Universal hydrodynamic flow in holographic planar shock collisions

We study the collision of planar shock waves in AdS$_5$ as a function of shock profile. In the dual field theory the shock waves describe planar sheets of energy whose collision results in the formation of a plasma which behaves hydrodynamically at late times. We find that the post-collision stress tensor near the light cone exhibits transient non-universal behavior which depends on both the shock width and the precise functional form of the shock profile. However, over a large range of shock widths, including those which yield qualitative different behavior near the future light cone, and for different shock profiles, we find universal behavior in the subsequent hydrodynamic evolution. Additionally, we compute the rapidity distribution of produced particles and find it to be well described by a Gaussian.


Introduction and summary
Holographic duality [18][19][20] has proven to be a useful tool to study the dynamics of strongly coupled quark-gluon plasma (for a review see for example [17]). One interesting problem is the collision of gravitational shock waves in AdS 5 , which can result in the formation of a black hole. In the dual field theory, which is N = 4 supersymmetric Yang-Mills (SYM), colliding gravitational waves are equivalent to colliding distributions of energy, which for brevity we simply refer to as shock waves. The formation of a black hole is dual to the formation of a quark-gluon plasma and the ring down of the black hole encodes the relaxation of the plasma to a hydrodynamic description. The complete evolution of the field theory stress tensor T µν -from pre-collision dynamics to far-from-equilibrium dynamics to hydrodynamics -is encoded in the dual classical relativity problem.
Unlike QCD, where nuclei are bound states whose energy distribution is fixed by the theory, in conformal SYM the energy distribution of colliding shocks is not fixed; shock energy distributions can have any desired shape and only must propagate at the speed of light. In the dual gravitational description this reflects the fact that gravitational waveforms are not fixed by Einstein's equations and gravitational waves always propagate at the speed of light. This has led to studies of a diverse range of energy profiles from planar shocks with δ-function longitudinal profiles [9,30,31] to planar shocks with finite longitudinal thickness [1,2,11,15] to shocks which are also localized in the plane transverse to the collision axis [8,14,21,22] to holographic models of proton-nucleus collisions [29].
The fact that SYM doesn't specify the energy distribution of colliding shocks begs the question: how much do details in different energy distributions imprint themselves on the future hydrodynamic evolution of the produced quark-gluon plasma? What features of JHEP11(2015)135 collisions in SYM are universal and what features depend on one's chosen energy profile for the shocks? Indeed, in [1], where the collision of planar shocks with Gaussian longitudinal profiles was studied, it was found that some qualitative features of the debris produced by the collision are sensitive to the thickness of the shocks. For suitably thin shocks remnants of the initial shocks can survive the collision event and propagate on the forward light cone and regions of negative energy density appear near the light cone. In contrast, when the shock thickness is suitably large no obvious remnants of the initial shocks survive the collision event and the energy density is everywhere smooth and positive in the forward light cone [1,15].
To begin to address the above questions we focus on the simple case of planar shock collisions in AdS 5 , where the shocks have no dependence on the coordinates transverse to the collision axis. The shocks move in the ±z direction at the speed of light and have energy density Hence, the energy per unit transverse area of the shocks is N 2 c 2π 2 µ 3 . We investigate the collision dynamics as a function of the shock width and the functional form of δ w . For simplicity we consider profiles δ w in which w is the only scale. The dimensionless measure of the shock width is µw for which we consider µw 1 2 . In contrast to the collisions studied in [1,15], which were in the background of a low temperature plasma, we study collisions at zero background temperature. This allows us to study long time evolution without pollution due to thermal regulators.
We find that the post-collision stress tensor near the light cone is non-universal and depends on both the shock width w and the precise functional form of the shock profile δ w . However, we observe that the non-universal behavior is transient: irrespective of w or δ w , long after the collision event, nearly all the energy lies inside the future light cone and the evolution of the stress tensor is governed by hydrodynamics. Over a large range of shock widths, including those which yield qualitative different behavior near the future light cone, and for different shock profiles δ w , we find universal behavior in the initial hydrodynamic data. On a surface of constant proper time τ = τ init t hydro , with t hydro ≈ 2/µ the hydrodynamization time, we find that the fluid velocity is well described by boost invariant flow and that the proper energy density is well described by where ξ is spacetime rapidity and f is a w-independent function with ξ FWHM its full width at half maximum. Therefore, the only w dependence in the hydrodynamic flow is in the proper energy's normalization and rapidity width. Aside from this w dependence, we observe that the hydrodynamic flow is otherwise insensitive to the precise functional form of δ w .

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Choosing τ init = 3.5/µ and the normalization condition f (0) = 1, we find A(µw) and ξ FWHM (µw) are well approximated by the quadratic functions Note dξ FWHM d(µw) ≈ −1, signifying appreciable w dependence. The function f is well described by a Gaussian with unit full width at half maximum, Given that the w → 0 limit of our collisions is that of colliding δ-functions, it is not surprising that the hydrodynamic evolution becomes insensitive to both w and the details of the shock profile when w is sufficiently small. Indeed, similar insensitivities were observed in [11]. However, we find it surprising that finite w effects merely alter the normalization and rapidity width of the proper energy, as opposed to changing its functional form altogether, and don't affect the fluid velocity. This is especially noteworthy given that the w-dependence of the rapidity width ξ FWHM in (1.4) is not weak.
Additionally, we construct a simple framework to extend the universal initial hydrodynamic data for planar shocks to initial hydrodynamic data for shocks with slowly varying transverse profiles. Using this framework, we employ our planar shocks to study axisymmetric collisions. We evolve our initial axisymmetric data forward in time using viscous hydrodynamics and then compute the rapidity spectrum of produced hadrons using a Cooper-Frye freeze out prescription [16]. We find that the rapidity distribution of massless particles is well approximated by a Gaussian with variance 1.9 and 2.1 for √ s N N energies of 200 GeV and 2.76 TeV respectively. An outline of our paper is as follows. In section 2 we develop the gravitational setup for our planar shock collisions. In section 3 we present our results for planar collisions, including early time non-universal transient effects, and universal late time hydrodynamic evolution. In section 4 we develop the framework for including slowly varying transverse dynamics and present hydrodynamic simulations for axisymmetric collisions. In section 5 we present results for the spectrum of produced hadrons. We conclude in section 6.

Setup
We construct initial data for Einstein's equations by superimposing the metric of gravitational shock waves moving in the ±z directions. In Fefferman-Graham coordinates, the metric of a single shock moving in the ±z direction is x ⊥ are the two coordinates transverse to the collision axis, r is the AdS radial coordinate, and

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The AdS curvature scale has been set to unity. The boundary of the asymptotically AdS spacetime lies at radial coordinate r = ∞. The single-shock metric (2.1) is an exact solution to Einstein's equations for any choice of H ± [8]. This geometry represents a state in the dual SYM theory with stress tensor, with non-zero components where H ± is the 2D transverse Fourier transform of H ± . Note that here and in what follows we used hats to denote quantities normalized by N 2 c 2π 2 . That is, for any function F we define where µ is an energy scale and δ w (z ∓ ) is a smeared δ-function which satisfies the normalization and variance conditions in eq. (1.2). We employ two different shock profiles We refer to the first shock profile in (2.6) as the Gaussian profile and the second as the non-Gaussian profile. These shock profiles are plotted in figure 1.
At early times, t −w, the functions h ± have negligible overlap and the pre-collision geometry can be constructed from (2.1) by replacing the last term with the sum of corresponding terms from left and right moving shocks. The resulting metric satisfies Einstein's equations, at early times, up to exponentially small errors.
To evolve the pre-collision geometry forward in time through the collision we use the characteristic formulation of gravitational dynamics in asymptotically AdS spacetimes discussed in detail in [2]. Our metric ansatz reads with Greek indices denoting spacetime boundary coordinates, x µ = (t, x ⊥ , z). Near the boundary, g µν = η µν + g (4) µν /r 4 + O(1/r 5 ). The sub-leading coefficients g (4) µν determine the boundary stress tensor, 00 . (2.9) To generate initial data for our characteristic evolution, we numerically transform the pre-collision metric in Fefferman-Graham coordinates to the metric ansatz (2.8); this requires computing a congruence of infalling radial null geodesics and is outlined in [10].
We then numerically solve Einstein's equations using the methods outlined in [2]. We measure dimensional quantities in units of µ. For the Gaussian profile we study collisions with shock widths w = nw o , n = 1, 2, . . . , 7 where 1 For the non-Gaussian profile we study collisions with w = w o and w = 5w o . We time evolve all collisions µt = 14 units after the collision event. After numerically solving Einstein's equations we extract the boundary stress tensor via (2.9) and (2.3).
3 Results for planar shocks

Early time dynamics and non-universal transient effects
Let us begin by focusing on the energy density produced by Gaussian shock collisions. In figure 2 we plot the rescaled energy density T 00 , for Gaussian shock profiles with widths w = 5w o (top) and w = w o (bottom). The shocks approach each other at the speed of light in the ±z direction and collide at z = 0 at time t = 0. For both cases, the debris leaving the collision event appears dramatically different than the initial incoming shocks. Nevertheless, comparing the two plots in figure 2, it is clear that there are qualitative differences in the energy density near the light cone. To highlight these differences, in figure 3 we plot T 00 at times µt = 1, 3, 5, 7, again for Gaussian shock profiles with widths w = 5w o and w =    Our numerics are consistent with a t −0.9 power law decay, which was also seen in [2]. Note that a power law decay is natural in a conformal theory, where there is no intrinsic scale. By time µt = 14 more than 90% of the initial shock energy lies in the forward light cone. Hence, our results suggest that irrespective of the shock width, the collision results in the complete annihilation of the shocks with essentially all energy lying well inside the forward light cone at late times. How sensitive is the stress near the forward light cone to perturbations in the functional form of the shock profile? In figure 5 we plot the energy density at time µt = 4 with width w = w o for both Gaussian and non-Gaussian shock profiles. Clearly, near the light cone the stress is different for the different shock profiles. The relative magnitude of the change should be compared to that of the different profiles shown in figure 1. Evidently, in addition to being sensitive to the width of the shocks, the stress tensor near the future light cone is also sensitive to the functional form of the shock profile. We note, however, that the amplitude of the decaying shock remnants also scales like t −0.9 for the non-Gaussian profile.

Universal initial data for hydrodynamics
According to fluid/gravity duality [3,4], at sufficiently late times the evolution of the stress tensor should be governed by hydrodynamics. How do the decaying shock remnants and negative energy density near the light cone imprint themselves on the hydrodynamic evolution? Are there qualitative differences between the hydrodynamic evolution for thick and thin shocks? To address these questions we first identify the domain R in spacetime in which hydrodynamics is a good description of the evolution of the stress and then study the w and δ w dependence of the hydrodynamic variables on a fixed Cauchy surface in R.
In relativistic neutral fluid hydrodynamics the hydrodynamic variables are typically taken to be the proper energy density and the fluid velocity u µ . The fluid velocity is defined to be the normalized time-like (u µ u µ = −1) future directed (u 0 > 0) eigenvector of the stress tensor,

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with the associated eigenvalue. In terms of and u µ the constitutive relations of fluid/gravity read T µν hydro = pg µν + ( + p)u µ u ν + Π µν , (3.2) where p = 3 is the pressure and Π µν is the viscous stress. The viscous stress satisfies u µ Π µν = 0 and g µν Π µν = 0 and at first order in gradients is given by Π µν = −ησ µν with η the shear viscosity and the shear tensor. The shear viscosity may be expressed in terms of the proper energy via [6,7] η = 1 3πT , with the temperature T given by The hydrodynamic equations of motion are given by the energy-momentum conservation equation ∂ µ T µν hydro = 0. Note that the hydrodynamic stress tensor is completely determined by four functions. In contrast, in general the exact (traceless) stress tensor contains nine independent function.
Instead of solving the hydrodynamic equations of motion for the evolution of and u µ , a simple way to compare the gravitational evolution to hydrodynamics is to extract the exact and u µ from the eigenvalue equation (3.1) with the exact stress tensor. In the domain where hydrodynamics is a good description this should yield the same time evolution for proper energy density and fluid velocity as hydrodynamics. With the exact proper energy and fluid velocity known, we can then construct T µν hydro from eqs. (3.2)-(3.5) and compare T µν and T µν hydro . To quantify the domain in which hydrodynamics is applicable we then define the residual measure The quantity ∆, evaluated in the local fluid rest frame, measures the relative difference between the spatial stress in T µν and T µν hydro . Regions of spacetime with ∆ 1 are evolving hydrodynamically.
Let us first focus on the hydrodynamic evolution produced by Gaussian shock collisions. In figure 6 we plot the hydrodynamic residual ∆ for Gaussian shock profiles with widths w = 5w o (top) and w = w o (bottom). Note that we only plot ∆ in the region R defined to be the largest connected region in spacetime where ∆ ≤ 0.15. Outside of R hydrodynamics is not a good description and the fluid velocity need not even be well defined [5] (i.e. the stress need not have a time-like eigenvector). The dashed line in the figure, which bounds the region R, is given by with µτ * = 1.5 and µ∆t = 0.58. We therefore conclude that the domain of applicability of hydrodynamics is approximately the same for both shock thicknesses. Figure 6 clearly shows that our planar shock collisions result in the formation of an expanding volume of fluid which is well described by hydrodynamics everywhere except near the light cone, where non-hydrodynamic effects become important. At mid-rapidity viscous hydrodynamics becomes a good approximation at time µt hydro ≈ 2. (3.8) With the applicability of hydrodynamics established, we now turn to the w dependence of the initial hydrodynamic data. We introduce proper time τ and rapidity ξ coordinates via and study the hydrodynamic variables u µ and on the τ = τ init Cauchy surface with µτ init = 3.5. (3.10) Note that in what follows we restrict the rapidity range to that bounded by eq. In figure 7 we plot and the τ -component of the fluid velocity, u τ , as a function of rapidity at τ = τ init for Gaussian shock profiles. Included in the figure are shock widths w = nw o , n = 1, 2, . . . , 7. Recall that for boost invariant flow u τ = 1. We see from the figure that u τ ≈ 1 with narrower shocks having u τ closer to 1 than wider shocks. We therefore conclude that for all shock widths shown the initial fluid velocity is very well described by boost invariant flow. Turning to the proper energy density, we see that thinner shocks lead to a broader rapidity profile with smaller amplitude than wider shocks. In figure 8 we plot the amplitude (ξ = 0) and the full width at half maximum rapidity ξ FWHM of as a function of shock thickness w, again for Gaussian shock profiles. Also included in the plots are the quadratic fits in eqs. (1.4), which clearly well-describe the numerical data. Note A striking feature of the initial hydrodynamic data presented in figure 7 is the absence of any distinct qualitative change in either the proper energy or fluid velocity as the shock thickness is varied from w = 7w o to w = w o . Indeed, there is very little quantitative change in u τ as the shock thickness is varied from w = 7w o to w = w o . This stands in stark contrast to the behavior of the stress near the forward light cone, which changes qualitatively as the shock width is varied. In figure 9 we plot the normalized proper energy / (ξ = 0) at proper time τ init as a function of normalized rapidity ξ/ξ FWHM for Gaussian shock profiles. Included in the figure are shock widths w = nw o , n = 1, 2, . . . , 7.
Remarkably, when rescaled all the proper energy curves in figure 9 collapse onto one single curve! Only at w = 7w o do we see a small discrepancy between the different curves. This observation implies that the initial hydrodynamic proper energy has the form of eq. (1.3), with all w dependence solely in the normalization and rapidity width of the proper energy.
Is the hydrodynamic evolution sensitive to perturbations in the shock profile? To answer this question, in figure 10 we plotˆ and u τ again at τ = τ init with widths w = 5w o (top) and w = w o (bottom) for both Gaussian and non-Gaussian shock profiles. As is evident from the figure, both and u τ are nearly identical for both shock profiles. This should be contrasted with figure 5, where the energy density near the light cone was seen to be sensitive to the shock profile. Evidently, the initial hydrodynamic data is insensitive to the precise functional form of the shock profile δ w .
What is the function f in eq. (1.3)? In figure 11 we plot as a function of ξ/ξ FWHM for w = 5w o together with the Gaussian (1.5), which has unit full width at half maximum. Evidently, the initial hydrodynamic data is well described by a boost invariant fluid velocity and a Gaussian proper energy rapidity profile. While we have restricted our numerical analysis to w o ≤ w ≤ 7w o , we note that we see no evidence of the above universal hydrodynamic behavior disappearing as w is further decreased. Why? First of all, in figure 9 we see no sign that the functional form (1.3) of the proper energy changes as w is decreased. Second, as shown in figure 8, both the width and amplitude of the proper energy are well-described by the quadratic fits (1.4) and show no signs of additional structure at small w. Last, as shown in figure 7, it appears that boost invariant flow becomes a better and better approximation to the initial fluid velocity as w decreases. A natural interpretation of these observations is that the produced hydrodynamic flow has a smooth w → 0 limit, in which the incoming shocks become δfunctions, with the initial proper energy being a Gaussian in rapidity and the initial fluid velocity given by boost invariant flow. Using the fits (1.4), we extrapolate to w = 0 and estimate the initial width and amplitude of the proper energy for δ-function collisions to be ξ FWHM | w=0 ≈ 2.25 andˆ (ξ = 0)| w=0 /µ 4 ≈ 0.14. numerics Gaussian Figure 11. The rescaled proper energy at τ = τ init as a function of the rescaled rapidity ξ/ξ FWHM . Also included is the Gaussian (1.5), which has unit full width at half maximum.

Including transverse dynamics during hydrodynamic evolution
Heavy ion collisions are of course not translationally invariant as our planar shock collisions are. Let us for simplicity focus on head-on collisions with zero impact parameter. A gravitational model of a heavy ion with non-trivial transverse profile is the shock metric (2.1) with the function H ± given by with the constant V fixed by the condition dzd 2 x ⊥ H ± = E (so E = N 2 c 2π 2 E is the total shock energy). The function (4.1) is simply a Woods-Saxon potential translating in the ±z direction at the speed of light. The parameter R is the nuclear radius and the parameter a is the nuclear surface thickness. The parameter γ mimics the effects of Lorentz contraction in the z-direction.
In the limit where transverse gradients are small, at each x ⊥ the stress tensor must be that of planar shock collisions. This happens when the nuclear radius R is much greater than the hydrodynamization time t hydro . Therefore, when R t hydro we can construct initial hydrodynamic data at some early time merely using planar shock collisions. The future evolution -including transverse dynamics -can then be studied using hydrodynamics. To this end we define the x ⊥ -dependent energy scale µ(x ⊥ ) and longitudinal width w(x ⊥ ) via As we shall see below, for energies at RHIC and the LHC the local width µ(x ⊥ )w(x ⊥ ) 1/2 and the initial hydrodynamic data falls within the domain of universality seen above in figures 7 and 9. In other words, the initial hydrodynamic data at some x ⊥ only depends on the local energy scale µ(x ⊥ ) and the local width w(x ⊥ ) and not on the precise longitudinal structure of the shock profile (4.1).

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x ⊥ /R Figure 12. Left: the normalized scale factor µ as a function of the transverse coordinate. Note that the scale over which µ varies is the nuclear radius R. Right: the dimensionless local shock thickness µw as a function of transverse coordinate for both RHIC and LHC simulations. Note that for our simulations µw < 0.51, which is in the range of universality for the hydrodynamic initial data seen in figures 7 and 9, where µw ≤ 0.525.
Let us henceforth denote the post-collision stress tensor for planar collisions by T µν planar . T µν planar can be written T µν planar (τ, ξ, w) = µ 4 T µν planar (µτ, ξ, µw), where T µν planar (·, ·, ·) is a dimensionless function of three dimensionless arguments and is independent of the structure of the colliding shocks. Therefore, in the limit where transverse gradients are small we can write the stress tensor as Eq. (4.4) is valid for times τ with the typical length scale over which µ(x ⊥ ) varies. Since the local hydrodynamization time is of order 1/µ, we may use (4.4) to construct initial data for hydrodynamics when µ 1. Before continuing let us first ask whether the parameters in the previous paragraph yield collisions with small transverse gradients and with suitably small longitudinal widths as to enjoy the universal features of the hydrodynamic flow discovered in this paper. First consider the size of transverse gradients of µ(x ⊥ ). Figure 12 shows a plot of µ/µ(x ⊥ = 0) as a function of transverse coordinate. Clearly the scale over which µ varies is ∼ R. sets of collisions, which justifies the use of the transverse gradient expansion. Turning now to the local shock thickness, also included in figure 12 are plots of µ(x ⊥ )w(x ⊥ ) for both energies E RHIC and E LHC . We see that µw takes its maximum value 0.51 for E = E RHIC . Therefore, the local widths are in the range of universality for the hydrodynamic initial data seen in figures 7 and 9, where µw ≤ 0.525. This justifies using the universal planar shock stress T µν planar to construct initial hydrodynamic data in (4.4). We construct our initial hydrodynamic data at time τ init = 5/µ(x ⊥ = 0) = 0.045R ≈ 0.3 fm/c for the RHIC simulation and τ init = 7.5/µ(x ⊥ = 0) = 0.028R ≈ 0.2 fm/c for the LHC simulation. We then evolve forward in time using Israel-Stewart hydrodynamics [26] with viscosity (3.4) and relaxation time τ Π = (2 − log(2))/(2πT ) [3,4]. For simplicity we focus on rapidity dependent observables only and leave a detailed analysis for future work. Figure 13 shows a plot of the proper energy at x ⊥ = ξ = 0 as a function of τ for the LHC simulation. Also included in the plot is the curve τ −4/3 . Note that for boost invariant flow the proper energy decays like τ −4/3 . At early times we see from the figure that ∼ τ −4/3 . However, as time progresses the rate of fall off grows faster than τ −4/3 . Figure 14 shows plots of and u τ at x ⊥ = 0 for the LHC simulation. As time progresses the rapidity width of broadens. Both the broadening and violation of the τ −4/3 scaling are due to the fact that has non-trivial rapidity dependence. Simply put, rapidity gradients drive longitudinal expansion faster than boost invariant flow, which results in the broadening of in ξ and correspondingly, less energy at smaller rapidities than there would be in the case of boost invariant flow. Moreover

Spectrum of produced particles
After the quark-gluon plasma produced in heavy ion collisions cools below the QCD deconfinement transition the system transitions from a quark-gluon liquid into a gas of hadrons. An interesting observable to study is the spectrum of produced hadrons. Using a Cooper-Frye freeze-out prescription, the spectrum of produced particles can be computed from the hydrodynamics evolution. The spectrum of hadrons of degeneracy d, four-momentum p µ = (E, p) is given in terms of the hydrodynamic variables and u µ by [16] with the + sign for Fermions and the − sign for Bosons. The integration in (5.1) is over the hypersurface of constant temperature T = T freeze ≈ 150 MeV with T given in terms of the proper energy by (3.5). In this simple study we assume all particles are massless bosons. As such, the number of particles produced per unit rapidity is given by where the transverse component of the particle's momentum is p T and y = tanh −1 pz E is its rapidity (which equals its pseudo-rapidity, as we are assuming massless particles). Figure 15 shows a plot of the normalized rapidity distribution of particles for our RHIC and LHC simulations together with Gaussian fits. Both distributions are well described by Gaussians with rapidity widths σ = 1.9 for the RHIC simulation and σ = 2.1 for the LHC simulation. Curiously, the rapidity spectrum of particles produced in 200 GeV collisions at RHIC is also well approximated by a Gaussian with a width just 15% larger than we observe in our holographic simulations [27].

Concluding remarks
Our results demonstrate that the hydrodynamic flow produced in strongly coupled collisions is insensitive to the detailed structure of the colliding shocks and has universal characteristics. Finite w effects, which can be appreciable in size, merely alter the normalization and rapidity width of the proper energy as opposed to changing its functional form altogether. One utility of this observation is that numerical simulations of collisions need not have asymptotically small shock widths in order to approach the δ-function limit. This observation is especially valuable for simulations without any spacetime symmetry, such as the off-center shock collisions of [14], where taking the shock thickness w → 0 can be computationally expensive. We note, however, that when the width w becomes of order the microscopic relaxation time in the produced plasma, which is t hydro ≈ 2/µ, the structure of the shock can imprint itself on the hydrodynamic evolution. Indeed, as seen in figure 10, when µw ∼ 1/2 the proper energy begins to develop small deviations from the universal behavior in eq. (1.3). Why does this happen? When w µ, the system cannot equilibrate until energy stops piling up in the collision plane, which happens for an amount of time of order w. The resulting hydrodynamic evolution must then become sensitive to the detailed structure of the shocks. Indeed, it was demonstrated in [1] that for sufficiently wide shock collision, the dynamics are well described by the Landau model of heavy ion collisions, where the nuclei are assumed to be thermalized at the time they overlap completely and the initial fluid velocity is small [23,24,28].
The observation that the proper energy rapidity width has a finite limit as the shock width w vanishes implies that at w = 0, the only source of rapidity broadening comes from hydrodynamic evolution alone, where rapidity gradients drive longitudinal expansion. This stands in contrast to asymptotically free QCD, where the longitudinal thickness of nuclei deceases due to Lorentz contraction and the rapidity width of the produced plasma grows larger and larger as the energy is increased [25]. It would be interesting to study finite coupling corrections to our strongly coupled collisions and see how they affect the initial hydrodynamic data.