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 completely degenerate modification to the inferred luminosity distance $d_L$. The customary method to address this is to increase the uncertainty on $d_L$ according to the dispersion of the magnification distribution at the source redshift, ${\sigma }_{\mu }\left(z\right)$. But this term is dependent on the cosmological parameters that are being constrained by gravitational wave "standard sirens," such as the Hubble parameter $H_0$, and the matter density fraction ${\Omega }_m$. The dispersion ${\sigma }_{\mu }\left(z\right)$ 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value of $H_0$ from independent datasets, and the present use of weak-lensing fitting functions calibrated using outdated cosmological simulations, suggest ${\sigma }_{\mu }\left(z\right)$ could be underestimated. This motivates an investigation into the consequences of mischaracterizing ${\sigma }_{\mu }\left(z\right)$. We consider two classes of standard siren, supermassive black hole binary and binary neutron star mergers. Underestimating $H_0$ and ${\Omega }_m$ when calculating ${\sigma }_{\mu }\left(z\right)$ increases the probability of finding a residual lensing bias on these parameters greater than $1\sigma$ by 1.5–3 times. Underestimating ${\sigma }_{\mu }\left(z\right)$ by using low resolution/small sky-area simulations can also significantly increase the probability of biased results; the probability of a $1\sigma \left(2\sigma \right)$ bias in $H_0$ and ${\Omega }_m$ found from binary neutron star mergers is 54% (19%) in this case. For neutron star mergers, the mean bias on $H_0$ caused by magnification selection effects is $\Delta H_0=-0.1\mathrm{km}{s}^{-1}{\mathrm{Mpc}}^{-1}$. The spread around this mean bias—determined by assumptions on ${\sigma }_{\mu }\left(z\right)$—is $\Delta H_0=\pm 0.25\mathrm{km}{s}^{-1}{\mathrm{Mpc}}^{-1}$, comparable to the forecasted uncertainty. These effects do not impact merging neutron stars' utility for addressing the $H_0$ tension, but left uncorrected they limit their use for precision cosmology. For supermassive black hole binaries, the spread of possible biases on $H_0$ is significant, $\Delta H_0=\pm 5\mathrm{km}{s}^{-1}{\mathrm{Mpc}}^{-1}$, but $O\left(200\right)$ observations are needed to reduce the variance below the bias. To achieve accurate subpercent 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.