Euclid preparation. XCVI. Cosmology Likelihood for Observables in Euclid (CLOE). 3. Inference and forecasts

The Euclid mission aims to measure the positions, shapes, and redshifts of over a billion galaxies to provide unprecedented constraints on the nature of dark matter and dark energy. Achieving this goal requires a continuous reassessment of the mission's scientific performance, particularly in terms of its ability to constrain cosmological parameters, as our understanding of how to model large-scale structure observables improves . In this study, we present the first scientific forecasts using the Cosmology Likelihood for Observables in Euclid ( a dedicated Euclid cosmological pipeline developed to support this endeavour. Using advanced Bayesian inference techniques applied to synthetic Euclid-like data, we sampled the posterior distribution of cosmological and nuisance parameters across a variety of cosmological models and Euclid primary probes: cosmic shear, angular photometric galaxy clustering, galaxy-galaxy lensing, and spectroscopic galaxy clustering. We validated the capability of to produce reliable cosmological forecasts, showcasing Euclid's potential to achieve a figure of merit for the dark energy parameters and which exceed 400 when all primary probes are combined. Furthermore, we illustrate the behaviour of the posterior probability distribution of the parameters of interest given different priors and scale cuts. Finally, we emphasise the importance of addressing computational challenges, proposing further exploration of innovative data science techniques to efficiently navigate the Euclid high-dimensional parameter space in upcoming cosmological data releases.