Hydrodynamic attractor in a Hubble expansion

We analytically investigate hydrodynamic attractor solutions in both Müller-Israel-Stewart (MIS) and kinetic theories in a viscous fluid system undergoing a Hubble expansion with a fixed expansion rate. We show that the gradient expansion for the MIS theory and the Chapman-Enskog expansion for the Boltzmann equation within the relaxation time approximation are factorially divergent. By resumming those asymptotic divergent series exactly via the Borel resummation technique (without using any approximations such as the Borel-Padé method), we obtain closed expressions for hydrodynamic attractor solutions. In both theories, we find that the hydrodynamic attractor solutions are globally attractive and only a single nonhydrodynamic mode exists. We also find that the hydrodynamic attractor solutions in the two theories disagree with each other when gradients become large, and that the speed of the attraction is different. Similarities and differences from hydrodynamic attractors in the Bjorken and Gubser flows are also discussed. Our results push the idea of far-from-equilibrium hydrodynamics in systems undergoing a Hubble expansion.

Figure 1

The hydrodynamic attractor solutions (black lines) for the bulk stress (20) (left panel) and the energy density (25) (right panel) as functions of time $t/{\tau }_{\mathrm{\Pi }}$. As a comparison, we also plot numerical solutions to the MIS theory (12) with various initial conditions (gray lines) and the low-order hydrodynamics (16) of the zeroth (red line), first (blue line), and second order (green line). The parameters are set as $\alpha =2/3$ and $c_s^2=1/3$. The initial conditions for the numerical solutions are set at $t_{\mathrm{in}}/{\tau }_{\mathrm{\Pi }}=1$ as $t_{\mathrm{in}}\mathrm{\Pi }(t_{\mathrm{in}})/\zeta =-7+m$ and $t_{\mathrm{in}}E(t_{\mathrm{in}})/\zeta =4$ with $m=1,2,\dots ,14$.

Figure 2

The effective viscosity coefficient ${\zeta }_B$ (30) versus time.

Figure 3

A comparison between the hydrodynamic attractor solutions for the energy density in the MIS (25) (red) and kinetic (56) (blue) theories. The horizontal axis is scaled with $\tau ={\tau }_{\mathrm{\Pi }}$ and ${\tau }_R$ for the MIS and kinetic theories, respectively. We choose the parameters as $\alpha =2/3$ and $c_s^2=1/3$.

Figure 4

The hydrodynamic attractor solution for the energy density in the kinetic theory (56) (black line), in comparison with the exact solutions to the Boltzmann equation (43) with various initial energies $E_{\mathrm{in}}$ (gray lines) and low-order results (53) (red, blue, and green lines). The parameters are set as $\alpha =2/3$ and $t_{\mathrm{in}}={\tau }_R$, and the initial energies are set as $t_{\mathrm{in}}{E}_{\mathrm{in}}/\zeta =2(m-1)$ ($m=1,2,\dots ,14$).