Abstract
We analyze the propagation of high-frequency gravitational waves (GW) in scalar-tensor theories of gravity, with the aim of examining properties of cosmological distances as inferred from GW measurements. By using symmetry principles, we first determine the most general structure of the GW linearized equations and of the GW energy momentum tensor, assuming that GW move with the speed of light. Modified gravity effects are encoded in a small number of parameters, and we study the conditions for ensuring graviton number conservation in our covariant set-up. We then apply our general findings to the case of GW propagating through a perturbed cosmological space-time, deriving the expressions for the GW luminosity distance dL (GW) and the GW angular distance dA (GW). We prove for the first time the validity of Etherington reciprocity law dL (GW) = (1+z)2 dA (GW) for a perturbed universe within a scalar-tensor framework. We find that besides the GW luminosity distance, also the GW angular distance can be modified with respect to General Relativity. We discuss implications of this result for gravitational lensing, focussing on time-delays of lensed GW and lensed photons emitted simultaneously during a multimessenger event. We explicitly show how modified gravity effects compensate between different coefficients in the GW time-delay formula: lensed GW arrive at the same time as their lensed electromagnetic counterparts, in agreement with causality constraints.
Highlights
We analyze the propagation of high-frequency gravitational waves (GW) in scalartensor theories of gravity, with the aim of examining properties of cosmological distances as inferred from GW measurements
Modified gravity effects are encoded in a small number of parameters, and we study the conditions for ensuring graviton number conservation in our covariant set-up
We explicitly show how modified gravity effects compensate between different coefficients in the GW time-delay formula: lensed GW arrive at the same time as their lensed electromagnetic counterparts, in agreement with causality constraints
Summary
We develop a covariant approach for investigating the dynamics of high-frequency modes in scalar-tensor theories of dark energy. Controlling the ratio among the typical (small) wavelength λ of the high-frequency fields versus the (large) scale LB of spatial variation of slowly-varying background quantities Among the latter, we include a dark energy scalar φ(x) whose time-like profile varies on scales of order LB. We expect that propagation effects are able to excite scalar modes with an amplitude suppressed by a factor of O( ) with respect to metric fluctuations: O(φ) ∼ O(hμν). Keeping a fixed high-frequency wavelength λ for the metric fluctuations hμν, in the limit ∇μφ → 0 (or equivalently LB → ∞) we expect the scalar Goldstone modes φ to be absent, since the symmetry is restored, and Goldstone bosons do not propagate. For definiteness, we concentrate on studying the dynamics of the transverse-traceless GW modes h(μTνT ), leaving the study of the independent scalar sector, when propagating, to a separate work
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