We generalize to reduced Horndeski theories of gravity, where gravitational
waves (GWs) travel at the speed of light, the expression of a statistically
homogeneous and unpolarized stochastic gravitational wave background (SGWB)
signal measured as the correlation between the individual signals detected by
two interferometers in arbitrary configurations. We also discuss some results
found in the literature regarding cosmological distances in modified theories,
namely, the simultaneous validity of a duality distance relation for GW signals
and of the coincidence between the gravitational wave luminosity distance,
based on the energy flux, and the distance inferred from the wave amplitude.
This discussion allows us to conclude that the spectral energy density per unit
solid angle of an astrophysical SGWB signal has the same functional dependency
with the luminosity of each emitting source as in General Relativity (GR).
Using the generalized expression of the GW energy-momentum tensor and the
modified propagation law for the tensor modes, we conclude that the energy
density of a SGWB maintains the same functional relation with the scale factor
as in GR, provided that the modified theory coincides with GR in a given
hypersurface of constant time. However, the relation between the detected
signal and the spectral energy density is changed by the global factor
${G}_{4}(\phi ({t}_{0}))$ , thus potentially serving as a probe for modified gravity
theories.

PREPRINT

# Gravitational wave stochastic background in reduced Horndeski theories

João C. Lobato, Isabela S. Matos, Maurício O. Calvão, Ioav Waga

Submitted on 2 November 2022

## Abstract

## Preprint

Comment: 10 pages

Subjects: General Relativity and Quantum Cosmology; Astrophysics - Cosmology and Nongalactic Astrophysics