  Until recently our knowledge of the primordial curvature perturbation was
relatively modest. Ever since COBE delivered its map of data we know the scalar
spectrum of primordial perturbations is approximately flat, with the power
being only slightly stronger at larger scales. Most inflationary models predict
an approximately scale-invariant spectrum, which therefore cannot be used as a
distinctive signature. To distinguish between different inflationary
microphysics we need to study higher point statistics of the primordial
perturbation, which can encode non-gaussian data. In the first part of this
thesis we study the bispectrum in all single-field models with a well-defined
quantum field theory during a quasi-de Sitter inflationary phase. Any
single-field models without ghost-like instabilities fall into this
description: from canonical, to Dirac-Born-Infeld inflation and galileon
inflation theories. We investigate the scale and shape- dependences of the
bispectrum to next-order in the slow-roll approximation. We illustrate our
results by applying them to different models and argue these corrections must
be taken into account to keep the theoretical error below the observational
precision set by the Planck satellite. We then explore the ability of using
bispectrum shapes to distinguish between inflationary models more efficiently.
We further extend the study of the bispectrum of single-field models beyond the
slow-roll approximation, demanding the spectral index to be close to, but not
exactly, unity. In the second part of this thesis we explore the process by
which the universe is repopulated with matter particles at the end of a
Dirac-Born-Infeld inflation phase. We place some mild bounds on the reheating
temperature of these models. We argue that the constraints arising from the
preheating analysis are complementary to those derived from the primordial
perturbation.
