Quadratic-Quartic Scalar Gauss-Bonnet

• QQSGB theory is a curvature-modified scalar-tensor model where a real scalar field with quadratic or quartic self-interaction couples nonminimally to the Gauss-Bonnet term to influence cosmic evolution and black hole physics.
• The framework enables analytic de Sitter inflation, bouncing cosmologies, and the emergence of scalarized black hole solutions through precise parameter choices in the coupling function and potential.
• Its rich structure, including anomaly-induced vertex identities and higher-dimensional extensions, offers insights into overcoming singularities and unifying early-universe and strong-field gravitational phenomena.

Quadratic-Quartic-Scalar-Gauss-Bonnet (QQSGB) theory encompasses a class of curvature-modified scalar-tensor gravitational models in which a real scalar field $\phi$ is nonminimally coupled to the Gauss-Bonnet (GB) term $R_{GB}^2$ through a coupling function $f(\phi)$, and the scalar field possesses a polynomial self-interaction potential—most commonly quadratic ($V(\phi) \propto \phi^2$) or quartic ($V(\phi) \propto \phi^4$). The GB term, a particular quadratic curvature invariant $$R_{GB}^2 = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} - 4 R_{\mu\nu} R^{\mu\nu} + R^2$$, is topological in $D=4$ but modifies the dynamics when coupled to $\phi$. Such theories are central in early-universe cosmology, black hole physics, and effective field theory approaches to quantum gravity, and relate closely to string-inspired and higher-dimensional gravity models.

1. Mathematical Structure and Coupling Functions

The generic action for QQSGB models is

$$S = \int d^4x \sqrt{-g} \left( \frac{1}{2} R -\frac{1}{2} (\nabla \phi)^2 - V(\phi) + \frac{1}{8} f(\phi) R_{GB}^2 \right)$$

where $V(\phi)$ is typically taken as a polynomial in $\phi$ (e.g., $V(\phi) = m^2\phi^2$ or $V(\phi) = \lambda \phi^4$) and $f(\phi)$ is typically a quadratic, quartic, or exponential function (e.g., $f(\phi) = \alpha \phi^2$, $f(\phi)=\beta\phi^4$, or $f(\phi) = \xi_0 e^{-\lambda\phi}$).

For $D=4$, $R_{GB}^2$ is a topological term and does not contribute to the equations of motion unless $f(\phi)$ is nonconstant. In $D>4$ the GB term is generically dynamical even for $f(\phi) = \rm const$, leading to additional modifications (see (Millano et al., 15 May 2024)).

The field equations contain higher-derivative terms but admit second-order equations for suitable $f(\phi)$ due to the Lovelock structure, avoiding Ostrogradsky ghosts (see e.g., (Sberna, 2017, Corianò et al., 2023)).

2. Early Universe Inflationary and Bouncing Solutions

In the strong-curvature regime of the early universe, the GB term can dominate over the Ricci scalar (Kanti et al., 2015, Kanti et al., 2015, Kanti, 2015, Sberna, 2017). For a quadratic coupling $f(\phi)=\lambda\phi^2$ and negative $\lambda$, the theory admits an exact de Sitter inflationary solution: $$a(t) = a_0 \exp\left(\sqrt{\frac{5}{24|\lambda|}} t\right), \quad \phi(t) = \phi_0 \exp\left(-\frac{5}{4}\sqrt{\frac{5}{6|\lambda|}}\, t\right)$$ with a graceful exit naturally provided by transition to a Milne universe (linear expansion), as determined by the integration constant $C_1$ in the solution. Only the quadratic case $f(\phi)\propto\phi^2$ admits such de Sitter and exit dynamics among monomial coupling choices (Kanti et al., 2015, Kanti et al., 2015).

For positive $\lambda$, the quadratic coupling yields singularity-free, expanding cosmologies with a minimum scale factor: $a^2\geq \tilde{\nu}^2$. Thus, the sign of $\lambda$ dictates whether the universe inflates or avoids the initial singularity (Kanti et al., 2015, Kanti, 2015). These properties extend when including quartic self-interaction ($V\propto \phi^4$), relevant for Higgs-like potentials (Pozdeeva et al., 2020, Mudrunka et al., 2 Apr 2025).

The Ricci scalar is dynamically negligible at early times, allowing clean analytic reductions, but recovers dominance at later times, enabling a natural transition to standard cosmology (Kanti et al., 2015).

3. Inflationary Observables, Constraints, and Deviations from Slow Roll

Coupling the inflaton to the GB term in quadratic or quartic potential models modifies inflationary predictions. For quadratic $V$, slow-roll is accurate even when the GB term is present, but for quartic $V$, the backreaction from the GB term can be strong, invalidating the slow-roll approximation and necessitating full dynamical evolution (Mudrunka et al., 2 Apr 2025).

The effective field equation for $\phi$ becomes

$$\ddot{\phi} + 3H\dot{\phi} + V'(\phi) + \frac{3}{2} f'(\phi) H^2 (\dot{H} + H^2) = 0$$

with additional friction from the $f'(\phi)$ term. For suitable $f(\phi)$, the extra term can flatten the potential, decreasing the tensor-to-scalar ratio $r$ and making otherwise ruled-out monomial models (e.g., Higgs inflation) compatible with Planck constraints (Bhattacharjee et al., 2016, Pozdeeva et al., 2020).

Constraints from CMB data ($n_s \sim 0.96$, $r < 0.07$), reheating dynamics, and tree-level unitarity restrict the allowed combination of coupling and potential parameters; quadratic GB couplings are more tightly constrained than linear ones (Bhattacharjee et al., 2016). In quartic models with strong GB coupling, the inflaton can become trapped in a local minimum of an effective potential induced by the GB term, leading to eternal inflation unless initial conditions are carefully chosen (Mudrunka et al., 2 Apr 2025).

4. Black Holes, Spontaneous Scalarization, and “Hairy” Solutions

QQSGB theory generically admits static, spherically symmetric black holes with nontrivial scalar hair provided that $f''(0)R_{GB}^2$ is sufficiently negative near the horizon to render the scalar field tachyonic, leading to spontaneous scalarization. In particular, for a quadratic coupling and a Schwarzschild background, the Klein-Gordon equation for the linear perturbation $\delta\phi$ becomes

$$(\Box + m_\mathrm{eff}^2(r)) \delta\phi = 0, \qquad m_\mathrm{eff}^2(r) = -96\alpha M^2/r^6$$

rendering the Schwarzschild solution unstable for $0 < r_+/\sqrt{\alpha} < 3.321$ (Myung et al., 2019). The fully nonlinear branch leads to black holes with scalar hair, which bifurcate from the GR solutions.

Recent work has constructed rotating and excited scalarized black holes in QQSGB with quadratic couplings. The bifurcation points for radially and angularly excited solutions follow a regular pattern in the $(n,l)$ plane, and the existence domain is constrained by the parameters of the model—rotation suppresses scalarization. Notably, in some parameter ranges, scalarized spinning black holes exhibit higher entropy than their Kerr counterparts of equal mass and spin, indicating thermodynamic preference (Liu et al., 17 Mar 2025, Collodel et al., 2019).

5. Nonperturbative and Nonlocal Structures: Anomaly Actions and Vertex Identities

A central theoretical advance is the recognition that in $D=4$, the GB density—while topological—induces nontrivial physics through its conformal (Weyl) anomaly structure when coupled to scalar fields (Corianò et al., 2022, Corianò et al., 2023). By extracting the conformal factor ($g_{\mu\nu} = \bar{g}_{\mu\nu}e^{2\phi}$) and implementing dimensional regularization with suitable subtractions, two effective formulations emerge:

• A local quartic-dilaton “Wess-Zumino” anomaly action,
• A nonlocal quadratic-dilaton action after integrating out $\phi$ (by inverting the conformal operator $\Delta_4$ and finite renormalization of the GB term with a $(d-4) R^2$ correction).

These formulations organize the three-point and four-point graviton interaction vertices into “pole” (longitudinal) and traceless parts, with the structure determined by the hierarchy of conformal Ward identities. For cubic and quartic interactions, the anomaly-induced pole terms (massless $1/p^2$ structures) are explicitly displayed in the vertex expansions, and at the quartic level the 4-graviton vertex splits into a pole plus totally traceless component. The identities and decomposition persist for all Lovelock densities in $d\neq4$, providing a unified link to higher-order topological invariants (Corianò et al., 2023).

6. Collapse, Compact Objects, and Regular Particle-like Solutions

Within QQSGB theory, the inclusion of quartic scalar self-interactions alters the collapse dynamics of scalar fields and the landscape of compact object solutions. For quadratic coupling and quartic potential $V(\phi)\propto\phi^4$, analytic techniques (e.g., invertible point transformation of the scalar field evolution equation) reveal two classes of fate for a collapsing scalar distribution (Chakrabarti, 2017):

• For certain initial conditions, evolution leads to a spacetime singularity with divergent scalar field and curvature invariants.
• For others (notably, different sign of the integration constant $a_0$), the system exhibits a “bounce” at finite radius and the scalar field disperses, generalizing the possible outcomes beyond canonical scenarios.

Similarly, QQSGB models admit both regular black holes (with nontrivial scalar “hair”) and regular particle-like solutions, depending on the scalar boundary behavior and the coupling function. Notably, regular traversable wormholes and particle-like configurations can emerge without introducing exotic matter, with scalarization and nontrivial energy conditions violated through the GB term (Kanti, 28 Dec 2024).

7. Generalizations: Higher-Dimensional Analogues and Effective Theories

In $D>4$, as in $D=5$, the GB term is not purely topological. With scalar couplings (even $f(\phi)=\text{const.}$), the GB term becomes dynamical and modifies cosmological evolution (Millano et al., 15 May 2024). Dynamical systems analyses employing dimensionless variables and phase space methods reveal the presence of “scaling solutions” and “super-collapsing” attractors, where the latter (characterized by $q=3$) can be avoided as late-time attractors by choosing the potential’s steepness parameter ($\lambda$) within a certain range. The existence of stable scaling or de Sitter fixed points suggests the higher-dimensional QQSGB models can naturally explain early-time inflation and late-time acceleration.

Dimensional reduction of higher-dimensional Lovelock actions, specifically in the GB combination, uniquely yields four-dimensional Horndeski-type scalar-tensor theories with second-order equations—providing a controlled embedding of QQSGB theory in a phenomenologically viable effective field theory framework (Bruck et al., 2018).

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In summary, quadratic–quartic–scalar–Gauss–Bonnet theory encompasses a rich spectrum of gravitational physics: unique inflationary and singularity-free cosmologies, spontaneous black hole scalarization and excited hairy black hole branches, regular wormhole and particle-like compact object solutions, nontrivial anomaly-induced vertex structures governed by conformal Ward identities, and extensions to higher dimensions and effective theory frameworks. The quadratic monomial coupling is singled out as privileged for supporting both viable inflation and regular solutions, while quartic scalar potentials and higher-order GB corrections further enrich the phenomenology and constraint structure. The interplay between coupling form, scalar self-interaction, and higher-curvature effects continues to inform advances in quantum gravity, early universe cosmology, and strong-field gravitational phenomenology.