• Cosmological perturbation theory
• geodesic light-cone gauge
• GLC
• luminosity distance

Nowadays the high sensibility of large scale structure surveys has allowed us to study the Universe on that scale and with unprecedented precision. In particular the data show how and to which extent the assumption that the Universe is homogeneous and isotropic can only be interpreted as a statistical property: our Universe contains structures, such as galaxies and galaxy clusters, and the light we measure
is affected by the local inhomogeneities distributed along its path.
Due to this fact in the recent years ever more effort has been done in order to develop a theory capable to give a precise enough description of the Universe, taking into account the relativistic effects generated by local inhomogeneities.
In developing the perturbative theory necessary to describe the Universe on such large scales, a problem that arises is that the metric used to describe the space-time is not gauge invariant, so generic quantities could take different values depending on which gauge choice we adopt. But this is not true for observables: general relativity in fact assures that the physics of a given system can not depend on which
coordinates system we chose to describe it. This means that observables has to be gauge invariant.
In this thesis we start from a background coordinate system adapted to the observations and build on it a perturbative theory in order to address the correspondent gauge issue. In such a coordinate system proper time, angles and the background past light-cone are used as space-time coordinates. This background system corresponds to the background of the so-called geodesic light-cone (GLC) gauge. The GLC gauge allows to describe physical observables from the point of view of the free-falling observer. Indeed its coordinates are the proper time as measured by the above-mentioned observer, a null coordinate chosen to label the arrival time of photons emitted by a given source (the non-perturbative past light-cone), and the last two as the observed angles of the arrival direction of photons in the observer frame.
The GLC gauge is especially useful because it is adapted to our past light-cone, that is the region where the physical information carried by light-like signals travels. In particular, if we fix the so-called GLC observational gauge on it, there are fully non-linear (exact) and simple expressions in the GLC gauge for the redshift and the Jacobi Map: from this last one we can then obtain an exact expression for the luminosity distance. Given these analytical expressions, a coordinate transformation from the GLC metric to a chosen coordinate system is enough to write the desired observable (redshift or luminosity distance) in terms of the new coordinate system.
The novelty of the work presented in this thesis consists of building a perturbative theory directly on top the GLC background coordinates. We have then related the metric perturbations on the GLC background metric to the standard FLRW metric perturbations, in order to switch from a gauge in a coordinate system to another one just thanks to a background coordinate transformation. In particular we have found how the expressions for some gauges widely used in literature, namely the uniform curvature gauge and the synchronous gauge, look in the new perturbative scheme.
We have also used the obtained results to write the luminosity distance at the first perturbative order with a new approach. Indeed the luminosity distance has an exact solution in GLC gauge whereas it can be computed only in a perturbative way in the perturbed FLRW metric. Therefore, by fixing the gauge on top of the GLC background coordinates to obtain the perturbative version of the GLC gauge and thanks to our map, we have written the d L to first order only by a background coordinates transformations in the FLRW gauge correspondents to the GLC gauge. This is an important result because this procedure can be generalized to all perturbative orders, simply substituting the perturbative coordinate trasformation previously used to obtain the d L using the GLC property with the construction of a higher-ordrer perturbative theory. Furthermore this allowed us to define a gauge invariant luminosity distance, in FLRW plus perturbation, as the one correspondent to the d L of the GLC gauge.