Bayesian inference on Calabi--Yau moduli spaces and the axiverse: experimental data meets string theory

We develop tools of Bayesian inference on the moduli space of Calabi–Yau (CY) manifolds. We sample from the invariant Weil–Petersson (WP) measure using Markov Chain Monte Carlo and normalising flows on \Kahler moduli space with dimension up to $h^{1,1}=30$, and present results on the spectrum of the CY volume and properties of divisors when the measure is restricted in physically meaningful ways. We furthermore present a theory-informed prior on axion masses and decay constants $(m_a,f_a)$ marginalised over the WP measure for all inequivalent CYs constructable from the Kreuzer–Skarke database with $h^{1,1}\le 5$. We then impose likelihoods based on axion physics. We demonstrate how detection of a relatively heavy QCD axion at small $h^{1,1}$, e.g. by ADMX, provides detailed information about CY geometry and topology. Finally, we compute a full forward model incorporating likelihoods from the cosmic microwave background and Lyman-alpha forest and find the maximum posterior probability region on the moduli space of a given CY favoured by a resolution of the tension in these data by an ultralight axion composing $\mathcal{O}(1\mathrm{\%})$ of the dark matter. This demonstration serves as a blueprint for future statistical analyses within string phenomenology.