IDEAL characterization of isometry classes of FLRW and inflationary spacetimes

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 $T[g,\varphi ]=0$ are allowed to also depend on the extra fields $\varphi$. 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.

v4: Fixed minor typos relative to published version. v3: 42 pages; restructured order of sections, fixed some inconsistent formulas; close to published version

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• Compatibility complex for black hole spacetimes
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• Approaches to linear local gauge-invariant observables in inflationary cosmologies
• IDEAL characterization of higher dimensional spherically symmetric black holes
• Relativistic kinematic approach to the classical ideal gas
• Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation
• Spatially-Homogeneous Cosmologies
• New characterization of Robertson-Walker geometries involving a single timelike curve
• On the flow of perfect energy tensors