My classmate and I are working on a project right now in which we use GWs data. We did our research through papers and different websites in order to understand well this topic: redshift derivation. Can somebody explain to me exactly and with transparency how the redshift has been derived from the data of the detector?

And also, we have seen that there is a catalog used for some sources: GLADE+. Is there any form to know which redshift sources came from this catalog and which ones didn’t?

The redshifts quoted for the detections in the GWTC papers (e.g., GWTC-3) and other LVK discovery papers are obtained from the luminosity distances that one can infer directly from analysis of the gravitational wave data (assuming general relativity), assuming a cosmology. Specifically, as mentioned in the GWTC-3 paper, this uses the Planck 2015 ΛCDM cosmology with Hubble constant of 67.9 km/s/Mpc and Omega matter of 0.3065.

As far as how one infers the luminosity distance from the detector data, see, e.g., the GW150914 parameter estimation paper for an overview of the methods used to obtain it.

Of course, if one wants to infer the cosmology, then one has to obtain the redshift some other way, and this is where GLADE+ is used–see the latest LVK cosmology paper (though note that the redshifts quoted in Table 1 of that paper still are inferred from the luminosity distances using a reference cosmology and are just given for reference). However, the GLADE+ analysis just gives constraints on cosmology–the sky and distance localizations from current gravitational wave observations are far too broad to allow one to obtain just a single galaxy in the catalogue, so one does not quote redshifts on individual sources using this method. The analysis of features in the mass distribution from that paper is similarly just used to obtain constraints on cosmology.

Of course, we do know the host galaxy of one gravitational wave detection, GW170817, and there the redshift of the host galaxy NGC 4993 is used in the gravitational wave data analysis (with some small uncertainties due to the motion of GW170817 relative to the galaxy as a whole) even when not inferring the cosmology–see this paper.