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Parameterizing Smooth Viscous Fluid Dynamics With a Viscous Blast Wave

Blast wave fits can capture essential features of global properties of systems near kinetic equilibrium. They usually provide temperature fields and collective velocity fields on a given hypersurface. We investigate how faithful the viscous blast wave introduced in [1] (Z. Yang and R. J. Fries, arXiv:1807.03410), can reproduce the given temperature and specific shear viscosity at freeze-out of a viscous fluid dynamic calculation, if the final spectrum and elliptic flow of several particle species are fitted. We focus here on fluid dynamic simulations appropriate for high energy nuclear collisions at current collider energies. We find that specific shear viscosities are reproduced to good accuracy by viscous blast wave fits while temperatures tend to be slightly underpredicted. We quantify the deviations of fitted from true quantities for some examples. The maps we obtain can be used to improve raw results obtained from viscous blast wave fits.

Blast wave fits can capture essential features of global properties of systems near kinetic equilibrium. They usually provide temperature fields and collective velocity fields on a given hypersurface. We investigate how faithful the viscous blast wave introduced in [1] can reproduce the given temperature and specific shear viscosity at freeze-out of a viscous fluid dynamic calculation, if the final spectrum and elliptic flow of several particle species are fitted. We focus here on fluid dynamic simulations appropriate for high energy nuclear collisions at current collider energies. We find that specific shear viscosities are reproduced to good accuracy by viscous blast wave fits while temperatures tend to be slightly underpredicted. We quantify the deviations of fitted from true quantities for some examples. The maps we obtain can be used to improve raw results obtained from viscous blast wave fits.

I. INTRODUCTION
Blast waves are simple and effective tools that provide snapshots of a system that is close enough to local kinetic equilibrium so that macroscopic concepts like temperature, local collective velocity, and shear and bulk stress can be used. The full dynamics of such systems is usually very well captured by viscous fluid dynamic equations of motion. If the equation of state and a sufficient number of transport coefficients (e.g. shear and bulk viscosity) of the system are known, fluid dynamics can evolve the system starting from a given initial state. In contrast, blast waves usually provide a static picture, typically the temperature field T (x) and collective flow field u µ (x) on a hypersurface Σ consisting of points r µ . E.g., if the hypersurface of kinetic freeze-out in an expanding system is chosen, these fields can often be directly related to observables of the system and can be fitted to data.
In high energy nuclear collisions, blast waves have routinely been used to analyze the properties of the fireball of hadrons at the time of kinetic freeze-out [2][3][4][5][6]. After kinetic freeze-out hadrons free stream to the detectors and thus directly carry information about the freeze-out hypersurface. Blast wave parameters like the freeze-out temperature and average radial flow velocity can be determined by fits to transverse momentum spectra of hadrons. Additional information, like elliptic deformations of the fireball in coordinate and momentum space at the time of freeze-out can be extracted from fits of elliptic flow coefficients v 2 . More recently, viscous corrections to blast waves have been considered [1,[7][8][9]. They can be used to extract the specific shear viscosity η/s of hadronic matter at the freeze-out temperature [1]. Viscous blast waves can also be used to extend the range of validity of ideal blast wave fits to larger transverse hadron momenta P T .
The question arises to what extent blast waves, which use certain simplifying approximations, can faithfully reproduce important properties of the full dynamical system. The blast wave used here is defined by an ansatz for the temperature, flow field and space-time structure of the hypersurface, as discussed below. The ansatz contains several parameters with physical meaning, like the freeze-out temperature and specific shear viscosity which can be fitted to data. It is important to understand and quantify the uncertainties in the fit results that are due to the simplifications made in the blast wave ansatz. Once quantified, the systematic deviations could be corrected for and the extraction of important parameters from blast waves could turn into a tool with improved precision. In this work we take a first step in this direction by comparing freeze-out conditions from smooth viscous fluid dynamics to blast wave fits. The global picture is one of two successive approximations real collision system fluid dynamic simulation blast wave fit where means "approximated by". Viscous fluid dynamic simulations are widely accepted to give accurate descriptions of the low transverse momentum region of high energy nuclear collisions, although analyses of experimental data with fluid dynamic simulations continue to be an active field of study [10][11][12]. There are several approximations that enter when describing the evolution of the system and its freeze-out with fluid dynamics. Some relevant ones are briefly discussed in [1], but their quantification is outside the scope of this work. Here we focus on the second step, and quantify uncertainties when approximating fluid dynamic systems by blast waves at freeze-out. We use smooth relativistic viscous fluid dynamics to create systems close to local equilibrium as they typically occur in the late stages of high energy nuclear collisions. Approximations used by deploying blast waves are discussed and then quantified in a specific case. We focus here on the accuracy of the extracted freeze-out temperature T fo and the specific shear viscosity η/s, i.e. the ratio of shear viscosity η and entropy density s, at freeze-out. The results from the analysis here can be used directly to improve the extraction of η/s from experimental data in [1].
In our work we use the viscous generalization of the Retiere-Lisa (RL) blast wave [4] introduced in [1]. We utilize the viscous fluid dynamics code MUSIC [10,13,14] to generate simulation pseudodata. The setup of the fluid dynamic calculations roughly reflect conditions in Au+Au collisions at the Relativistic Hadron Collider (RHIC) and Pb+Pb collisions at the Large Hadron Collider (LHC), as described below. For direct comparisons of MUSIC simulations to data we refer the reader to [13,14]. We will introduce a map from the (T fo -η/s)-plane of "true" values set in fluid dynamic simulations to the corresponding (T fo -η/s)-plane extracted from viscous blast wave fits of fluid dynamic spectra and elliptic flow. We are providing a parameterization of this map and its inverse. The inverse map can be used to improve blast wave fits by removing the bias introduced by the blast wave approximations. We will also estimate the uncertainties from this procedure.
The paper is organized as follows. In section II, we review the viscous Retiere-Lisa blast wave and discuss the approximations made in blast waves. In section III, we describe the setup of the MUSIC hydrodynamic calculations and the pseudodata that is fitted. In section IV, we provide the results of the viscous blast wave fits. In section V, we discuss the relation between fluid dynamic parameters and blast wave fit parameters and utilize it to improve a previous extraction of hadronic shear viscosity.

II. FLUID DYNAMICS AND VISCOUS BLASTWAVE
In this section we briefly review some basic concepts shared by both fluid dynamics freeze-out and blast waves. We will then discuss the particular ansatz for the viscous blast wave in [1], based on the work by Retiere and Lisa [4]. For a system close enough to local kinetic equilibrium one can assign a local temperature field T (r) and a flow field u µ (r) to describe the temperature and collective motion as a function of position 4-vector r µ . The particle distribution in the local rest frame of a fluid cell can then be written as where f 0 is the equilibrium Bose or Fermi-distribution as a function of particle momentum p µ = (E, p) with the local temperature T , and δf is a small correction that accounts for the out-of-equilibrium behavior. We neglect chemical potentials in the distribution but their inclusion is straight forward [1]. The general form of the correction term is [1,15,16] δf (r, p) = η s Here the shear stress tensor π µν has been expressed by its Navier-Stokes approximation, π µν = 2ησ µν , where σ µν is the traceless gradient tensor, defined as We have used the notation ∇ µ = ∆ µν ∂ ν , with ∆ µν = g µν − u µ u ν , for the derivative perpendicular to the flow field vector u µ . In the following we will use the standard choice λ = 2 for the residual momentum dependence of the correction as in [1]. When solving the viscous fluid dynamics equations of motion, numerical stability requires second order gradient terms to be included, leading to equations of motion for the shear stress tensor π µν and bulk stress Π [17][18][19][20][21]. At freeze-out, it is then convenient to compute δf directly from the shear stress tensor π µν . On the other hand, for the blast wave it is more practical to utilize the Navier-Stokes approximation π µν = 2ησ µν and to compute viscous corrections using Eq. (3). In that case δf is calculated simply from the flow field, which can be independently constrained by fits to flow data and the specific shear viscosity η/s of nuclear matter. The differences between the two approaches of calculating δf at freeze-out, π µν vs Navier-Stokes, are parametrically small in situations of small gradients towards the end of the time evolution. However, they could still be noticeable at freeze-out in realistic systems and are part of the uncertainties to be accounted for.
In both blast wave and fluid dynamic freeze-out, the invariant particle momentum spectrum emitted from a hypersurface Σ in Minkowski space is given by the Cooper-Frye formula [22] dN where g is the degeneracy factor for a given particle and dΣ µ is the forward normal vector on the freeze-out hypersurface. The momentum vector in the laboratory frame is written as usual, p µ = (M T cosh y, P T cos ψ, P T sin ψ, M T sinh y), in terms of the transverse momentum P T , the longitudinal momentum rapidity y and the azimuthal angle ψ in the transverse plane. M 2 T = P 2 T + M 2 defines the transverse mass M T for a hadron of mass M . The final particle spectrum at freeze-out is usually calculated on a hypersurface at constant temperature T = T fo . In contrast, in fluid dynamics this isothermal hypersurface, as well as the flow field u µ on it can be computed in the simulation itself. For the blast wave we have to choose ansätze for both.
Following Ref. [4] we assume that freeze-out from an isothermal hypersurface at temperature T can be approximated by freeze-out from a hypersurface at constant proper longitudinal time τ . We enforce boost invariance, which is a good approximation for nuclear collisions around midrapidity at top RHIC and LHC energies and is also often found in fluid dynamic calculations. To keep the blast wave simple we have to restrict ourselves to describing smooth fluid dynamics which corresponds to the averaging over many events. We can then assume that the hypersurface in the x − y-plane is approximately an ellipse with semi-axes R x and R y in x-and y-directions, respectively. We define the coordinate axes such that the impact parameter b of the collision is measured along the x-axis. In the following we use the reduced radius ρ = x 2 /R 2 x + y 2 /R 2 y together with the azimuthal angle θ, with tan θ = (R x y)/(R y x), and space time rapidity η s = 1/2 log[(t + z)/(t − z)] to carry out the integral over the hypersurface. Restricting ourselves to hadrons measured at midrapidity y = 0 and changing to convenient coordinates we obtain the final expression for the particles from the blastwave: Next we have to make an ansatz for the collective flow field. The general parameterization is where η T is the transverse rapidity in the x−y-plane, and φ u is the azimuthal angle of the flow vector in the transverse plane. Boost invariance fixes the longitudinal flow rapidity to be equal to the space time rapidity η s . We follow Retiere and Lisa and choose to model the transverse flow velocity v T = tanh η T as [4] v T = ρ n (α 0 + α 2 cos(2φ u )) which encodes a Hubble-like velocity ordering with an additional shape parameter n. α 0 is the average velocity on the boundary ρ = 1, and α 2 parameterizes the elliptic deformation of the flow field built up from the initial elliptic spatial deformation of the system. Flow vectors tend to be tilted towards the smaller axis of the ellipse. In the RL approach they are chosen to be perpendicular to the elliptic surface at ρ = 1, i.e. tan φ u = R 2 x /R 2 y tan φ, where φ = arctan y/x is the azimuthal angle of the position r µ .
With a parameterization of the flow field at hand, the next step is the calculation of the gradient tensor σ µν . This has been carried out in [1] and we refer the reader to the details in that reference. We want to point out that temporal derivatives are calculated using ideal fluid dynamic equations of motion rather than the free-streaming approximation [7,8]. This introduces the nuclear matter equation of state, specifically the speed of sound squared c 2 s into our calculation of δf .
The blast wave ansatz has several parameters which allow us to adjust the flow field and the hypersurface, as well as the temperature, specific shear viscosity, and speed of sound squared at freeze-out. The full set of parameters is P = (τ fo , R x , R y , T fo , n, α 0 , α 2 , η/s, c 2 s ). We will drop c 2 s from this list and rather use guidance on the hadronic matter equation of state from existing literature [27]. Note that the viscous blast wave depends on the parameters τ fo , R x and R y separately, and not just on the total volume τ R x R y and the ellipticity R y /R x [1], as is the case for the ideal blast wave.
Despite the large number of parameters it is clear that the blast wave has introduced significant simplifications compared to a realistic freeze-out calculated in fluid dynamics. They come mostly from the simplified shape of the hypersurface (constant proper time) and the spatial structure of the flow field. Two more major approximations are made for sake of simplicity. First, resonance decays are usually neglected in blast wave calculations, and only hadrons stable under strong decays are taken into account (see [28] for recent work on implementing decays). Secondly, correction terms to the particle distribution f due to bulk stress have been ignored. They could in principle be added and we plan to do so in the future. We summarize these five major approximations compared to fluid dynamic freeze-out in the following list: • Navier-Stokes approximation used for δf .
• Certain aspects of the shape of the hypersurface are fixed.
• General shape of the flow field is fixed.
• Lack of resonance production and decay.
• Missing bulk stress effects on particle distributions.
Since we have eliminated event-by-event fluctuations from the comparison, the effects of event-by-event fluid dynamic simulations compared to smooth fluid dynamics need to be considered separately. They are not included in the study below. The same is true for deviations of state-of-the-art 3+1D fluid dynamics from the boost-invariant 2+1D fluid dynamics used here. The effects of fluctuations and breaking of boost invariance have already been studied within fluid dynamics [23,24] and can be added to the considerations in this work. b (fm) (Au+Au) 5 5 6 6.5 7 8 9 10.5 10.5 b (fm) (Pb+Pb) 5.3 6.3 6.9 7.4 8.5 9.6 11.

III. GENERATION OF MUSIC PSEUDODATA
We use the viscous hydrodynamics code MUSIC to simulate averaged nuclear collisions at RHIC and LHC energies at various impact parameters. MUSIC is a relativistic second-order viscous hydrodynamics code for heavy ion collisions [13,14,23]. We choose boost-invariant (2+1)D mode consistent with the boost-invariant blast wave set-up. We use the built-in optical Glauber model to generate initial conditions with the appropriate nucleon-nucleon cross section and an overall normalization roughly consistent with pertinent multiplicity data from RHIC and LHC. We use the equation of state (EOS) s95p-v1.2 in MUSIC. We set a constant shear viscosity η/s and the default MUSIC bulk viscosity. We freeze out at pre-determined temperatures T fo and compute the final hadron spectra and elliptic flow, including resonance decays and including viscous corrections to freeze-out. The detailed MUSIC settings are documented in Appendix A. With later applications in mind, we focus on Au+Au collisions at the top RHIC energy of √ s N N = 200 GeV, as well as Pb+Pb collisions at the LHC energy of √ s N N = 5.02 GeV.
As discussed above, our main goal here are the relations of the temperature T (extr) fo and specific shear viscosity (η/s) (extr) extracted from a blast wave fit of the pseudodata to the true values T (true) fo and (η/s) (true) used in the generation of these pseudodata sets. Generally, we expect η/s to decrease with temperature in the hadronic phase, and T fo to increase with impact parameter b [1]. To focus on the physical region we investigate a band in the T (extr) fo -(η/s) (extr) -plane which covers the values extracted from RHIC and LHC data in Ref. [1]. For the case of simulations at RHIC energy we choose 27 points, in the T (true) fo -(η/s) (true) -plane, organized in 3 sets of nine points each. For LHC we choose a similar set of points. For each point we run MUSIC and perform a blast wave fit to the resulting hadron spectra and elliptic flow. The value of the impact parameter b for the fluid dynamic simulation for each T (true) fo -(η/s) (true) -point is chosen using guidance from the fits to data carried out in [1]. The nine points (eight points for Pb+Pb) of the first set (Set I) used in the T (true) fo -(η/s) (true) -plane are shown in Tab. I together with the impact parameter for each simulation.
We briefly discuss the details of this process for Set I for Au+Au collisions as an example. For a given parameters values MUSIC runs and computes the transverse momentum spectra and elliptic flow v 2 at rapidity y = 0 for pions, kaons and protons. The P T -range of the pseudodata generated is from 0 to 3 GeV/c. The fit ranges used for blast wave fits of the pseudodata in Set I are shown in Tab. II. The rationale behind the choice of fit ranges was discussed in detail in Ref. [1]. To summarize briefly, very low momenta tend to be not directly described by blast wave fits as they are dominated by resonance decays, while higher momenta receive viscous corrections larger than what can be reliably described by the Navier-Stokes approximation. Overall, the fit ranges for fitting the simulation pseudodata are very similar to the ranges used for good quality fits of experimental data in [1].
The fluid dynamic simulations do not provide useful uncertainty estimates on the pseudodata. For this study we choose uncertainties in line with uncertainties in the pertinent available experimental data. We assign 5% uncertainty and 2% uncertainty to pseudodata spectra and v 2 , respectively. We add a pedestal of 0.002 to the uncertainty of v 2 for realistic error bars at smaller P T where v 2 is very small. To quantify the uncertainty from this choice we vary the uncertainties, similar to the procedure in [1].
For fits we use the statistical analysis package from the Models and Data Analysis Initiative (MADAI) project [25,26]. The MADAI package includes a Gaussian process emulator and a Markov Chain Monte Carlo for Bayesian analyses. One example of such a Bayesian analysis can be found in Fig. 1. We use N = 500 training points for each Gaussian emulator. We check the Gaussian emulators and find their errors to be acceptably low.

IV. BLAST WAVE FITS
We fit the following blast wave parameters in the Bayesian analysis: P = (T fo , R y /R x , α 0 , α 2 , η/s). Chemical potentials µ are set to zero in MUSIC. To simplify the analysis we fix the radial shape parameters n and the freezeout time τ by choosing values close to those extracted from RHIC and LHC data in Ref. [1], see Tab. III and Tab. IV. As in [1] we also use geometric arguments to determine R x and R y independently from the fitted ratio. As an example, let us take a brief look at the case of parameters T  Interestingly, the scaling between true and extracted events does not seem to depend very much on the initial energy of the fireball. and (η/s) (true) under a map M . Before we discuss this map in detail in the next section let us discuss uncertainties for the extracted values. At the end of Sec. II we had summarized important approximations made by the blast wave approach. These approximations lead to differences between the true and extracted values whose quantification is the goal of this and the following section. The extracted values themselves already come with a set of uncertainties. We have quantified three of them: (i) Uncertainties due to using a fixed value of the shape parameter n in our analysis. (ii) Uncertainties due to the errors assigned to MUSIC pseudodata; while our choice of errors for the pseudodata is motivated by experiment, the size of the error is expected to influence fit results. Finally, (iii) uncertainties of the Bayesian analysis itself, expressed in the posterior distributions. The latter ones can be extracted directly from the analysis and can be seen in Fig.  1. We have studied the uncertainties from sources (i) and (ii) by systematically varying n and the error given to pseudodata. We combine all three sources into combined uncertainties in temperature and specific shear viscosity for each point. An example for this analysis is given in appendix B.
The final extracted freeze-out temperature and specific viscosities for the Au+Au points from Tab. I are shown in Fig. 4 (left panel) with their combined uncertainties. We find that the extracted specific shear viscosities are mostly consistent with true values within uncertainties. However the extracted temperatures underestimate the true temperatures significantly at higher values of T fo . The extracted T fo is about 15 MeV smaller for hydro events at high temperature (or peripheral collisions). For events at low freeze-out temperature (or central collisions), the extracted T fo are within 5 MeV of the true temperature. The deviations in freeze-out temperature for peripheral collisions are significant with respect to the estimated uncertainties also shown in Fig. 4. A very similar picture emerges for the results for Pb+Pb points also shown in Fig. 4 (right panel).

V. QUANTIFYING THE ACCURACY OF BLAST WAVE FITS
We are now ready to quantify the deviations between true and fitted values. We define a map M as We first focus on Au+Au collisions. We find that a linear map M in this 2-dimensional parameter space has sufficient accuracy given the uncertainties established for the fitted values. For a given set of points we apply the method of normal equations which gives the least square approximate solution to M . For Au+Au Set I we find, see Appendix C, M = 0.8365 3.90 0.0011 0.98 .      This M is only an approximate solution to Eq. (9). We define the predictions from the map to be and show them together with the true and extracted values in Fig. 4 (left panel). Overall, the map M provides a good approximation of extracted blast wave values. For Pb+Pb collisions, we obtain a similar map through the same technique, We are now in a position to remove biases that come with the extraction of properties of nuclear collisions using blast wave fits. As an example, we will attempt to correct the bias introduced by the blast wave fit from the analysis of the specific shear viscosity in [1]. Although our work has focused on those two fit parameters, the analysis in this work could be repeated for other quantities extracted from blast wave fits in a straight forward way. We define corrected blast wave fit values as Since Eq. (9) assures us that we postulate that this correction step takes the blast wave fitted values closer to physical reality. We will revisit this argument in the discussion in the next section.
As an example, for Au+Au collisions at RHIC the inverse matrix is In Ref. [1] T and η/s have been extracted from data with carefully quantified Gaussian uncertainties. These uncertainties are transported by M −1 to Gaussian uncertainties on T (corr) , (η/s) (corr) , see left panel of Fig. 7 for an example. We add the uncertainty on the map M −1 itself which we do by applying additional Gaussian smearing using the matrices introduced in Appendix C. The right panel of Fig. 7 gives an impression of the size of the effect by plotting the variation of the center value under these inverse maps.
The final results of this correction, including all uncertainties combined, is shown in Fig. 6. We notice that after correction the temperature dependence is less steep, and values close to 4πη/s = 1 are only reached around T ≈ 150 MeV. For a discussion of the implications of this particular result we refer the reader to reference [1].

VI. SUMMARY
We have discussed the shortcomings of blast wave fits compared to fluid dynamic simulations applied to high energy nuclear collisions. We have carried out a systematic study of parameters at kinetic freeze-out set in fluid dynamic simulations which have subsequently been extracted from blast wave fits to hadron spectra and elliptic flow. We find that blast wave fits correctly reproduce trends of the true values set in the simulations, and they qualitatively agree with the numerical values. However, some deviations between extracted and true values exist are significant. The quality of blast wave fits can be improved by understanding and quantifying these deviations. We have done so for both the freeze-out temperature and specific shear viscosity at freeze-out and have presented the results in this paper.
We have established a map from true parameter values to blast wave fitted parameters. This map can be inverted and can be applied to correct the biases in blast wave fits. This has been demonstrated for the case of the extraction of the specific shear viscosity as a function of temperature from data. We have also estimated uncertainties for the correction which have been added to the uncertainties of the final corrected results. There are good reasons to expect that the corrected results are closer to the true physical values than the raw blast wave fit values. A rigorous argument can only be made for the correction to improve fits to hydrodynamic pseudodata. This leaves the questions how close hydrodynamic modeling is to the true physical situation in high energy nuclear collisions. We can only approach an answer to that by continued studies of nuclear collisions with fluid dynamic models, which are outside of the scope of this work.
One interesting question regards the universality and systematic behavior of the corrections discussed here. We found a large dependence on the temperature, which might be a reflection of changing system size. On the other hand we found very little variation going from RHIC to LHC energies. The systematic study of the deviations as a function of various changes that can be made to the fluid dynamic calculations would be a worthwhile goal for a follow up study. Future improvements to the blast wave itself can be made by adding bulk stress corrections, and to include some resonances and their decays. However, simplicity is key to blast wave fits and perhaps a simpler model with known corrections and a study of uncertainties is more useful. Blast waves with corrections applied are still orders of magnitude cheaper in labor and computational cost compared to fluid dynamics. They remain a quick tool for data analysis. The mechanism to reduce bias laid out here should inspire confidence in their ability, as long as realistic uncertainty estimates are added to the analysis.   If ones assume the blast wave model as given, there are uncertainties to consider when it is deployed to describe MUSIC pseudodata. As described in the main text they mainly come from three sources: (i) Uncertainties due to using a fixed value of the radial shape parameter n in our analysis. (ii) Uncertainties due to the errors assigned to MUSIC pseudodata. (iii) Uncertainties of the fit itself as encoded in the posterior distributions. We have studied the uncertainties from source (i) by systematically varying n. An example for one point in Au+Au Set I is given in Tab. VI. Similarly, we vary the error given to the pseudodata before the fit. Tab. VII gives an example for the same point in Set I when the error on the hadron spectra is varied. In general, we find that T fo varies only within a few MeV while η/s shows larger changes when varying n. Variations in the assigned pseudodata error lead to small changes of both the extracted temperature and η/s. Uncertainties from the fit can be taken directly from the MADAI code. We treat the sources of uncertainties as independent and add them quadratically to arrive at estimates of the total uncertainty assigned to the extracted values of T fo and η/s. For the point selected for this example these errors are summarized in Tab. VIII. We proceed to process Sets II and III of parameter points in the same way as Set I. The true parameter values and the MUSIC pseudodata and extracted blast wave fit parameters are listed in Tabs. IX and X. Sets II and III are generated by varying η/s compared to the values of Set I. The parameterizations of the maps from true to extracted values for these two sets are