  In three dimensions, non-interacting bosons undergo Bose-Einstein
condensation at a critical temperature, $T_{c}$, which is slightly shifted by
$\Delta T_{\mathrm{c}}$, if the particles interact. We calculate the excitation
spectrum of interacting Bose-systems, \sup{4}He and \sup{87}Rb, and show that a
roton minimum emerges in the spectrum above a threshold value of the gas
parameter. We provide a general theoretical argument for why the roton minimum
and the maximal upward critical temperature shift are related. We also suggest
two experimental avenues to observe rotons in condensates. These results, based
upon a Path-Integral Monte-Carlo approach, provide a microscopic explanation of
the shift in the critical temperature and also show that a roton minimum does
emerge in the excitation spectrum of particles with a structureless,
short-range, two-body interaction.
