Viscous hydrodynamics with bulk viscosity -- uncertainties from relaxation time and initial conditions
aa r X i v : . [ nu c l - t h ] S e p Viscous hydrodynamics with bulk viscosity –uncertainties from relaxation time and initial conditions
Huichao Song and Ulrich Heinz
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
Abstract
Bulk viscosity suppresses elliptic flow v , as does shear viscosity. It can thus not be neglectedwhen extracting the shear viscosity from elliptic flow data. We here explore uncertainties inthe bulk viscous contribution to viscous v suppression that arise from presently uncontrolleduncertainties in the initial value of the bulk viscous pressure and its microscopic relaxation time.
1. Introduction
Recently, causal viscous hydrodynamics for relativistic heavy ion collisions has been expe-riencing rapid development. Several groups independently developed 2 + ff ects caused by shear viscosity and on developingstrategies for constraining the QGP shear viscosity to entropy density ratio η/ s from experimen-tal data [1, 2, 3, 4, 5, 6, 7, 8, 9]. It was found that elliptic flow v is very sensitive to η/ s andthat, due to the rapid expansion of the fireballs created in heavy-ion collisions, even the mini-mal KSS bound η/ s = / π [10] leads to a strong suppression of v compared to the ideal fluidcase [1, 2, 3, 4, 5, 6]. A first attempt by Luzum and Romatschke [5] to extract the QGP viscos-ity from experimental elliptic flow data, using viscous hydrodynamics simulations, indicate that η s | QGP < × π . To obtain a more precise value requires additional theoretical e ff ort in at least thefollowing four directions (see [11] for references): (1) connecting viscous hydrodynamics to ahadron cascade to properly account for e ff ects from the highly viscous hadronic stage; (2) includ-ing the e ff ects from bulk viscosity; (3) employing a more realistic equation of state (EOS) thatuses the latest lattice QCD data above T c matched to a hadron resonance gas in partial chemicalequilibrium below T c , to properly account for chemical freeze-out at T chem ≃ −
170 MeV;and (4) a better treatment of the initial conditions that not only aims to eliminate presently largeuncertainties in the initial fireball eccentricity but also properly accounts for pre-equilibriumtransverse flow and fluctuations in the initial fireball deformation and orientation.In this contribution we focus on point (2) and study e ff ects from bulk viscosity. A longeraccount of this work can be found in Ref. [12]. In [11], where we reported first results andto which we refer the reader for additional details, we constructed and used a function for thespecific bulk viscosity ζ/ s ( T ) that interpolates between a ”minimal” value well above T c , basedon strong-coupling calculations using the AdS / CFT correspondence [13], to a zero value in thehadron resonance gas well below T c , using a Gaussian function between these limits that peaks at T c . (For a discussion of uncertainties in ζ/ s below T c see [14].) We call this function ”minimalbulk viscosity”. To study larger bulk viscosities, we made comparison runs where this entire Preprint submitted to Nuclear Physics A August 30, 2018 unction was multiplied with a coe ffi cient C >
1. For twice the minimal bulk viscosity wefound that the viscous v suppression increases from 20% (for a fluid that has only minimalshear viscosity η s = π ) to 30% (for a fluid that additionally features bulk viscosity at twice the”minimal” level, C = v , accounting for such a 50% relativeincrease in the viscous v suppression translates into a reduction of the extracted shear viscosity η/ s by roughly a factor . Consequently, bulk viscous e ff ects cannot be ignored when extracting η/ s from experimental data.The analysis [11] suggests that the main uncertainty stems from insu ffi cient knowledge ofthe peak value of ζ/ s near the phase transition . However, this is only part of the story. Addi-tional complications arise from the fact that the expected peak in ζ/ s near T c is due to rapidlygrowing correlation lengths, associated with a ”critical slowing down” of microscopic relaxationprocesses near the phase transition. This probably leads near T c to a much larger relaxation time τ Π for the bulk viscous pressure than commonly used for the shear viscous pressure (which near T c is of the order of 0 . / c , as obtained from kinetic theory [18, 19, 20] or AdS / CFT [21]).Large relaxation times can lead to strong memory e ff ects, i.e. strong sensitivity to the initialconditions for the bulk viscous pressure. This is what we discuss here [12].
2. Setup
We use the code
VISH2+1 [2, 3] to solve (2 + T mn and the bulk viscous pressure Π : d m T mn = , T mn = eu m u n − ( p + Π ) ∆ mn , (1) D Π = − τ Π ( Π + ζ∂ · u ) − Π ζ T τ Π d k τ Π ζ T u k ! . (2)Here, m , n denote components in ( τ, x , y , η s ) coordinates, with covariant derivative d m (for detailssee [22]), D = u m d m and ∇ m = ∆ ml d l (where ∆ mn = g mn − u m u n is the projector transverse to theflow vector u m ) are the time derivative and spatial gradient in the local comoving frame, ζ is bulkviscosity, and τ Π is the corresponding relaxation time. For ζ s ( T ) we use the phenomenologicalconstruction described in [11]. For τ Π we consider three choices: the constant values τ Π = . / c , and the temperature dependent function τ Π ( T ) = max[˜ τ · ζ s ( T ) , . / c ] with ˜ τ =
120 fm / c . The last choice implements phenomenologically the concept of critical slowing down;it yields τ Π ≈ . / c at T =
350 MeV and τ Π ≈ / c at T c .To study memory e ff ects, we explore two di ff erent initializations for the bulk viscous pres-sure: (a) Navier-Stokes (N-S) initialization, Π ( τ ) = − ζ∂ · u , and (b) zero initialization, Π ( τ ) = τ = . / c . For all other inputs we make standard choices as discussed in Refs. [3, 7]and listed in the figure below.
3. Bulk viscosity e ff ects: uncertainties from relaxation time and bulk pressure initialization The left panel of Fig. 1 shows the di ff erential elliptic flow v ( p T ) of directly emitted pions(without resonance decays) for non-central Au + Au collisions at b = Currently, theoretical uncertainties for the peak value of ζ s near T c are very large. Extraction from lattice QCDsimulations gives a peak value around 0.7 [15]. This is more than 10 time larger than the string theory prediction basedon holographic models [16]. A critical discussion of the lattice QCD based extraction can be found in [17]. T (GeV)00.020.04 v Glauber initialization:
Au+Au, b=7 fmEOS L e = 30 GeV/fm τ = 0.6 fm/cT dec = 130 MeV ideal hydro viscous hydro (min. bulk viscosity only) τ−τ (fm/c)-0.10 < Π > ( G e V / f m ) τ Π = 0.5 fm/c τ Π =(ζ/ s)(T) (120 fm/c) Π = 0Π = −ζ ( ∂ u) Π = −ζ ( ∂ u) Π = 0 { { η /s=0, ζ /s = minimal -- min. bulk viscosity only..viscous hydrodynamics τ Π = 5 fm/c { Π = −ζ ( ∂ u) Π = 0 .. Figure 1: (color online) Left: Di ff erential pion elliptic flow v ( p T ) from ideal and viscous hydrodynamics, includingonly bulk viscosity. Right: Time evolution of the bulk pressure h Π i averaged over the transverse plane (weighted bythe energy density) from viscous hydrodynamics. Di ff erent curves correspond to di ff erent initializations and relaxationtimes, as indicated (see text for discussion). hydrodynamics and minimally bulk viscous hydrodynamics with identical initial and final con-ditions. The di ff erent lines from viscous hydrodynamics correspond to di ff erent relaxation times τ Π and di ff erent initializations Π ( τ ). One sees that these di ff erent inputs can lead to large un-certainties for the bulk viscous v suppression. For minimal bulk viscosity, the v suppression at p T = . ∼
2% to ∼
10% compared to ideal hydrodynamics (blue dashed linein the left panel).For the shorter relaxation time, τ Π = . / c, the bulk viscous v suppression is insensitiveto the initialization of Π , and both N-S and zero initializations show ∼ v suppression relativeto ideal fluids. The reason behind this becomes apparent in the right panel showing the timeevolution of the average bulk pressure h Π i . For short relaxation times, h Π i quickly loses allmemory of its initial value, relaxing in both cases to the same trajectory after about 1 − / c(i.e. after a few times τ Π ). This is similar to what we found for shear viscosity where themicroscopic relaxation times are better known and short ( τ π ( T c ) ≃ . − . / c) and where theshear pressure tensor π mn therefore also loses memory of its initialization after about 1 fm / c [3].This changes if one accounts for the critical slowing down of the evolution of Π near T c . Ifone simply multiplies the constant relaxation time by a factor 10, setting τ Π = / c , one obtainsthe dotted and solid magenta lines in Fig. 1. Now the bulk viscous v suppression relative to theideal fluid becomes very sensitive to the initialization of the bulk viscous pressure: For zeroinitialization Π ( τ ) =
0, the viscous v suppression is very small (only ∼
2% at p T = . / c ).The right panel shows that in this case the magnitude of the (transversally averaged) bulk pressureevolves very slowly and always stays small, leading to almost ideal fluid evolution. On the otherhand, if Π is initialized with its Navier-Stokes value, which initially is large due to the stronglongitudinal expansion, it decays initially more slowly than for the shorter relaxation time. Itsbraking e ff ect on the flow evolution is therefore bigger, resulting in much stronger suppressionof v than for zero initialization, at ∼
10% slightly exceeding even the viscous v suppressionseen for the tenfold shorter relaxation time.The ”critical slowing down” scenario with temperature-dependent τ Π ( T ) (black lines) inter-polates between the short and long relaxation times. As for the fixed larger value τ Π = / c , v depends sensitively on the initialization of Π , but for N-S initialization the viscous v suppressionis somewhat smaller than for both short and long fixed relaxation times. The reasons for this are3ubtle since now, at early times, the bulk viscous pressure Π evolves on very di ff erent time scalesin the dense core and dilute edge regions of the fireball. As a result, for N-S initialization theaverage value h Π i is smaller in magnitude than for both short and long fixed τ Π , throughout thefireball evolution (right panel, black lines).
4. Conclusions
Relaxation times and initial values for the dissipative flows (bulk and shear pressure) arerequired inputs in viscous hydrodynamic calculations, in addition to the transport coe ffi cientsand the EOS. Near T c , the bulk viscosity ζ can exceed the shear viscosity η of the stronglyinteracting matter. If the relaxation time τ Π for the bulk viscous pressure Π is short, it quicklyloses memory of its initial valaue, but the relatively large peak value of ζ/ s near T c can lead toa significant viscous suppression of the elliptic flow v , competing with shear viscous e ff ects.If τ Π grows rapidly near T c , due to critical slowing down, the bulk viscous suppression e ff ectson v depend crucially on the initial value of Π : If Π is zero initially, bulk viscous e ff ects on v are almost negligible; if Π is initially large, however, as for the case of the N-S initialization,it remains relatively large throughout the evolution, suppressing the buildup of elliptic flow at alevel that again competes with shear viscous e ff ects. Additional research on initial conditions andrelaxation times for the bulk viscous pressure is therefore necessary for a quantitative extractionof η/ s from measured data. Acknowledgments
We thank K. Dusling and P. Petreczky for enlightening discussions. This work was supportedby the U.S. Department of Energy under grant DE-FG02-01ER41190.
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