  This thesis presents research exploring aspects of inflation and cosmology in
the context of inflation models in which an inflaton is non-minimally coupled
to the Ricci scalar, or is considered in conjunction with a term quadratic in
the Ricci scalar. We consider a $\phi^{2}$ Palatini inflation model in $R^{2}$
gravity and investigate whether this model can overcome some of the problems of
the original $\phi^{2}$ chaotic inflation model. We investigate the
compatibility of this model with the observed CMB when treated as an effective
theory of inflation in quantum gravity by examining the constraints on the
model parameters arising due to Planck-suppressed potential corrections and
reheating. Additionally, we consider two possible reheating channels and assess
their viability in relation to the constraints on the size of the coupling to
the $R^{2}$ term. We present an application of the Affleck-Dine mechanism, in
which quadratic $B$-violating potential terms generate the asymmetry, with a
complex inflaton as the Affleck-Dine field. We derive the $B$ asymmetry
generated in the inflaton condensate analytically and numerically. We use the
present-day asymmetry to constrain the size of the $B$-violating mass term and
derive an upper bound on the inflaton mass in order for the Affleck-Dine
dynamics to be compatible with non-minimally coupled inflation in the metric
and Palatini formalisms. We demonstrate the existence of a new class of
inflatonic Q-balls in a non-minimally coupled Palatini inflation model, through
an analytical derivation of the Q-ball equation and numerical confirmation of
the existence of solutions, and derive a range of the inflaton mass squared
within which the model can inflate and produce Q-balls. We derive analytical
estimates of the properties of these Q-balls, explore the effects of curvature,
and discuss observational signatures of the model.
