  We derive an exact expression for the photo-current of photo-emission
spectroscopy using time-dependent current density functional theory (TDCDFT).
This expression is given as an integral over the Kohn-Sham spectral function
renormalized by effective potentials that depend on the exchange-correlation
kernel of current density functional theory. We analyze in detail the physical
content of this expression by making a connection between the
density-functional expression and the diagrammatic expansion of the
photo-current within many-body perturbation theory. We further demonstrate that
the density functional expression does not provide us with information on the
kinetic energy distribution of the photo-electrons. Such information can, in
principle, be obtained from TDCDFT by exactly modeling the experiment in which
the photo-current is split into energy contributions by means of an external
electromagnetic field outside the sample, as is done in standard detectors. We
find, however, that this procedure produces very nonlocal correlations between
the exchange-correlation fields in the sample and the detector.
