Gravitational wave propagation in $f(R)$ models: New parametrizations and observational constraints

Modified gravity (MG) theories predict, in general, that the ratio of gravitational wave (GW) to electromagnetic (EM) luminosity distances, $\Xi$, differs from its general relativity (GR) value of unity at cosmological scales, thus providing another perturbative probe to MG. In this paper, we introduce new phenomenological parametrizations for both the Friedmann-Lema\^itre-Robertson-Walker (FLRW) background evolution of $f(R)$ models, via the dark energy equation of state parameter, $w_{\text{DE}}$, and for $\Xi$ in this class of theories. We simulate a mock dataset for the Einstein Telescope (ET) of 1000 GW signals from binary neutron star (BNS) mergers and redshift information from their EM counterpart, exploring the consequent constraints on the relevant gravitational, cosmological and phenomenological parameters. As a model of particular interest, we take $\gamma$-gravity theory and investigate whether it could be distinguished from GR's $\Lambda$CDM model. We then combine our results with actual data from type Ia supernovae (SNIa) and combined baryon acoustic oscillations (BAO) and cosmic microwave background (CMB) observations. Additionally, we also investigate the potential bounds to $f_{R0}$ for any viable $f(R)$ whose background evolution is indistinguishable from the standard model of cosmology above a certain redshift, showing that, for a $\Lambda$CDM fiducial model, ET data would provide $|f_{R0}|<10^{-2}$ at a 95\% level. We conclude altogether that probing the redshift evolution of the GW luminosity distance from detections of the ET in its first running decade will not substantially help constraining $f(R)$ theories of gravity.