Dark Sirens

For most of the GW events, electromagnetic counterparts are not observed, classifying these events as dark sirens. In the absence of electromagnetic counterparts, alternative methods are required to estimate the redshift of the GW sources. These methods include translating the redshift mass distribution of the GW signal to the source-frame mass distribution using astrophysical models that also include pair-instability supernovae (PISN) to constrain the redshift mass distribution. Another approach involves statistical associations of GW events with galaxy catalogs to estimate the source's redshift. Additionally, spatial clustering of GW sources with galaxies, or prior knowledge of the redshift probability distribution of merging sources, derived from intrinsic merger rates estimated by population synthesis models, can be used to infer the redshift. We want to investigate the capability of ET to constrain the cosmological model and the astrophysical population distribution jointly.

We have used the four catalogs of GW events listed in the right Table, and generated by adopting the fiducial model described above, whose parameters were set to the values listed in the left Table, to forecast the precision on mass and redshift distributions and the cosmological parameters. To achieve such a result, we used ICAROGW, a Python-based tool developed to infer the population distributions of BBHs observed through GWs. In each run, we always recover the fiducial value of each parameter at 68% of the credible level. 

Looking at the cosmological parameters, the matter density parameter shows a factor of ten improvement in precision as the SNR decreases from 200 to 80. Using only events with SNR>200, the accuracy is ~59% while, as the threshold decreases to 80, the precision increases dramatically, resulting in a much narrower credible interval and an accuracy of approximately 13.5%. This trend suggests that data at lower SNRs, possibly due to a higher number of detected events up to redshift ~6,  provides more reliable constraints. Similarly, the Hubble constant exhibits relative accuracies of approximately 39%, 23%, 10%, and 6% as the SNR threshold decreases from 200 to 80. Although these results still do not reach the precision of either current analysis based on SNeIa and Cosmic Microwave Background, nor future bright siren cosmological analysis, they represent a significant step forward in refining these estimates, in particular considering that we are constraining the cosmological model jointly with the underlying mass and redshift distributions of the BBH population.

Looking at the astrophysical parameters involved in the definitions of the mass distribution of the BBH systems, the power law indices show relative accuracies of approximately 6.6%, 4.5%, 3.4%, and 2.6% for α, and 18%, 9.5%, 7.9%, and 14.8% for β, across the SNR thresholds, respectively. The mass distribution is also characterized by the minimum and maximum masses of the binary systems, whose relative accuracies are estimated to be approximately 18.2%, 5.3%, 4.7%, and 4.6%, and 10.8%, 7.9%, 5.2%, and 4.0%, respectively.

The parameters related to the Gaussian component of the mass distribution are constrained with better precision when the SNR threshold is 80. Thus, we argue that these observations will provide a better resolution for the Gaussian distribution within the mass population, achieving a relative accuracy of approximately 1.7% on the mean and 11.6% on the variance of the Gaussian distribution, respectively. 

In Figure 3, we depict the fiducial mass distribution with fiducial parameters (black solid line) and the contours at the 68% credible level (shaded regions)  for the inferred mass distributions for the different SNR thresholds considered in our analysis. 

Finally, in Figure 4, we highlight the degeneracy between the Hubble constant and some astrophysical parameters related to the mass distribution. More importantly, a correlation between the mass distribution and cosmological parameters would prevent constraining them separately without resulting in biased results.

Lastly, the astrophysical parameters involved in the definitions of the redshift distribution, are more challenging to constrain accurately. The power to constrain these parameters is closely related to the fiducial maximum redshift of our dataset, and hence to the SNR threshold.

Bibliography:

1. M. Califano, I. de Martino, D. Vernieri, "Joint estimation of the cosmological model and  the mass and redshift distributions of the binary black hole population  with Einstein Telescope", 2025, Physical Review D, 111, 123535