# Hydrodynamic attractors, initial state energy and particle production
in relativistic nuclear collisions

Giuliano Giacalone<EMAIL_ADDRESS>Institut de physique théorique,
Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France  Aleksas
Mazeliauskas<EMAIL_ADDRESS>Theoretical Physics Department,
CERN, CH-1211 Genève 23, Switzerland  Institut für Theoretische Physik,
Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany  Sören
Schlichting<EMAIL_ADDRESS>Fakultät für Physik,
Universität Bielefeld, D-33615 Bielefeld, Germany

###### Abstract

We exploit the concept of hydrodynamic attractors to establish a macroscopic
description of the early-time out-of-equilibrium dynamics of high energy
heavy-ion collisions. One direct consequence is a general relation between the
initial state energy and the produced particle multiplicities measured in
experiments. When combined with an ab initio model of energy deposition, the
entropy production during the pre-equilibrium phase naturally explains the
universal centrality dependence of the measured charged particle yields in
nucleus-nucleus collisions. We further estimate the energy density of the far-
from-equilibrium initial state and discuss how our results can be used to
constrain non-equilibrium properties of the quark-gluon plasma.

††preprint: INT-PUB-19-036

Introduction.— Understanding the equilibration of isolated quantum systems is
a fundamental question that touches physical phenomena across vastly different
energy scales, from micro kelvin temperatures in cold atom experiments to
trillion kelvin temperatures in the dense strong-interaction matter produced
in ultra-relativistic nuclear collisions Busza  _et al._ (2018); Schäfer and
Teaney (2009); Adams  _et al._ (2012); Schlichting and Teaney (2019). One
outstanding discovery made in the field of heavy-ion collisions is that the
system created about $1\,\rm{fm}/c$ ($\approx 3\cdot 10^{-24}\,\text{s})$
after the impact of two relativistic nuclei can be described as a deconfined
plasma of quarks and gluons (QGP) with macroscopic properties of temperature
and velocity Adams  _et al._ (2005); Adcox  _et al._ (2005); Back  _et al._
(2005); Arsene  _et al._ (2005). Such “unreasonable effectiveness of
hydrodynamics” in describing the violent expansion of the QGP droplets
triggered a new research area in mathematical physics devoted to the study of
_hydrodynamic attractors_ , that emerge in out-of-equilibrium systems
experiencing very fast memory loss of initial conditions and exhibiting a
universal approach towards thermal equilibrium Florkowski  _et al._ (2018);
Romatschke and Romatschke (2019).

In this article we show that hydrodynamic attractors can be used to describe
entropy production in relativistic nuclear collisions and to make robust
estimates of initial-state energy _before_ the onset of equilibration. We
derive a simple formula, Eq. (6), that relates the energy density of the
initial state to the measured charged particle multiplicity, $dN_{\rm
ch}/d\eta$, and point out two important phenomenological consequences of this
result. We show that the universal centrality dependence of $dN_{\rm
ch}/d\eta$ across a wide range of collision systems can be naturally
reproduced by combining the initial-state energy deposition in high-energy
quantum chromodynamics (QCD) with the non-linear entropy production during the
equilibration process. Secondly, we determine the initial energy per unit
space-time rapidity, $dE_{0}/d\eta_{s}$, for different collision centralities
at the Relativstic Heavy-Ion Collider (RHIC) and the Large Hadron Collider
(LHC). By comparing our results to the experimentally measured
$dE_{\text{final}}/dy$ in the final state, we estimate the work performed
during the expansion of the system Gyulassy and Matsui (1984) and discuss how
such an analysis constrains non-equilibrium and transport properties of the
QGP.

Hydrodynamic attractors & Entropy production.— We describe the early time
dynamics ($\tau\lesssim 1\,\text{fm}/c$) of the high-temperature QCD plasma
created in high-energy heavy-ion collisions by the out-of-equilibrium
evolution of a boost-invariant and transversely homogeneous conformal system
Bjorken (1983). Energy-momentum conservation dictates that the energy density,
$e=T^{\tau\tau}$, evolves according to

$\partial_{\tau}e=-\frac{e+P_{L}}{\tau}\;,$ (1)

where $P_{L}\equiv\tau^{2}T^{\eta_{s}\eta_{s}}$ is the longitudinal pressure
and we use proper time $\tau\equiv\sqrt{t^{2}-z^{2}}$ and space-time rapidity
$\eta_{s}\equiv\text{atanh}\,z/t$ coordinates. In (local) thermal equilibrium,
the longitudinal pressure is directly related to the energy density via an
equation of state, e.g. $P_{L}=e/3$ for a conformal system. While for small
deviations around equilibrium the longitudinal pressure is determined by
hydrodynamic constitutive relations in terms of the gradient expansion
$P_{L}/e=\frac{1}{3}-\frac{16}{9}\frac{\eta/s}{\tau T}$, where $\eta/s$ is the
specific shear viscosity Florkowski  _et al._ (2018); Romatschke and
Romatschke (2019), this is generally not the case far from equilibrium, where,
for instance, at early times after the collision of heavy nuclei the system is
highly anisotropic $P_{L}\ll e$ Schlichting and Teaney (2019). Nevertheless,
new insights from microscopic equilibration studies Heller  _et al._ (2012);
Heller and Spalinski (2015); Romatschke (2018); Heller  _et al._ (2018);
Strickland  _et al._ (2018); Blaizot and Yan (2018); Strickland (2018);
Behtash  _et al._ (2019a, 2018, b); Kurkela and Zhu (2015); Keegan  _et al._
(2016a, b); Kurkela  _et al._ (2019a, b); Kurkela and Mazeliauskas (2019a, b);
Kurkela  _et al._ (2019c) point to the existence of a hydrodynamic attractor
Heller and Spalinski (2015), where the far-from-equilibrium system displays an
effective constitutive equation $P_{L}/e=f(\tilde{w})$ well before reaching
local thermal equilibrium. Such attractor behavior has been established for a
number of different microscopic theories (QCD Kinetic Theory Kurkela  _et al._
(2019a, b); Kurkela and Mazeliauskas (2019a, b), Boltzmann RTA Heller  _et
al._ (2018); Strickland  _et al._ (2018); Blaizot and Yan (2018); Strickland
(2018); Behtash  _et al._ (2019a) and AdS/CFT Heller  _et al._ (2012); Heller
and Spalinski (2015); Romatschke (2018)), where the time evolution on the
attractor is controlled by a single scaling variable, $\tilde{w}={\tau T_{\rm
eff}(\tau)}/{(4\pi\eta/s)}$, where $T_{\rm eff}(\tau)$ is an effective
temperature such that $e(\tau)\equiv\frac{\pi^{2}}{30}\nu_{\rm eff}T_{\rm
eff}^{4}(\tau)\;$ ($\nu_{\text{eff}}$ is the number of effective degrees of
freedom, e.g., $\nu_{\text{eff}}=16$ for ideal gluonic gas).

Figure 1: Hydrodynamic attractor for pre-equilibrium evolution of the energy
density obtained from QCD and Yang Mills (YM) kinetic theory Kurkela  _et al._
(2019a, b); Kurkela and Mazeliauskas (2019a, b), AdS/CFT Heller  _et al._
(2012); Heller and Spalinski (2015); Romatschke (2018) and Boltzmann RTA
Heller  _et al._ (2018); Strickland  _et al._ (2018); Blaizot and Yan (2018);
Strickland (2018); Behtash  _et al._ (2019a). Solid lines show the asymptotic
behavior of the attractor curves given by Eq. (4).

Based on these insights, the conservation law in Eq. (1) can be integrated,
yielding a universal relation between the initial state energy density $e_{0}$
at very early times $\tilde{w}(\tau_{0})\ll 1$, and the energy density
$e(\tau_{\text{hydro}})$ of the near thermal system at later times
$\tilde{w}(\tau_{\text{hydro}})\gg 1$

$\displaystyle
e(\tau_{\text{hydro}})=e_{0}\exp\left(-\int_{\tilde{w}_{0}}^{\tilde{w}_{\text{hydro}}}\frac{d\tilde{w}}{\tilde{w}}\frac{1+f(\tilde{w})}{\frac{3}{4}-\frac{1}{4}f(\tilde{w})}\right)\;.$
(2)

Close to equilibrium $f(\tilde{w}_{\text{hydro}})\approx 1/3$ and the energy
density of the longitudinally expanding plasma follows the Bjorken scaling
$e(\tau)=e_{\rm hydro}\left(\tau/\tau_{\text{hydro}}\right)^{-4/3}$, while the
entropy density per unit rapidity, $s\tau$, remains constant Bjorken (1983).
Eventually, for $\tau\gtrsim R/c$, where $2R$ denotes the transverse extent of
the system, the QGP fireball starts expanding in the transverse plane and
ultimately freezes out in color neutral hadrons Teaney (2010). During the
transverse expansion the QGP remains close to equilibrium and the total
entropy per unit rapidity ${dS}/{d\eta_{s}}=A_{\bot}~{}(s\tau)_{\rm hydro}$
(where $A_{\bot}=\pi R^{2}$) is approximately conserved onwards from the time
$\tau_{\text{hydro}}$ when the QGP can be described as an almost ideal fluid.
Ultimately, on the freeze-out surface ${dS}/{d\eta_{s}}$ becomes proportional
to the produced charged hadron multiplicity, $dN_{\text{ch}}/d\eta$. The
multiplicity of final-state particles emitted from the QGP is therefore a
sensitive probe of the entropy production during the pre-equilibrium phase.

Strikingly, the correspondence between initial-state energy density and
charged hadron multiplicity can be quantified further using the theory of
hydrodynamic attractors. By factoring out the late time Bjorken scaling from
Eq. (2) the evolution of the energy density during the pre-equilibrium phase
can be characterized by an attractor curve $\mathcal{E}(\tilde{w})$

$\displaystyle\frac{e(\tau)\tau^{4/3}}{e_{\rm
hydro}\tau_{\text{hydro}}^{4/3}}=\mathcal{E}\left(\tilde{w}=\frac{T_{\rm
eff}(\tau)\tau}{4\pi\eta/s}\right)\;.$ (3)

As can be seen from Fig. 1, the function $\mathcal{E}(\tilde{w})$ smoothly
interpolates between an early free-streaming and late-stage viscous
hydrodynamics Romatschke (2018); Kurkela  _et al._ (2019a)

$\displaystyle\begin{split}\mathcal{E}(\tilde{w}\ll
1)&=C_{\infty}^{-1}\tilde{w}^{4/9}\qquad(\text{free streaming})\;,\\\
\mathcal{E}(\tilde{w}\gg 1)&=1-\frac{2}{3\pi\tilde{w}}\qquad\;\;(\text{viscous
hydro})\;,\end{split}$ (4)

where $C_{\infty}$ is a constant of order unity. Even though the evolution at
intermediate times can be different for different microscopic theories, the
overall similarity between different theories is remarkable. Most importantly
for our purpose, all curves have the same universal characteristics, Eq. (4),
at early and late times, irrespective of the underlying microscopic theory.

Based on Eq. (3), we can immediately establish a quantitative relation between
the energy densities $e(\tau)$ at various stages, which upon use of the
thermodynamic relations $Ts=e+p$ and $p=e/3$ once the system is close to
equilibrium turns into an estimate of the entropy density per unit rapidity

$\displaystyle(s\tau)_{\rm hydro}=\frac{4}{3}\left(\frac{\pi^{2}}{30}\nu_{\rm
eff}\right)^{1/4}\Bigg{(}\lim_{\tau\to
0}\frac{e(\tau)\tau^{4/3}}{\mathcal{E}\big{(}\frac{T_{\rm
eff}(\tau)\tau}{4\pi\eta/s}\big{)}}\Bigg{)}^{3/4}\\!\\!.$ (5)

Evaluating the limit according to Eq. (4) one arrives at the central result of
this paper, namely the relation

$\displaystyle(s\tau)_{\rm hydro}$
$\displaystyle=\frac{4}{3}C_{\infty}^{3/4}\left(4\pi\frac{\eta}{s}\right)^{1/3}\left(\frac{\pi^{2}}{30}\nu_{\rm
eff}\right)^{1/3}\left(e\tau\right)^{2/3}_{0},$ (6)

from which one can directly estimate the charged particle multiplicity as
discussed above:

$\displaystyle\frac{dN_{\rm
ch}}{d\eta}\approx\frac{1}{J}A_{\bot}~{}(s\tau)_{\rm hydro}\frac{N_{\rm
ch}}{S}\;.$ (7)

Here
$S/N_{\text{ch}}\equiv\left(dS/dy\right)/\left(dN_{\text{ch}}/dy\right)\approx
6.7\text{--}8.5$ is the entropy per charged particle at freeze-out Hanus  _et
al._ (2019) and $J\approx 1.1$ is a Jacobian factor between particle rapidity
$y$ and pseudo rapidity $\eta$ Adam  _et al._ (2016).

Equations (6) and (7) establish a one-to-one correspondence between the
initial-state energy per unit rapidity $dE_{0}/d\eta_{s}\approx
A_{\bot}(e\tau)_{0}$ and the charged particle multiplicity ${dN_{\rm
ch}}/{d\eta}$. One crucial feature of this result is that it accounts for the
entropy production during the pre-equilibrium phase, which gives rise to a
nontrivial dependence on the initial-state energy density $(e\tau)^{2/3}_{0}$
as well as on the transport coefficient $(\eta/s)^{1/3}$. Our estimate
includes all relevant pre-factors, in particular, the constant $C_{\infty}$,
which is the property of the hydrodynamic attractor, Eq. (3), and depends on
the microscopic physics of equilibration. However it is striking to observe
that for the different theories shown in Fig. 1 the variation of $C_{\infty}$
is only at $\sim 10\%$ level. We emphasize that Eq. (6) is entirely based on
the macroscopic evolution described by a generic hydrodynamic attractor, which
sets it apart from previous parametric estimates of entropy production based
on particular microscopic scenarios Baier  _et al._ (2002, 2011); van der
Schee and Schenke (2015); Berges  _et al._ (2017).

Figure 2:  The effect of pre-equilibrium dynamics on the centrality dependence
of the charged-particle multiplicity, $dN_{\rm ch}/d\eta$, can be appreciated
by comparing the theoretical estimate on the gluon number (blue dashed line),
given by Eq. (11), to that on the charged-particle multiplicity after
equilibration (red solid line), given by Eq. (12). The green dot-dashed line
includes as well fluctuations in the initial state energy within a Glauber
Monte Carlo approach. Experimental data points are shown for different
collision systems: Xe-Xe Acharya  _et al._ (2019), Pb-Pb Aamodt  _et al._
(2011), U-U Adare  _et al._ (2016), Au-Au Alver  _et al._ (2011), and Cu-Cu
Alver  _et al._ (2011) collisions. All curves are normalized to present the
same value of multiplicity as ALICE Pb-Pb data in the 10-20% centrality bin.
The bottom panel shows the ratio between 2.76 TeV Pb-Pb data and the theory
curves.

Centrality dependence of particle multiplicity.—One important phenomenological
consequence of the entropy production in the pre-equilibrium phase concerns
the determination of initial conditions for hydrodynamic simulations of heavy
ion collisions (see e.g. Heinz and Snellings (2013)). While strictly speaking
our estimate of the entropy density in Eq. (6) was derived assuming a one
dimensional expansion, the influence of transverse gradients can be neglected
over the short pre-equilibrium times and we can directly promote Eq. (6) to an
estimate for the _local_ entropy density, $\left.\tau
s(\tau,\mathbf{x}_{\bot})\right|_{\tau=\tau_{\text{hydro}}}$. Specifically,
the pre-flow $v_{\bot}\sim\tau_{\text{hydro}}\mathbf{\nabla}_{\bot}e/e$ is
negligible as long as gradients $\mathbf{\nabla}_{\bot}e/e$ are small on the
scale of hydrodynamization time $(c\tau_{\text{hydro}})$ Keegan  _et al._
(2016b); Kurkela  _et al._ (2019a) and the one dimensional constitutive
relation for $P_{L}/e$ approximately remains valid, as explicitly confirmed in
Kurkela  _et al._ (2020). Effectively, Eq. (6) then provides a non-linear map
of the initial-state energy density profile to the entropy density profile at
later times $\tau\sim\tau_{\text{hydro}}$.

Now, in order to illustrate the impact of the pre-equilibrium phase, we will
study the effects on the centrality dependence of the charged particle
multiplicity, within a simple initial state model based on the color-glass
condensate effective theory of high-energy QCD Gelis  _et al._ (2010). Within
the dilute-dense formulation of this theory Dumitru and McLerran (2002); Lappi
(2006); Blaizot  _et al._ (2004); Blaizot and Mehtar-Tani (2009); Blaizot  _et
al._ (2010), both the initial energy density per unit rapidity $(e\tau)_{0}$
as well as the initial gluon multiplicity per unit rapidity $(n\tau)_{0}$ can
be calculated from $\mathbf{k}_{\bot}$-factorization, and are given in terms
of convolutions of unintegrated gluon distributions 111We note that explicit
comparisons of dilute-dense and dense-dense calculations in Ref. Blaizot  _et
al._ (2010) show that in the momentum range $p_{T}\gtrsim Q_{s}$, which gives
the dominant contribution to ${dE}/{dy}$, the dilute-dense $k_{T}$
factorization calculations accurately describe dense-dense Classical Yang-
Mills results.. Essentially, one finds that (up to logarithmic corrections)
$(e\tau)_{0}$ and $(n\tau)_{0}$ are proportional to the (local) saturation
scales $Q_{s}(\mathbf{x}_{\bot})$ of the two colliding nuclei Dumitru and
McLerran (2002); Lappi (2006):

$\displaystyle(e\tau)_{0}(\mathbf{x}_{\bot})$ $\displaystyle\propto$
$\displaystyle(Q_{s}^{<})^{2}(\mathbf{x}_{\bot})Q_{s}^{>}(\mathbf{x}_{\bot})\;,$
(8) $\displaystyle(n\tau)_{0}(\mathbf{x}_{\bot})$ $\displaystyle\propto$
$\displaystyle(Q_{s}^{<})^{2}(\mathbf{x}_{\bot}),$ (9)

where $Q_{s}^{>/<}$ is the saturation scale of the nucleus representing
larger/smaller $Q_{s}$ at position $\mathbf{x}_{\bot}$.

Since the saturation scale locally characterizes the longitudinally integrated
density of color charge inside the nucleus, it is generically proportional to
the nuclear thickness

$\displaystyle Q_{s}^{2}(\mathbf{x}_{\bot})\propto T(\mathbf{x}_{\bot}),$ (10)

whose definition is recalled in the supplemental material 222The supplemental
material provides the description of the Monte Carlo Glauber model of nucleus-
nucleus collisions used to determine the nuclear thickness functions, which
are needed for the calculation of the initial-state energy profile. In
addition, we provide detailed formulas for calculating the centrality
dependence of the averaged charged-particle multiplicity and the initial-state
energy per rapidity, i.e. the lines and shaded bands in Figs. 2 and 3 of the
manuscript. The supplemental material includes additional Refs. De Vries  _et
al._ (1987); Albacete and Marquet (2014). Based on these considerations, one
can then try to estimate the charged particle multiplicity per unit rapidity
from the initial gluon multiplicity $(n\tau)_{0}$ (i.e. w/o pre-equilibrium)
according to

$\displaystyle\frac{dN_{\rm ch}}{d\eta}\propto\int
d^{2}\mathbf{x}_{\bot}~{}T^{<}(\mathbf{x}_{\bot})\;,$ (11)

as was done, for example, in Mace  _et al._ (2018); Schenke  _et al._ (2012).
However, such an estimate is appropriate only when there is no significant
amount of particle production in the final state. Conversely, if the initial
state evolves into an almost ideal QGP fluid, one needs to account for the
entropy production during the pre-equilibrium phase. By employing Eqs. (6) and
(7) the charged particle multiplicity is then estimated from the initial state
energy density $(e\tau)_{0}$, (i.e. w/ pre-equilibrium) as

$\displaystyle\frac{dN_{\rm ch}}{d\eta}\propto\int
d^{2}\mathbf{x}_{\bot}~{}\left(T^{<}(\mathbf{x}_{\bot})\sqrt{T^{>}(\mathbf{x}_{\bot})}\right)^{2/3}\;.$
(12)

We illustrate the difference between the two estimates in the upper panel of
Fig. 2, where we compare the centrality dependence of the multiplicity
${dN_{\rm ch}}/{d\eta}$ from Eq. (12) (solid line) and Eq. (11) (dashed line),
with the nuclear thickness and centrality quantilies determined from the
optical Glauber model (see the supplemental material for details Note (2)).
Both estimates are normalized to reproduce the experimentally measured value
of ${dN_{\rm ch}}/{d\eta}$ in the $10-20\%$ centrality class of Pb–Pb
collisions at $\sqrt{s_{\text{NN}}}=2.76\,\text{TeV}$. Different trends in the
centrality dependence of ${dN_{\rm ch}}/{d\eta}$ are clearly visible,
indicating the importance of the pre-equilibrium phase when comparing
observables of this type to experimental data.

Since ${dN_{\rm ch}}/{d\eta}$ is accurately measured in experiment for a wide
variety of colliding systems and energies, we can also compare the two
estimates directly to experimental data, which are reported as symbols in the
upper panel of Fig. 2. It is interesting to note that the average
$\left<{dN_{\rm ch}}/{d\eta}\right>$ as a function of centrality possesses a
remarkable degree of universality, such that—up to an overall normalization
factor for each collision system—data points for different colliding species
(Au, Cu, Pb, U, Xe) at RHIC and LHC energies all collapse onto a single curve
to high accuracy. Despite the simplicity of our theoretical estimate, we find
that the curve including pre-equilibrium effects provides a rather good
description of the experimental data, except for the more peripheral bins,
where fluctuations play an important role (see below). Due to the non-trivial
geometry dependence in Eq. (12), the calculation including pre-equilibrium
dynamics provides a much better description of the data than the estimate in
Eq. (11), which is based solely on the initial state.

Even though Eq. (12) can be clearly justified from theoretical calculations,
our description is by no means unique. Other phenomenological models Drescher
and Nara (2007); Alver  _et al._ (2008); Gale  _et al._ (2013); Niemi  _et
al._ (2016); Bernhard  _et al._ (2016) successfully reproduce the centrality
dependence seen in Fig. 2 by introducing various sources of event-by-event
fluctuations such as number and positions of participant nucleons, their
interaction strength, etc. However, it is important to emphasize in this
context that the pre-equilibrium phase also modifies the statistics of
fluctuations, such that for the long wavelength perturbations Keegan  _et al._
(2016b); Kurkela  _et al._ (2019a)

$\displaystyle\frac{\delta s_{\rm hydro}}{s_{\rm
hydro}}=\frac{2}{3}\frac{\delta e_{0}}{e_{0}},$ (13)

which follows from the linearization of Eq. (6). While Eq. (12) over-predicts
particle production in peripheral collisions, it is therefore not surprising
that one can restore agreement with peripheral data by including event-by-
event fluctuations. This is demonstrated by the dot-dashed line in Fig. 2,
where the average of the nuclear thickness in Eq. (12) has been determined
from a Glauber Monte Carlo model Miller  _et al._ (2007) (see the supplemental
material for details Note (2)).

Figure 3: Estimate of the initial energy per unit space-time rapidity
$dE_{0}/d\eta_{s}$ (shaded area) determined from the measured particle
multiplicity in $\sqrt{s_{\text{NN}}}=200\,\text{GeV}$ Au-Au (left) and
$\sqrt{s_{\text{NN}}}=2.76\,\text{TeV}$ Pb-Pb collisions (right). Bands
correspond to variations of $\eta/s=0.08\text{--}0.24$ and
$C_{\infty}=0.80\text{--}1.15$ while keeping all other parameters fixed as in
Eq. (14), and similarly the dashed lines correspond to specific values of
$\eta/s=0.08,0.16$. For comparison we show data points for the experimentally
measured final state energy $dE_{\text{final}}/dy$ Adam  _et al._ (2016);
Adare  _et al._ (2016), which is smaller due to the work performed against the
longitudinal expansion.

Estimating the initial-state energy density.—So far we illustrated the utility
of Eq. (6) for describing entropy production in the pre-equilibrium phase of
high-energy heavy-ion collisions. However, an equally important application
concerns the inverse problem, namely the estimation of the energy density
$e_{0}$ of the non-equilibrium state at very early time $\tau_{0}$ from
experimental measurements of hadrons in the final state.

By inverting Eqs. (6) and (7) (and inserting typical values of
$dN_{\text{ch}}/d\eta\sim 1600$ and $A_{\bot}\approx\pi
R_{\text{Pb}}^{2}\approx 138\,\text{fm}^{2}$) we obtain the following estimate
for the initial energy density for central Pb-Pb collisions at
$\sqrt{s_{NN}}=2.76~{}{\rm TeV}$:

$\displaystyle e_{0}\approx 270~{}{\rm
GeV/fm^{3}}~{}\left(\frac{\tau_{0}}{0.1{\rm
fm}/c}\right)^{-1}\left(\frac{C_{\infty}}{0.87}\right)^{-9/8}\\!\left(\frac{\eta/s}{2/4\pi}\right)^{-1/2}$
$\displaystyle\left(\frac{A_{\bot}}{138{\rm
fm}^{2}}\right)^{-3/2}\left(\frac{dN_{\rm
ch}/d\eta}{1600}\right)^{3/2}\left(\frac{\nu_{\rm
eff}}{40}\right)^{-1/2}\left(\frac{S/N_{\rm
ch}}{7.5}\right)^{3/2}\\!\\!\\!\\!,$ (14)

at a time $\tau_{0}=0.1\,\text{fm}/c$, which should be at least of the order
of the formation time $1/Q_{s}\approx 0.1\text{fm}/c$ Schlichting and Teaney
(2019), but small compared to the equilibration time $\tau_{\rm hydro}\
\approx 1\,{\rm fm}/c$ Kurkela  _et al._ (2019b, a) for the estimate in Eq.
(14) to be valid. One finds that the initial energy density quoted in Eq. (14)
is nearly three orders of magnitude higher than the energy density at the QCD
cross-over $e_{c}\approx 0.346(41)~{}{\rm GeV}/\text{fm}^{3}$ (for 2+1 flavor
QCD) Ding  _et al._ (2015); Bazavov  _et al._ (2019).

We emphasize that, unlike the usual Bjorken estimate based on the measured
final-state energy $e_{0}=\frac{1}{\tau_{0}A_{\bot}}{dE_{\text{final}}}/{dy}$
Bjorken (1983), our estimate in Eq. (14) includes the work done during the
expansion of the QGP Gyulassy and Matsui (1984). We demonstrate this effect in
Fig. 3 where we compare the experimentally measured $dE_{\text{final}}/dy$ in
the final state to the initial-state energy per unit rapidity
$dE_{0}/d\eta_{s}=\int d{\mathbf{x}}_{\perp}(e\tau)_{0}$ reconstructed from
the measured particle multiplicities as in Eq. (14). Note that to better
account for the non-trivial transverse geometry, we have estimated the
transverse area $A_{\bot}$ from our fluctuating initial state model as
described in the supplemental material Note (2).

Based on this analysis, we find that, especially in central collisions at high
energies, the initial state $dE_{0}/d\eta_{s}$ can easily exceed the measured
${dE_{\text{final}}}/{dy}$ in the final state by a factor of two to three.
Evidently the exact amount of work done during the expansion is subject to
uncertainties in the non-equilibrium and transport properties of the QGP,
which we quantify by uncertainty bands in Fig. 3, corresponding to variations
of $\eta/s$ and $C_{\infty}$ within the anticipated margins
($\eta/s=0.08-0.24$ and $C_{\infty}=0.80-1.15$). Vice versa, the size of the
uncertainty bands in Fig. 3 also demonstrates the fact that, if the initial
state energy density can be determined precisely, e.g. from theoretical
calculations, then the experimentally measured $dN_{\rm ch}/d\eta$ will impose
strong constraints on the non-equilibrium evolution of the QGP characterized
by $C_{\infty}$ and $\eta/s$. Since, the $dE_{0}/d\eta_{s}$ should always be
larger than $dE_{\text{final}}/dy$ due to the work performed against the
longitudinal expansion, one can further rule out values in the parameter
space, where the initial $dE_{0}/d\eta_{s}$ falls below the experimental data
points. Specifically, large values of $\eta/s$ and $C_{\infty}$ can already be
ruled out, because the estimated $dE_{0}/d\eta_{s}$ in peripheral collisions
turns out to be unphysically small, i.e. below the experimental points of
$dE_{\text{final}}/dy$, which provide a lower bound on the initial state
energy.

Discussion.—Entropy production in high-energy heavy-ion collisions occurs
predominantly during the earliest stages, when the system is significantly
out-of-equilibrium; therefore measurements of the charged particle
multiplicities—reflecting the total amount of entropy produced in the
collision—provide a highly sensitive probe of the pre-equilibrium dynamics.
Based on the concept of hydrodynamic attractors, which give a macroscopic
description of the early time dynamics of the QGP, we established for the
first time a direct relation between the initial-state energy and the final-
state entropy. This relation, Eq. (6), is remarkably insensitive to the
microscopic details of the approach to equilibrium (see Fig. 1).

By combining the information from $dN_{\text{ch}}/d\eta$ on entropy production
and $dE_{\text{final}}/dy$ on the work performed against the longitudinal
expansion, we demonstrate that a precise calculation of the initial state
energy can impose stringent constraints on the shear-viscosity to entropy
density ratio $\eta/s$. Based on our extraction of $dE_{0}/d\eta_{s}$, which
assumes a scenario of (nearly-)complete equilibration, we obtain an upper
limit for $\eta/s\lesssim 0.4$ for the most favorable choice of all other
parameters. Conversely, for $\eta/s\gtrsim 0.4$ we can not expect the QGP to
equilibrate in peripheral nucleus-nucleus collisions (see also Bhalerao  _et
al._ (2005); Kurkela  _et al._ (2019c)) and our estimates need to be revised.
Non-trivial modifications due to incomplete equilibration will arise in this
context, which should be investigated further, for instance by means of the
KøMPøST pre-equilibrium package Kurkela  _et al._ (2019a, b, 2018). We expect
such effects to become particularly important in collisions of smaller nuclei,
e.g., p–A or O–O, which may therefore provide even deeper insights into the
fascinating out-of-equilibrium dynamics of the QGP.

Acknowledgments.—We thank A. Andronic, J. Berges, N. Borghini, M. Heller, C.
Klein-Bösing, A. Kurkela, T. Lappi, M. Martinez, J.Y. Ollitrault, K. Reygers,
C. Schmidt, D. Teaney, and R. Venugopalan for stimulating discussions and P.
Romatschke for also providing the AdS/CFT attractor curve. This work was
supported in part by the German Research Foundation (DFG) through the
Collaborative Research Centres “CRC-TR 211: Strong-interaction matter under
extreme conditions” (S.S.) and “SFB 1225: Isolated quantum systems and
universality in extreme conditions (ISOQUANT)” (A.M.). We thank the Institute
for Nuclear Theory at the University of Washington for its kind hospitality
and stimulating research environment.

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##  Appendix A Supplemental material

Below we explain the details of the implementation of the initial state model
described in the main text. When referring to the optical Glauber
calculations, which yield the dashed and solid lines in Fig. 2, we determine
the nuclear thickness $T(\mathbf{x}_{\bot})$ of each nucleus according to the
longitudinal integral of the (average) nuclear matter density distribution
$T(\mathbf{x}_{\bot})=\int_{z}dz~{}\rho(\sqrt{\mathbf{x}_{\bot}^{2}-z^{2}})$.
We follow previous works and parametrize $\rho(r)$ by a two-parameter Fermi
distribution

$\rho(r)=\rho_{0}\biggl{[}1~{}+~{}\exp\biggl{(}\frac{1}{a}\bigl{[}r-R\bigr{]}\biggr{)}\biggr{]}^{-1}\;,$
(15)

with $a=0.55$ fm and $R=6.62$ fm for 208Pb nuclei, and $a=0.53$ fm, $R=6.40$
fm for 197Au nuclei De Vries  _et al._ (1987), while $\rho_{0}$ is such that
$\rho(r)$ is normalized to the total number of nucleons. The thickness
functions of two nuclei (A and B) colliding at a finite impact parameter,
$\mathbf{b}$, are given by
$T^{\text{A}/\text{B}}(\mathbf{x}_{\bot})=T(\mathbf{x}_{\bot}\pm\mathbf{b}/2)$.
The centrality of a given collision is obtained from the geometric relation
$centrality=\pi|\mathbf{b}|^{2}/\sigma_{\text{tot}}$ Teaney (2010), where
$\sigma_{\text{tot}}$ is the total inelastic nucleus-nucleus cross section,
and we use $\sigma_{\text{tot}}=767$ fm2 for Pb-Pb collisions, and
$\sigma_{\text{tot}}=685$ fm2 in Au-Au collisions.

When referring to fluctuating initial conditions, which yield the dot-dashed
line in Fig. 2, we introduce nucleonic degrees of freedom following the Monte
Carlo version of the Glauber model Miller  _et al._ (2007). In both colliding
nuclei we sample the positions of the nucleons according to $\rho(r)$, which
we then shift by $\pm\mathbf{b}/2$ to take into account the finite impact
parameter. Subsequently, we determine the participants in a black-disk
approximation as those nucleons of nucleus A that are located within a
distance $\sqrt{\sigma_{\rm nn}/\pi}$ from at least one nucleon in nucleus B,
and vice versa. Here $\sigma_{\rm nn}$ is the nucleon-nucleon cross-section,
which is equal to $4.2$ fm2 for $\sqrt{s_{\rm NN}}=200$ GeV and to $6.4$ fm2
for $\sqrt{s_{\rm NN}}=2.76$ TeV. The thickness $T(\mathbf{x}_{\bot})$ of each
nucleus is then determined on an event-by-event basis by summing the density
profiles of all its participating nucleons, where the density of each
participant nucleon is taken as a Gaussian profile of width $0.5$ fm.

Explicitly restoring the constants in Eqs. (10) and (8), the saturation scale
$Q_{s}(\mathbf{x})$ of the nucleus is determined as

$\displaystyle
Q_{s}^{2}(\mathbf{x_{\perp}})=Q_{sp}^{2}(\sqrt{s})A_{p}~{}T(\mathbf{x_{\perp}})\;,$
(16)

where $Q_{sp}(\sqrt{s})$ and $A_{p}=\pi R_{p}^{2}$ denote the saturation scale
and (transverse) size of a proton, and the initial state energy density per
unit rapidity is then given by

$(e\tau)_{0}(\mathbf{x}_{\bot})=\mathcal{N}_{e}~{}\left(Q_{sp}^{2}(\sqrt{s})A_{p}\right)^{3/2}~{}\,T^{<}(\mathbf{x}_{\bot})\sqrt{T^{>}(\mathbf{x}_{\bot})},$
(17)

where $\mathcal{N}_{e}$ is a dimensionless proportionality coefficient (see
e.g. Dumitru and McLerran (2002)). By use of Eqs. (6) and (7), the
multiplicity in an event is then given by

$\displaystyle\frac{dN_{\text{ch}}}{d\eta}$
$\displaystyle=\frac{4}{3J}C_{\infty}^{3/4}\left(4\pi\frac{\eta}{s}\right)^{1/3}\left(\frac{\pi^{2}}{30}\nu_{\rm
eff}\right)^{1/3}~{}\frac{N_{\rm ch}}{S}$
$\displaystyle\mathcal{N}_{e}^{2/3}~{}Q_{sp}^{2}(\sqrt{s})A_{p}~{}\int
d\mathbf{x}_{\bot}~{}\left(T^{<}(\mathbf{x}_{\bot})\sqrt{T^{>}(\mathbf{x}_{\bot})}\right)^{2/3},$
(18)

while the initial-state energy per unit rapidity is simply given by the
transverse integral of $(e\tau)_{0}$ in Eq. (17) as

$\displaystyle\frac{dE_{0}}{d\eta_{s}}=\mathcal{N}_{e}~{}\left(Q_{sp}^{2}(\sqrt{s})A_{p}\right)^{3/2}~{}\int
d\mathbf{x}_{\bot}~{}T^{<}(\mathbf{x}_{\bot})\sqrt{T^{>}(\mathbf{x}_{\bot})}.$
(19)

Statistical averages of $\frac{dN_{\text{ch}}}{d\eta}$ and
$\frac{dE_{0}}{d\eta_{s}}$ at a given centrality percentile are then obtained
by sampling many events at the corresponding impact parameter. By matching the
multiplicity in Eq. (18) to experimental data, we find that the dimensionless
combination of pre-factors

$\frac{4}{3J}C_{\infty}^{3/4}\left(4\pi\frac{\eta}{s}\right)^{1/3}\left(\frac{\pi^{2}}{30}\nu_{\rm
eff}\right)^{1/3}~{}\frac{N_{\rm
ch}}{S}\mathcal{N}_{e}^{2/3}~{}Q_{sp}^{2}(\sqrt{s})A_{p}$ (20)

in Eq. (18) should be equal to $10.7$ for 2.76 TeV Pb-Pb collisions $4.8$ for
200 GeV Au-Au collisions, which is compatible with the expected increase of
the saturation scale $Q_{s}^{2}(\sqrt{s})\propto\sqrt{s}^{\lambda}$ with
$\lambda\approx 0.3$ Albacete and Marquet (2014).

Vice versa, to determine the initial state energy per unit rapidity shown in
Fig. 3, we can use Eq. (18) for the multiplicity to eliminate the pre-factor
$N_{e}~{}\left(Q_{sp}^{2}(\sqrt{s})A_{p}\right)^{3/2}$ from Eq. (19) and
express $\frac{dE_{0}}{d\eta_{s}}$ in the form of Eq. (14) as

$\displaystyle\frac{dE_{0}}{d\eta_{s}}$
$\displaystyle=\left(\frac{4}{3J}\right)^{-3/2}C_{\infty}^{-9/8}\left(4\pi\frac{\eta}{s}\right)^{-1/2}\left(\frac{\pi^{2}}{30}\nu_{\rm
eff}\right)^{-1/2}$ $\displaystyle~{}A_{\bot}^{-1/2}\left(\frac{S}{N_{\rm
ch}}\right)^{3/2}\left(\frac{dN_{\text{ch}}}{d\eta}\right)^{3/2},$ (21)

where the effective area, $A_{\bot}$, is given by

$\displaystyle\sqrt{A_{\bot}}=\frac{\left(\int
d\mathbf{x}_{\bot}~{}\left(T^{<}(\mathbf{x}_{\bot})\sqrt{T^{>}(\mathbf{x}_{\bot})}\right)^{2/3}\right)^{3/2}}{\int
d\mathbf{x}_{\bot}~{}T^{<}(\mathbf{x}_{\bot})\sqrt{T^{>}(\mathbf{x}_{\bot})}}.$
(22)

Evaluating Eq. (22) within our Monte-Carlo calculations yields
$A_{\bot}\approx 120$ fm2 in central Pb-Pb collisions, which is in fair
agreement with the value anticipated in Eq. (14). Note that, while we could
have used the experimental data for $dN_{\rm ch}/d\eta$ to produce Fig. 3, we
conveniently make use of the model estimate in Eq. (18) to interpolate between
experimental data points; as shown in Fig. 2 the model is within $5\%$ of all
experimental data.