  We investigate the effect of electron-phonon interactions (EPI) in systems
exhibiting one or more flat electron bands close to the Fermi level and a
comparatively large phonon energy scale. After solving the self-consistent
full-bandwidth Eliashberg equations, we compute Angular Resolved Photo Emission
Spectroscopy and Scanning Tunneling Spectroscopy/Microscopy spectra. We obtain
a sequence of quasiparticle replica bands in both the normal and
superconducting states that originate from frequency dependent features of the
electron mass renormalization function. We show that these replica bands can be
used to extract the relevant phonon energy scale from experiments. Focusing in
particular on twisted bilayer graphene, we predict replica-band formation
which, when observed, will shed light on the role of EPI in this archetypal
flat-band system.
