Likelihood-free Cosmological Constraints with Artificial Neural Networks: An Application on Hubble Parameters and SN Ia

The uncertainty of cosmological data generated from complex processes, such as observational Hubble parameter data (OHD) and the Type Ia supernova (SN Ia) data, cannot be accurately modeled by simple analytical probability distributions, e.g. Gaussian distribution. This necessitates the use of likelihood-free inference, which bypasses the direct calculation of likelihood. In this paper, we propose a new procedure to perform likelihood-free cosmological inference using two artificial neural networks (ANN), the Masked Autoregressive Flow (MAF) and denoising autoencoder (DAE). Our procedure is the first to use DAE to extract features from data in order to simplify the structure of MAF needed to estimate the posterior. Tested on simulated Hubble parameter data with a simple Gaussian likelihood, the procedure shows the capability of extracting feature from data and estimating posteriors without the need for tractable likelihood. We demonstrate that it can accurately approximate the real posterior, and achieves a performance comparable to the traditional MCMC method. We also discuss the application of the proposed procedure on OHD and Pantheon SN Ia data, and use them to constrain cosmological parameters from the non-flat $\Lambda$CDM model. With little prior knowledge added, we found constraints on $H_0, \Omega_m, \Omega_\Lambda$ similar to relevant work. In addition, this work is also the first to use Gaussian process in the procedure of OHD simulation.