Temperature dependence of the collective mode and its influence on the band splitting in bilayer cuprates

The recently observed bilayer splitting in high-T${}_c$ cuprates is analyzed within a model where the charge carriers are coupled to a phenomenological bosonic spectrum which interpolates between the marginal Fermi liquid structure and collective mode type behavior as a function of temperature. We argue that the origin of the collective mode is probably associated with dynamic incommensurate charge density waves. Moreover it is shown that the resulting temperature dependence of the self-energy $\mathrm{\Sigma }$ is in good agreement with $\mathrm{\Sigma }$ as extracted from angle-resolved photoemission data.