Beyond Horndeski Scalar-Tensor Gravity

• Beyond Horndeski action is a generalization of scalar-tensor theories that extends Horndeski gravity by incorporating higher-derivative terms while avoiding ghost instabilities.
• Its formulation leverages degenerate operators and disformal invariance to maintain a healthy propagating degree of freedom despite third-order field equations.
• The framework finds practical applications in modeling non-singular cosmologies, scalarised compact objects, and traversable wormholes with distinctive observational signatures.

Beyond Horndeski Action is a generalization of the original Horndeski scalar–tensor gravity, introducing higher-derivative operators while maintaining a healthy (ghost-free) propagating degree-of-freedom content. It extends the framework of modified gravity beyond the restriction of strictly second-order field equations, allowing for a wider class of interactions and physical phenomena in cosmology and strong gravity regimes. The theoretical construction is motivated by higher-dimensional reductions, symmetry principles, and phenomenological requirements such as self-tuning and stability, and finds applications ranging from cosmological bounces and singularity resolution to the construction of scalarised black holes and wormholes.

1. Theoretical Construction and Structure

The original Horndeski action is the most general four-dimensional scalar–tensor theory yielding second-order equations of motion, constructed from functions $G_2$, $G_3$, $G_4$, and $G_5$ of a scalar $\phi$ and its kinetic term $X = -\frac{1}{2}\nabla_\mu\phi\nabla^\mu\phi$ (Kobayashi, 2019). The beyond Horndeski extension introduces additional terms (commonly denoted $F_4$ and $F_5$) that allow for third-order field equations off unitary gauge but are degenerate, ensuring the absence of Ostrogradsky ghosts and preserving the number of dynamical degrees of freedom (Kobayashi et al., 2014, Crisostomi et al., 2016).

An explicit form of the beyond Horndeski Lagrangian density is: $$\mathcal{L}_{bH} = G_4(\phi, X) R + F_4(\phi, X)\,\epsilon^{\mu\nu\rho}_{\ \ \ \ \sigma} \epsilon^{\mu'\nu'\rho'\sigma} \partial_\mu\phi \partial_{\mu'}\phi \nabla_\nu \nabla_{\nu'}\phi \nabla_\rho \nabla_{\rho'}\phi + \cdots$$ Here, the $F_4$-type terms are not captured by Horndeski's construction and appear naturally via Kaluza–Klein reductions from higher-dimensional Lovelock or Gauss–Bonnet gravity (Charmousis, 2014, Jana et al., 2020).

Theories in this class can also be reformulated using the extrinsic curvature of the constant–$\phi$ hypersurfaces, leading to compact expressions and clarifying the constraint structure crucial for ghost-freedom (Crisostomi et al., 2016).

2. Degrees of Freedom, Constraints, and Disformal Invariance

Beyond Horndeski maintains three dynamical degrees of freedom (two tensor, one scalar), even though its equations are, generically, of third order. The critical mechanism is a hidden primary constraint in the Hamiltonian formalism, constructed from antisymmetrized combinations of the metric and scalar momenta (Crisostomi et al., 2016): $$A_{*}(2 \hat{A}^2 - A_{*}^2) \pi_{*} - \hat{A}^\mu \hat{A}_\nu \pi^\nu_\mu \approx 0$$ This constraint removes the would-be Ostrogradsky ghost, allowing for healthy propagation even with higher-derivative couplings.

Conversely, the theory is structurally “isolated” in the space of theories. Combinations of beyond Horndeski terms of different derivative order, or with Horndeski terms of different order, typically spoil degeneracy and reintroduce ghosts. Mixing terms of the same order (e.g., quartic beyond Horndeski with quartic Horndeski) allows the theory to be mapped—via a generalized disformal transformation

$$\bar{g}_{\mu\nu} = g_{\mu\nu} + \Gamma(\phi,X) \nabla_\mu\phi\nabla_\nu\phi$$

—to a Horndeski theory, but for pure beyond Horndeski this mapping becomes singular and the theory is disformally invariant and isolated (Crisostomi et al., 2016, Mironov et al., 2022).

Crucially, this disformal mapping is singular precisely when constructing non-singular, stable cosmological solutions, explaining why such backgrounds exist in beyond Horndeski but are forbidden in Horndeski due to a no-go theorem (Mironov et al., 2022).

3. Covariant Structure, Effective Field Theory, and Quantum Consistency

Beyond Horndeski terms are best understood within the effective field theory framework of higher-derivative scalar–tensor gravity, supported by symmetry and power-counting arguments (Santoni et al., 2018). Theories with “weakly broken Galileon invariance” enforce a natural hierarchy

$$\mathcal{L} \sim (∇\phi)^{2n}/\Lambda_2^{4(n-1)} (∇∇\phi)^m/\Lambda_3^{3m}$$

and restrict the allowed contraction structures (through coefficients $\alpha_i$) such that quantum corrections do not generate unsuppressed dangerous operators. The quartic beyond Horndeski sector [containing $F_4$] arises as the unique combination ensuring perturbative unitarity and naturalness in this EFT.

Quantum effects universally generate certain beyond Horndeski interactions at one-loop, such as

$$-4\kappa^4 (\nabla_\mu\phi)(\nabla_\nu\phi) T^{\mu\nu} + 2\kappa^4 \phi^2 T$$

with model-independent, UV-finite coefficients (Latosh, 2021). These radiatively-induced terms are suppressed by $G_N^2$, but their existence emphasizes that even classically “Horndeski” theories are inevitably “contaminated” by quantum-generated beyond Horndeski operators.

4. Phenomenological Signatures in Cosmology and Astrophysics

The effective field theory of cosmological perturbations in beyond Horndeski can be parametrized by five functions, including $\alpha_H$, exclusively nonzero for beyond Horndeski (Traykova et al., 2019). This parameter controls the mixing between scalar and matter degrees of freedom and modifies the growth of structures, CMB lensing, and ISW effect. Specifically:

• $\alpha_H$ induces damping of the matter power spectrum on all scales,
• enhances large-angle CMB temperature power via the integrated Sachs–Wolfe effect,
• decreases the lensing potential (Weyl) and associated CMB lensing strength,
• and is tightly constrained by CMB, BAO, RSD, and GW170817 (luminal gravity wave propagation implies $G_{4,X} = G_5 = 0$ for the viable sector) (Kobayashi, 2019, Traykova et al., 2019).

At the nonlinear level, the matter bispectrum kernel is time-dependent—a feature impossible in Horndeski—and particularly alters folded triangle configurations, providing a unique observational signature (Hirano et al., 2018).

In the Vainshtein mechanism, screening of extra scalar forces persists outside and near massive bodies, but inside the source the gravitational potentials $\Phi \neq \Psi$ and the effective gravitational coupling depends on local density gradients, breaking the standard screening and modifying astrophysical structure (Kobayashi et al., 2014).

5. Black Holes, Stars, and Exotic Compact Objects

Scalarised black holes arise naturally in beyond Horndeski, either via time-dependent scalars (with shift symmetry) or through direct couplings to the Gauss–Bonnet invariant or energy-momentum tensor. Analytic and numeric studies show that scalar “hair” modifies both the background metric and the effective potential for perturbations, with observable consequences:

• Quasinormal mode (QNM) frequencies acquire lower real parts and longer damping times,
• Greybody (GB) factors and absorption cross-sections are suppressed or display non-monotonic features,
• Effective potentials develop multiple extrema as the hair parameter increases (Antoniou et al., 23 Jul 2025, Babichev et al., 2016, Babichev et al., 2017).

In neutron stars, the mass–radius relation is affected, with scalar–tensor stars being heavier or lighter than their GR counterparts depending on model parameters, and screening is incomplete inside matter (Babichev et al., 2016).

Traversable wormholes—static, spherically symmetric and free of ghosts in parity-even perturbations—can be constructed using beyond Horndeski terms ($F_4$), circumventing the no-go theorem that forbids stable wormholes in Horndeski gravity (Mironov et al., 2018, Mironov et al., 2018). However, these solutions require fine-tuning and may still suffer from gradient or tachyonic instabilities.

6. Non-Singular and Complete Cosmologies

Beyond Horndeski provides the only currently known local scalar–tensor framework allowing completely stable, non-singular cosmological bounces and Genesis scenarios in four dimensions (Kolevatov et al., 2017, Mironov et al., 2022). The essential mechanism is the additional “beyond Horndeski” contribution to stability conditions, permitting kinetic coefficients (such as $\mathcal{G}_T + D$) to cross zero without encountering ghost or gradient instability, a transition forbidden by a no-go theorem in Horndeski.

Explicit model constructions exhibit smooth transitions between early-time NEC violation (driven by full-fledged beyond Horndeski dynamics) and late-time conventional scalar field cosmology, ensuring stability throughout. The same structural features enable geodesically complete cosmologies in otherwise singular backgrounds (Santoni et al., 2018).

7. Geometric and Fluid Equivalence Perspectives

Dimensional reduction of higher-dimensional Lovelock or Gauss–Bonnet gravity is a geometric origin for beyond Horndeski terms, providing a constructive method to generate ghost-free scalar–tensor Lagrangians with arbitrary coupling functions and matching the subset compatible with observational constraints (e.g., luminal gravitational waves) (Jana et al., 2020, Charmousis, 2014).

On cosmological backgrounds, the energy-momentum tensor of Horndeski models corresponds to an imperfect fluid (with nonzero anisotropic stress), while in the beyond Horndeski case, all higher-derivative contributions precisely cancel such that the effective energy-momentum tensor matches that of a perfect fluid (Quiros et al., 2019). This mapping aids in understanding cosmological perturbations and the observational imprints of these theories.

---

In summary, the beyond Horndeski action generalizes scalar–tensor gravity by allowing higher-derivative, degenerate operators and encompasses sectors motivated by dimensional reduction, effective field theory, and quantum corrections. It underpins stable non-singular cosmologies, provides a ghost-free construction for wormholes and scalarised compact objects, and exhibits rich phenomenology in cosmology and strong gravity environments, with observational signatures distinct from both GR and standard Horndeski theory.