Extending the Dynamical Systems Toolkit: Coupled Fields in Multiscalar Dark Energy

TL;DR

This work develops a unified dynamical-systems approach to multifield dark-energy models comprising axion-saxion pairs with both kinetic and potential couplings, a setting motivated by string theory. By introducing a tailored, closed set of dimensionless variables, it derives a compact expression for the non-geodesicity parameter $oldsymbol{\boldsymbol{\omega}}$ at fixed points and uncovers genuine non-geodesic fixed points ${\mathcal{NG}}_{U\pm}$ in exponential-coupling scenarios, though these attract only on invariant submanifolds. The analysis clarifies that, in the shift-symmetric (flat) axion limit, previously reported non-geodesic fixed points are not physical when full dynamics are accounted for. The framework is further extended to power-law axion potentials and embedded into string-inspired supergravity with a concrete example, highlighting the potential for non-geodesic trajectories to drive late-time acceleration and informing multifield inflationary scenarios. Overall, the paper provides a robust toolkit and UV-motivated realizations for exploring multifield dark energy and early-universe dynamics in theories with coupled axion–saxion sectors.

Abstract

We study the dynamics of a two-field scalar model consisting of an axion-saxion pair with both kinetic and potential couplings, as motivated by string theory compactifications. We extend the dynamical systems (DS) toolkit by introducing a new set of variables that not only close the system and enable a systematic stability analysis, but also disentangle the role of the kinetic coupling. Within this framework we derive a compact, general expression for the non-geodesicity (turning-rate) parameter evaluated at fixed points, valid for arbitrary couplings. This provides a transparent way of diagnosing non-geodesic dynamics, with direct applications to both dark energy and multifield inflation. We first consider exponential coupling functions to establish analytic control and facilitate comparison with previous literature. In this case, we uncover a pair of genuinely non-geodesic fixed points, which act as attractors within a submanifold of the full system. In contrast, when the axion shift symmetry remains unbroken, our analysis shows that the apparent non-geodesic fixed point reported previously does not persist once the full dynamics are taken into account. Finally, we illustrate how our approach naturally extends to more realistic string-inspired models, such as power-law axion potentials combined with exponential saxion couplings, and present an explicit supergravity realisation.

Figures (1)

• Figure 1: Phase space diagram for the 3D subspace $x_1=y_2=y_k=0$, for the parameters $(\beta,\gamma,\lambda_1,\lambda_2)=(8, -5, 3, 3)$, for which the ${\mathcal{NG}}_{U\pm}$ points are attractors as described in the main text. The yellow area denotes the region of the phase space where the universe is accelerating. Left: The diagram shows a trajectory with initial conditions $(x_1,x_f,x_2,y_2,y_2,y_f)=(0,-2\times10^{-10},-6\times10^{-8}, 2.6\times 10^{-6},0,0)$, starting in kination domination $(\mathcal{K_{\pm}})$, followed by the scaling domination $(\mathcal{S_\gamma})$ and landing in the non-geodesic domination $\mathcal{NG}_{U-}$, in an accelerating universe. Right: The diagram depicts a trajectory with initial conditions $(x_1,x_f,x_2,y_2,y_2,y_f)=(0,2.5\times10^{-12},-0.2, 0.2,0,0)$, that starts at kination domination $(\mathcal{K_{-}})$, followed by fluid domination $(\mathcal{F})$, passing by a scaling epoch $\mathcal{S_\gamma}$ and ending at the non-geodesic $\mathcal{NG}_{U+}$ point, in the accelerating region.