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 .
We present a slow-roll formalism generalizing the four-dimensional one ; full towers of slow-roll parameters are constructed and the dynamics of the inflaton field explored in detail .
The inflationary attractor condition , exact cosmological solutions , and perturbation 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 within an extended version of patch cosmological braneworlds , starting from a maximally invariant generalization of the action for scalar and tensor perturbations to a noncommutative brane embedded in a commutative bulk .
Slow-roll expressions and consistency relations for the cosmological observables are provided , both in the ultraviolet and infrared region of the spectrum .
The effects of noncommutativity are 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 .

PACS numbers : 98.80.Cq , 04.50.+h , 98.70.Vc