  Slow-roll inflation is studied in theories where the inflaton field is
conformally coupled to the Ricci scalar. In particular, the case of Higgs field
inflation in the context of the noncommutative spectral action is analyzed. It
is shown that while the Higgs potential can lead to the slow-roll conditions
being satisfied once the running of the self-coupling at two-loops is included,
the constraints imposed from the CMB data make the predictions of such a
scenario incompatible with the measured value of the top quark mass. We also
analyze the role of an additional conformally coupled massless scalar field,
which arises naturally in the context of noncommutative geometry, for
inflationary scenarios.
