  The goal of this article is to parametrise solutions to Einstein's equations
with big bang singularities and quiescent asymptotics. To this end, we
introduce a notion of initial data on big bang singularities and conjecture
that it can be used to parametrise quiescent solutions. A mathematical
statement of the conjecture presupposes a precise definition of the class of
quiescent solutions as well as a proof of existence and uniqueness of
developments corresponding to initial data on a big bang singularity. We
provide one definition of quiescence here. We also appeal to existing results
in order to illustrate that, in certain cases, there are unique developments
corresponding to initial data on the singularity. However, our perspective
leads to a large class of open problems corresponding to the general
conjecture. An additional benefit of the notion of initial data developed here
is that it can be used to give a unified perspective on the existing results
concerning quiescent singularities. In fact, we provide several examples of how
existing results can be considered to be special cases of the framework
developed here. A second, potential, application is to oscillatory and
spatially inhomogeneous big bang singularities. Considering the existing
arguments in the spatially homogeneous setting, a crucial first step in the
study of oscillatory behaviour is to understand how solutions approach the
Kasner circle along a stable manifold, and then depart via an unstable
manifold. In order to carry out a similar analysis in the spatially
inhomogeneous setting, it is of central importance to first identify the stable
manifold. Building on the work of Fournodavlos and Luk, we here propose such an
identification.
