Initial data on big bang singularities

We introduce a geometric notion of initial data on the singularity for the Einstein-scalar field equations and demonstrate that previous notions of data on the singularity constitute special cases. The definition thus gives a unified geometric perspective on existing results. The formulation also leads to a natural geometric initial value formulation of Einstein’s equations with initial data on the singularity. However, in order for any notion of initial data on the singularity to parametrise convergent solution, the crucial question is: do convergent solutions necessarily induce such data? There are several notions of data on the singularity for the Einstein-scalar field equations and corresponding existence results. The existence results demonstrate that the corresponding notions are sufficient to ensure convergent behaviour. However, none of the existing notions of data on the singularity are necessary. A central part of the article is therefore to demonstrate the necessity of our requirements.