Multifield dark energy: Interplay between curved field space and curved spacetime

Abstract

Exponential quintessence models motivated by string compactifications naturally involve both a dilatonic scalar and its axionic partner evolving on a curved field space, while spatial curvature enlarges the cosmological phase space and may affect late-time dynamics. We perform a systematic analysis of the minimal two-field exponential system in a curved FLRW background including radiation and matter components, combining a complete dynamical systems classification with analytical approximations and numerical integration. In the scalar-dominated limit, non-geodesic trajectories can sustain accelerated expansion on steep potentials, and in curved universes a scaling-curvature fixed point can in principle soften the requirements for acceleration. However, we show that these mechanisms arise in distinct invariant manifolds and cannot be simultaneously realized in the presence of a background fluid: no non-geodesic scaling fixed point exists within an open region of parameter space. As a consequence, in the observationally viable thawing regime the axion does not track the background fluid and spatial curvature becomes dynamically subdominant, leading to an effectively single-field evolution. We further identify a degeneracy between curvature effects and scalar-field dynamics that limits their independent impact on late-time cosmology. Confronting the model with current cosmological background data (Planck 2018 distance priors, Pantheon+, BAO, and cosmic chronometers), we obtain an upper bound $λ\lesssim 0.75$ (95 percent CL) on the potential slope. Our results demonstrate that even in the minimal multifield setup with spatial curvature, sustained late-time acceleration requires a sufficiently flat potential, so that the tension between cosmic acceleration and quantum gravity expectations persists within this framework.

Figures (14)

• Figure 1: Attractor regions for the geodesic (${\cal G}$), non-geodesic (${\cal NG}$) and scaling (${\cal S}$) fixed points for the common realizations of the barotropic fluid: curvature (left), matter (middle), and radiation (right), respectively. These regions meet at the triple point.
• Figure 2: Fixed point configuration for $\alpha = 2/3, 1$ and $4/3$, corresponding to curvature (top), matter (middle), and radiation (bottom) fluids. Each case is evaluated for $(\lambda,\nu)=\{(\sqrt{2},\sqrt{2}),~(2,1),~(\sqrt{6},\sqrt{2/3})\}$. The green region corresponds to the acceleration phase. The interior of the shell corresponds to $\Omega_\alpha > 0$, whilst the shell itself is the $\Omega_\alpha = 0$ invariant manifold.
• Figure 3: Stability phase diagram for the multifield system in a curved universe ($\alpha = 2/3$). The shaded regions indicate the parameter space where the Geodesic ($\text{FP}_\mathcal{G}$, blue), Non-Geodesic ($\text{FP}_\mathcal{NG}$, red), and Scaling ($\text{FP}_\mathcal{S}$, green) points are stable. The discrete data points represent string theory realizations for varying integer $p \in [1, 12]$. The F-term scenario (black dots) drives the system towards the scaling solution, while D-term scenarios (dark yellow and orange dots) may fall into the geodesic and/or scaling regions. Notably, none of the standard string realizations naturally populate the non-geodesic region, suggesting that the turning mechanism is disfavored in these minimal setups.
• Figure 4: Parametric phase diagram for the single scalar field limit ($x_2=0$, or $\nu=0$). The geodesic point $\text{FP}_{\cal G}$ is the attractor in Region ($I$), while the scaling point $\text{FP}_\mathcal{S}$ dominates in Regions ($II$) and ($III$). The shaded areas denote the parameter combinations allowing for acceleration in the respective attractors (vertical shading for $\text{FP}_{\cal G}$ and horizontal shading for $\text{FP}_{\cal S}$). Dashed lines indicate the specific cuts for radiation, matter, and curvature fluids.
• Figure 5: 2D phase portraits for the single-field subsystem ($x_2=0$). From top to bottom, the rows correspond to backgrounds with increasing EoS: $\alpha=4/3$ (radiation, top), $\alpha=1$ (matter, middle), and $\alpha=2/3$ (curvature, bottom). The green area represents the acceleration region ($q<0$). Red dots indicate the scaling point $\text{FP}_\mathcal{S}$, while magenta dots denote the field-dominated point $\text{FP}_\mathcal{G}$. Note how the acceleration region shrinks and the scaling attractor moves out of it as $\alpha$ increases. In all cases shown for $\lambda = 1$ (left column), the scaling point lies in the $\Omega_\alpha<0$, not allowed for the matter and radiation cases. For $\alpha = 2/3$, the inner region of the circle represents a closed universe, while the outer region represents an open universe.