An introduction to inflation after Planck: from theory to observations

These lecture notes have been written for a short introductory course on the status of inflation after Planck and BICEP2, given at the Xth Modave School of Mathematical Physics. The first objective is to give an overview of the theory of inflation: motivations, homogeneous scalar field dynamics, slow-roll approximation, linear theory of cosmological perturbations, classification of single field potentials and their observable predictions. This includes a pedagogical derivation of the primordial scalar and tensor power spectra for any effective single field potential. The second goal is to present the most recent results of Planck and BICEP2 and to discuss their implications for inflation. Bayesian statistical methods are introduced as a tool for model analysis and comparison. Based on the recent work of J. Martin et al., the best inflationary models after Planck and BICEP2 are presented. Finally a series of open questions and issues related to inflation are mentioned and briefly discussed.