Hydrodynamic Attractor in a Hubble Expansion

TL;DR

This paper analytically explores hydrodynamic attractor solutions in a Hubble-expanding viscous fluid, revealing their divergence, global attractiveness, and differences between theories, advancing understanding of far-from-equilibrium hydrodynamics.

Contribution

It demonstrates the factorial divergence of gradient expansions, applies Borel resummation to find attractors, and compares attractor behaviors in MIS and kinetic theories under Hubble expansion.

Findings

Hydrodynamic attractors are globally attractive in both theories.

Only the first non-hydrodynamic mode is present in attractors.

Attractors differ significantly at large gradients.

Abstract

We analytically investigate hydrodynamic attractor solutions in both M\"{u}ller-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 and obtain hydrodynamic attractor solutions by applying the Borel resummation technique to those asymptotic divergent series. In both theories, we find that the hydrodynamic attractor solutions are globally attractive and only the first-order non-hydrodynamic 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.