Causal viscous hydrodynamics in 2 + 1 dimensions for relativistic heavy-ion collisions

We explore the effects of shear viscosity on the hydrodynamic evolution and final hadron spectra of Cu + Cu collisions at ultrarelativistic collision energies, using the newly developed (2 + 1)-dimensional viscous hydrodynamic code VISH2+1. Based on the causal Israel-Stewart formalism, this code describes the transverse evolution of longitudinally boost-invariant systems without azimuthal symmetry around the beam direction. Shear viscosity is shown to decelerate the longitudinal and accelerate the transverse hydrodynamic expansion. For fixed initial conditions, this leads to a longer quark-gluon plasma (QGP) lifetime, larger radial flow in the final state, and flatter transverse momentum spectra for the emitted hadrons compared to ideal fluid dynamic simulations. We find that the elliptic flow coefficient $v_2$ is particularly sensitive to shear viscosity: even the lowest value allowed by the AdS/CFT conjecture $\eta /s⩾1/4\pi$ suppresses $v_2$ enough to have significant consequences for the phenomenology of heavy-ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC). A comparison between our numerical results and earlier analytic estimates of viscous effects within a blast-wave model parametrization of the expanding fireball at freeze-out reveals that the full dynamical theory leads to much tighter constraints for the specific shear viscosity $\eta /s$, thereby supporting the notion that the quark-gluon plasma created at RHIC exhibits almost “perfect fluidity.”

Figure 1

Equations of state EOS Q and SM-EOS Q (modified EOS Q).

Figure 2

Time evolution of the hydrodynamic source terms in Eqs. (7)–(9), averaged over the transverse plane, for central Cu + Cu collisions, calculated with EOS I and SM-EOS Q. The small insets blow up the vertical scale to show more detail. The dashed blue lines are for ideal hydrodynamics with $e_0=30$ GeV/fm${}^3$ and ${\tau }_0=0.6$ fm/$c$. Solid red lines show results from viscous hydrodynamics with identical initial conditions and $\frac{\eta }{s}=\frac{1}{4\pi }\approx 0.08,{\tau }_{\pi }=\frac{3\eta }{\mathit{sT}}\approx 0.24\left(\frac{200\hspace{1em}\mathrm{MeV}}{T}\right)$ fm/$c$. The positive source terms drive the transverse expansion, while the negative ones affect the longitudinal expansion.

Figure 3

Time evolution of the local temperature in central Cu + Cu collisions, calculated with EOS I and SM-EOS Q, for the center of the fireball ($r=0$) and a point near the edge ($r=9$ fm). Same parameters as in Fig. 2.

Figure 4

Time evolution of the local entropy density for central Cu + Cu collisions, calculated with EOS I and SM-EOS Q, for the center of the fireball ($r=0$) and a point at $r=3$ fm. Same parameters and color coding as in Fig. 3.

Figure 5

Surfaces of constant temperature $T$ and constant transverse flow velocity $v_{\perp }$ for central Cu + Cu collisions, evolved with EOS I and SM-EOS Q. In each panel, results from viscous hydrodynamics in the left half are directly compared with the corresponding ideal fluid evolution in the right half. (The thin isotherm contours in the right halves of each panel are reflected from the left halves, for easier comparison.) The lines of constant $v_{\perp }$ are spaced by intervals of 0.1, from the inside outward, as indicated by the numbers near the top of the figures. The right panel contains two isotherms for $T_c=164$ MeV, one separating the mixed phase (MP) from the QGP at energy density $e_Q=1.6$ GeV/fm${}^3$, the other separating it from the hadron resonance gas (HRG) at energy density $e_H=0.45$ GeV/fm${}^3$. See text for discussion.

Figure 6

Time evolution of the average radial flow velocity $\langle v_T\rangle \equiv \langle v_{\perp }\rangle$ in central Cu + Cu collisions, calculated with EOS I and SM-EOS Q, comparing ideal and viscous fluid dynamics. The initially faster rate of increase reflects large positive shear viscous pressure in the transverse direction at early times. The similar rates of increase at late times indicate the gradual disappearance of shear viscous effects. In the right panel, the curves exhibit a plateau from 2 to 4 fm/$c$, reflecting the softening of the EOS in the mixed phase.

Figure 7

Midrapidity particle spectra for central Cu + Cu collisions, calculated with EOS I (gluons) and with SM-EOS Q ( ${\pi }^{-},K^{+}$, and $p$) for ideal and viscous hydrodynamics. The dotted lines show viscous hydrodynamic spectra that neglect the viscous correction $\delta f_i$ to the distribution function in Eq. (12); i.e., they include only the effects from the larger radial flow generated in viscous hydrodynamics.

Figure 8

Ratio of the viscous correction $\delta N$, resulting from the nonequilibrium correction $\delta f$, Eq. (16), to the distribution function at freeze-out, to the equilibrium spectrum $N_{\mathrm{eq}}\equiv {\mathit{dN}}_{\mathrm{eq}}/(\mathit{dy}\hspace{1em}{d}^2{p}_T)$ calculated from Eq. (12) by setting $\delta f=0$. The gluon curves are for evolution with EOS I, the curves for ${\pi }^{-},K^{+},$ and $p$ are from calculations with SM-EOS Q.

Figure 9

Surfaces of constant temperature $T$ and constant transverse flow velocity $v_{\perp }$ for semiperipheral Cu + Cu collisions at $b=7$ fm, evolved with SM-EOS Q. In the top row, we contrast ideal and viscous fluid dynamics, with a cut along the $x$ axis (in the reaction plane) shown in the right half of each panel, while the left half shows a cut along the $y$ axis (perpendicular to the reaction plane). In the bottom row, we compare ideal and viscous evolution in the each panel, with cuts along the $x$ ($y$) direction shown in the left (right) panel. See Fig. 5 for comparison with central Cu + Cu collisions.

Figure 10

Time evolution of the transverse flow anisotropy $\langle \left|v_x\right|-\left|v_y\right|\rangle$ (top row) and of the anisotropy in the transverse source term $\langle \left|{\mathcal{S}}^{\tau x}\right|-\left|{\mathcal{S}}^{\tau y}\right|\rangle$ (bottom row). Both quantities are averaged over the transverse plane, with the Lorentz-contracted energy density ${\gamma }_{\perp }$ as weight function. The left (right) column shows results for EOS I (SM-EOS Q), comparing ideal and viscous fluid dynamical evolution.

Figure 11

Time evolution for the spatial eccentricity ${\epsilon }_x$, momentum anisotropy ${\epsilon }_p,$ and total momentum anisotropy ${\epsilon }_p^{\text{'}}$ (see text for definitions), calculated for $b=7$ fm Cu + Cu collisions with EOS I (left column) and SM-EOS Q (right column). Dashed lines are for ideal hydrodynamics, while the solid and dotted lines show results from viscous hydrodynamics.

Figure 12

Differential elliptic flow $v_2(p_T)$ for Cu + Cu collisions at $b=7$ fm.

Figure 13

Left: Time evolution of the various components of the shear viscous pressure tensor, normalized by the enthalpy and averaged in the transverse plane over the thermalized region inside the freeze-out surface.Note that the normalizing factor $e+p~T^4$ decreases rapidly with time. Right: Comparison of the full viscous hydrodynamic source terms ${\mathcal{S}}^{\mathit{mn}}$, averaged over the transverse plane, with their approximations given in Eqs. (7)–(9), as a function of time. The thin gray lines indicate the corresponding source terms in ideal fluid dynamics.

Figure 14

Time evolution of ${\pi }^{\tau \tau },\Sigma$, and Δ, as well as that of the transverse velocity, along the decoupling surface for $b=7$ fm Cu + Cu collisions in viscous hydrodynamics with SM-EOS Q, as shown in Fig. 9. Solid and dashed lines represent cuts along, respectively, the $x$ ($\varphi =0$) and $y$ $(\varphi =\frac{\pi }{2})$ directions.

Figure 15

Viscous corrections to the azimuthally averaged pion spectrum resulting from individual components of the viscous pressure tensor ${\pi }^{\mathit{mn}}$ as indicated, as well as the total correction $\delta N/N_{\mathrm{eq}}$, for Cu + Cu collisions at $b=7$ fm with SM-EOS Q. The horizontal dashed line at $-50\%$ indicates the limit of validity.

Figure 16

Ratio of nonequilibrium and equilibrium contributions to particle production (solid line) and to its momentum anisotropy (dashed line), as a function of $p_T$ for pions from Cu + Cu collisions at $b=7$ fm with SM-EOS Q.

Figure 17

Similar to Fig. 13, but now comparing runs with different initial conditions. The thick lines reproduce the results from Fig. 13, obtained with ${\pi }^{\mathit{mn}}=2\hspace{1em}\eta {\sigma }^{\mathit{mn}}$at initial time ${\tau }_0$, while thin lines of the same type show the corresponding results obtained by setting initially ${\pi }^{\mathit{mn}}=0$. The right panel shows the full viscous source terms, without approximation: $\langle \left|{\mathcal{S}}^{\tau x}\right|\rangle$ (dashed), $\langle \left|{\mathcal{S}}^{\tau y}\right|\rangle$ (dotted), and $\langle {\mathcal{S}}^{\tau \tau }\rangle$ (dash-dotted).

Figure 18

Differential elliptic flow $v_2(p_T)$ for pions from $b=7$ fm Cu + Cu collisions with SM-EOS Q. Thick lines reproduce the pion curves from Fig. 12, obtained with ${\pi }^{\mathit{mn}}=2\hspace{1em}\eta {\sigma }^{\mathit{mn}}$at initial time ${\tau }_0$, while thin lines of the same type show the corresponding results obtained by setting initially ${\pi }^{\mathit{mn}}=0$.

Figure 19

Time evolution of the two independent viscous pressure tensor components ${\pi }^{\tau \tau }$ and $\Sigma ={\pi }^{\mathit{xx}}+{\pi }^{\mathit{yy}}$ for central Cu + Cu collisions (solid lines), compared with their Navier-Stokes limits $2\eta {\sigma }^{\tau \tau }$ and $2\eta ({\sigma }^{\mathit{xx}}+{\sigma }^{\mathit{yy}})$ (dashed lines), for two values of the relaxation time, ${\tau }_{\pi }=3\eta /\mathit{sT}$ (thick lines) and ${\tau }_{\pi }=1.5\hspace{1em}\eta /\mathit{sT}$ (thin lines). All quantities are scaled by the thermal equilibrium enthalpy $e+p$ and transversally averaged over the thermalized region inside the decoupling surface.

Figure 20

Differential elliptic flow $v_2(p_T)$ for ${\pi }^{-}$ from $b=7$ fm Cu + Cu collisions with SM-EOS Q, calculated from viscous hydrodynamics with two different values for the relaxation time ${\tau }_{\pi }$. Thick lines reproduce the pion curves from Fig. 12, thin lines show results obtained with a twice shorter relaxation time. For the standard (twice larger) classical relaxation time value ${\tau }_{\pi }=6\eta /\mathit{sT}$ [26, 30] deviations from ideal hydrodynamics would exceed those seen in the thick lines.

Figure 21

Characteristic transverse momentum $p_T^{*}$ where the viscous corrections to the final pion spectrum become so large ($>50\%$) that the spectrum becomes unreliable, as a function of the initial energy density in the center of the fireball. Stars are for central Cu + Cu collisions, open circles for peripheral Cu + Cu collisions at $b=7$ fm. Note that identical $e(r=0)$ values correspond to higher collision energies in peripheral than in central collisions. $p_T^{*}$ values are higher for more massive hadrons (see Fig. 8), and they also increase for smaller relaxation times ${\tau }_{\pi }$ (see discussion of Fig. 20).

Figure 22

Left: Differential elliptic flow $v_2(p_T)$ for ${\pi }^{-}$ from $b=4$ fm Cu + Cu collisions and $b=7$ fm Au + Au collisions, using EOS Q. Results from VISH2+1 for $\eta =0$ and ${\pi }^{\mathit{mn}}=0$ are compared with the ideal fluid code azhydro. Right:$v_2(p_T)$ for ${\pi }^{-}$ from Cu + Cu collisions at impact parameters $b=4$ and 7 fm, comparing VISH2+1 evolution with EOS Q and SM-EOS Q in the ideal fluid limit $\eta =0,{\pi }^{\mathit{mn}}=0$.

Figure 23

Comparison between the analytical temperature evolution for (0 + 1)-d boost-invariant Navier-Stokes viscous hydrodynamics and numerical results from VISH2+1 with homogeneous transverse initial energy density profiles.

Figure 24

Differential elliptic flow $v_2(p_T)$ for gluons from $b=7$ fm Cu + Cu collisions, calculated with ideal hydrodynamics, relativistic Navier-Stokes (NS) hydrodynamics, and Israel-Stewart (IS) viscous hydrodynamics with $\frac{\eta }{s}=\frac{T}{2\hspace{1em}\mathrm{GeV}}$ and ${\tau }_{\pi }=0.03$ fm/$c$, using EOS I. The lines for NS and IS viscous hydrodynamics are almost indistinguishable. Solid lines show the full results from viscous hydrodynamics, dotted lines neglect viscous corrections to the spectra and take only the flow anisotropy effect into account.

Figure 25

Top row: Velocity profiles from the blast wave model (left) and from the hydrodynamic model with EOS I at fixed times (middle) and along the decoupling surface (right). Bottom row: The corresponding profiles for the transverse shear viscous pressure ${\pi }^{\mathit{rr}}$ in the Navier-Stokes limit, ${\pi }^{\mathit{rr}}=2\eta {\nabla }^{\langle \mu }{u}^{\nu \rangle }$. Calculations are for central Cu + Cu collisions, and the curves in the middle panels correspond to the times $\tau =1$, 2, 4, and 6 fm/$c$.