Tests of inflation from the largest to the smallest scales

Inflation is arguably the leading candidate theory for the origin of the Big Bang. In the simplest realization, the scalar field(s) driving inflation slowly-rolls along a smooth path, leading to a near scale invariant spectrum of primordial density perturbations. Models beyond this vanilla scenario often imprint specific scale-dependent 'features' in the primordial power spectrum, which are highly informative as they record the mechanism which generates them. In the first part of my talk, I will discuss how deviations from slow-roll are tightly constrained by the extremely accurate measurements of the Cosmic Microwave Background anisotropies, but can nevertheless intriguingly explain some of the anomalies in the Planck data, providing a clear target for future CMB experiments that will be able to decisively detect such primordial signals. Then, I will move on to explore the uncharted territory at smaller scales, where direct, but rather loose, bounds can only be set by Spectral Distortions. The lack of constraints together with the promise of the detection of a Stochastic Gravitational Wave Background in the next decades, have stimulated theorists to build models where deviations from slow-roll are much stronger than those observationally allowed at very large scales. I will discuss current strategies to detect such deviations indirectly using gravitational waves, as well as perturbativity issues that may threaten the predictivity of such models, for which observables are often derived using linear cosmological perturbation theory.