Cosmology from small scales to the horizon: relativistic effects in structure formation

The LCDM is the successful standard model of cosmology. Alternatives to
the cosmological constants include models of dark energy, modified
theories of gravity, as well as general relativistic (GR) models that
either weaken the symmetry assumptions of the cosmological principle or
try to construct an average universe to explain acceleration as a
back-reaction effect. At a time were we are going to have observational
data allowing measurements with unprecedented precision, it is however
also worth reconsidering the fine details of how we model structure
formation in LCDM cosmology, and how this may affect how we interpret
observations. By and large, we model very large scales with relativistic
perturbation theory, small scales where non-linearity is important with
Newtonian N-body simulations, and we interpret observations (e.g.
supernovae) as if light was propagating in a homogeneous-isotropic
background. In the first part of this talk I will illustrate an example
of relativistic effects, on how redshift and distances are affected if
we propagate light in a inhomogeneous LCDM universe described by an
exact GR solution. In the second part of the talk I will present a new
non-linear post-Friedmannian scheme, which is a sort of generalisation
to cosmology of the post-Newtonian approximation. Using a 1/c expansion
of Einstein equations, a set of non-linear approximate equations are
obtained in the Poisson gauge, which include the full non-linearity of
the Newtonian regime at small scales, and when linearised give standard
scalar and vector linear relativistic perturbations. The scheme thus
provides a unified framework to deal with small as well as large scales,
to the horizon and beyond. Just analysing these equations two main
results emerge: in the Newtonian regime, a frame-dragging vector
potential cannot be neglected in the metric; in the post-Friedmannian
non-linear regime, there are two gravitational scalar potentials.