Scalar-Gauss-Bonnet Coupling Models

• Scalar-Gauss-Bonnet coupling is a nonminimal interaction between a scalar field and the Gauss-Bonnet invariant, introducing ghost-free higher-derivative curvature corrections in gravitational theories.
• It plays a pivotal role in diverse phenomena such as inflation, compact object scalarization, and braneworld stabilization, with the coupling function form (e.g., quadratic or exponential) critically affecting the dynamics.
• Theoretical and numerical analyses impose strict constraints on the coupling parameters to ensure stability and consistency with cosmological and astrophysical observations.

A scalar-Gauss-Bonnet (SGB) coupling refers to the interaction between a scalar field and the Gauss-Bonnet curvature invariant, typically through a nonminimal coupling function in the gravitational action. The fundamental property of such couplings is their ability to introduce higher-derivative curvature corrections to the gravitational field equations without producing ghost degrees of freedom in four-dimensional spacetime. SGB couplings can drive novel phenomena—such as spontaneous scalarization of compact objects, modifications to cosmological inflation, braneworld stabilization, modified black hole thermodynamics, and quantum particle production—while being highly constrained by theoretical consistency conditions, stability, and empirical data.

1. Action Formalism and Coupling Structure

In four-dimensional spacetime, the generic SGB action is written as

$$S = \int d^4x\,\sqrt{-g}\Big[\tfrac12M_P^2 R - \tfrac12 (\nabla\phi)^2 - V(\phi) + f(\phi)\mathcal{G}\Big],$$

where $R$ is the Ricci scalar, $\phi$ is a real scalar field, $V(\phi)$ its potential, and $$\mathcal{G} = R^2 - 4R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$ is the Gauss-Bonnet invariant. The coupling function $f(\phi)$ determines the nonminimal interaction. In $D>4$, $\mathcal{G}$ is dynamical even for constant $f(\phi)$, while in $D=4$ the SGB term is topological unless $f(\phi)$ is non-constant.

Common choices for $f(\phi)$ include polynomial (e.g. $f(\phi) = \alpha\phi^2$) (Peng, 2019), exponential (e.g. $f(\phi) = e^{\lambda\phi}$), or more general forms motivated by string theory and topological constructions (Herdeiro et al., 2021). The coupling parameter is often subject to normalization with the scalar mass or Planck mass, leading to dimensionless combinations such as $m^2\alpha$ (Peng, 2019) or $\beta = f''/M_P^2$ (Bhattacharjee et al., 2016).

2. Field Equations and Critical Phenomena

Variation yields a scalar field equation:

$\Box\phi - V'(\phi) + f'(\phi)\mathcal{G} = 0,$

and modified Einstein equations with additional $f(\phi)$-dependent curvature terms. The SGB coupling $f'(\phi)\mathcal{G}$ can act effectively as a position- or curvature-dependent mass, leading (for suitable sign and magnitude) to tachyonic instabilities that trigger spontaneous scalarization (Peng, 2019Herdeiro et al., 2021). Solutions demonstrating this effect commonly exhibit a lower bound on the coupling constant—for quadratic couplings, e.g.,

$\alpha > m^2 r_s^6 / (48 M^2),$

where $r_s$ is the compact object radius and $M$ its mass (Peng, 2019).

For generic $f(\phi)$, the scalar can develop nontrivial configurations only above such critical values. In compact star and black hole scenarios, these lead to discrete bands of "scalarized" objects existing only for specific couplings ("no-hair" below threshold) (Peng, 2019Herdeiro et al., 2021).

3. Cosmological Dynamics and Inflationary Models

SGB couplings influence early and late-universe cosmology in several ways:

• Inflation: SGB terms can modify slow-roll dynamics, alter the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$, suppress or enhance primordial gravitational waves, and even allow for blue-tilted tensor spectra when $f'(\phi)$ grows sufficiently fast (Koh et al., 2016Fomin, 2020).
• Power-law and de Sitter solutions: A quadratic coupling $f(\phi) = \lambda\phi^2$ gives rise to exact de Sitter inflation for $\lambda < 0$, interpolating inflation-Milne phases, and singularity-free bouncing cosmologies for $\lambda > 0$ (Kanti et al., 2015).
• Braneworld and higher dimensions: In $D=5$ or higher, SGB couplings control the emergence of warped AdS bulk geometries and stabilize thick branes; the sign and magnitude of $f(\phi)$ set bounds on the cosmological constant and the localization properties of the massless graviton (Silva et al., 29 Aug 2025Millano et al., 15 May 2024).
• Fractional SGB cosmology: Introduction of fractional calculus in the SGB action enables new classes of scaling and late-time attractor solutions compatible with current observation, mimicking dark energy behavior (Micolta-Riascos et al., 1 Oct 2024).

A table summarizing key cosmological SGB model classes:

Model Type	Coupling $f(\phi)$	Key Phenomenon
Quadratic	$\lambda\phi^2$	Exact inflation, no singularity
Exponential	$e^{\lambda\phi}$	String-motivated inflation, collapse instabilities
Cubic, Topological	$\phi(1-\phi^2/3)$	Scalarization from Chern-Simons
Fractional	General, with $\mu\neq1$	Power-law/attractor cosmology

4. Compact Object Scalarization and Black Hole Physics

SGB couplings are central to the phenomenon of spontaneous scalarization in compact stars and black holes:

• Discrete spectrum and criticality: Only for discrete couplings above a threshold can horizonless stars or black holes carry scalar hair; outside these "bands", objects remain "bald" (Peng, 2019).
• Backreaction and non-linearities: Numerical studies demonstrate that higher-order couplings regulate the nonlinear growth and quenching of scalar hair. Existence and stability of scalarized branches depend on the detailed form of $f(\phi)$ (Doneva et al., 2019).
• Thermodynamics: SGB-modified black holes exhibit shifts in the area, entropy (via Wald's formula $S_H = A_H/4 + 4\pi \lambda^2 f(\phi_H)$), and temperature; scalar field mass suppresses scalar hair and contracts the domain of hairy solutions (Doneva et al., 2019).
• Extension to vectorization: Certain topological SGB extensions induce non-minimal couplings for vector fields, leading to "vectorized" black holes via the same mechanism (Herdeiro et al., 2021).

5. Constraints, Numerical Implementation, and EFT Bounds

The parameter space of SGB models is tightly constrained by empirical data and theoretical consistency:

• Cosmological and astrophysical bounds: Planck data, reheating considerations, and unitarity restrict parameters to narrow bands (e.g., for quadratic coupling, $\beta m^2 / M_P^2 \sim 10^{-8}$, essentially excluded; linear coupling survives for $\beta \lesssim 10^3$ and $m\sim10^{-3} M_P$ (Bhattacharjee et al., 2016)).
• EFT and positivity bounds: Requiring a Lorentz-invariant, causal, local, unitary UV completion for the SGB action leads to strict constraints on the Taylor coefficients $f^{(n)}(\phi_0)$: all but quadratic couplings are suppressed below the Planck scale, implying that essentially all non-quadratic forms (those rich enough to stably quench scalarization) are incompatible with standard EFT expectations unless Planckian physics is invoked (Herrero-Valea, 2021).
• Numerical implementations: Nonperturbative 3+1 decompositions of the SGB field equations reveal significant complexity in constraint structure, hyperbolicity, and the presence of $\alpha$, $\alpha^2$-level terms coupling derivatives of curvature and scalar (Witek et al., 2020). The invertibility of the evolution system and preservation of hyperbolicity require careful monitoring for dynamical simulations.

6. Quantum and Holographic Effects

SGB coupling impacts quantum field theory in curved spacetime and holographic settings:

• Particle creation: In FRW cosmology, $f(\phi) = \zeta \phi^2$ produces a time-dependent effective mass in the Klein-Gordon equation; for $\zeta m^2 \ll 1$, effects on particle creation are subdominant and spectra remain controlled by standard Compton/horizon scales (Pavlov, 2014).
• Holographic chaos and pole skipping: In AdS/CFT setups, a generic SGB coupling alters the locations of nonhydrodynamic pole-skipping points and the shear diffusion constant in the boundary theory, with the Lyapunov exponent and butterfly velocity remaining unchanged at leading order (Baishya et al., 2023).

7. Summary and Physical Implications

SGB couplings furnish a robust, ghost-free mechanism for introducing nontrivial curvature-scalar field dynamics in both four and higher dimensions. Their predictive power in scalarization (including black holes, compact stars, and braneworlds), inflationary model-building, and quantum corrections is balanced by severe constraints arising from consistency, causality, and data. Quadratic couplings, in particular, are empirically and theoretically favored for their analytic tractability and EFT compatibility, while more complex forms are often excluded at sub-Planckian scales. SGB extensions thus remain a fertile yet tightly circumscribed sector of gravitational modeling, with ongoing work (theoretical, numerical, and observational) focused on refining the landscape of viable couplings and their physical consequences (Peng, 2019, Kanti et al., 2015, Herdeiro et al., 2021, Witek et al., 2020, Bhattacharjee et al., 2016, Doneva et al., 2019, Herrero-Valea, 2021, Silva et al., 29 Aug 2025, Millano et al., 15 May 2024, Koh et al., 2016).