The Weyl tensor correlator in cosmological spacetimes

Abstract

We give a general expression for the Weyl tensor two-point function in a general Friedmann-Lemaître-Robertson-Walker spacetime. We work in reduced phase space for the perturbations, i. e., quantize only the dynamical degrees of freedom without adding any gauge-fixing term. The general formula is illustrated by a calculation in slow-roll single-field inflation to first order in the slow-roll parameters and δ, and the result is shown to have the correct de Sitter limit as , δ → 0. Furthermore, it is seen that the Weyl tensor correlation function in slow-roll does not suffer from infrared divergences, unlike the two-point functions of the metric and scalar field perturbations. Lastly, we show how to recover the usual tensor power spectrum from the Weyl tensor correlation function.

Gupta-Bleuler quantization for linearized gravity in de Sitter spacetime

“…However, as already pointed out, thanks to the mathematical structure of the formulation we are using, inspired by the ambient space approach, it can be checked quite easily that the aforementioned break down in the normalization factor is because of a degeneracy for L = 0 mode reflecting the gauge-like symmetry (2). This point explicitly reveals that, quite contrary to the authors claim in [33][34][35][36][37][38][39][40][41][42][43][44][45][46], such a construction in the STT gauge conditions (L ≥ 2), by ignoring the L = 0 mode and consequently the gauge-like symmetry (2) reflected by it, does not transform correctly under the whole symmetries of the classical theory, and therefore, even obtaining an infrared free graviton twopoint function in this way is not physically significant since it is not covariant anyway. To see other criticism to this method, one can refer to [51][52][53][54][55].…”

“…|c Llm | 2 < ∞ , (38) in which E 2 g 2 µν stands for the projection tensor associated with the invariant subspace V 2 /V g2 , denoted by the cindependent part in (19). Again, the zero mode (L = 0) does not contribute to the physical space of solutions because if it was included, the set of physical modes would be transformed into modes of negative frequency violating unitarity.…”

Causality implies inflationary back-reaction

“…One consequence is that gravitons do not possess a fully de Sitter invariant vacuum state, just like the massless, minimally coupled scalar [83], whose plane wave mode functions are identical to those of dynamical gravitons [84]. There has been a long and confusing debate about this [52][53][54][55][56][57]. All agree that the graviton mode functions approach a constant at late times (that is what causes the tensor power spectrum) and that this freezing-in endows the completely gauge fixed graviton propagator with a de Sitter-breaking time dependence which takes the form of a linearized gauge transformation.…”

“…Physicists accept that the freezing in of scalars can mediate effects because the value of a scalar field is observable; for example, the expectation value of the Higgs field determines masses in the Standard Model. However, physicists are conditioned to dismiss nonzero constant values of the graviton field as gauge artifacts which could be eliminated by an appropriate choice of coordinates [52][53][54][55][56][57].…”

‘Hidden’ symmetry of linearized gravity in de Sitter space

“…From the very beginning of this scientific dispute, a firm reasoning in favor of covariant quantization of the graviton field in the natural dS vacuum state (the Bunch-Davies vacuum) was put forward in Ref. [4], and during recent three decades, it has been dynamically subject to scrutiny in a number of works (see, for instance, [5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Let us make the idea lying behind of this reasoning explicit, using the so-called conformal (global) coordinates, x = (x 0 = H −1 tan ρ, (H cos ρ) −1 u), ρ ∈] −π 2 , π 2 [, u ∈ S 3 .…”