# IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

April 18, 2017

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic)
characterization of a reference spacetime metric $g_0$ consists of a set of
tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$,
its Riemann curvature and their derivatives, that are satisfied if and only if
$g$ is locally isometric to the reference spacetime metric $g_0$. The same
notion can be extended to also include scalar or tensor fields, where the
equations $T[g,\phi]=0$ are allowed to also depend on the extra fields $\phi$.
We give the first IDEAL characterization of cosmological FLRW spacetimes, with
and without a dynamical scalar (inflaton) field. We restrict our attention to
what we call regular geometries, which uniformly satisfy certain identities or
inequalities. They roughly split into the following natural special cases:
constant curvature spacetime, Einstein static universe, and flat or curved
spatial slices. We also briefly comment on how the solution of this problem has
implications, in general relativity and inflation theory, for the construction
of local gauge invariant observables for linear cosmological perturbations and
for stability analysis.

Keywords:

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