Approaches to linear local gauge-invariant observables in inflationary cosmologies

Abstract

We review and relate two recent complementary constructions of linear local gauge-invariant observables for cosmological perturbations in generic spatially flat single-field inflationary cosmologies. After briefly discussing their physical significance, we give explicit, covariant and mutually invertible transformations between the two sets of observables, thus resolving any doubts about their equivalence. In this way, we get a geometric interpretation and show the completeness of both sets of observables, while previously each of these properties was available only for one of them.

Compatibility Complex for Black Hole Spacetimes

“…Checking their equivalence with a systematically constructed complete set of linear gauge invariants can accomplish a double goal: provide the complete gauge invariants with a geometric interpretation, and show that the set {Ṫ g i } is indeed complete. Such an exercise has already been successfully carried out for a class of FLRW geometries [18]. It would be worth while to complete the comparison, already initiated in [2], of the K 1 operator with the linearized IDEAL characterization of the Kerr spacetime given by Ferrando and Sáez [15].…”

“…Proofs of completeness of a set of gauge invariants are a relatively recent development. They have been given for a small number of other spacetime reference backgrounds including the spherically symmetric Schwarzschild spacetime and the conformally flat Friedmann-Robertson-Walker spacetimes [17,18,23]. Nevertheless, this paper is the first to fully demonstrate completeness for a set of gauge invariants for the Kerr spacetime.…”

“…Ours is not the first attempt to describe perturbative gauge invariants on black hole spacetimes. See [2,18,20,23,26,27,34,38] and references therein for earlier work. See also [36] for a discussion of coordinate and tetrad gauge dependence.…”

Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation

“…In principle, the methods of [13,14] could have been used to generate the compatibility complex on locally symmetric spacetimes (those with a covariantly constant Riemann tensor), but to our knowledge they have never been explicitly elaborated in the Lorentzian setting [6]. In addition, complete sets of local gauge-invariant observables are known only for cosmological (inflationary FLRW) spacetimes in any dimension, due to the recent construction in [11,8,12], and for the 4-dimensional Kerr black hole, as recently highlighted in [1]. Full proofs of the results announced in [1] will appear in [2] and will be based on the methods to be presented in this work.…”

The Case for Nonlocal Modifications of Gravity

“…The correct generalization of the power spectrum is still unclear [63] but the answer may be gauge invariant correlators. There is interesting recent work on these by Markus Fröb and collaborators [64][65][66]. An invariant measure of the local expansion rate has also been proposed [67], and its one loop renormalization on de Sitter background has been accomplished [68].…”