Cutting out the cosmological middle man: General Relativity in the
  light-cone coordinates
  Analytical computations in relativistic cosmology can be split into two sets:
time evolution relating the initial conditions to the observer's light-cone and
light propagation to obtain observables. Cosmological perturbation theory in
the FLRW coordinates constitutes an efficient tool for the former task, but the
latter is dramatically simpler in light-cone-adapted coordinates that
trivialize the light rays towards the observer world-line. Here we point out
that time evolution and observable reconstruction can be combined into a single
computation that relates directly initial conditions to observables. This is
possible if one works uniquely in such light-cone coordinates, thus completely
bypassing the FLRW "middle-man" coordinates. We first present in detail these
light-cone coordinates, extending and generalizing the presently available
material in the literature, and construct a particularly convenient subset for
cosmological perturbation theory. We then express the Einstein and
energy-momentum conservation equations in these coordinates at the fully
non-linear level. This is achieved through a careful 2+1+1 decomposition which
leads to relatively compact expressions and provides good control over the
geometrical interpretation of the involved quantities. Finally, we consider
cosmological perturbation theory to linear order, paying attention to the
remaining gauge symmetries and consistently obtaining gauge-invariant
equations. Moreover, we show that it is possible to implement statistical
homogeneity on stochastic fluctuations, despite the fact that the coordinate
system privileges the observer world-line.
