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Joint tests of cosmology and modified gravity in light of GW170817

In this Letter we constrain for the first time both cosmology and modified gravity theories conjointly, by combining the GW and electromagnetic observations of GW170817. We provide joint posterior distributions for the Hubble constant $H_0$, and two physical effects typical of modified gravity: the gravitational wave (GW) friction, encoded by the parameter $α_M$, and several GW dispersion relations. Among the results of this analysis, we can improve by 15\% the bound of the graviton mass with respect to measurement using the same event, but fixing $H_0$. We obtain a value of $m^2_g=2.08_{-4.25}^{+13.90} \cdot 10^{-44} \rm{eV^2/c^4}$ at 99.7\% confidence level (CL), when marginalising over the Hubble constant and GW friction term $α_M$. We find poor constraints on $α_M$, but demonstrate that for all the GW dispersions relations considered, including massive gravity, the GW must be emitted $\sim$ 1.74s before the Gamma-ray burst (GRB). Furthermore, at the GW merger peak frequency, we show that the fractional difference between the GW group velocity and $c$ is $\lesssim 10^{-17}$.