Anisotropic matching principle for the hydrodynamic expansion

Following the recent success of anisotropic hydrodynamics, I propose here a new, general prescription for the hydrodynamic expansion around an anisotropic background. The anisotropic distribution fixes exactly the complete energy-momentum tensor, just like the effective temperature fixes the proper energy density in the ordinary expansion around local equilibrium. This means that momentum anisotropies are already included at the leading order, allowing for large pressure anisotropies without the need of a next-to-leading-order treatment. The first moment of the Boltzmann equation (local four-momentum conservation) provides the time evolution of the proper energy density and the four-velocity. Differently from previous prescriptions, the dynamic equations for the pressure corrections are not derived from the zeroth or second moment of the Boltzmann equation, but they are taken directly from the exact evolution given by the Boltzmann equation. As known in the literature, the exact evolution of the pressure corrections involves higher moments of the Boltzmann distribution, which cannot be fixed by the anisotropic distribution alone. Neglecting the next-to-leading-order contributions corresponds to an approximation, which depends on the chosen form of the anisotropic distribution. I check the the effectiveness of the leading-order expansion around the generalized Romatschke-Stricklad distribution, comparing with the exact solution of the Boltzmann equation in the Bjorken limit with the collisional kernel treated in the relaxation-time approximation, finding an unprecedented agreement.

Figure 1

Time evolution of the pressure anisotropy ${\mathcal{P}}_L/{\mathcal{P}}_T$ (left) and the bulk corrections time the proper time $\tau \mathrm{\Pi }$ (right). The black solid line corresponds to the exact solution of the Boltzmann equation in RTA, the red dotted one to the set of equation presented in this paper, the purple dashed line is the solution of second-order viscous hydrodynamics, while the blue dashed-dotted line is a previous formulation of anisotropic hydrodynamics. The initial anisotropy is vanishing (top) or ${\xi }_0=100$ (bottom) in the two rows.

Figure 2

Same as Fig. 1, except here I take $m=1$ GeV.