Primordial power spectrum and cosmology from black-box galaxy surveys

We propose a new, likelihood-free approach to inferring the primordial matter power spectrum and cosmological parameters from arbitrarily complex forward models of galaxy surveys where all relevant statistics can be determined from numerical simulations, i.e. black-boxes. Our approach, which we call simulator expansion for likelihood-free inference (SELFI), builds upon approximate Bayesian computation using a novel effective likelihood, and upon the linearisation of black-box models around an expansion point. Consequently, we obtain simple "filter equations" for an effective posterior of the primordial power spectrum, and a straightforward scheme for cosmological parameter inference. We demonstrate that the workload is computationally tractable, fixed a priori, and perfectly parallel. As a proof of concept, we apply our framework to a realistic synthetic galaxy survey, with a data model accounting for physical structure formation and incomplete and noisy galaxy observations. In doing so, we show that the use of non-linear numerical models allows the galaxy power spectrum to be safely fitted up to at least $k_\mathrm{max} = 0.5$ $h$/Mpc, outperforming state-of-the-art backward-modelling techniques by a factor of $\sim 5$ in the number of modes used. The result is an unbiased inference of the primordial matter power spectrum across the entire range of scales considered, including a high-fidelity reconstruction of baryon acoustic oscillations. It translates into an unbiased and robust inference of cosmological parameters. Our results pave the path towards easy applications of likelihood-free simulation-based inference in cosmology. We have made our code pySELFI and our data products publicly available at http://pyselfi.florent-leclercq.eu.