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, μ, 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, σ_μ. 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 Ω_m. σ_μ 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 Tension in the measured value of H_0 from independent datasets, and the present use of outdated cosmological simulations, suggest σ_μ could be underestimated. We consider two classes of standard siren, supermassive black hole binary and binary neutron star mergers. Underestimating H_0 and Ω_m when calculating σ_μ increases the probability of finding a residual lensing bias on these parameters greater than 1σ by 1.5-3 times. Underestimating σ_μ 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.