CMB in the river frame and gauge invariance at second order
  GAUGE INVARIANCE: The Sachs-Wolfe formula describing the Cosmic Microwave
Background (CMB) temperature anisotropies is one of the most important
relations in cosmology. Despite its importance, the gauge invariance of this
formula has only been discussed at first order. Here we discuss the subtle
issue of second-order gauge transformations on the CMB. By introducing two
rules (needed to handle the subtle issues), we prove the gauge invariance of
the second-order Sachs-Wolfe formula and provide several compact expressions
which can be useful for the study of gauge transformations on cosmology. Our
results go beyond a simple technicality: we discuss from a physical point of
view several aspects that improve our understanding of the CMB. We also
elucidate how crucial it is to understand gauge transformations on the CMB in
order to avoid errors and/or misconceptions as occurred in the past. THE RIVER
FRAME: we introduce a cosmological frame which we call the river frame. In this
frame, photons and any object can be thought as fishes swimming in the river
and relations are easily expressed in either the metric or the covariant
formalism then ensuring a transparent geometric meaning. Finally, our results
show that the river frame is useful to make perturbative and non-perturbative
analysis. In particular, it was already used to obtain the fully nonlinear
generalization of the Sachs-Wolfe formula and is used here to describe
second-order perturbations.
