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, upsilon(2)/epsilon, and the produced entropy fraction, Delta S/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 upsilon(2)/epsilon and Delta S/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 eta/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 sornewhat larger scale-breaking effects that increase with increasing ills and decreasing system size. and initial energy density. We propose to use precision studies of these scaling violations to help constrain the shear viscosity ills of the quark-gluon plasma created in relativistic heavy ion collisions.