tokens: within the assumed cosmological prior for the parameters . As specified above, in this investigation we assume very conservative prior h ∈ [0.6, 0.85] and Ω m ∈ [0.04, 1] (fixing ). Therefore, F − ( z ) = 0.6[0.04(1 + z ) 3 + 0.96] 1/2 and F + ( z ) = 0.85(1 + z ) 3/2 . Secondly, because of peculiar velocities v p , the apparent redshift z obs of the GW host might be different from the cosmological redshift z determined by the Hubble flow, thus adding a further uncertainty term to the estimated redshift. To account for this effect, for each event defined by D l ± Δ D l , we assign a possible host redshift range . We assume v p = 500 km s −1 , consistent with the standard deviation of the radial peculiar velocity distribution observed in the Millennium run (Springel et al. 2005 ). We neglect additional contributions to the apparent redshift of the GW event coming from kicks at the compact binary formation, which can be as high as 360 km s −1 (Kelley et al. 2010 ), thus comparable with our choice of the peculiar velocity. The support Θ that enters in the evaluation of the galaxy likelihood function in equation ( 8 ) is the error-box defined by . To perform the proposed statistical D l – z reconstruction, we now need to attach a realistic distribution of galaxy to each of those error-boxes. We use the simulated observed light cones of Henriques et al. ( 2012 ) constructed from the Millennium run. The simulation assumes a Λ CDM cosmology with (which justifies the cosmological parameters assumed above to construct the BHB population). Because of the resolution of the simulation, galaxy distributions extracted from it are reliable only down to M * = 10 9 M ⊙ . We checked that at the magnitude cut imposed by Henriques et al. ( 2012 ), their reconstruction of the all-sky galaxy map is complete down to this mass limit out to z ≈ 0.15. The all-sky galaxy map provides sky location, stellar mass, cosmological redshift z , and observed redshift z obs (i.e. inclusive of peculiar velocities) for each galaxy. To select galaxy counterpart candidates to each GW event, we proceed as follow: we fix the cosmological parameters to the values of the Millennium run and we select all galaxies with a cosmological redshift (i.e. excluding peculiar velocities) in the volume shell z ± Δ z consistent with the LISA measurement D l ± Δ D l ; we randomly choose one of those galaxies to be the ‘true host’ of the GW signal. Note that this amounts to say that the true host can be anywhere in the interval D l ± Δ D l , i.e. we are not centring the error-box around the true host; we consider the error-box defined by , where ΔΩ is centred at the sky location of the selected ‘true host’; and we list all the galaxies with M * > 10 9 M ⊙ that fall within the error-box. In doing this, we make sure that their observed redshift z obs lies within the selected range. Each galaxy z obs is the relevant z j that enters in equation ( 8 ). In fact, in a real cosmological measurement one can
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