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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.