Testing theories of Gravity and Supergravity with inflation and observations of the cosmic microwave background

Abstract: Many extensions of Einstein's theory of gravity have been studied and proposed with various motivations like the quest for a quantum theory of gravity to extensions of anomalies in observations at the solar system, galactic and cosmological scales. These extensions include adding higher powers of Ricci curvature $R$, coupling the Ricci curvature with scalar fields and generalized functions of $R$. In addition when viewed from the perspective of Supergravity (SUGRA) many of these theories may originate from the same SUGRA theory interpreted in different frames. SUGRA therefore serves as a good framework for organizing and generalizing theories of gravity beyond General Relativity. All these theories when applied to inflation (a rapid expansion of early Universe in which primordial gravitational waves might be generated and might still be detectable by the imprint they left or by the ripples that persist today) can have distinct signatures in the Cosmic Microwave Background radiation temperature and polarization anisotropies. In this review we give a detailed discussion on the standard model of cosmology ($\Lambda$CDM), inflation and cosmological perturbation theory. We survey the theories of gravity beyond Einstein's General Relativity, specially which arise from SUGRA, and study the consequences of these theories in the context of inflation and put bounds on the theories and the parameters therein from the observational experiments like Planck, Keck/BICEP. The possibility of testing these theories in the near future in CMB observations and new data coming from colliders like the LHC, provides an unique opportunity for constructing verifiable models of particle physics and General Relativity.