Bright Siren Cosmology Beyond Fisher: DALI Forecasts for Third-Generation Gravitational Wave Observatories

In this work we assess non-Gaussian extensions of Fisher forecasting based on the Derivative Approximation for Likelihoods (DALI). We simulate ensembles of bright siren catalogs for third-generation observatories, adopting a Cosmic Explorer (CE) and a triangular Einstein Telescope (ET) network, frequency-domain inspiral waveforms, and selection-motivated priors for counterpart detection. We reconstruct approximate likelihoods using Fisher and the first DALI correction (Doublet), and propagate the inferred distance information to cosmological inference in a w0waCDM model. We find that the Doublet approximation provides tighter cosmological constraints than the Fisher approximation, reducing the inferred uncertainties by approximately a factor of two in agreement with non-approximate likelihood (called Exact) results. For our GW network onfiguration, 300 bright-siren detections up to z=1 yield an H0 precision of ~2.7% with the Doublet approximation, compared with ~2.5% for the exact likelihood. The adopted non-Gaussian GW priors induce a systematic displacement of the inferred luminosity distances from the fiducial values, which is also present in the exact GW reconstruction and propagates into the cosmological parameters. Compared with the exact cosmological likelihood, the Doublet results differ by only ~10% on average in the parameter uncertainties, compared with ~195% for the traditional Fisher approximation. The Doublet approximation also substantially reduces the computational cost, by a factor of ~55 for the GW analysis and by ~12 for cosmological likelihood evaluations, yielding an estimated reduction of ~55 in the total computational cost of the combined analysis. Our results show the potential of DALI-based approximations to provide accurate and computationally efficient tools for forecasting in bright siren cosmology.