  We develop a frame-covariant formulation of inflation in the slow-roll
approximation by generalizing the inflationary attractor solution for
scalar-curvature theories. Our formulation gives rise to new generalized forms
for the potential slow-roll parameters, which enable us to examine the effect
of conformal transformations and inflaton reparameterizations in
scalar-curvature theories. We find that cosmological observables, such as the
power spectrum, the spectral indices and their runnings, can be expressed in a
concise manner in terms of the generalized potential slow-roll parameters which
depend on the scalar-curvature coupling function, the inflaton wavefunction,
and the inflaton potential. We show how the cosmological observables of
inflation are frame-invariant in this generalized potential slow-roll
formalism, as long as the end-of-inflation condition is appropriately extended
to become frame-invariant as well. We then apply our formalism to specific
scenarios, such as the induced gravity inflation, Higgs inflation and $F(R)$
models of inflation, and obtain more accurate results, without making
additional approximations to the potential. Our results are shown to be
consistent to lowest order with those presented in the literature. Finally, we
outline how our frame-covariant formalism can be naturally extended beyond the
tree-level approximation, within the framework of the Vilkovisky--DeWitt
effective action.
