In recent years new types of coordinate transformations have appeared in
cosmology on top of the standard gauge transformations, such as the dilatations
and special conformal transformations, or the ones leading to (conformal) Fermi
coordinates. Some of these can remove effects that are invariant under the
standard gauge transformations and also affect asymptotic boundary conditions,
thus introducing a non-trivial ambiguity in our cosmological modeling. In this
short note we point out that this ambiguity is irrelevant for the quantities we
use to compare our model with observations — the cosmological observable
relations — as they are invariant under all of these transformations.
Importantly, this invariance holds only if one takes into account all the
relativistic contributions to an observable, which is not the case in the
literature in general. We finally also show that the practically-relevant
property of conformal Fermi coordinates (a FLRW metric up to second order in
distance) can be achieved through a globally-defined standard gauge
transformation.
