Gauge-Invariant Scalar-Induced Gravitational Waves from Physical Observables

Abstract: This paper discusses the gauge issue touching the gravitational waves induced at the second order by the scalar modes of cosmological perturbations. These waves are known to depend on the gauge used for their calculation. In this paper, we propose a simple method of obtaining physically meaningful expressions for such scalar-induced gravitational waves at the leading order. The method is centred on well-defined observables, such as the magnetic part of the Weyl tensor, or the Cotton tensor of a slicing of spacetime, which vanish in the background and do not depend linearly on the scalar perturbations. Generalizing the Stewart-Walker lemma, it is shown that the gravitational waves contributing to such observables at the second order are automatically gauge-invariant, even when the observable itself does not vanish at the first order. In each case, the scalar-induced gravitational waves are related to the ones computed in the Newtonian gauge, first for a general background, and then for the particular case of a spacetime dominated by either radiation or cold matter.