Weak lensing of bright standard sirens: prospects for $σ_8$

Abstract: Gravitational wave events with electromagnetic counterparts provide direct measurements of the Hubble diagram. We demonstrate that incorporating weak lensing into bright standard siren analyses allows measurements of cosmological parameters that do not influence the mean luminosity distance-redshift relation but do impact the cosmic structures. In particular, we examine the prospects for measuring the standard deviation of matter perturbations $σ_8$, in addition to the Hubble constant and the matter abundance. We find that a $10\%$ measurement of $σ_8$ would be feasible with ET, provided a population of $300$ neutron star binaries with electromagnetic counterparts is observed. With LISA, the measurement of $σ_8$ would have $30\%$ accuracy, assuming a population of $12$ massive black hole binaries with electromagnetic counterparts is observed.