This paper compares two ways of learning about the Universe from future gravitational-wave observations. The authors study how well the traditional Hierarchical Bayesian Inference (HBI) method performs against a Simulation-Based Inference (SBI) method when applied to large catalogs of “dark sirens” — gravitational-wave events without an electromagnetic counterpart. Using a mock catalogue of about 10^4 binary black hole mergers expected for the Einstein Telescope (ET), they test recovery of the Hubble constant H0 and the matter density Ωm in a flat ΛCDM (Lambda Cold Dark Matter) cosmology.

To make this comparison they build a synthetic ET data set. The ET design they use is the triangular ET-D reference, with each nested interferometer having 10 km arms. They model the redshift distribution of merging binaries from a convolved star-formation-rate density and simulate noisy distance measurements for each event. They then run two analyses on the same mock catalog. One uses the standard hierarchical analytical likelihood from HBI. The other uses Marginal Neural Ratio Estimation (MNRE), a form of SBI that learns the relationship between parameters and simulated data from many forward simulations.

The main result is agreement between the two methods. SBI reproduces the HBI posteriors on H0 and Ωm to high accuracy for this mock ET catalogue. The authors also report that, once the cost of running simulations and training the neural networks is amortized, SBI requires orders of magnitude less computation than the direct HBI likelihood evaluation. They further show SBI can be extended easily to a joint analysis that fits cosmology together with astrophysical parameters (for example, parameters describing the star formation rate density) at little extra computational cost. In contrast, adding those extra parameters greatly increases the complexity of the HBI approach.