  In view of cosmological parameters being measured to ever higher precision,
theoretical predictions must also be computed to an equally high level of
precision. In this work we investigate the impact on such predictions of
relaxing some of the simplifying assumptions often used in these computations.
In particular, we investigate the importance of slow-roll corrections in the
computation of multi-field inflation observables, such as the amplitude of the
scalar 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 separate
universes approach and $\delta N$ formalism, which allows us to consider
slow-roll corrections to the non-Gaussianity of the primordial curvature
perturbation as well as corrections to its two-point statistics. In the context
of the $\delta N$ expansion, we divide slow-roll corrections into two
categories: those associated with calculating the correlation functions of the
field perturbations on the initial flat hypersurface and those associated with
determining the derivatives of the e-folding number with respect to the field
values 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-roll
approximation in solving for the super-horizon evolution, which in turn
corresponds to using different levels of slow-roll approximation in the
background equations of motion. We consider four different levels of
approximation and apply the results to a few example models. The various
approximations are also compared to exact numerical solutions.
