Exact solution of the Einstein-scalar-Gauss-Bonnet model with Noether symmetry constraints

By applying Noether symmetry methods, analytic solutions are obtained for a generalized Einstein-scalar-Gauss-Bonnet model with a $ξ(ϕ)f(G)$ component. Variation with respect to the metric, supplemented by small perturbations, produces the equations of motion and the terms that determine the propagation speed of tensor perturbations. The resulting Hubble parameter incorporates contributions from stiff matter and dark energy, the last originating from a scalar field non-minimally coupled to the Gauss-Bonnet invariant. The viability of the model is assessed by using Cosmic Chronometers, Baryon Acoustic Oscillations, and type Ia supernovae data. Best model selection based on information criteria indicates a slight preference for this new framework over the $Λ$ Cold Dark Matter model. Stability of the model follows from the positive speed of sound and absence of ``Ostrogradsky ghosts’’. The total equation of state parameter hints towards the presence of a transition from decelerated to accelerated expansion at $z\approx 0.66$, corresponding to the transition from matter to dark energy dominance. Early Universe dynamics, derived from the slow-roll parameters, spectral indices, and the tensor-to-scalar ratio, are found to be perfectly consistent with observations from Planck 2018 and the Atacama Cosmology Telescope.

💡 Research Summary

This paper presents a significant advance in modified gravity theories by deriving exact analytical solutions for a generalized Einstein-scalar-Gauss-Bonnet (EsGB) model. The model extends the standard EsGB action by generalizing the non-minimal coupling term to ξ(ϕ)f(G), aiming to capture richer phenomenology while avoiding theoretical pitfalls like Ostrogradsky ghosts and deviations in gravitational wave speed.

The core achievement lies in the application of Noether symmetry methods. Instead of resorting to the reconstruction approach—which assumes specific functional forms for ξ(ϕ) and f(G)—the authors employ Noether’s theorem to derive conservation laws directly from the Lagrangian constructed in a Friedmann-Robertson-Walker (FRW) spacetime. By imposing the symmetry condition on the Lagrangian, they obtain a system of equations for the symmetry generators and the unknown functions. Solving this system yields the key results: the scalar potential V(ϕ) is constant (V₀), and the non-minimal coupling combination takes the simple linear form ξ(ϕ)f(G) = - (φ₀/δ₀) G φ + φ₁, where φ₀, δ₀, and φ₁ are integration constants. This solution naturally contains the standard EsGB model (where ξ∝ϕ and f∝G) as a special case (φ₁=0).

A major theoretical concern for such models is the propagation speed of gravitational waves (c_GW). The paper meticulously derives the modified wave equation by perturbing the metric and identifies the terms that could cause c_GW to differ from the speed of light. It shows that in the homogeneous and isotropic FRW background, a specific condition can nullify the offending terms, suggesting compatibility with the tight constraints from the GW170817 event.

The derived solution is then used to obtain an expression for the Hubble parameter H(z), which incorporates contributions from matter, stiff matter (Ω_s), and a scalar-field-based dark energy component. The model’s viability is rigorously tested against a suite of modern cosmological observations: Cosmic Chronometers, Baryon Acoustic Oscillations (BAO), and Type Ia supernovae data. Bayesian analysis and model comparison using information criteria (AIC, BIC) indicate a slight statistical preference for this new EsGB framework over the standard ΛCDM model. The model predicts a transition from decelerated to accelerated expansion at redshift z ≈ 0.66, consistent with the matter-to-dark-energy dominance transition.

Furthermore, the paper demonstrates the model’s theoretical stability (positive speed of sound, no ghosts) and its consistency with early-Universe physics. Predictions for the slow-roll parameters, spectral indices, and the tensor-to-scalar ratio align well with measurements from Planck 2018 and the Atacama Cosmology Telescope. In summary, this work successfully leverages Noether symmetry as a powerful mathematical tool to construct an exact, observationally viable, and theoretically sound solution within a generalized modified gravity framework, offering a compelling alternative to the ΛCDM paradigm.