The Cosmological Perturbation Theory on the Geodesic Light-Cone
  background
  Inspired by the fully non-linear Geodesic Light-Cone (GLC) gauge, we consider
its analogous set of coordinates which describes the unperturbed Universe.
Given this starting point, we then build a cosmological perturbation theory on
top of it, study the gauge transformation properties related to this new set of
perturbations and show the connection with standard cosmological perturbation
theory. In particular, we obtain which gauge in standard perturbation theory
corresponds to the GLC gauge, and put in evidence how this is a useful
alternative to the standard Synchronous Gauge. Moreover, we exploit several
viable definitions for gauge invariant combinations. Among others, we build the
gauge invariant variables such that their values equal the ones of linearized
GLC gauge perturbations. This choice is motivated by two crucial properties of
the GLC gauge: i) it admits simple expressions for light-like observables, e.g.
redshift and angular distance, at fully non-linear level and ii) the GLC proper
time coincides with the one of a free-falling observer. Thanks to the first
property, exact expressions can then be easily expanded at linear order to
obtain linear gauge invariant expression for the chosen observable. Moreover,
the second feature naturally provides gauge invariant expressions for physical
observables in terms of the time as measured by such free-falling observer.
Finally, we explicitly show all these aspects for the case of the linearized
angular distance-redshift relation.
