In this thesis we study models of inflation with a curved field-space metric. We concern ourselves with the calculation of the statistics of curvature perturbations which is essential for connecting models to observations. To begin, we review the standard model of cosmology. We then reflect on cosmological perturbation theory and quantization procedures for inflationary fields in a flat field-space. With these tools we inspect how curvature perturbations, seeded from inflation, generate observables. We then extend this framework so that we can calculate observables from models with a curved field-space metric. To do this we extend the transport method for numerically evaluating the statistics in multifield inflation. This allows us to calculate the power spectrum and bispectrum in multifield inflation in the case of a curved field-space metric. This method naturally accounts for all sub- and super-horizon tree level effects, including those induced by the curvature of the field-space. We present an open source implementation of our equations in an extension of the publicly available PyTransport code. Next we apply our numerical methods to models of inflation with field-space metrics that produce interesting observables. We investigate the attractor behaviour of multifield models of inflation where the fields are coupled non-minimally to gravity for two theories of gravity, metric and Palatini gravity. It is conjectured that the two formalisms will have different attractor behaviour. We present the results, illustrating this attractor behaviour, using our numerical approach and Monte Carlo methods. Finally we analyze a class of models that undergo what is called the geometric destabilization of inflation. We study the observable consequences of these models after this instability occurs. In particular we calculate the bispectrum with our numerical approach, finding large non-Gaussianities of equilateral and orthogonal shapes.Frederick Perren Fund of the University of Londo
