  We investigate a model based on a generalised version of the Fourier
transform for curved space-time manifolds. This model is possible if the metric
has an asymptotic flat region which allows a duality to be implement between
coordinates and momenta, hence, the models's name, trans-Planckian. The theory
and the action are based on the postulate of the absolute egalitarian relation
between coordinates $x$ and momenta $p$. We show how to implement this
construction in a cosmological setting, on a Friedman-Robertson-Walker metric
background, where the asymptotic time infinity plays the role of the required
asymptotic flat region. We discuss the effect of gravity, and, in particular,
of the Hubble expansion of the universe scale factor on the Fourier map. The
dual inflationary stage is responsible for making the dual sector of the action
inaccessible at ordinary low energies. We propose a scenario in which an
effective positive cosmological constant is caused by how the dual sector of
the theory affects the equation of state for matter particles.
