Aspects of Inflation and Cosmology in Non-Minimally Coupled and $R^{2}$ Palatini Gravity

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.