Cosmological perturbations of brane-induced gravity and the vDVZ discontinuity on FLRW space-times

TL;DR

This paper demonstrates that the vDVZ discontinuity in brane-induced gravity models disappears on FLRW backgrounds, showing a well-defined limit of cosmological perturbations and extending previous results beyond maximally symmetric space-times.

Contribution

It proves the vanishing of the vDVZ discontinuity for general FLRW space-times and clarifies the conditions under which linear perturbation theory remains valid.

Findings

vDVZ discontinuity disappears on FLRW backgrounds

Linearized Einstein equations hold when Hubble factor is non-zero

Linear perturbation theory breaks down when Ricci scalar vanishes

Abstract

We investigate the cosmological perturbations of the brane-induced (Dvali-Gabadadze-Porrati) model which exhibits a van Dam-Veltman-Zakharov (vDVZ) discontinuity when linearized over a Minkowski background. We show that the linear brane scalar cosmological perturbations over an arbitrary Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time have a well defined limit when the radius of transition between 4D and 5D gravity is sent to infinity with respect to the background Hubble radius. This radius of transition plays for the brane-induced gravity model a role equivalent to the Compton wavelength of the graviton in a Pauli-Fierz theory, as far as the vDVZ discontinuity is concerned. This well defined limit is shown to obey the linearized 4D Einstein's equations whenever the Hubble factor is non vanishing. This shows the disappearance of the vDVZ discontinuity for general FLRW background, and extends the previously know result for maximally-symmetric space-times of non vanishing curvature. Our reasoning is valid for matter with simple equation of state such as a scalar field, or a perfect fluid with adiabatic perturbations, and involves to distinguish between space-times with a vanishing scalar curvature and space-times with a non vanishing one. We also discuss the validity of the linear perturbation theory, in particular for those FLRW space-times where the Ricci scalar is vanishing only on a set of zero measure. In those cases, we argue that the linear perturbation theory breaks down when the Ricci scalar vanishes (and the radius of transition is sent to infinity), in a way similar to what has been found to occur around sources on a Minkowski background.