Cosmological milestones, conformal frameworks and quiescent cosmology.

To understand the nature of the birth of our Universe and its eventual demise is a driving force in theoretical physics and astronomy and, indeed, for humanity. A zoo of definitions has appeared in the literature to catalogue different types of cosmological milestones such as 'Big Bangs', 'Big Crunches', 'Big Rips', 'Sudden Singularities', 'Bounces' and 'Turnarounds'. is the notion that the Universe commenced in a Big Bang that was highly regular and smooth, and evolved away from this initial isotropy and homogeneity due to gravitational attraction. The concept meshes well with Penrose's ideas regarding gravitational entropy and the clumping of matter, and the associated Weyl Curvature Hypothesis. Conformal frameworks, such as the Isotropic Past Singularity (IPS), have been devised to encapsulate initial and final states for the Universe which are in accordance with these programmes. These geometric definitions are independent of models, coordinates and the equation of state of the source of the gravitational field. Much of the research on cosmological milestones has been focussed on the FRW solutions, many of which possess initial singularities which are isotropic Big Bangs. We analyse here the relationship between cosmological milestones and conformal frameworks for these solutions. We establish the general properties of FRW models which admit these conformal frameworks, including whether they satisfy various energy conditions, and are therefore physically reasonable. These results inform future development of the program. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.