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Recent classical-statistical numerical simulations have established the "bottom-up" thermalization scenario of Baier et al. [Phys. Lett. B 502, 51 (2001)] as the correct weak coupling effective theory for thermalization in ultrarelativistic heavy-ion collisions. We perform a parametric study of photon production in the various stages of this bottom-up framework to ascertain the relative contribution of the off-equilibrium "glasma" relative to that of a thermalized quark-gluon plasma. Taking into account the constraints imposed by the measured charged hadron multiplicities at Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), we find that glasma contributions are important especially for large values of the saturation scale at both energies. These nonequilibrium effects should therefore be taken into account in studies where weak coupling methods are employed to compute photon yields. [ABSTRACT FROM AUTHOR]