The topic of this thesis is about cosmic inflations, including the early-universe inflation that seeds the initial inhomogeneities of our universe, and the late-time cosmic acceleration triggered by dark energy. The two inflationary epochs have now become part of the standard ΛCDM cosmological model. In the standard paradigm, dark energy is a cosmological constant or vacuum energy, while the early-universe inflation is driven by a slowly rolling scalar field. Currently the minimal ΛCDM model with six parameters agrees well with cosmological observations.
If the greatest achievement of the last twenty golden years of cosmology is the ΛCDM model, the theme of future precision cosmology will be to search for deviations from the minimal ΛCDM paradigm. It is in fact expected that the upcoming breakthroughs of cosmology will be achieved by observing the subdominant anomalies, such as non- Gaussianities in the Cosmic Microwave Background map. The aim of this thesis is then to make theoretical predictions from models beyond ΛCDM, and confront them with cosmological observations. These models include: 1) a new dark energy parametrization based on quintessence models; 2) reconstructing early-universe inflationary trajectories, going beyond the slow-roll assumption; 3) non-Gaussian curvature fluctuations from preheating after the early-universe inflation; 4) infra-red cascading produced by particle production during inflation; 5) preheating after Modular inflation; 6) decaying cold dark matter. We update the cosmological data sets – Cosmic Microwave Background, Type Ia supernova, weak gravitational lensing, galaxy power spectra, and Lyman-$alpha; forest – to the most current catalog, and run Monte Carlo Markov Chain calculations to obtain the likelihood of parameters. We also simulate mock data to forecast future observational constraints.