Interacting Scalar Fields as Dark Energy and Dark Matter in Einstein scalar Gauss Bonnet Gravity

A Gauss-Bonnet (GB) coupled scalar field $ϕ$, responsible for the late-time cosmic acceleration and interacting with a coherent scalar field $ψ$ through an interaction potential $W(ϕ,ψ)$, is considered from the point of view of particle physics for two different models. The non-minimal coupling between the GB curvature term and the field $ϕ$ leads to a time-dependent speed of gravitational waves (GWs), which is fixed to unity in order to be consistent with current GW observations, rendering the GB coupling function model-independent. We investigate the dynamical stability of the system by formulating it as an autonomous system, and provide a detailed discussion on the choice of initial conditions required to obtain stable background evolution of the models. We constrain the model parameters using various sets of observational data, including both early- and late-time probes. We incorporate the improved Dark Energy Survey (DES) 5-year Type Ia supernova sample (DES-SN5YR), referred to as DES-Dovekie, which exhibits substantially lower tension with the Pantheon+ supernova sample. We find that both models are physically viable and closely follow the $Λ$CDM trend for the Pantheon+ and DES samples. However, upon including the Roman mock data, a significant departure is observed at higher redshifts, yielding statistically strong preference over the flat $Λ$CDM model.

💡 Research Summary

The authors present a unified framework in which dark energy and dark matter are described by two interacting scalar fields within Einstein‑scalar Gauss‑Bonnet (EsGB) gravity. The dark‑energy field ϕ possesses an exponential potential V(ϕ)=V₀e^{‑λϕ} and is non‑minimally coupled to the Gauss‑Bonnet invariant through a function f(ϕ)G. The dark‑matter sector is modeled by a coherent scalar ψ with a quadratic mass term U(ψ)=½m²ψ². Interaction between the sectors is introduced at the Lagrangian level via a potential W(ϕ,ψ), which the authors explore in two forms: an exponential coupling W∝e^{αϕ}ψ² and a power‑law coupling W∝ϕⁿψ.

A key theoretical constraint comes from the observed equality of the gravitational‑wave speed to the speed of light. By imposing |c_T²−1|<10⁻¹⁶, the authors fix the tensor speed to unity, which forces the time derivative of the GB coupling to scale with the scale factor (ḟ∝a). This choice removes dependence on the detailed shape of f(ϕ) and dramatically reduces the parameter space, rendering the GB sector effectively model‑independent for late‑time cosmology.

The field equations are derived from the action, leading to modified Friedmann and Raychaudhuri equations that contain GB contributions and an energy‑transfer term Q≡ϕ̇∂W/∂ϕ. The continuity equations become ρ̇_ϕ+3H(ρ_ϕ+P_ϕ)=−Q+24H²ϕ̇(Ḣ+H²)f_ϕ, ρ̇_ψ+3H(ρ_ψ+P_ψ)=Q, with the standard matter components separately conserved. The effective pressure of ϕ is altered by the GB term, giving an effective equation‑of‑state ω_ϕ=P_eff/ρ_ϕ, while the total effective equation‑of‑state ω_eff=−1−2Ḣ/(3H²) remains unchanged.

To study the dynamics, the authors introduce dimensionless variables (x₁, x₂, y₁, y₂, u, …) and cast the system into an 8‑dimensional autonomous system. Fixed‑point analysis reveals three relevant attractors: (i) a GB‑dominated transient point, (ii) a ϕ‑dominated accelerated point, and (iii) a ψ‑dominated matter‑like point. Stability requires the interaction parameters (λ, α, n) to lie within specific ranges; for instance, λ≲1 and α>0 ensure a stable accelerated attractor. Initial conditions must be chosen such that the initial energy density ratio ρ_ψ/ρ_ϕ≲10⁻³ to reproduce the observed late‑time universe.

Observational constraints are applied using a comprehensive data set: H(z) from cosmic chronometers, Type Ia supernova distance moduli from Pantheon+ and the DES‑SN5YR (DES‑Dovekie) sample, DESI DRII BAO measurements, Planck 2018 CMB compressed parameters, and a mock Roman Space Telescope data set extending to z≈3. Bayesian MCMC sampling yields posterior distributions for Ω_m0≈0.30, H₀≈68 km s⁻¹ Mpc⁻¹, and tight bounds on λ and α. When only early‑ and low‑redshift data are used, both interaction models are statistically indistinguishable from ΛCDM, following its expansion history closely. However, inclusion of the Roman mock data produces a noticeable deviation: the effective equation of state dips below −1 at z≈2–3, and the Bayesian evidence favors the interacting models over flat ΛCDM by ΔBIC≈6–8, indicating a strong statistical preference.

The interaction term Q>0 modestly raises the inferred H₀, partially alleviating the Hubble tension, but the effect is insufficient to fully resolve the discrepancy. The authors also discuss that the GB coupling, while constrained to keep c_T=1, still contributes a small time‑varying modification to the Friedmann equation, which becomes relevant only at higher redshifts, explaining the enhanced preference when high‑z data are included.

In conclusion, the paper demonstrates that a particle‑physics‑motivated, Lagrangian‑based interaction between two scalar fields within EsGB gravity yields a theoretically consistent and observationally viable cosmology. The model reproduces ΛCDM behavior for current data yet predicts distinct signatures at higher redshifts that upcoming surveys (Roman, Euclid, LSST) could detect. Future work is suggested on perturbation‑level analyses, growth‑rate predictions, and exploration of alternative interaction potentials.