# IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

## TL;DR

This paper provides the first covariant, tensorial characterization of FLRW and inflationary spacetimes in general relativity, aiding in the analysis of cosmological models and perturbations.

## Contribution

It introduces an IDEAL characterization for FLRW and inflationary spacetimes, including scalar fields, using covariant tensor equations, which was not previously available.

## Key findings

- First covariant IDEAL characterization of FLRW spacetimes
- Includes scalar (inflaton) fields in the characterization
- Implications for gauge-invariant perturbation analysis

## Abstract

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.