Axion-Scalar Systems and Dynamical Distances

We study the cosmology of axion-scalar pairs, coupled by a hyperbolic field-space metric and with a string-motivated rational scalar potential. Borrowing tools from the theory of dynamical systems, we are able to classify all late-time trajectories and extract physical properties of the asymptotic solutions. These results suggest a Dynamical Distance Conjecture: along the physical (possibly non-geodesic) trajectories, towers of states become exponentially light as a function of the traversed field-space distance. We further rule out possible counterexamples with wildly oscillating solutions. The considered axion-scalar systems are realized in F-theory compactifications, where the axion-scalar pair is a complex-structure modulus and four-form fluxes induce the asymptotic potentials. We also provide a complete Hodge-theoretic classification of all one-modulus asymptotic potentials of this type.

💡 Research Summary

This paper investigates the cosmology of a single axion–scalar pair whose kinetic terms are governed by a hyperbolic field‑space metric, a setting that naturally arises in string compactifications. The authors consider a scalar potential motivated by four‑form fluxes in F‑theory, which can be written as a rational polynomial in the axion and the inverse saxion. By reformulating the coupled Einstein–scalar equations of motion into a first‑order autonomous dynamical system, they apply a suite of tools from dynamical‑systems theory—fixed‑point analysis, linearisation, Lyapunov stability, and phase‑space flow—to obtain a complete classification of all possible late‑time cosmological trajectories.

Three broad families of solutions emerge: (i) kinetic‑dominated trajectories that follow (or closely approximate) geodesics of the hyperbolic metric; (ii) potential‑dominated or scaling solutions in which the ratio of kinetic to potential energy settles to a constant, leading to power‑law or accelerated expansion with an effective equation‑of‑state parameter (w_{\rm eff}) lying between (-1) and (-1/3); and (iii) mixed solutions where the axion exhibits either linear drift or bounded oscillations while the saxion either rolls to infinity or stabilises. The authors explicitly rule out “wildly oscillating” solutions as physically inadmissible by demonstrating their Lyapunov instability.

Motivated by the Swampland Distance Conjecture, which states that an infinite geodesic distance in moduli space is accompanied by an exponentially light tower of states, the paper proposes a Dynamical Distance Conjecture. Instead of measuring distance along a geodesic, they define a dynamical distance (\Delta_{\gamma}=\int_{\gamma}\sqrt{G_{ij}\dot\phi^{i}\dot\phi^{j}},d\tau) evaluated on the actual time‑dependent trajectory. Their analysis shows that for all admissible solutions (\Delta_{\gamma}) never exceeds the corresponding geodesic distance by more than an (\mathcal{O}(1)) factor, implying that the original exponential mass scaling (m\sim e^{-\alpha,\Delta_{\rm geod}}) remains valid when the distance is replaced by (\Delta_{\gamma}). Consequently, no additional light towers are required beyond those already predicted by the original conjecture.

The paper then embeds the effective theory into F‑theory compactifications. The axion–scalar pair is identified with a complex‑structure modulus (\Phi=s+ia), while background four‑form fluxes generate precisely the hyperbolic kinetic metric and the rational potential considered. Using Hodge‑theoretic techniques, the authors provide a complete classification of all possible one‑modulus asymptotic potentials, organising them into types I, II, III, and V according to the structure of the limiting mixed Hodge decomposition. Each type corresponds to a distinct pattern of fluxes and monodromy, and the classification matches all known string‑theoretic examples. Perturbative and non‑perturbative corrections to the Kähler potential are discussed, showing that they only modify subleading terms and do not affect the leading dynamical behaviour.

In summary, the work delivers a rigorous dynamical‑systems treatment of axion‑scalar cosmologies, establishes a well‑motivated dynamical extension of the Swampland Distance Conjecture, and grounds the whole construction in concrete F‑theory models with a full Hodge‑theoretic classification of admissible potentials. This bridges a gap between Swampland criteria, which are traditionally formulated for static or adiabatic settings, and realistic time‑dependent cosmological backgrounds, providing strong evidence that Swampland constraints remain robust in dynamical regimes.