• cosmology
• gravitational wave
• galaxy catalog
• inference

One of the greatest limitations in astrophysics and cosmology is the distance measurement. Before the first gravitational wave (GW) detection, the only solution to this problem was the cosmic distance ladder, which is a combination of several distance indicators. On the other hand, a gravitational signal carries the information on the luminosity distance, giving access to direct distance measurement on cosmologically relevant scale.
These events can be used as standard sirens to calibrate the relation between luminosity distance and redshift, the generalized Hubble law. Once the luminosity distance and the host galaxy redshift is known, one can apply the Hubble law to measure the cosmological parameters such as H0 and ΩM .
Unfortunately, our inability to localize the source makes a direct, precise and solely GW-based cosmological parameters determination impossible. This is because, in general, we are completely unaware of the host’s redshift. To date, the only exception is GW170817, which is the first gravitational event with associated electromagnetic counterpart.
In a realistic scenario, where no electromagnetic counterpart can be identified and no redshift can be measured, two different paths have been proposed. The first, described by Li, Del Pozzo and Messenger, makes use of the equation of state of neutron stars to measure z by tidal deformability. even if applicable only to binary neutron star merger events, could help to achieve a completely independent cosmological parameters inference.
The second, first proposed by Schutz, uses Bayesian statistics to infer the host galaxy. Schutz proposed a statistical approach to the problem, taking into account all the galaxies within the volume reconstructed from a GW posterior probability distribution. Hence the galaxy catalog method name. The effectiveness of this method, however, relies on the completeness of the considered catalog: the more complete the catalog is, the more likely the host to be included in the list.
While working within small distances, the incompleteness of the catalog can be neglected, but in a more realistic scenario - for instance in binary black hole mergers - the number of undetected galaxies can be comparable to or even exceed the number of objects in the catalog: the statistical description of the phenomena must properly treat this feature.
The method was demonstrated by Del Pozzo and applied in Abbott et al. as described in Gray et al. With the events detected during the second observing run, the standard siren measurement of the Hubble constant leads to the necessity of computing a selection function.
The aim of this thesis is to present a different approach to the galaxy-catalog method; we show that it is possible, giving a more general definition of galaxy catalog and rigorously applying probability theory rules, to explicitly derive the cosmological parameters posterior probability distribution given a galaxy catalog and a gravitational signal - without making use of a selection function.
We apply the method to a set of simulated signals and demonstrate that, at least in a simplified scenario, the recovered posterior distribution is consistent with expectations. Furthermore, we analyze GW170817 with and without electromagnetic counterpart, comparing our results with Abbott et al. and Fishback et al.
Finally, we show how the formalism we developed for the cosmological parameters inference, which relies on the assumption of perfect knowledge of the galaxy population, can be inverted: assuming a fiducial cosmology, we demonstrate that, in principle, it is possible to infer the properties of gravitational wave hosts.