  We investigate the gauge-invariant cosmological perturbations in the gravity
and matter frames in the general scalar-tensor theory where two frames are
related by the disformal transformation. The gravity and matter frames are the
extensions of the Einstein and Jordan frames in the scalar-tensor theory where
two frames are related by the conformal transformation, respectively. First, it
is shown that the curvature perturbation in the comoving gauge to the scalar
field is disformally invariant as well as conformally invariant, which gives
the predictions from the cosmological model where the scalar field is
responsible both for inflation and cosmological perturbations. Second, in case
that the disformally coupled matter sector also contributes to curvature
perturbations, we derive the evolution equations of the curvature perturbation
in the uniform matter energy density gauge from the energy (non)conservation in
the matter sector, which are independent of the choice of the gravity sector.
While in the matter frame the curvature perturbation in the uniform matter
energy density gauge is conserved on superhorizon scales for the vanishing
nonadiabatic pressure, in the gravity frame it is not conserved even if the
nonadiabatic pressure vanishes. The formula relating two frames gives the
amplitude of the curvature perturbation in the matter frame, once it is
evaluated in the gravity frame.
