  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.
