E. Razzoli, F. Boschini, M. Zonno, M. X. Na, M. Michiardi, M. Schneider, E. H. da Silva Neto, S. Gorovikov, R. D. Zhong, J. Schneeloch, G. D. Gu, S. Zhdanovich, A. K. Mills, G. Levy, D. J. Jones, C. Giannetti, A. Damascelli

The 40-70 meV band-structure renormalization (so-called kink) in high-temperature cuprate superconductors - which has been mainly interpreted in terms of electron-boson coupling - is observed to be strongly suppressed both above the superconducting transition temperature and under optical excitation. We employ equilibrium and time- and angle-resolved photoemission spectroscopy, in combination with Migdal-Eliashberg simulations, to investigate the suppression of the near-nodal kink in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. We show that the $\sim$30$\%$ decrease of the kink strength across the superconducting-to-normal-state phase transition can be entirely accounted for by the filling of the superconducting gap, without additional consideration of temperature-dependent electron-boson coupling. Our findings demonstrate that consideration of changes in the density of states is essential to quantitatively account for the band structure renormalization effects in cuprates.