(2+1)-d vs. (3+1)-d viscous hydrodynamics from RHIC and LHC*

Using (2+1)-d viscous hydrodynamics with a state-of-the-art equation of state (s95p-PCE), we present comparisons with recent ALICE measurements of the charged hadron spectra and elliptic flow, as well as successful predictions of the differential elliptic flow coefficient $v_2(p_T)$ for identified pions, kaons and protons from 2.76\,$A$\,GeV Pb+Pb collisions at the Large Hadron Collider (LHC) [1]. We also study how the ``universal'' curves describing the dependence of the eccentricity-scaled charged elliptic flow $v^{\mathrm{ch}}/\bar{\epsilon}$ on the charged multiplicity density per unit area $(1/S)(dN_{\mathrm{ch}}/dy)$ change from RHIC to LHC energies. In (2+1)-d viscous hydrodynamics we find a tendency of producing less $v^{\mathrm{ch}}/\bar{\epsilon}$ at higher collision energies, which contradicts the opposite tendency found by Hirano {\it et al.}[2] for (3+1)-d ideal hydrodynamics coupled to a hadron cascade. By comparing (2+1)-d with (3+1)-d viscous hydrodynamics we explore to what extend these different tendencies may indicate a collision energy dependent gradual breakdown of longitudinal boost-invariance near midrapidity when going from higher to lower collision energies. \\[4pt] [1] C. Shen, U. Heinz, P. Huovinen, and H. Song, arXiv:1105.3226 \\[0pt] [2] T. Hirano, P. Huovinen, Y. Nara, arXiv:1010.6222