early fireball signatures) it can be easily measured with high statistical accuracy since it affects all final state particles.

Microscopic kinetic models show (see Fig. 19 below) that, for a given initial spatial deformation, the induced momentum space anisotropy is a monotonically rising function of the strength of the interaction among the matter constituents.[77, 78, 109] The maximum effect should thus be expected if their mean free path approaches zero, i.e. in the hydrodynamic limit.[4, 13] Within this limit, we will see that the magnitude of the effect shows some sensitivity to the nuclear equation of state in the early collision stage, but the variation is not very large. This implies that, since the initial spatial deformation can be computed from the collision geometry (the average impact parameter can be determined, say, from the ratio of the observed multiplicity in the event to the maximum multiplicity from all events), the observed magnitudes of the momentum anisotropies, and in particular their dependence on collision centrality,[109, 111] provide valuable measures for the degree of thermalization reached early in the collision.

Experimentally this program was first pursued at the SPS in 158 GeV
Pb+Pb collisions.[112]
These data still showed significant sensitivity to details of the
analysis procedure[113] and thus remained somewhat
inconclusive.[23]
Qualitatively, the SPS data (where the directed and elliptic flow
coefficients, and , can both be measured) confirmed
Ollitrault’s 1992 prediction[70] that near midrapidity
the preferred flow direction is into the reaction plane,
supporting the conclusions from earlier measurements in Au+Au
collisions at the AGS[114] where a transition from
out-of-plane to in-plane elliptic flow had been found between 4
and 6 GeV beam energy.
A comprehensive quantitative discussion of elliptic flow became first
possible with RHIC data, because of their better statistics and
improved event plane resolution (due to the larger event multiplicities)
and also as a result of improved[111] analysis techniques.^{1}^{1}1One of the important experimental issues is the
separation[115, 116] of collective “flow” contributions
to the observed momentum anisotropies from “non-flow” angular
correlations, such as Bose-Einstein correlations,[117]
correlations arising from momentum conservation,[118, 119]
and two-particle momentum correlations from resonance decays and jet
production.[120]
Many of the data shown in this review have not yet been corrected for
non-flow contributions, but subsequent analysis[121]
has shown that for the RHIC data and in the -range of interest
for this review these corrections are small ().
In the meantime the latter have also been re-applied to SPS data
and produced very detailed results from Pb+Pb collisions at this
lower beam energy.[122, 123]

##### 4.1.4 Elliptic flow at RHIC

The second published and still among the most important results from Au+Au collisions at RHIC was the centrality and dependence of the elliptic flow coefficient at midrapidity.[124] For central to midperipheral collisions and for transverse momenta GeV/ the data were found to be in stunning agreement with hydrodynamic predictions,[4, 23] as seen in Fig. 18. In the left panel, the ratio of the charged particle multiplicity to the maximum observed value is used to characterize the collision centrality, with the most central collisions towards the right near 1. corresponds to an impact parameter fm.[121] Up to this value the observed elliptic flow is found to track very well the increasing initial spatial deformation of the nuclear overlap zone,[121] as predicted by hydrodynamics.[4]

Following this discovery it soon became clear that the agreement of the data with the hydrodynamic calculations could not be accidental and in fact allows to draw a number of very strong and important conclusions. These conclusions refer to soft particle production, that is to mesons with transverse momenta up to about 1.5 GeV/ and baryons with GeV/. This momentum range covers well over 99% of the produced particles. This means that we are talking about the global dynamical features of the bulk of the fireball matter. Of course, the small fraction of particles emitted with larger transverse momenta carry very important information themselves, but their behavior is not expected to be controlled by hydrodynamics in the first place, and they are not the subject of our discussion here. It should be noted, however, that interpreting the behavior of high- particles does require a prior understanding of the global fireball dynamics which is the subject of this review. In the following we discuss the aforementioned conclusions, as well as a number of additional theoretical and experimental aspects.

Strong rescattering:

It was quickly realized[78] that the measured[124] almost linear rise of the charged particle (i.e. predominantly pionic) elliptic flow with requires strong rescattering among the fireball constituents. Figure 19 shows the results from microscopic simulations which describe the dynamics of the early expansion stage by solving a Boltzmann equation for colliding on-shell partons.[78] The different curves are parametrized by the transport opacity involving the product of the parton rapidity density and cross section in the early collision stage. As the opacity is increased, the elliptic flow is seen to approach the data (and the hydrodynamic limit) monotonically from below. Whereas the hydrodynamic limit predicts a continuous rise of , the elliptic flow from the parton cascade saturates at high , as also seen in the data[125] (see Fig. 26 below). This is due to incomplete equilibration at high : the critical at which the cascade results cease to follow the hydrodynamic rise shifts to higher (lower) values as the transport opacity is increased (decreased).

It is interesting to observe that stronger rescattering manifests itself in this way, i.e. by following the hydrodynamic curve with the full slope to higher and not by producing a hydro-like quasi-linear rise with a reduced slope. In view of this the elliptic flow data at large impact parameters (see Fig. 20 below) and at lower collision energies[23, 112, 122, 123], which show a linear rise of with a smaller slope than hydrodynamically predicted, pose an unresolved puzzle which is not simply explained by incomplete local thermalization.

Figure 19 also shows that very high transport opacities are necessary if the parton cascade is required to follow the data to GeV/. The necessary values exceed perturbative expectations by at least a factor 30,[78] raising the question which microscopic interaction mechanism is responsible for the large observed elliptic flow.[13, 14, 126] However, it was recently discovered[127] that the partonic elliptic flow may not necessarily have to follow the hydrodynamic prediction all the way out to GeV/: If the elliptic flow of the partons gets transferred to the hadrons by a momentum-space coalescence mechanism,[128] it is sufficient if it behaves hydrodynamically up to GeV/ for the pion and proton to “look hydrodynamic” up to GeV/ and 2.2–2.4 GeV/, respectively.[127] This takes away some of the pressure for anomalously large partonic transport opacities.[127]

Centrality dependence of elliptic flow:

Figure 18 showed that in peripheral collisions the -integrated elliptic flow lags behind the hydrodynamic predictions. This may reflect incomplete thermalization in the smaller fireballs created in these cases. To study this in more detail, Fig. 20 shows the -differential elliptic flow for pions and protons for three centrality bins.[129]

Due to limitations for particle identification, the data cover only the low- region up to about 800 MeV/. In this region, the left panel shows good agreement of for pions with the hydrodynamic predictions[81] for central and midcentral collisions, but smaller elliptic flow than predicted for the most peripheral bin (45–85% centrality, corresponding to an average impact parameter of about 11 fm[121]). The graph clarifies that for peripheral collisions the smaller-than-predicted -integrated elliptic flow seen in Fig. 18 arises mostly from a smaller-than-predicted slope of the -differential elliptic flow for pions. For the most peripheral collisions this slope is about 20% less than expected if also there the reaction zone were able to fully thermalize. In view of the large average impact parameter in this centrality bin, it is rather surprising that the discrepancy to the hydrodynamic predictions is not larger.

The right panel in Fig. 20 shows a similar comparison for protons and antiprotons. Due to the limited statistics of the data, which also were not fully corrected for feeddown from weak decays, no strong conclusions can be drawn from the plot, but the data seem to be generally on the low side compared to the hydrodynamic curves. However, this can have other reasons than a breakdown of hydrodynamics, due to a specific sensitivity of the elliptic flow of heavy hadrons to the nuclear equation of state (see below).

In hydrodynamic calculations the finally observed elliptic flow is essentially proportional to the initial spatial eccentricity of the reaction zone (Section 2.4). This is displayed in the left panel of Fig. 21, where the elliptic flow scaled by the initial eccentricity is plotted as a function of impact parameter[4] (note the suppressed zero on the vertical axis). The slight variation of the ratio with impact parameter reflects changes in the stiffness of the equation of state, resulting from the quark-hadron phase transition, which are probed as the impact parameter (and thus the initial energy density in the center of the reaction zone) is varied.[4] This will be discussed in more detail when we describe the beam energy dependence of elliptic flow.

The right panel of Fig. 21 shows RHIC data from the PHENIX Collaboration[130] for the same ratio, at low and high transverse momenta. While the low- data agree with the hydrodynamic prediction of an approximately constant ratio , at high the scaled elliptic flow is seen to decrease for more peripheral collisions. This is consistent with the earlier discussion of a gradual breakdown of hydrodynamics for increasing and impact parameter.

Elliptic flow of different hadron species:

Hydrodynamics predicts a clear mass-ordering of elliptic flow.[81] As the collective radial motion boosts particles to higher average

velocities, heavier particles gain more momentum than lighter ones, leading to a flattening of their spectra at low transverse kinetic energies.[83] When plotted against this effect is further enhanced by a kinematic factor arising from the transformation from to (see Fig. 14 and discussion below Fig. 12). This flattening reduces the momentum anisotropy coefficient at low ,[81] and the heavier the particle the more the rise of is shifted towards larger (see top left panel in Fig. 22).

This effect, which is a consequence of both the thermal shape of the single-particle spectra at low and the superimposed collective radial flow, has been nicely confirmed by the experiments: Figure 22 and the right panel of Fig. 23 show that the data[129, 131, 132] follow the predicted mass ordering out to transverse momenta of about 1.5 GeV/. For and much more accurate data than those shown in Fig. 22 (top right) have recently become available from 200 GeV Au+Au collisions,[133] again in quantitative agreement with hydrodynamic calculations up to GeV/ for kaons and up to GeV/ for . The inversion of the mass-ordering in the data at large is caused by the mesons whose breaks away from the hydrodynamic rise and begins to saturate at GeV/. In contrast, baryons appear to behave hydrodynamically to GeV/, breaking away from the flow prediction and saturating at significantly larger than the mesons. This is consistent with the idea that the partonic elliptic flow established before hadronization exhibits a hydrodynamic rise at low followed by saturation above MeV/, and that these features are transferred to the observed hadrons by quark coalescence, manifesting themselves there at twice resp. three times larger -values.[127]

Sensitivity to the equation of state:

The experimental determination of the nuclear equation of state at high densities relies on detailed studies of collective flow patterns generated in relativistic heavy-ion collisions.[11] Since elliptic flow builds up and saturates early in the collision, it is more sensitive to the high density equation of state than the azimuthally averaged radial flow.[75] Hydrodynamic calculations allow to study in the most direct way the influence of the phase transition and its strength (the latent heat ) on the generated flow patterns. This was investigated systematically and in great detail by Teaney et al.,[18, 19, 20] using hydrodynamics to describe the early quark-gluon plasma expansion stage (including the quark-hadron phase transition), followed by a kinetic afterburner which simulates the subsequent hadronic evolution and freeze-out with the relativistic hadron cascade RQMD.[134]

One of their important results is shown in the left panel of Fig. 23 which (similar to Fig. 18) gives the -integrated elliptic flow as a function of the normalized particle yield as a centrality measure. The three curves correspond to equations of state with a first order quark-hadron transition, with latent heat values of 0.4, 0.8 and 1.6 GeV/fm, respectively.[20] Comparison with the results from the STAR Collaboration shows that a phase transition of significant strength ( GeV/fm) is necessary to reproduce the data. Without the softening of the equation of state induced by the phase transition, the single-particle spectra are too flat and the -integrated elliptic flow comes out too large, even though has roughly the correct slope for pions.

On the other hand, if one eliminates the “QGP push” entirely and replaces the hard quark-gluon plasma equation of state above the phase transition by a softer hadron resonance gas without phase transition (EOS H), one underpredicts the hydrodynamic mass-splitting of the elliptic flow.[14, 135] This is seen in the right panel of the figure which shows the -differential elliptic flow for pions and protons both with a realistic equation of state (EOS Q) and for a pure hadron resonance gas (EOS H). The proton data[129] shown in this plot obviously favor EOS Q, irrespective of moderate variations of the freeze-out temperature, indicated by the three lines labelled by EOS Q.[14, 135] Teaney et al.[19] came to similar conclusions; this implies that the details of how the hadronic rescattering stage is described (hydrodynamically with Cooper-Frye freeze-out[14, 135] or kinetically via a hadron cascade[18, 19, 20]) do not matter.

Rapid thermalization:

We can summarize the comparison between RHIC data and the hydrodynamic model up to this point by stating that hydrodynamics provides a good description of all aspects of the single particle momentum spectra, from central and semicentral Au+Au collisions up to impact parameters fm, and for transverse momenta up to 1.5 GeV/ for mesons and up to 2.5 GeV/ for baryons. Since this -range covers well over 95% of the emitted particles, it is fair to say that the bulk of the fireball matter created at RHIC behaves hydrodynamically, with little indication for non-ideal (viscous) effects. As explained, the successful description of the data by the hydrodynamic model requires starting the hydrodynamic evolution no later than about 1 fm/ after nuclear impact. This estimate is even conservative since it does not take into account any transverse motion of the created fireball matter between the time when the nuclei first collide and when the fireball has thermalized and the hydrodynamic expansion begins. We will now give an independent argument[4, 111] why thermalization must happen very rapidly in order for the elliptic flow signal to be as strong as observed in the experiments.

As shown earlier in this Section, the hydrodynamically predicted elliptic flow is proportional to the initial spatial eccentricity at the beginning of the hydrodynamic evolution. If thermalization is slow, the matter will start to evolve in the transverse directions even before , following its initial locally isotropic transverse momentum distribution. Even if no reinteractions among the produced particles occur, this radial free-streaming motion dilutes the spatial deformation, although not quite as quickly as in the opposite limit of complete thermalization where it decreases faster due to anisotropic hydrodynamic flow (see Fig.