Gauss-Bonnet Quintessence: Background Evolution, Large Scale Structure and Cosmological Constraints

TL;DR

This work investigates a string-inspired dark energy scenario in which a canonical scalar φ is non-minimally coupled to the Gauss–Bonnet invariant via f(φ)R^2_{GB}. With exponential forms for V(φ) and f(φ), the authors perform a background and linear perturbation analysis, revealing a late-time transition from a scaling matter era to accelerated expansion when α ≥ λ, potentially accompanied by a transient phantom phase without a Big Rip. They derive a scale-dependent but often to first order scale-invariant growth equation ̈δ + 2Hδ̇ = 4π G_* ρ δ, where G_* depends on the background evolution; perturbations may diverge in the future, signaling possible breakdowns of the linear regime. Confronting multiple datasets (SNe Ia, CMB shift parameter, BAO, nucleosynthesis, and Solar System bounds), they find that while SNe Ia and CMB can be accommodated, BAO and early-universe constraints pose strong tensions, and the necessity to fine-tune the Newtonian limit further constrains the model. Overall, Gauss–Bonnet quintessence offers an appealing, minimal modification to explore dark energy physics and potential connections to string theory, but its viability hinges on resolving perturbative instabilities and reconciling with small-scale and early-universe constraints.

Abstract

We investigate a string-inspired dark energy scenario featuring a scalar field with a coupling to the Gauss-Bonnet invariant. Such coupling can trigger the onset of late dark energy domination after a scaling matter era. The universe may then cross the phantom divide and perhaps also exit from the acceleration. We discuss extensively the cosmological and astrophysical implications of the coupled scalar field. Data from the Solar system, supernovae Ia, cosmic microwave background radiation, large scale structure and big bang nucleosynthesis is used to constrain the parameters of the model. A good Newtonian limit may require to fix the coupling. With all the data combined, there appears to be some tension with the nucleosynthesis bound, and the baryon oscillation scale seems to strongly disfavor the model. These possible problems might be overcome in more elaborate models. In addition, the validity of these constraints in the present context is not strictly established. Evolution of fluctuations in the scalar field and their impact to clustering of matter is studied in detail and more model-independently. Small scale limit is derived for the perturbations and their stability is addressed. A divergence is found and discussed. The general equations for scalar perturbations are also presented and solved numerically, confirming that the Gauss-Bonnet coupling can be compatible with the observed spectrum of cosmic microwave background radiation as well as the matter power spectrum inferred from large scale surveys.

Figures (10)

• Figure 1: A phase portrait of the model. The saddle point $E$ attracts the field to the scaling regime but later always passes it, along the same track, to the fixed point $C$ which corresponds to an accelerated expansion. The requirement for the fixed point $C$ to be reached and be stable is $\alpha \ge \lambda$. For the acceleration to begin only after a sufficiently long scaling matter era, the scale $f_0$ has to bet set suitably.
• Figure 2: The fractional energy densities for matter, $\Omega_m$ (dash-dotted line), the scalar field $\Omega_\phi$ (dashed line) and Gauss-Bonnet term, $\Omega_f$ (dotted line). The solid line is the total equation of state $w_{eff}$. The upper panel is for $\lambda=4$ and the lower panel for $\lambda=8$. For both cases $\Omega_m^0=0.35$ and $\alpha=20$. The transient phantom era in the upper plot is caused by the dynamics of the coupling.
• Figure 3: Evolution of effective gravitational constant $G_*$ (top panel) and of the dimensionless growth rate $d\log{\delta_m}/(d\log{a})$ (lower panel) as functions of redshift. We see that a divergence is possible for some parameter combinations. The thick lines are numerical solutions of the full linearized equations, and the thin dotted lines in the lower panel are solutions to the approximative equation (\ref{['d_evol']}). The agreement is excellent in most of the parameter space, though in the two extreme cases depicted here (dashed line corresponding to very slow transition, the dash-dotted line corresponding to instability) deviation is visible.
• Figure 4: The effect of matter density on the CMBR and matter power spectra. Here $\lambda=6$ and $\alpha=20$. Dotted lines are for $\Omega^0_m=0.3$, dashed line for $\Omega^0_m=0.4$), and dash-dotted for $\Omega^0_m=0.5$. The solid line is $\Lambda$CDM model.
• Figure 5: The effect of the potential slope on the CMBR and matter power spectra. Here $\Omega_m^0=0.4$ and $\alpha=20$. Dotted lines are for $\lambda=4.5$, dashed line for $\lambda=6.0$), and dash-dotted for $\lambda=8.0$. The solid line is $\Lambda$CDM model.