Multiplicity scaling in ideal and viscous hydrodynamics

Using numerical results from ideal and viscous relativistic hydrodynamic simulations with three different equations of state, for Au+Au and Cu+Cu collisions at different centralities and initial energy densities, we explore the dependence of the eccentricity-scaled elliptic flow, v_2/ɛ, and the produced entropy fraction, ΔS/S_0, on the final charged hadron multiplicity density dN_ch/dy per unit transverse overlap area S,(1/S)dN_ch/dy. The viscous hydrodynamic simulations are performed with two different versions of the Israel-Stewart kinetic evolution equations, and in each case we investigate the dependence of the physical observables on the kinetic relaxation time. We find approximate scaling of v_2/ɛ and ΔS/S_0 with (1/S)dN_ch/dy, with scaling functions that depend on the EOS and, in particular, on the value of the specific shear viscosity η/s. Small scaling violations are seen even in ideal hydrodynamics, caused by a breaking of the scale invariance of ideal fluid dynamics by the freeze-out condition. Viscous hydrodynamics shows somewhat larger scale-breaking effects that increase with increasing η/s and decreasing system size and initial energy density. We propose to use precision studies of these scaling violations to help constrain the shear viscosity η/s of the quark-gluon plasma created in relativistic heavy ion collisions