Einstein-Scalar-Gauss-Bonnet Gravity

• Einstein-scalar-Gauss-Bonnet gravity is a higher-curvature extension of GR that couples a scalar field to the Gauss–Bonnet invariant, enabling dynamical corrections in high-curvature regimes.
• The theory yields de Sitter inflation, bouncing cosmologies, and singularity-free solutions, illustrating its potential to describe early-universe dynamics without relying on conventional potentials.
• It also predicts novel black hole features such as spontaneous scalarization and regular, non-singular solutions, with significant implications for gravitational collapse and compact star configurations.

Einstein-scalar-Gauss-Bonnet (EsGB) gravity is a higher-curvature extension of general relativity characterized by coupling a scalar field to the Gauss–Bonnet (GB) invariant, $$\mathcal{G} = R^2 - 4R_{\mu\nu}R^{\mu\nu} + R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$$. While the GB term is topological and does not affect dynamics in pure four-dimensional gravity, introducing a scalar-dependent coupling $f(\phi)\mathcal{G}$ renders its contribution dynamical. This framework is motivated by effective low-energy theories from string theory and Lovelock gravity. It has been investigated for its implications in cosmology, black hole physics, gravitational collapse, and the phenomenology of compact objects. EsGB gravity yields second-order field equations, avoids Ostrogradsky instabilities, and accommodates a broad spectrum of solutions with novel features, including singularity resolution, spontaneous scalarization, new black hole solutions, and nonstandard compact star configurations.

1. Action, Symmetries, and Field Equations

The canonical EsGB action in four dimensions takes the form

$$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2} R - \frac{1}{2} (\nabla\phi)^2 + f(\phi) \mathcal{G} - V(\phi) + \mathcal{L}_\text{matter} \right],$$

where $R$ is the Ricci scalar, $\phi$ is a real (or, in some studies, complex or multi-component) scalar field, $V(\phi)$ is a scalar potential, and $f(\phi)$ is a model-dependent coupling function. The theory accommodates extensions to higher dimensions, generalized scalar target manifolds, or compactified scenarios (Bruck et al., 2018, Cardenas et al., 2023).

String theory provides strong motivation for EsGB gravity: the Gauss–Bonnet term appears as the leading $\alpha'$ correction in heterotic string effective actions, often coupled exponentially or quadratically to the dilaton or moduli fields (Sberna, 2017, Kanti et al., 2015). Dimensional reduction of Lovelock or Chern–Simons gravity yields EsGB-type scalar–tensor actions, which, under specific choices of the coupling and field content, fall into the class of Horndeski theories, ensuring second-order field equations and absence of Ostrogradsky ghosts (Bruck et al., 2018, Cardenas et al., 2023, Fernandes, 2021).

The field equations are:

• Metric: the Einstein tensor is modified by curvature–scalar interactions,
• Scalar: the Klein–Gordon equation acquires a source term $f'(\phi) \mathcal{G}$, causing scalar field dynamics and backreaction in high-curvature regions.

2. Cosmological Solutions and Early Universe Implications

EsGB gravity has been extensively studied as a model for early-universe inflation, singularity avoidance, and dark energy-like acceleration (Kanti et al., 2015, Kanti et al., 2015, Sberna, 2017, Chakraborty et al., 2018, Bousder et al., 2023). A particular focus has been on scalar-GB couplings $f(\phi) = \lambda \phi^2$ (quadratic) and string-inspired exponential couplings. Key findings include:

• Inflation without potential: For $f(\phi)=\lambda\phi^2$ and $\lambda<0$, the pure scalar–GB sector admits exact de Sitter (inflationary) solutions. The field equations for the scale factor $a(t)$ and scalar field $\phi(t)$ yield $a(t) = a_0 e^{Ht}$, with $H^2 = -5/(24\lambda)$, and rapid exponential decay of $\phi$ (Kanti et al., 2015, Kanti et al., 2015).
• Graceful exit: Allowing nonzero integration constants, inflationary solutions naturally interpolate to a Milne (linearly expanding) phase without additional reheating mechanisms (Kanti et al., 2015).
• Singularity avoidance and bounces: For $\lambda>0$ or appropriate exponential couplings, the cosmological solution can be singularity-free, exhibiting an initial scale factor bounded away from zero and replacing the Big Bang with a bounce (Kanti et al., 2015, Sberna, 2017).
• Dominance of the GB term: In high-curvature regimes (early times), the GB correction dominates over the Ricci scalar ($|\dot{f} H| \gg 1$, $|\ddot{f}| \gg 1$), allowing the Einstein–Hilbert term to be neglected (Kanti et al., 2015, Chakraborty et al., 2018).
• Effective potential bounds: The effective scalar–GB interaction potential remains bounded during inflation, ameliorating large-field and fine-tuning issues (Kanti et al., 2015).
• Two-vacua structure: In maximally symmetric spacetimes, EsGB gravity may admit two branches of effective cosmological constants. One branch is associated with dark energy, the other is matter-induced and interpretable as a chameleonic dark matter sector (Bousder et al., 2023).

A summary table of crucial cosmological features:

Coupling $f(\phi)$	Parameter Sign/Type	Solution Class	Notable Features
$\lambda \phi^2$	$\lambda < 0$	de Sitter inflation	Exponential expansion, decay of $\phi$
$\lambda \phi^2$	$\lambda > 0$	Bounce, non-singular	Minimum $a(t)$, no Big Bang singularity
Exponential	string-motivated	Inflation, bounces	Both singular and non-singular possible

The stability of these cosmological solutions is nontrivial. The presence of the GB term can cause catastrophic tensor-mode instabilities (negative sound speed squared for tensor perturbations $c_T^2 < 0$) in pure GB models (Sberna, 2017, Chakraborty et al., 2018). Introducing a suitable potential $V(\phi)$ for the scalar field may cure this instability for generic slow-roll scenarios, restoring compatibility with observations (Sberna, 2017, Chakraborty et al., 2018). However, tensor perturbation constraints and the GW170817 bound on $c_T$ enforce severe restrictions on coupling strengths (Fier et al., 3 Mar 2025).

3. Gravitational Collapse and Black Hole Physics

The impact of the scalar–GB term on gravitational collapse, black hole solutions, and the cosmic censorship hypothesis has been a major focus (Taves et al., 2011, Banerjee et al., 2017, Chakrabarti, 2017, Doneva et al., 2020, Heydari-Fard et al., 2020, Nojiri et al., 2023). Key results:

• Hamiltonian formulation for collapse: The collapse of a massless scalar in EsGB gravity can be reduced to a two-dimensional Hamiltonian system via ADM decomposition and judicious gauge-fixing (e.g., Painlevé–Gullstrand–like coordinates). The gravitational sector simplifies to evolution in terms of a generalized Misner–Sharp mass function, and the scalar sector remains regular at horizon formation. This formulation is particularly crucial for numerical simulations (Taves et al., 2011).
• Singularity formation and horizon structure: In spherically symmetric scalar field collapse, EsGB gravity may produce spacetime singularities (divergent Kretschmann scalar), but generically such singularities remain hidden within an apparent horizon, preserving weak cosmic censorship (Banerjee et al., 2017). The interior solution often cannot be matched to Schwarzschild, indicating nontrivial modifications outside the horizon.
• Spontaneous scalarization: For quadratic or suitable $f(\phi)$, black holes may undergo spontaneous scalarization: vacuum GR solutions become unstable, and the end-state is a black hole with nontrivial scalar hair (Doneva et al., 2020, Heydari-Fard et al., 2020). This mechanism critically depends on the sign and second derivative of $f(\phi)$ at a constant scalar field background.
• No-go theorems: In the case of complex scalar fields, static regular black hole solutions with complex scalar hair are excluded for arbitrary scalar potentials and coupling functions by a local (near-horizon) argument: regularity and positive temperature require a trivial scalar profile at the horizon, regardless of the Gauss–Bonnet or cosmological constant contributions (Lin et al., 2020).
• Regular black holes and singularity resolution: By appropriate coupling to $f(\phi) \mathcal{G}$, regular black holes such as the Hayward solution (having no curvature singularity and two horizons) can be realized, evading the classical information loss paradox (Nojiri et al., 2023). Absence of ghosts in these models is guaranteed by specific choices of coupling and integration constants.
• Multi-scalar models: Extending to multi-component scalars on maximally symmetric target spaces (e.g., $\mathbb{S}^3$, $\mathbb{H}^3$, $\mathbb{R}^3$) leads to black holes with zero scalar charge, thereby strongly suppressing dipole scalar gravitational radiation and alleviating constraints from binary pulsar timing (Doneva et al., 2020).

4. Compact Stars and Exotics

EsGB gravity supports a heterogeneous spectrum of compact object solutions, many without analogues in GR (Q. et al., 18 Aug 2025):

• Branch structure: There exist both “regular” star solutions (with finite scalar field at the center, smoothly connected to GR), and “pure” scalar stars with a divergent scalar field at the center, yet with regular geometry and curvature invariants.
• Hybrid configurations: Compact stars consisting of both a regular scalar and fermionic matter (modeled via a polytropic equation of state) can reproduce neutron-star-like masses and radii.
• Phenomenology: Pure scalar stars can exhibit mass functions that briefly become negative near the center, and realize high compactness values exceeding those typical of GR fluid stars (even beyond the classical Buchdahl limit), but remaining below black hole compactness. Some configurations potentially act as “gravitational wave super-emitters,” emitting more gravitational radiation during mergers than binary black holes of the same mass.
• Test particle motion: Non-minimal scalar couplings for test particles result in novel effects, such as the existence of static circular orbits for zero angular momentum and orbits inside the GR innermost stable circular orbit (ISCO).

5. Mathematical Structure, UV Constraints, and Effective Theory

EsGB gravity occupies a well-defined slot in the landscape of effective field theories beyond GR (Herrero-Valea, 2021, Fernandes, 2021, Bruck et al., 2018):

• Horndeski class: The full scalar–GB interaction is a special case of the Horndeski scalar–tensor action. Dimensional reduction from higher-dimensional Lovelock or Chern–Simons gravity robustly yields EsGB gravity as a 4D effective field theory (Bruck et al., 2018, Cardenas et al., 2023), with all field equations remaining second-order.
• Generalized conformal and “geometric” extension: The theory can be further generalized by demanding only conformal invariance of the scalar field equations (rather than of the action), leading to purely geometric field equations whose solutions can be constructed in closed analytic form for both black holes and cosmologies. Such theories naturally incorporate EsGB gravity as a limiting case (Fernandes, 2021).
• UV completion and positivity bounds: Imposing requirements of analyticity, causality, unitarity, and Lorentz invariance in a potential UV completion, positivity bounds from graviton–scalar scattering restrict the allowed functional form of the scalar–GB coupling. For example, all higher even derivatives of $F(\phi)$ in $F(\phi)\,\mathcal{G}$ must be Planck-suppressed. This excludes many popular scalarization scenarios with cubic, quartic, or exponential couplings, leaving only simple quadratic functions as viable at astrophysical scales (Herrero-Valea, 2021).
• Constraints from GW speed and observed binary mergers: The stringent observational constraints set by the measured near-equality of the tensor GW propagation speed ($c_T$) and light limit the allowed present value of the coupling parameter ($$0 \leq \alpha\,\dot{f}(\phi_0) \lesssim 9 \times 10^{-24}$$ km) (Fier et al., 3 Mar 2025). This bounds the influence of EsGB corrections in current cosmological and astrophysical processes.

6. Gravitational Waves, Black Hole Binaries, and Astrophysical Observables

EsGB gravity can modify gravitational wave emission and propagation, with important implications for GW cosmology and binary dynamics (Fier et al., 3 Mar 2025, Julié et al., 2022, Heydari-Fard et al., 2020):

• Tensor and scalar modes: The gravitational spectrum comprises a standard spin-2 transverse-traceless mode (propagating along null geodesics with amplitude decay mirroring GR when $c_T = 1$) and an extra spin-0 scalar mode. The amplitude of the scalar GW may remain constant during radiation and matter domination, but exhibits distinct growth in the dark energy epoch under strict $c_T = 1$ (Fier et al., 3 Mar 2025).
• Compact binary dynamics and sensitivities: EsGB black holes possess “sensitivities”—parameters quantifying the mass response to a background scalar. In binaries, scalar fields sourced by companions can dynamically drive a black hole’s parameters outside the theory’s domain of existence, terminating its stationary solution branch before merger (Julié et al., 2022).
• Astrophysical constraints: GW emission, accretion disk spectra, and test particle dynamics enable constraints on the EsGB parameter space. For example, scalarized black holes predicted by spontaneous scalarization may host hotter, more luminous thin accretion disks with shifted ISCO radii and conversion efficiencies compared to Schwarzschild black holes (Heydari-Fard et al., 2020). Observational analysis methods (e.g., $\chi^2$ fitting) can in principle discriminate between GR and EsGB solutions using electromagnetic and GW signals.

7. Open Directions and Theoretical Significance

EsGB gravity provides a unifying framework for investigating several outstanding challenges in gravitational theory:

• Singularity regularization: The presence of a dynamical scalar–GB sector replaces classical singularities with regular cores or bounces under a range of couplings and initial conditions (Nojiri et al., 2023, Kanti et al., 2015).
• Cosmological sector unification: The two-vacua structure may offer a unified account of dark energy and dark matter, with the scalaron mass and effective gravitational constant dependent on local matter density via the chameleon mechanism (Bousder et al., 2023).
• Constraints on modified gravity: Tight constraints from GW observations, effective field theory considerations, and positivity bounds combine to heavily restrict viable scalar–GB couplings and suppress observable deviations from GR at non-Planckian scales (Herrero-Valea, 2021, Fier et al., 3 Mar 2025).
• Numerical relativity and strong-field dynamics: The ADM/Lagrangian/Hamiltonian reduction of EsGB gravity clarifies challenges in numerical evolution, including multivalued Hamiltonians, quasilinear evolution equations, and breakdowns in highly dynamical or strongly curved regimes (Taves et al., 2011, Julié et al., 2020).

Overall, EsGB gravity bridges effective field theory, string-inspired corrections, and phenomenological applications to strong gravity, while being subject to stringent theoretical and observational constraints. It continues to serve as a testbed for new physics in cosmology, black hole dynamics, and gravitational wave science.