  Some classes of inflationary models naturally introduce two distinct
metrics/frames, and their equivalence in terms of observables has often been
put in question. D-brane inflation proposes candidates for an inflaton embedded
in the string theory and possesses descriptions on the brane and bulk
metrics/frames, which are connected by a conformal/disformal transformation
that depends on the inflaton and its derivatives. It has been shown that
curvature perturbations generated by the inflaton are identical in both frames,
meaning that observables such as the spectrum of cosmic microwave background
(CMB) anisotropies are independent of whether matter fields---including those
in the standard model of particle physics---minimally couple to the brane or
the bulk metric/frame. This is true despite the fact that the observables are
eventually measured by the matter fields and that the total action including
the matter fields is different in the two cases. In contrast, in curvaton
scenarios, the observables depend on the frame to which the curvaton minimally
couples. Among all inflationary scenarios, we focus on two models motivated by
the KKLMMT fine-tuning problem: a slow-roll inflation with an inflection-point
potential and a model of a rapidly rolling inflaton that conformally couples to
gravity. In the first model, the difference between the frames in which the
curvaton resides is encoded in the spectral index of the curvature
perturbations, depicting the nature of the frame transformation. In the second
model, the curvaton on the brane induces a spectral index significantly
different from that in the bulk and is even falsified by the observations. This
work thus demonstrates that two frames connected by a conformal/disformal
transformation lead to different physical observables such as CMB anisotropies
in curvaton models.
