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

Olga Razina, Dauren Rakhatov, Pyotr Tsyba, Emilio Elizalde

TL;DR

The paper derives an analytic, ghost-free solution to a generalized Einstein-scalar-Gauss-Bonnet model with a non-minimal $\xi(\phi)f(G)$ coupling using Noether symmetry to fix functional forms and obtain a conserved quantity. The resulting Hubble parameter naturally includes a stiff-matter component and a scalar-field–driven dark energy, yielding a background evolution closely resembling $\Lambda$CDM while enabling early-Universe features that align with Planck 2018 and ACT data. Stability is demonstrated via a positive speed of sound and absence of Ostrogradsky ghosts, with a deceleration-to-acceleration transition around $z\approx0.66$ and no phantom behavior. A comprehensive Bayesian analysis with CC, BAO, Pantheon+, Planck, and ACT shows the EsGB model to be competitive with $\Lambda$CDM, offering modest improvements on some criteria and robust compatibility with observational data. The work highlights Noether-symmetry methods as a powerful tool for constructing consistent, higher-derivative cosmologies that reconcile early- and late-time cosmic evolution.

Abstract

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.

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

TL;DR

The paper derives an analytic, ghost-free solution to a generalized Einstein-scalar-Gauss-Bonnet model with a non-minimal coupling using Noether symmetry to fix functional forms and obtain a conserved quantity. The resulting Hubble parameter naturally includes a stiff-matter component and a scalar-field–driven dark energy, yielding a background evolution closely resembling CDM while enabling early-Universe features that align with Planck 2018 and ACT data. Stability is demonstrated via a positive speed of sound and absence of Ostrogradsky ghosts, with a deceleration-to-acceleration transition around and no phantom behavior. A comprehensive Bayesian analysis with CC, BAO, Pantheon+, Planck, and ACT shows the EsGB model to be competitive with CDM, offering modest improvements on some criteria and robust compatibility with observational data. The work highlights Noether-symmetry methods as a powerful tool for constructing consistent, higher-derivative cosmologies that reconcile early- and late-time cosmic evolution.

Abstract

By applying Noether symmetry methods, analytic solutions are obtained for a generalized Einstein-scalar-Gauss-Bonnet model with a 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 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 , 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.

Paper Structure

This paper contains 11 sections, 59 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Field Equations and the Propagation of Gravitational Waves
  3. Application of Noether Symmetry Methods
  4. Observational Constraints and Model Testing
  5. Late Universe Observational datasets
  6. Early Universe constraints
  7. Information Criteria
  8. Results and Discussion
  9. Parameter estimation and comparison
  10. Equation of State and Stability Analysis
  11. Conclusion

Figures (4)

  • Figure 1: Posterior parameter distributions for the generalized EsGB derived from the CC (blue), BAO (SDSS+DESI) (green), and Pantheon+ with SH0ES (red) datasets, showing 1D marginalized posteriors (diagonal) and 1$\sigma$/2$\sigma$ confidence contours (off-diagonal)
  • Figure 2: Constraints on the tensor-to-scalar ratio $r$ and scalar spectral index $n_s$ from Planck 2018 (green) and ACT DR6.02 (blue) data, compared with the predictions of the EsGB model (red). The shaded regions correspond to the $68\%$ and $95\%$ confidence levels
  • Figure 3: The equation of state parameter $\omega(z)$ (blue) and the deceleration parameter $q(z)$ (red) of the model. The corresponding dashed lines indicate the parameter values at the transition from decelerated to accelerated expansion
  • Figure 4: The square of the speed of sound $C_S^2(z)$ as a condition for the stability of the model (blue) in comparison with the results of observations Planck 2018 (red)