Lecture notes on inflation and primordial black holes

These lecture notes are based on those presented at the Theoretical Aspects of Astroparticle Physics, Cosmology and Gravitation School at the Galileo Galilee Institute in Florence in 2021, https://agenda.infn.it/event/24368/. They aim to provide a pedagogical introduction and basic working knowledge of single-field inflation including the ultra-slow-roll regime, where the perturbations grow exponentially. This rapid growth is connected to the formation of primordial black holes (PBHs), a special dark matter candidate and probe of the initial conditions of the early universe. Although there are many textbooks and introductory texts about inflation, to the best of our knowledge there is no comparable introduction to ultra-slow-roll inflation. Furthermore, given their recent surge in popularity, there are numerous research articles and reviews on primordial black holes, however these notes aim to be more accessible for graduate students and those brand new to the topic. Some problems and solutions to primordial black hole-related calculations are also included. The reader of these lecture notes should come away being able to calculate the present-day abundance of primordial black holes produced from the density fluctuations left over at the end of single-field inflation with an ultra-slow-roll phase, and understand how this abundance compares with current observational constraints.