Learning Theory Informed Priors for Bayesian Inference: A Case Study with Early Dark Energy

Cosmological models are often motivated and formulated in the language of particle physics, using quantities such as the axion decay constant, but tested against data using ostensibly physical quantities, such as energy density ratios, assuming uniform priors on the latter. This approach neglects priors on the model from fundamental theory, including from particle physics and string theory, such as the preference for sub-Planckian axion decay constants. We introduce a novel approach to learning theory-informed priors for Bayesian inference using normalizing flows (NF), a flexible generative machine learning technique that generates priors on model parameters when analytic expressions are unavailable or difficult to compute. As a test case, we focus on early dark energy (EDE), a model designed to address the Hubble tension. Rather than using uniform priors on the $\textit{phenomenological}$ EDE parameters $f_{\rm EDE}$ and $z_c$, we train a NF on EDE cosmologies informed by theory expectations for axion masses and decay constants. Our method recovers known constraints in this representation while being $\sim 300,000$ times more efficient in terms of total CPU compute time. Applying our NF to $\textit{Planck}$ and BOSS data, we obtain the first theory-informed constraints on EDE, finding $f_{\rm EDE} \lesssim 0.02$ at $95\%$ confidence with an $H_0$ consistent with $\textit{Planck}$, but in $\sim 6\sigma$ tension with SH0ES. This yields the strongest constraints on EDE to date, additionally challenging its role in resolving the Hubble tension.