Necessary and sufficient conditions for big bangs, bounces, crunches, rips, sudden singularities and extremality events

Until recently, the physically relevant singularities occurring in FRW cosmologies had traditionally been thought to be limited to the 'big bang', and possibly a 'big crunch'. However, over the last few years, the zoo of cosmological singularities considered in the literature has become considerably more extensive, with 'big rips' and 'sudden singularities' added to the mix, as well as renewed interest in nonsingular cosmological events such as 'bounces' and 'turnarounds'. In this paper we present an extensive catalogue of such cosmological milestones, both at the kinematical and dynamical level. First, using generalized power series, purely kinematical definitions of these cosmological events are provided in terms of the behaviour of the scale factor a(t). The notion of a 'scale-factor singularity' is defined, and its relation to curvature singularities (polynomial and differential) is explored. Second, dynamical information is extracted by using the Friedmann equations (without assuming even the existence of any equation of state) to place constraints on whether or not the classical energy conditions are satisfied at the cosmological milestones. We use these considerations to derive necessary and sufficient conditions for the existence of cosmological milestones such as bangs, bounces, crunches, rips, sudden singularities and extremality events. Since the classification is extremely general and, modulo certain technical assumptions, is complete, the corresponding results are to a high degree model independent: in particular, we provide a characterization of the class of bangs, crunches and sudden singularities for which the dominant energy condition is satisfied.