General Relativity (GR), which forms the basic framework for understanding cosmological and astrophysical phenomena, is based solely on general covariance. Therefore, the theory admits extensions regarding various phenomena related to inflation, massive gravity, dark energy etc. In this thesis work we study Born-Infeld type extensions of the GR. There are mainly two parts in the thesis: The extension based on Ricci tensor (already known in literature) and a novel extension based on Riemann tensor. We call them respectively Born-Infeld-Einstein (BIE) and Born-Infeld-Riemann (BIR) extensions. The BIR formalism is being proposed and studied in this thesis work. In a comparative fashion, we study these two extensions for determining their implications for 1. Mixing between Abelian and Non-Abelian gauge fields, and 2. Inflationary phase of cosmic evolution. As we prove explicitly, the two approaches yield distinct predictions for these phenomena. We emphasize that a slow-roll inflationary dynamics is naturally realized in BIR. The mixing between Abelian and Non-Abelian sectors enables cosmic photon production in inflationary phase.