  In this work we develop the patch formalism, an approach providing a very
simple and compact description of braneworld-motivated cosmologies with
nonstandard effective Friedmann equations. In particular, the Hubble parameter
is assumed to depend on some power of the brane energy density, H^2 \propto
\rho^q. The high-energy limit of Randall-Sundrum (q=2) and Gauss-Bonnet (q=2/3)
braneworlds are considered, during an accelerating era triggered by a single
ordinary or tachyonic scalar field. The inflationary dynamics, solutions, and
spectra are provided. Using the latest results from WMAP and other experiments
for estimates of cosmological observables, it is shown that future data and
missions can in principle discriminate between standard four-dimensional and
braneworld scenarios. The issue of non-Gaussianity is also studied within
nonlinear perturbation theory. The introduction of a fundamental energy scale
reinforces these results. Several classes of noncommutative inflationary models
are considered and their features analyzed in a number of ways and energy
regimes. Finally, we establish dual relations between inflationary,
cyclic/ekpyrotic and phantom cosmologies, as well as between scalar-driven and
tachyon-driven cosmologies. The exact dualities relating the four-dimensional
spectra are broken in favour of their braneworld counterparts. The dual
solutions display new interesting features because of the modification of the
effective Friedmann equation on the brane.
