Escher in the Sky

We give a brief review of the history of inflationary theory and then concentrate on the recently discovered set of inflationary models called cosmological alpha-attractors. These models provide an excellent fit to the latest observational data. Their predictions n(s) approximate to 1 - 2/N and r approximate to 12 alpha/N-2 are very robust with respect to the modifications of the inflaton potential. An intriguing interpretation of a-attractors is based on a geometric moduli space with a boundary: a Poincare disk model of a hyperbolic geometry with the radius,root 3 alpha, beautifully represented by the Escher's picture Circle Limit IV. In such models, the amplitude of the gravitational waves is proportional to the square of the radius of the Poincare disk. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.