Implicit Likelihood Inference of the Neutrino Mass Hierarchy from Cosmological Data

In this paper, we turn to the Learning the Universe Implicit Likelihood Inference (LtU-ILI) pipeline to perform a multi-round ILI of the neutrino mass hierarchy from cosmological data, including $TT$, $TE$, $EE$ power spectra of Planck 2018 and distance ratios of DESI DR2. More precisely, we first embed the CMB power spectra simulator $\mathtt{CLASS}$ into the LtU-ILI pipeline. And then, opting for Sequential Neural Likelihood Estimation (SNLE), we sequentially train neural networks using $2$ rounds of $5000$ simulations to target a ``black box'' likelihood of our forward model with one additional neutrino mass hierarchy parameter $\tilde{\Delta}$ and six base cosmological parameters. We find that $\tilde{\Delta}=0.12216^{+0.26193}_{-0.29243}~(68\%{\rm CL})$ which slightly prefers $\tilde{\Delta}>0$, hence the normal hierarchy.