  Ultrarelativistic heavy-ion collisions at the Relativistic Heavy Ion Collider
(RHIC) are thought to have produced a state of matter called the quark-gluon
plasma, characterized by a very small shear-viscosity to entropy-density ratio
$\eta/s$, near the lower bound predicted for that quantity by AdS/CFT methods.
As the produced matter expands and cools, it evolves through a phase described
by a hadron gas with rapidly increasing $\eta/s$. We calculate $\eta/s$ as a
function of temperature in this phase both in and out of chemical equilibrium
and find that its value poses a challenge for viscous relativistic
hydrodynamics, which requires small values of $\eta/s$ in order to successfully
describe the collective flow observables at RHIC. We therefore conclude that
the origin of the low viscosity matter at the RHIC must be in the partonic
phase of the reaction.
