Non-Gaussianity in two-field inflation beyond the slow-roll approximation

We use the long-wavelength formalism to investigate the level of bispectral non-Gaussianity produced in two-field inflation models with standard kinetic terms. Even though the Planck satellite has so far not detected any primordial non-Gaussianity, it has tightened the constraints significantly, and it is important to better understand what regions of inflation model space have been ruled out, as well as prepare for the next generation of experiments that might reach the important milestone of Delta f_NL(local) = 1. We derive an alternative formulation of the previously derived integral expression for f_NL, which makes it easier to physically interpret the result and see which types of potentials can produce large non-Gaussianity. We apply this to the case of a sum potential and show that it is very difficult to satisfy simultaneously the conditions for a large f_NL and the observational constraints on the spectral index n_s. In the case of the sum of two monomial potentials and a constant we explicitly show in which small region of parameter space this is possible, and we show how to construct such a model. Finally, the new general expression for f_NL also allows us to prove that for the sum potential the explicit expressions derived within the slow-roll approximation remain valid even when the slow-roll approximation is broken during the turn of the field trajectory (as long as only the epsilon slow-roll parameter remains small).