Hydrodynamic attractor in a Hubble expansion
Zhiwei Du, Xu-Guang Huang, Hidetoshi Taya
Abstract
We analytically investigate hydrodynamic attractor solutions in both M\"uller-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-Pad\'e 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.