tokens: i.e., the prior distribution of BBH source parameters θ conditional upon some hyperparameters Λ. For this work, we are concerned with the prior distribution of ( D L , m 1 , m 2 ) ∈ θ that is dependent on cosmological hyperparameters ( H 0 , Ω m ) ∈ Λ via Equation ( 6 ). The conditional priors is denoted as π ( θ ∣Λ). The hyper-likelihood is related to the regular likelihood by:where h denotes the GW data. In hierarchical inference, we have access to the posterior distribution of parameters of individual BBHs P ( θ ∣ h ), which is related to the regular likelihood through the Bayes’ theoremwhere is the evidence and π ( θ ) is prior used for the parameter estimation of individual events. Rewriting Equation ( 8 ) by replacing the integral with the summation over discrete posterior samples (e.g., Hogg Foreman-Mackey 2018 ), we obtainwhere n is the number of posterior samples for individual events. Combining N independent events, we obtain the total likelihoodwhere h denotes the collection of data { h i }. The hyper-posteriors are given by , with π (Λ) being the prior distribution of hyperparameters. In order to perform model selection, it is also necessary to calculate the hyper-evidence given a model M The Bayes factor (BF) between model M 1 and model M 2 isWe impose a threshold of the natural logarithm of BF at as the point when one model is significantly favored against another (e.g., Mackay 2003 ). 4. Results In this section, we present results of hyperparameter estimation and cosmological model selection using the simulated BBH population that is expected to be detected by third-generation detectors such as ET. 4.1. Hyperparameter Estimation Figure 3 shows the joint posterior distribution of H 0 and Ω m using 10 3 BBH events detected with ET, while ignoring the delay time between binary formation and binary merger and assuming we know the cosmic star formation rate and black hole mass distribution a priori. In our analysis, uniform priors are used: H 0 ∈ [40, 105] km s −1 Mpc −1 and Ω m ∈ [0, 0.75]. In this example, the Hubble constant is measured with a precision of 5.6%. By performing this analysis for a range of N (the number of BBH events) assuming zero measurement uncertainty of luminosity distance and black hole masses (which we call zero-error injections), we find the measurement precision of H 0 scales linearly with . We expect that one year operation of ET, yielding ∼10 5 BBH detections, will allow H 0 to be measured within ≈ 0.6%. However, this result is too optimistic as it does not account for uncertainties in cosmic star formation rate, delay time distribution and black hole mass distribution. In Figure 6 of the Appendix , we show that the estimates of ( H 0 , Ω m ) are biased if an incorrect model of star formation rate is used. Figure 3. Joint posterior distribution of the Hubble constant ( H 0 ) and matter density (Ω m ) in the ΛCDM model estimated using 10 3 BBH events. The 2D contour regions denote the 1 σ , 2 σ, and 3
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O