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In view of cosmological parameters being measured to ever higher precision,theoretical predictions must also be computed to an equally high level ofprecision. In this work we investigate the impact on such predictions ofrelaxing some of the simplifying assumptions often used in these computations.In particular, we investigate the importance of slow-roll corrections in thecomputation of multi-field inflation observables, such as the amplitude of thescalar spectrum $P_\zeta$, its spectral tilt $n_s$, the tensor-to-scalar ratio$r$ and the non-Gaussianity parameter $f_{NL}$. To this end we use the separateuniverses approach and $\delta N$ formalism, which allows us to considerslow-roll corrections to the non-Gaussianity of the primordial curvatureperturbation as well as corrections to its two-point statistics. In the contextof the $\delta N$ expansion, we divide slow-roll corrections into twocategories: those associated with calculating the correlation functions of thefield perturbations on the initial flat hypersurface and those associated withdetermining the derivatives of the e-folding number with respect to the fieldvalues on the initial flat hypersurface. Using the results of Nakamura &Stewart '96, corrections of the first kind can be written in a compact form.Corrections of the second kind arise from using different levels of slow-rollapproximation in solving for the super-horizon evolution, which in turncorresponds to using different levels of slow-roll approximation in thebackground equations of motion. We consider four different levels ofapproximation and apply the results to a few example models. The variousapproximations are also compared to exact numerical solutions.

Slow-roll corrections in multi-field inflation: a separate universes approach