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The advent of explicit Dirac-Born-Infeld (DBI) inflationary models withinstring theory has drawn renewed interest to the cosmological role of unusualscalar field dynamics, usually referred to as k-inflation. In this situation,the standard method used to determine the behavior of cosmologicalperturbations breaks down. We present a generic method, based on the uniformapproximation, to analytically derive the power spectra of scalar and tensorperturbations. For this purpose, a simple hierarchy of parameters, related tothe sound speed of the cosmological fluctuations and its successivederivatives, is introduced in a k-inflation analogue of the Hubble flowfunctions. The scalar spectral index and its running are obtained up to next tonext to leading order for all k-inflationary models. This result relies on theexistence of a well-motivated initial state, which is not trivial in thepresent context: having the wavelength of the Fourier mode smaller than thesonic horizon is indeed not enough and some conditions on the dynamics of thesound speed are also required. Our method is then applied to various modelsencountered in the literature. After deriving a generic slow-roll trajectoryvalid for any DBI model, simple formulae for the cosmological observables areobtained. In particular, the running, as the spectral index, for the so-calledUV and IR brane inflationary models is found to be uniquely determined by the't Hooft coupling. Finally, the accuracy of these cosmological predictions isassessed by comparing the analytical approximations with exact numericalintegrations.

K-inflationary Power Spectra in the Uniform Approximation