Gravitational wave luminosity distance in viscous cosmological models

We study the so-called Gravitational Wave luminosity distance-redshift relation d _L ^GW(z) during cosmological eras driven by non-perfect fluids. In particular, we show that the presence of a shear viscosity in the energy momentum tensor turns out to be the most relevant effect. Within this scenario, a constant shear viscosity imprints the gravitational wave propagation through a friction term δ(z) with a uniquely given redshift dependence. This peculiar evolution predicts a specific shape for the ratio d _L ^GW/d _L ^EM which tends to a constant value when the sources are at z ≳ 1, whereas scales linearly with the shear viscosity at lower redshifts, regardless of the value of Ω _m0. According to our final discussion, the predicted redshift dependence δ(z) provided by a shear viscosity could be tested by upcoming surveys of multi-messenger sources against analogous scenarios provided by some widely studied theories of modified gravity.