Relativistic viscous hydrodynamics for heavy-ion collisions with ECHO-QGP

Abstract

We present ECHO-QGP, a numerical code for (3+1)-dimensional relativistic viscous hydrodynamics designed for the modeling of the space-time evolution of the matter created in high-energy nuclear collisions. The code has been built on top of the Eulerian Conservative High-Order astrophysical code for general relativistic magneto-hydrodynamics (Del Zanna et al. in Astron. Astrophys. 473:11, 2007] and here it has been upgraded to handle the physics of the Quark–Gluon Plasma. ECHO-QGP features second-order treatment of causal relativistic viscosity effects both in Minkowskian and in Bjorken coordinates; partial or complete chemical equilibrium of hadronic species before kinetic freeze-out; initial conditions based on the Glauber model, including a Monte-Carlo routine for event-by-event fluctuating initial conditions; a freeze-out procedure based on the Cooper–Frye prescription. The code is extensively validated against several test problems and results always appear accurate, as guaranteed by the combination of the conservative (shock-capturing) approach and the high-order methods employed. ECHO-QGP can be extended to include evolution of the electromagnetic fields coupled to the plasma.

Appendices

Appendix A: Fluid description in Minkowski and Bjorken coordinates

In this appendix we summarize the essential formulas establishing the link between the fluid description in Minkowski and Bjorken coordinates. While what contained in the text is already sufficiently self-consistent, here we wish to establish a mapping with the notation adopted by other authors [100, 101] and widely employed in phenomenological studies, such as blast-wave fits. For the sake of clarity in this appendix four-vectors in Bjorken coordinates will be denoted by a “prime” and components of the three-velocity by a “tilde”; in the text such a distinction will be neglected.

1.1 A.1 Minkowski coordinates

Here we specify the notation employed for the different rapidities entering into our RHD setup.

• Fluid rapidity:

  $$ Y\equiv\frac{1}{2}\ln\frac{1+v^z}{1-v^z}\longrightarrow v^z=\tanh Y $$

  (A.1)

  with v _z the longitudinal component of the fluid velocity v;

• Space-time rapidity:

  $$ \eta_s\equiv\frac{1}{2}\ln\frac{t+z}{t-z}. $$

  (A.2)

• Particle rapidity (of the emitted hadron):

  $$ y\equiv\frac{1}{2}\ln\frac{E+p_z}{E-p_z}; $$

  (A.3)

The particle four-momentum is conveniently expressed in terms of its transverse mass m _⊥ and rapidity:

$$ p^\mu\equiv(m_\perp\cosh y,{\boldsymbol{p}}_\perp,m_\perp \sinh y) $$

(A.4)

The fluid four-velocity, defined as

$$ u^\mu\equiv\gamma\,(1,{\boldsymbol{v}})=\gamma\,(1,{\boldsymbol {v}}_\perp,v_z) ,\quad{\rm with}\ \gamma\equiv1/ \sqrt{1-{\boldsymbol{v}}^2}, $$

(A.5)

can be recasted in terms of the fluid-rapidity as

$$ u^\mu=\frac{1}{\sqrt{1-\cosh^2Y\,{\boldsymbol{v}}_\perp^2}}(\cosh Y,\cosh Y\,{\boldsymbol{v}} _\perp,\sinh Y). $$

(A.6)

This suggest to define the “transverse velocity”

$$ {\boldsymbol{u}}_\perp\equiv\cosh Y\,{\boldsymbol{v}}_\perp, $$

(A.7)

so that

$$ u^\mu\equiv\gamma_\perp\,(\cosh Y,{\boldsymbol{u}}_\perp, \sinh Y) ,\quad{\rm with}\ \gamma_\perp\equiv1/\sqrt{1-{ \boldsymbol{u}}_\perp^2}. $$

(A.8)

The scalar product between the particle momentum and the fluid velocity, entering into the Cooper–Frye decoupling prescription, reads (with metric [−,+,+,+])

$$ -p\cdot u=\gamma_\perp\bigl[m_\perp\cosh(y-Y)-{ \boldsymbol{p}}_\perp \cdot{\boldsymbol{u}}_\perp \bigr], $$

(A.9)

expressing the fact that particles tend to be emitted with rapidity close to the one of the fluid-cell.

In the general case the velocity field of the fluid depends on all the four space-time coordinates: one has u _⊥≡u _⊥(τ,r _⊥,η _s ) and Y≡Y(τ,r _⊥,η _s ).

In the case of longitudinal boost-invariance of the fluid profile one has always

$$ v_z=\frac{z}{t}\longrightarrow Y\equiv \eta_s, $$

(A.10)

so that the fluid velocity reduces to

$$ u^\mu=\gamma_\perp\,(\cosh\eta_s,{ \boldsymbol{u}}_\perp,\sinh \eta_s), $$

(A.11)

which only depends on u _⊥(τ,r _⊥).

1.2 A.2 Bjorken coordinates

One can go from the Minkowski coordinates

$$ u^\mu\equiv\bigl(u^0,u^1,u^2,u^3 \bigr) $$

(A.12)

to the Bjorken (sometimes also known as Milne) coordinates

$$ u'^m\equiv\bigl[u'^\tau,u'^x,u'^y,u'^\eta \bigr] $$

(A.13)

through the transformation

$$ u'^m\equiv\frac{\partial x'^m}{\partial x^\nu}u^\nu. $$

(A.14)

One has

$$\begin{aligned} &u'^\tau=\frac{d\tau}{dt}u^0+ \frac{d\tau}{dz}u^3=\gamma_\perp \cosh(Y- \eta_s) \end{aligned}$$

(A.15a)

$$\begin{aligned} &u'^\eta=\frac{d\eta_s}{dt}u^0+ \frac{d\eta_s}{dz}u^3=\gamma_\perp \frac{1}{\tau} \sinh(Y-\eta_s). \end{aligned}$$

(A.15b)

Hence

$$ u'^m=\gamma_\perp \biggl[\cosh(Y- \eta_s),{\boldsymbol{u}}_\perp ,\frac{1}{\tau }\sinh(Y- \eta_s) \biggr] $$

(A.16)

The four-momentum in Milne coordinates analogously reads

$$ p'^m= \biggl[m_\perp\cosh(y- \eta_s),{\boldsymbol{p}}_\perp,\frac {1}{\tau }m_\perp \sinh(y-\eta_s) \biggr]. $$

(A.17)

Its contraction with u′^m and the hypersurface element in Eqs. (A.16) and (66) allows one to recover the freeze-out formula employed in the text in Eq. (65) and expressed in terms of the output variables of ECHO-QGP. Notice that its contraction with the expression of the fluid four-velocity in Eq. (A.16) provides the result in Eq. (A.9).

After a further change of variables, introducing the definitions

$$ \widetilde{v}^\eta\equiv\frac{\tanh(Y-\eta_s)}{\tau}\quad{\rm and}\quad \widetilde{v}^{x/y}\equiv\frac{u_\perp^{x/y}}{\cosh (Y-\eta_s)}, $$

(A.18)

one has

$$ u'^m=\widetilde{\gamma} \bigl[1,\widetilde{v}^x, \widetilde {v}^y,\widetilde{v}^\eta \bigr], $$

(A.19)

where

$$ \widetilde{\gamma}\equiv\gamma_\perp\,\cosh(Y-\eta_s)= \frac {1}{\sqrt{1-(\widetilde{v}^x)^2-(\widetilde{v}^y)^2-\tau ^2(\widetilde{v}^\eta)^2 }} $$

(A.20)

in agreement with the definition $\widetilde{\gamma} \equiv (1-g_{ij}\tilde{v}^{i}\tilde{v}^{j})^{-1/2}$ quoted in the text.

In the case of a longitudinal boost-invariant expansion one has Y≡η _s , so that

$$ u'^m=\gamma_\perp[1,{\boldsymbol{u}}_\perp,0]. $$

(A.21)

Appendix B: Source terms

In the present appendix, we write down explicitly the source terms needed for the evolution of the set of balance laws as in Eq. (26). For the momentum and energy equations we have, respectively

$$\begin{aligned} \mathbf{S}(S_i) & = |g|^{1/2} \biggl[ \frac{1}{2} T^{\mu\nu} \partial_i g_{\mu\nu} \biggr], \end{aligned}$$

(B.22)

$$\begin{aligned} \mathbf{S}(E) & = |g|^{1/2} \biggl[ - \frac{1}{2} T^{\mu\nu} \partial_0 g_{\mu\nu} \biggr], \end{aligned}$$

(B.23)

whereas for the evolution of the bulk viscous pressure Π and for the spatial components of the viscous stress tensor π ^ij more terms are required. We recall here the general definitions for the expansion scalar, shear tensor, and vorticity, which are

$$\begin{aligned} &\theta = d_\mu u^\mu= \partial_\mu u^\mu+ \varGamma^\mu_{\mu\nu }u^\nu, \end{aligned}$$

(B.24)

$$\begin{aligned} & \begin{aligned}[b] \sigma^{\mu\nu} & = \frac{1}{2}\bigl(d^\mu u^\nu+ d^\nu u^\mu\bigr) + \frac{1}{2} \bigl(u^\mu Du^\nu+ u^\nu Du^\mu\bigr) \\ &\quad - \frac{1}{3}\bigl(g^{\mu\nu} + u^\mu u^\nu \bigr)\theta, \end{aligned} \end{aligned}$$

(B.25)

$$\begin{aligned} &\omega^{\mu\nu} = \frac{1}{2}\bigl(d^\mu u^\nu- d^\nu u^\mu\bigr) + \frac{1}{2} \bigl(u^\mu Du^\nu- u^\nu Du^\mu \bigr), \end{aligned}$$

(B.26)

with

$$\begin{aligned} &d^\mu u^\nu = g^{\mu\alpha} \bigl( \partial_\alpha u^\nu+ \varGamma^\nu _{\alpha\beta} u^\beta\bigr), \end{aligned}$$

(B.27)

$$\begin{aligned} &Du^\nu = u^\alpha\bigl(\partial_\alpha u^\nu+ \varGamma^\nu_{\alpha \beta} u^\beta \bigr). \end{aligned}$$

(B.28)

Moreover, the $\mathcal{I}^{\mu\nu}$ terms are provided by

$$\begin{aligned} \mathcal{I}^{\mu\nu}_0 & = - u^\alpha \bigl( \varGamma^\mu_{\alpha\beta}\pi^{\nu\beta} + \varGamma^\nu_{\alpha \beta} \pi^{\mu\beta}\bigr), \end{aligned}$$

(B.29)

$$\begin{aligned} \mathcal{I}^{\mu\nu}_1 & = g_{\alpha\beta}\bigl( \pi^{\mu\alpha}u^\nu + \pi^{\nu\alpha}u^\mu\bigr) Du^\beta, \end{aligned}$$

(B.30)

$$\begin{aligned} \mathcal{I}^{\mu\nu}_2 & = - \lambda\ g_{\alpha\beta}\bigl( \pi^{\mu \alpha}\omega^{\nu\beta} + \pi^{\nu\alpha}\omega^{\mu\beta} \bigr). \end{aligned}$$

(B.31)

In the remainder we shall specify to either Minkowski or Bjorken coordinates.

2.1 B.3 Minkowski coordinates

The simplest case is that of Minkowskian Cartesian coordinates (t,x,y,z), with metric g _μν =g ^μν=diag(−1,1,1,1) (|g|^1/2=1), and vanishing Christoffel symbols. Thus, no source terms are needed for the evolution of the energy-momentum tensor. Covariant derivatives simply become

$$\begin{aligned} d^t u^\nu& = - \partial_t u^\nu, \end{aligned}$$

(B.32)

$$\begin{aligned} d^i u^\nu& = \partial_i u^\nu, \end{aligned}$$

(B.33)

$$\begin{aligned} D u^\nu& = u^t \partial_t u^\nu+ u^i \partial_i u^\nu, \end{aligned}$$

(B.34)

so for instance the expansion scalar is

$$ \theta= \partial_t u^t + \partial_i u^i, $$

(B.35)

with i=x,y,z. Notice that $\mathcal{I}_{0}^{\mu\nu}\equiv0$, while for $\mathcal{I}_{1}^{\mu\nu}$ and $\mathcal{I}_{2}^{\mu\nu}$ the standard definitions apply.

2.2 B.4 Bjorken coordinates

Bjorken coordinates (τ,x,y,η _s ) have still a diagonal metric g _μν =diag(−1,1,1,τ ²), and g ^μν=diag(−1,1,1,1/τ ²) (|g|^1/2=τ), but here ∂ _τ g _ηη =2τ≠0 leading to the non-vanishing Christoffel symbols $\varGamma^{\tau}_{\eta\eta}=\tau$ and $\varGamma^{\eta}_{\eta\tau}=1/\tau$. Then, while the source term for the momentum equation is still zero, that for the energy equation becomes

$$ \mathbf{S}(E) = -\tau^2 T^{\eta\eta}. $$

(B.36)

Non-Minkowskian covariant derivatives are

$$\begin{aligned} &d^\tau u^\eta = - \bigl(\partial_\tau u^\eta+ u^\eta/\tau\bigr), \end{aligned}$$

(B.37)

$$\begin{aligned} &d^\eta u^\tau = \bigl(\partial_\eta u^\tau+ \tau u^\eta\bigr)/\tau^2, \end{aligned}$$

(B.38)

$$\begin{aligned} &d^\eta u^\eta = \bigl(\partial_\eta u^\eta+ u^\tau/\tau\bigr)/\tau^2, \end{aligned}$$

(B.39)

$$\begin{aligned} &D u^\tau = u^\tau\partial_\tau u^\tau+ u^i\partial_i u^\tau+ \tau u^\eta u^\eta, \end{aligned}$$

(B.40)

$$\begin{aligned} &D u^\eta = u^\tau\partial_\tau u^\eta+ u^i\partial_i u^\eta+ 2u^\tau u^\eta/\tau, \end{aligned}$$

(B.41)

with i=x,y,η, and the expansion scalar is now

$$ \theta= \partial_\tau u^\tau+ \partial_i u^i + u^\tau/\tau. $$

(B.42)

In Bjorken coordinates the non-vanishing $\mathcal{I}_{0}^{\mu\nu }\equiv0$ terms are

$$\begin{aligned} \mathcal{I}_0^{x\eta} & = -\bigl(u^\tau \pi^{x\eta} + u^\eta\pi^{\tau x}\bigr)/\tau, \end{aligned}$$

(B.43)

$$\begin{aligned} \mathcal{I}_0^{y\eta} & = -\bigl(u^\tau \pi^{y\eta} + u^\eta\pi^{\tau y}\bigr)/\tau, \end{aligned}$$

(B.44)

$$\begin{aligned} \mathcal{I}_0^{\eta\eta} & = -2\bigl(u^\tau \pi^{\eta\eta} + u^\eta \pi^{\tau\eta}\bigr)/\tau, \end{aligned}$$

(B.45)

while $\mathcal{I}_{1}^{\mu\nu}$ and $\mathcal{I}_{2}^{\mu\nu}$ are defined in the usual way.