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

Mudit Jain, Elijah Sheridan, David J. E. Marsh, Elli Heyes, Keir K. Rogers, Andreas Schachner

TL;DR

The paper develops a rigorous Bayesian framework to probe Calabi–Yau moduli spaces using the Weil–Petersson prior, enabling efficient sampling of Kähler moduli up to $h^{1,1}=30$ via MCMC and normalising flows. It builds theory-informed priors for axion parameters $(m_a,f_a)$ marginalized over the WP measure on Kreuzer–Skarke CYs, and integrates axion and cosmological likelihoods to constrain CY topology, divisor spectra, and moduli regions. Key advances include scalable WP-space sampling, geometry-driven statistics of divisor volumes, and demonstrations that a QCD-like axion detection can localize the moduli and topology, while Planck + Ly$\alpha$ data can pinpoint moduli-space regions favored by ultralight axion dark matter. The framework provides a concrete blueprint for statistical model testing in string phenomenology with current and forthcoming observational data, enabling future Bayesian evidence-based comparisons of CY-based models against EFTs and $\Lambda$CDM."

Abstract

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 $h^{1,1}\leq 5$. 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 $\mathcal{O}(1\%)$ of the dark matter. This demonstration serves as a blueprint for future statistical analyses within string phenomenology.

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

TL;DR

The paper develops a rigorous Bayesian framework to probe Calabi–Yau moduli spaces using the Weil–Petersson prior, enabling efficient sampling of Kähler moduli up to via MCMC and normalising flows. It builds theory-informed priors for axion parameters marginalized over the WP measure on Kreuzer–Skarke CYs, and integrates axion and cosmological likelihoods to constrain CY topology, divisor spectra, and moduli regions. Key advances include scalable WP-space sampling, geometry-driven statistics of divisor volumes, and demonstrations that a QCD-like axion detection can localize the moduli and topology, while Planck + Ly data can pinpoint moduli-space regions favored by ultralight axion dark matter. The framework provides a concrete blueprint for statistical model testing in string phenomenology with current and forthcoming observational data, enabling future Bayesian evidence-based comparisons of CY-based models against EFTs and CDM."

Abstract

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 , 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 marginalised over the WP measure for all inequivalent CYs constructable from the Kreuzer--Skarke database with . We then impose likelihoods based on axion physics. We demonstrate how detection of a relatively heavy QCD axion at small , 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 of the dark matter. This demonstration serves as a blueprint for future statistical analyses within string phenomenology.

Paper Structure

This paper contains 31 sections, 70 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Kähler moduli spaces of Calabi--Yau threefolds
  3. Kähler moduli and the Kähler metric
  4. Parametrising the Kähler moduli space
  5. Topological and Geometric Prerequisites
  6. Sampling Moduli Space
  7. Classical Markov Chain Monte Carlo
  8. Implementation and convergence tests
  9. Comparison of sampling performance
  10. Normalising flows
  11. Geometric results
  12. Application to axions
  13. Theory informed priors on $(m_a,f_a)$
  14. Inferring topology with haloscopes
  15. Blow-up divisors
  16. ...and 16 more sections

Figures (19)

  • Figure 1: Joint posterior distribution of $\log_{10}\mathcal{V}$ and $\log_{10}(\tau_{\rm fuzzy})$ obtained by sampling the Kähler moduli of a Calabi--Yau with $h^{1,1}=7$, where $\tau_{\rm fuzzy}$ is the volume of the prime toric divisor hosting the instanton associated with the ultralight (fuzzy) axion. We show three cases: (i) the Weil--Petersson prior alone (with IR/UV cutoffs from EFT control and the KK scale), (ii) the Weil--Petersson prior supplemented by the requirement that at least one prime toric divisor attains a volume suitable for a QCD-like gauge coupling, and (iii) the same prior further weighted by CMB and Lyman--$\alpha$ likelihoods. The inset zooms into the small region where the cosmology-selected posterior concentrates. The right panel shows the corresponding marginal posterior of $\log_{10}\mathcal{V}$ for all three cases, illustrating how cosmological data lead to a sharply peaked posterior and a strongly localised region in moduli space.
  • Figure 2: Convergence times for various values of $h^{1,1}$ and different numbers of generators for hyperplane sampling using emcee. The full likelihood consists solely of the Weil--Petersson prior, together with an upper bound on the Calabi--Yau volume fixed at $10^{10}\times\mathcal{V}_{\mathrm{tip}}$. We declare convergence once the total number of MCMC steps performed for the slowest converging parameter, exceeds $75$ times its autocorrelation length.
  • Figure 3: Marginal one-dimensional and two-dimensional distributions of the modified positive semi-definite Kähler parameters, $\bar{t}^i \in \mathbb{R}_+$ basis as defined in \ref{['eq:tbarbasis']}, for a cone with $N_g = 28$ extremal generators, at $h^{1,1} = 9$ obtained using hyperplane sampling of the Weil--Petersson prior.
  • Figure 4: Converged histograms of CY volume for example at $h^{1,1} = 9$ with a Kähler cone with $N_g = 28$ extremal generators. The likelihood is simply just the WP prior, with maximum CY volume set at $\mathcal{V}_{\rm max} = 10^{10} \times \mathcal{V}_o$ where $\mathcal{V}_o$ is the volume of the CY at the TSKC.
  • Figure 5: Distributions of the Kähler parameters inside the DSKC and the associated distribution of Calabi--Yau volumes, for a cone with $N_{g}=4$, at $h^{1,1}=3$ obtained using MCMC (red) sampling with WP prior, and the trained NF model (cyan).
  • ...and 14 more figures