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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Instrument O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Instrument O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 O O O O O O 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 O O O O O O O O O O O O O O O O B-Instrument O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: that model A is strongly favored by the data, while model B is preferred for . It is somewhat indicative to relate values of the odds ratio to σ -levels of Gaussian measurements:where erf is the error function. 4.1.1. Mock Observations We want to assess the number of observations N obs needed to discriminate between the two models, assuming that one of them is a faithful representation of reality. To understand how each parameter impacts the analysis, we run our statistical pipeline assuming we only measure either , , or . In practice, this implies that the integral in Equation ( 18 ) is evaluated on the selected variables while marginalizing over the others. Regarding the production of mock data: we generated 10 3 sets of observations for each value of N obs , in order to produce a statistical estimation on our results. Each of the mock observations was sampled from the model and included in the observation set with probability p det . In Figure 2 , we show values of the odds ratio O AB where the set of observed events is generated from model A, so that O AB is expected to increase as N obs grows. We present results for observations generated from the dynamical (orange) and isolated (blue) models, both for aLIGO/aVirgo (left panels) and ET (right panels). Figure 2. Evolution of the odds ratio as a function of the number of simulated observations generated from the dynamical (orange) and isolated model (blue). The odds ratios are defined such that the posterior of the true model is at the denominator. Equivalent σ levels are reported with dotted lines. The four rows correspond to analysis done with M t , q , , and (from top to bottom); the left and right panels refer to aLIGO/aVirgo and ET, respectively. For each scenario, the thick lines are the median values, while the surfaces represent the 90% credible regions. Let us focus first on the top row, where only the total mass is considered in the analysis. In the case in which the dynamical model is assumed to be true (orange) with aLIGO/aVirgo, the upper limit of the 90% credible interval presents high values even for small N obs . This is because the isolated model only predicts merging BBHs with M t 70 M ⊙ . As a result, any observation where the entire support of the total mass posterior distribution is above 70 M ⊙ can only be described by the dynamical model. Another interesting feature is that the lower bound of the credible interval has values of starting at , which is consistent with our models. In fact, as the number of dynamical BBHs with M t > 70 M ⊙ in our dynamical catalog is equal to 16 (out of 229), the probability of not observing any massive BBHs in 50 observations generated from the dynamical model is equal to p = 1.9 × 10 −3 . It is also interesting to see that the results obtained are similar when observing with ET, suggesting that the differences in total mass range of our models is the dominant effect in the analysis. In contrast, the evolution of the odds ratio for