The averaging problem on the past null cone in inhomogeneous dust cosmologies

We investigate average properties of light fronts as they propagate from a finite patch of a source (e.g., the last scattering surface) to a finite patch of a telescope pointing in generic irrotational dust cosmologies. In this setting we formulate Einstein's field equations on the light fronts and provide constraint equations and propagation equations along an observer's past null cone. For the resulting covariant system of area-averaged equations on the light fronts, we discuss integral-geometric properties together with closure conditions for the underdetermined set of equations. We formulate light front averages of observable quantities, including the effective angular diameter distance and the cosmological redshift drift. The `backreaction' in the effective angular diameter distance is quantified in terms of cosmological area-averaged variables on the screen space of observation, as well as a `memory function' containing information on the auto-correlation of the expansion rate of an incoming geodesic null ray bundle along the observer's past null cone