Viscosity, nonconformal equation of state, and sound velocity in Landau hydrodynamics

We find an analytical solution to relativistic viscous hydrodynamics for a ($1+1$)-dimensional Landau flow profile. We consider relativistic Navier-Stokes form of the dissipative hydrodynamic equation, for a nonconformal system with a constant speed of sound, and employ the obtained solution to fit rapidity spectrum of observed pions in $\sqrt{s_{NN}}=200$, 17.3, 12.3, 8.76, 7.62, 6.27, 4.29, 3.83, 3.28, and 2.63 GeV collision energies. We find that at the freeze-out hypersurface with improved Landau's freeze-out prescription, the viscous corrections do not affect the rapidity spectra. We demonstrate that the solution of the nonconformal Landau flow leads to a better agreement with the experimental data compared to the conformal ideal solution. We also extract speed of sound from fit to the rapidity spectra for various collision energies and find a monotonous decrease with decreasing collision energies. Appealing to the fact that viscosity has negligible effect on rapidity spectra for Landau's freeze-out scenario, we argue that our calculations provides a framework for extracting the average value of speed of sound in relativistic heavy-ion collisions.

Figure 1

Rapidity spectrum of pions fitted using Eq. (17) (red solid curves) and Eq. (18) (green dashed-dotted curves), for three representative collision energies: $\sqrt{s_{NN}}=200$, 17.3, and 4.29 GeV. Also shown are the fit result using conformal solution of Landau hydrodynamics [21] (blue dashed curves) and the experimental results (black symbols). Experimental data are from Refs. [17, 52, 53, 54].

Figure 2

Squared speed of sound, extracted by fitting the pion rapidity spectra using the rapidity distribution obtained using nonconformal solution given in Eq. (17) (red solid line) and also using the Gaussian distribution given in Eq. (18) (green dashed-dotted line), over various collision energies. The error bars corresponds to standard error from least-squares fit of the fit parameters.