  Some of the consequences for inflationary cosmology of a scale dependence
(running) in the tilt of the scalar perturbation spectrum are considered. In
the limit where the running is itself approximately scale-invariant, a
relationship is found between the scalar and tensor perturbation amplitudes,
the scalar spectral index and its running. This relationship is independent of
the functional form of the inflaton potential. More general settings, including
that of braneworld cosmological models, are also considered. It is found that
for the Randall-Sundrum single braneworld scenario, the corresponding relation
between the observables takes precisely the same form as that arising in the
standard cosmology. Some implications of the observations failing to satisfy
such a relationship are discussed.
