  In this work we have investigated the synergy between Stage-IV galaxy surveys
and future GW observatories for constraining the underlying cosmological model
of the Universe, focussing on photometric galaxy clustering, cosmic shear and
GW magnification as cosmological probes. We have implemented a Fisher matrix
approach for the evaluation of the full $6\times2$pt statistics composed by the
angular power spectra of the single probes together with their combination. For
our analysis, we have in particular considered dynamical dark energy and
massive neutrino scenarios. We have found that the improvement to galaxy survey
performance is below 1\%, in the case $\ell^{\rm GW}_{\rm max}=100$ and a
luminosity distance error of $\sigma_{d_L}/d_L=10\%$. However, when extending
the analysis to $\ell^{\rm GW}_{\rm max}=1000$, we find that the GW
magnification improves the galaxy survey performance on all the cosmological
parameters, reducing their errors by $3\%$-$5\%$, when $\sigma_{d_L}/d_L=10\%$,
and by $10\%$-$18\%$ when $\sigma_{d_L}/d_L=1\%$, especially for $M_\nu$, $w_0$
and $w_a$. However, here our analysis is unavoidably optimistic: a much more
detailed and realistic approach will be needed, especially by including
systematic effects. But we can conclude that, in the case of future
gravitational wave observatories the inclusion of the gravitational wave
magnification can improve Stage-IV galaxy surveys performance on constraining
the underlying cosmological model of the Universe.
