Title: A robust cosmic standard ruler from the cross-correlations of galaxies and dark sirens
Authors: João Ferri, Ian L. Tashiro, L. Raul Abramo, Isabela Matos, Miguel Quartin, Riccardo Sturani
First author institution: Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo
Status: Published in the Journal of Cosmology and Astroparticle Physics (JCAP), open access
The detection of gravitational waves, of which the first was GW150914, has led to entirely new fields of astrophysics and cosmology. They are propagating waves that lead to distortions in space time, created by any changing mass distribution that has the right asymmetry (called a quadrupole moment). This means they can be produced in various ways, but the ones we currently are able to detect are produced by the merging of massive compact stellar objects including black holes and neutron stars. While gravitational waves are interesting just on their own, they are also useful for cosmologists, who will take anything that can help them work out the distances to galaxies – or any other far away objects. Any information that gives us the distances to objects, that we can also compare to the velocity of that object due to expanding space, allows us to test the expansion rate of the Universe, \(H_0\). As it turns out, the signal (called the strain) that we detect from gravitational wave (GW) events is able to give us information about the distance to them; more distant events have a smaller amplitude in the strain.
There are various ways we can compare the distances of GW events to a cosmological model; ultimately, we need some kind of estimate for the redshift (which tells us the expansion velocity) of the host galaxy of the event to do so. For GW events with an electromagnetic counterpart (such as the merging of a black hole with a neutron star which creates a bright explosion), the redshift can be obtained from the electromagnetic spectrum – see an example of this being done here. However, the merging of two black holes (a binary black hole event, or BBH) won’t have this counterpart, so another approach which has been used is to try and infer the host galaxy by comparing the localization of the GW event on the sky with a host catalogue of galaxies. This method is pretty complicated, because it requires one to model the chance of the event being from any galaxy in a catalogue, as well as some information about the rate at which the GW events occur and in what kinds of galaxies. Another approach that has been used is to take the properties of the mass distribution of binary black hole events to help infer the redshift of the host galaxy.
However, the authors of this paper propose a new approach to use GW events to test cosmology. In this approach, they cross-correlate information from catalogues of galaxies, which trace the large-scale structure of the Universe (the way matter is spread out over space), to BBH events, and can use this to make a \(H_0\) measurement. This approach has the advantage in that it doesn’t actually require one to even really know the exact galaxies the BBH events occurred in. The galaxies and the BBHs have a correlation because they are connected by the local large-scale structure; we should expect BBH events to come from places where there is more matter, and thus more galaxies. This method takes BBH events and splits them up into thin shells based on their distances in space. Then, it takes the galaxies from a catalogue, and splits them up into thin shells of redshift. See Figure 1 below. Then, they measure the correlation between the BBH shells and the galaxy shells in space; where the correlation is maximised between the different shells allows a connection between redshift and distance, and this can be used to infer a best cosmological model and thus \(H_0\).
Creating simulations to test the methodology
After some development of a formalism to implement this method (calculating the cross-correlations) the authors want to test its potential with simulated data. Some simulations are actually always required in their approach, in order to compare the measured cross-correlations from theoretical expectations of different models (i.e. different \(H_0\)) to data. This can be computationally expensive; the data needs to be compared to many different models, requiring many simulations. However, they develop a shortcut, using the fact that the main impact of changing cosmological models is to shift the ‘peak’ or the location where the shells have maximum correlation; without going into the details, it is not entirely necessary for the theoretical cross-correlations to be fully recomputed from scratch for all possible cosmological models they want to compare the data to. This allows for their method to be computationally feasible.
An independent set of simulations are used to test their method of measuring the expansion rate and thus determine the ability of the method to measure \(H_0\); this can later be applied to real data. They generate ‘mock’ shells of galaxy redshifts and BBH distances, and use this to mimic measuring a cross-correlation between the galaxies and BBHs. They do this while accounting for the ability to detect BBHs from GW events with upcoming GW detectors and their observations; they consider the observations possible from the LVK (Ligo-Virgo-Kagra) network, which will have an observing run starting in 2027. They also consider the potential from the proposed Einstein Telescope, and the proposed Cosmic Explorer, in combination with each other as a network, but also the Einstein Telescope + LVK as a combined network. The creation of their mock data also requires several steps, which is briefly summarized here with a few dot points:
- To forecast the ability to measure the BBHs from these detectors, they use something called the Fisher information; this is a manner in which one can estimate the information an observable contains about a parameter of interest. In this case, it is a way for the authors to estimate the uncertainty on the measured distances and positions in the sky of BBHs, based on their GW signal; this also includes other properties of the GW events.
- They account for the expected distribution of events that the GW detectors will be sensitive to across the sky, and make sure that the observed BBH events are correlated with the underlying matter distribution, by producing a map from which to ‘select’ observed BBH events from galaxies.
- The details of the uncertainty of measurements of BBH signals also depend on how they are spread across the sky; they account for this and incorporate the appropriate correlations between uncertainties and sky positions into their simulations.
- Finally, weak gravitational lensing causes the apparent distances to the GW events to be distorted, and they include this impact in their simulations – see more on this phenomenon in this bite.
Results: applying the method to simulations
The authors applied their method to the simulated data and can compare the Hubble constant \(H_0\) determined from their method to the input of their simulated data; this is shown in Figure 2 and Figure 3 below. The contours show the most likely values of the parameters affecting the expansion rate that are inferred from the data, and individual PDFs are shown for each parameter also. The cross-hairs show the ‘truth’ of the simulation; this overlaps with the contours, indicating the method gives an unbiased measurement.
These constraints are expected to be informative in the future, allowing for a precise measurement of \(H_0\); given the Hubble tension, this is useful to have. They also find their method is able to potentially allow for one to constrain more complicated cosmological models that have an expansion history involving time-varying dark energy. Another benefit to their method is that the cosmology used in the simulations doesn’t really impact the results; this was mentioned briefly before, but they check this is true and find the method to be unbiased by using a different cosmological model to generate simulated galaxy data. They also check and verify that not actually having any of the original galaxy hosts for any GW event in their catalogue doesn’t impact the results.
Overall, the new method proposed in this work is promising and helps to overcome various challenges of using GW events to measure the expansion rate of the Universe.
Edited by Ansh Gupta
Featured image credit: A cropped version of Figure 1 from the paper, showing the shells of galaxy redshifts and BBH distances.
