Gravitational-wave cosmological distances in scalar-tensor theories of gravity

TL;DR

This paper investigates how scalar-tensor theories of gravity affect the propagation of gravitational waves and their inferred cosmological distances, confirming the Etherington reciprocity law and exploring implications for gravitational lensing.

Contribution

It derives general expressions for GW distances in scalar-tensor theories, confirming Etherington law validity and analyzing modifications to lensing time-delays in these theories.

Findings

Etherington reciprocity law holds in scalar-tensor cosmology

GW angular distance can differ from GR predictions

Modified gravity effects influence GW lensing time-delays

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 $d_L^{({\rm GW})}$ and the GW angular distance $d_A^{({\rm GW})}$. We prove for the first time the validity of Etherington reciprocity law $d_L^{({\rm GW})}\,=\,(1+z)^2\,d_A^{({\rm GW})}$ 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.