Averaging the luminosity distance in a perturbed FLRW Universe: from theory to observations.
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.