The cosmological perturbation theory on the Geodesic Light-Cone background

Abstract

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.

Cosmological Perturbations and Invariant Observables in Geodesic Lightcone Coordinates

“…Interestingly, the linearised gauge-fixing conditions (3.55) are closely related to the "observational synchronous gauge" proposed in Ref. [38]. Indeed, the main difference between the perturbative corrections to the GLC coordinates (3.41) that we define and the ones of Ref.…”

“…Indeed, the main difference between the perturbative corrections to the GLC coordinates (3.41) that we define and the ones of Ref. [38] is the choice of integration limits. In our case, as explained above we have fixed the remaining freedom in the choice of GLC coordinates by requiring the perturbed GLC coordinates to coincide with the background ones in the absence of perturbations, in particular when going to spatial or past infinity.…”

“…On the other hand, the choice of Ref. [38] corresponds to a more observer-centric view, and is better suited if one considers the effects fluctuations have on signals that reach a fixed observer. There is then still the freedom of fixing the GLC frame at the observer, corresponding to free integration constants, which can of course be chosen in such a way as to make their choice equal to ours.…”

“…Moreover, the invariant observables we will define are somewhat different from the ones proposed in Ref. [38]; we will discuss their exact relation later on in this paper. We will also address the time evolution of the observables at first order by deriving the linearised Einstein's equations when the spacetime is sourced by a scalar field (single-field inflation).…”

“…The issue of constructing gauge-invariant observables using the GLC coordinates has been recently addressed in Refs. [38,39], at first order in perturbation theory. The method we use allows us to more easily obtain explicit expressions for invariant observables beyond linear order.…”