Implicit Likelihood Inference of the Neutrino Mass Hierarchy from Cosmological Data

TL;DR

This work reframes neutrino mass hierarchy inference from cosmology as an implicit likelihood inference problem using the LtU-ILI pipeline with SNLE. By embedding the CLASS CMB solver into the pipeline and modeling realistic Planck-like noise and cosmic variance, the authors train an ensemble of neural density estimators over two rounds to learn the forward model for a seven-parameter cosmology extended by the hierarchy parameter Δ̃. An amortized posterior is obtained from Planck 2018 TT, TE, EE data and DESI DR2 distance ratios, yielding Δ̃=0.12216^{+0.26193}_{-0.29243} (68% CL), which mildly favors the normal hierarchy (Δ̃>0). The study validates the approach with calibration diagnostics and outlines extensions such as incorporating the CMB lensing-potential spectrum and additional statistics to tighten the hierarchy constraints.

Abstract

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 $\tildeΔ$ and six base cosmological parameters. We find that $\tildeΔ=0.12216^{+0.26193}_{-0.29243}~(68\%{\rm CL})$ which slightly prefers $\tildeΔ>0$, hence the normal hierarchy.

Figures (6)

• Figure 1: The total neutrino mass $\sum m_{\nu}$ as a function of $\Delta$ (left) or $\tilde{\Delta}=-{\rm sign}(\Delta)\times\lg(|\Delta|)$ (right), according to which an upper limit on $\sum m_{\nu}$ can be derived from a U-shaped posterior of $\Delta$ or a $\bigcap$-shaped posterior of $\tilde{\Delta}$.
• Figure 2: Comparison among $C_{\ell}^{XX'}(\bm{\theta}^0)$, $\bar{C}_{\ell}^{XX'}(\bm{\theta}^0)$ and $\hat{C}_{\ell}^{XX'}(\bm{\theta}^0)$ respectively, where $\bm{\theta}^0$ does not include the neutrino mass hierarchy parameter $\tilde{\Delta}$.
• Figure 3: Rank statistic of each parameter, where the number of samples is $N=400$ and the testing data set $\mathcal{D_{\rm test}}$ includes $500$ data-parameter pairs.
• Figure 4: Marginal percentile coverage test for cosmological parameters, where the empirical percentile from uniform distribution $\mathcal{U}(0,1)$ and the predicted percentile from the distribution of all CDF values calculated at $({\bm x}_i,{\bm \theta}_i)\in\mathcal{D_{\rm test}}$ as Eq. (\ref{['eq:CDF']}).
• Figure 5: Comparison between true and predicted value of cosmological parameters.