Impact of weak lensing on bright standard siren analyses

Gravitational waves from binary mergers at cosmological distances will experience weak lensing by large scale structure. This causes a (de-)magnification, $\mu$, of the wave amplitude, and a degenerate modification to the inferred luminosity distance $d_L$. To address this the uncertainty on $d_L$ is increased according to the dispersion of the magnification distribution at the source redshift, $\sigma_\mu$. But this term is dependent on cosmological parameters that are being constrained by gravitational wave "standard sirens", such as the Hubble parameter $H_0$, and the matter density fraction $\Omega_m$. $\sigma_\mu$ is also sensitive to the resolution of the simulation used for its calculation. Tension in the measured value of $H_0$ from independent datasets, and the present use of outdated cosmological simulations, suggest $\sigma_\mu$ could be underestimated. We consider two classes of standard siren, supermassive black hole binary and binary neutron star mergers. Underestimating $H_0$ and $\Omega_m$ when calculating $\sigma_\mu$ increases the probability of finding a residual lensing bias on these parameters greater than $1\sigma$ by 1.5-3 times. Underestimating $\sigma_\mu$ by using low resolution/small sky-area simulations can also significantly increase the probability of biased results. For neutron star mergers, the spread of possible biases is 0.25 km/s/Mpc, comparable to the forecasted uncertainty. Left uncorrected this effect limits the use of BNS mergers for precision cosmology. For supermassive black hole binaries, the spread of possible biases on $H_0$ is significant, 5 km/s/Mpc, but $O(200)$ observations are needed to reduce the variance below the bias. To achieve accurate sub-percent level precision on cosmological parameters using standard sirens, first much improved knowledge on the form of the magnification distribution and its dependence on cosmology is needed.