  We present ab initio calculations of the photoemission spectra of bulk sodium
using different flavors of the cumulant expansion approximation for the Green's
function. In particular, we study the dispersion and intensity of the plasmon
satellites. We show that the satellite spectrum is much more sensitive to many
details than the quasi-particle (QP) spectrum, which suggests that the
experimental investigation of satellites could yield additional information
beyond the usual studies of the band structure. In particular, a comparison to
the homogeneous electron gas (HEG) shows that the satellites are influenced by
the crystal environment, although the crystal potential in sodium is weak.
Moreover, the temperature dependence of the lattice constant is reflected in
the position of the satellites. Details of the screening also play an important
role; in particular, the contribution of transitions from 2s and 2p semi-core
levels influence the satellites, but not the QP. Moreover, inclusion of
contributions to the screening beyond the RPA has an effect on the satellites.
Finally, we elucidate the importance of the coupling of electrons and holes by
comparing the results of the time-ordered (TOC) and the retarded (RC) cumulant
expansion approximations. Again, we find small but noticeable differences.
Since all the small effects add up, our most advanced calculation yields a
satellite position which is improved with respect to previous calculations by
almost one eV. This stresses the fact that the calculation of satellites is
much more delicate than the calculation of a QP band structure.
