Title:
Averaging the luminosity distance in a perturbed FLRW Universe: from
theory to observations
Abstract:
What happens to the luminosity distance when we consider a
stochastically inhomogeneous Universe and, at a fixed redshift, we
average it over the sky ?

I will try to give the answer to this physically relevant question
by presenting a recent formalism, namely the "covariant averaging over
the past light-cone of a geodesic observer", in the context of a
perturbed FLRW Universe. I will use it in a system of coordinates 
(the "geodesic light-cone coordinates") very well adapted to such kind
of calculations and will arrive to the luminosity distance at second
order in the Poisson gauge (for scalars, vectors and tensors) necessary
to this computation.
The application range of this last relation is broad: going from SW and
ISW studies to lensing and redshift space distortions.

Using these derived tools, I will show how the luminosity distance
is affected, describing a small shift (~10^{-3}) in its average but a
much broader standard-deviation (depending on the power spectrum used
to describe the inhomogeneities), leading to a potentially
"precision threshold" at percent level on the measurement of d_L and
thus of the dark energy parameter. I will then conclude (if time
authorize it) on the importance of the observables themselves when
averaged over stochastic inhomogeneities, a distinction not present in 
homogeneous models.