Beyond Horndeski: Extended Scalar-Tensor Theories

• Beyond Horndeski is a class of scalar–tensor theories that extends Horndeski by allowing quartic and quintic operators while preserving three propagating degrees of freedom via degeneracy constraints.
• It alters gravitational-wave propagation, scalar–matter mixing, and large-scale structure, yielding distinctive signatures in cosmology, dark-matter phenomenology, and non-singular solutions.
• Its formulation leverages disformal transformations and effective field theory approaches, with GW170817 constraints enforcing luminal tensor speeds and ghost-free interactions.

Searching arXiv for recent and foundational papers on Beyond Horndeski to ground the article. Beyond Horndeski denotes a class of scalar–tensor theories that extends the original Horndeski construction by allowing higher-derivative equations of motion while preserving the same physical number of propagating degrees of freedom, namely two tensor modes and one scalar mode, through degeneracy or constraint structures rather than manifest second-order field equations (Gleyzes et al., 2014). In the covariant Gleyzes–Langlois–Piazza–Vernizzi (GLPV) formulation, these extensions are generated by the quartic and quintic operators weighted by functions $F_4(\phi,X)$ and $F_5(\phi,X)$, added to the usual Horndeski sector built from $G_2,G_3,G_4,G_5$ (Gleyzes et al., 2014). They occupy a distinguished place within the broader degenerate higher-order scalar–tensor (DHOST) framework, where higher derivatives are permitted provided the Lagrangian is degenerate enough to avoid an Ostrogradski ghost (Kobayashi, 2019). Across cosmology, modified gravity phenomenology, and compact-object physics, beyond-Horndeski interactions are notable because they alter scalar–metric mixing, can modify gravitational-wave propagation and matter clustering, and can evade several no-go theorems that constrain pure Horndeski models (Mironov et al., 2022).

1. Covariant structure and relation to Horndeski

Horndeski theory is the most general four-dimensional scalar–tensor theory with one scalar field and second-order equations of motion. In the notation common to the literature, its action is built from four pieces $$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$ involving arbitrary functions $G_i(\phi,X)$, where $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$ (Gleyzes et al., 2014). Beyond Horndeski extends the quartic and quintic sectors by adding the Levi–Civita-tensor structures

$${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$

or equivalently $L^{\rm bH}_4$ and $L^{\rm bH}_5$ in shift-symmetric notation, with $F_4$ and $F_5(\phi,X)$0 arbitrary functions of $F_5(\phi,X)$1 or of $F_5(\phi,X)$2 in shift-symmetric sectors (Mironov et al., 2022). Ordinary Horndeski is recovered by setting

$F_5(\phi,X)$3

(Gleyzes et al., 2014).

The defining conceptual shift is that second-order equations are no longer treated as necessary for consistency. Beyond-Horndeski theories allow equations of motion with higher derivatives, but the Hamiltonian or ADM structure is degenerate, so the extra Ostrogradski mode does not propagate (Gleyzes et al., 2014). This degeneracy-based viewpoint later became the organizing principle of DHOST theory, of which beyond Horndeski is a prominent subclass (Kobayashi, 2019). A recurrent formulation employs the ADM decomposition in unitary gauge, where the scalar defines the preferred foliation. In that language, the action depends on the lapse $F_5(\phi,X)$4, the spatial metric $F_5(\phi,X)$5, the extrinsic curvature $F_5(\phi,X)$6, and the intrinsic spatial curvature $F_5(\phi,X)$7, and Horndeski corresponds to specific relations among the ADM coefficient functions, while beyond Horndeski arises when those relations are relaxed (Gleyzes et al., 2014).

A distinct but related terminology is the $F_5(\phi,X)$8 formulation, introduced as a systematic realization of healthy scalar–tensor theories beyond Horndeski. There the action takes a simple ADM form in the foliation aligned with constant-$F_5(\phi,X)$9 hypersurfaces, with arbitrary functions $G_2,G_3,G_4,G_5$0 and $G_2,G_3,G_4,G_5$1 for the terms $G_2,G_3,G_4,G_5$2 through $G_2,G_3,G_4,G_5$3. Horndeski appears when the quartic and quintic functions satisfy specific relations such as

$G_2,G_3,G_4,G_5$4

while dropping these relations yields the beyond-Horndeski extension with two additional functional freedoms (Gleyzes et al., 2014).

2. Degrees of freedom, degeneracy, and disformal relations

A central result of the foundational literature is that beyond-Horndeski theories propagate only three gravitational degrees of freedom despite their higher-order covariant equations (Gleyzes et al., 2014). In the ADM treatment of the quartic and quintic sectors, the lapse and shift remain nondynamical, spatial diffeomorphism constraints remain first class, and the lapse sector yields a second-class constraint pair. The resulting counting leaves precisely two tensor polarizations and one scalar mode (Gleyzes et al., 2014). This same conclusion was re-examined in a fully gauge-invariant way by constructing the theory from the geometry of constant-scalar hypersurfaces and identifying the primary constraint that removes the would-be Ostrogradski mode (Crisostomi et al., 2016).

The relation to Horndeski is subtle. Disformal metric transformations of the form

$G_2,G_3,G_4,G_5$5

or, in restricted cases,

$G_2,G_3,G_4,G_5$6

map parts of Horndeski into beyond Horndeski and vice versa (Mironov et al., 2022). When the transformation is invertible, the number of physical degrees of freedom is preserved. This explains why some beyond-Horndeski theories can be viewed as field-redefined Horndeski theories (Crisostomi et al., 2016). However, the equivalence is not unrestricted. A generic beyond-Horndeski theory with arbitrary $G_2,G_3,G_4,G_5$7 cannot necessarily be mapped back to Horndeski by a regular disformal transformation; the covariant relation must satisfy a constraint such as

$G_2,G_3,G_4,G_5$8

(Mironov et al., 2022).

The structure of the equivalence classes is further constrained by mixing different sectors. A gauge-invariant analysis found that pure beyond Horndeski is healthy, and mixing beyond Horndeski with Horndeski of the same order can still preserve a primary constraint and may be mapped to Horndeski by a generalized disformal transformation. By contrast, mixing beyond Horndeski with Horndeski of a different order obstructs the construction of the primary constraint and likely reintroduces a ghost (Crisostomi et al., 2016). This suggests that beyond Horndeski is not merely Horndeski in disguise; rather, it is a healthy but structurally restricted extension.

3. Cosmological EFT description and perturbation variables

At the level of linear cosmological perturbations, beyond Horndeski is efficiently characterized by the Bellini–Sawicki or EFT-of-dark-energy parameterization. In that language, the background and linear perturbations are specified by the effective Planck mass $G_2,G_3,G_4,G_5$9 together with functions

$$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$0

representing braiding, kineticity, Planck-mass running, tensor speed excess, and the beyond-Horndeski parameter (Traykova et al., 2019). The parameter $$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$1 vanishes in Horndeski and is nonzero only in beyond Horndeski. In one standard convention,

$$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$2

(Traykova et al., 2019). In the shift- and mirror-symmetric GLPV sector used to study dark-matter-like behavior, one finds instead

$$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$3

so in that particular model $$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$4 and $$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$5 coincide (Diez-Tejedor et al., 2018).

The role of $$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$6 is not merely classificatory. It encodes new scalar–matter mixing even when matter is minimally coupled to the metric (Traykova et al., 2019). In the original “Healthy theories beyond Horndeski” analysis, the scalar degree of freedom was found to affect the speed of sound of matter even when matter is minimally coupled to gravity (Gleyzes et al., 2014). In the quadratic action for coupled scalar and matter perturbations, the high-$$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$7 dispersion relation contains a mixing term between the scalar gravitational mode and the matter fluctuation. In Horndeski that mixing vanishes in the short-scale limit, but in beyond Horndeski it survives and modifies the matter sound speed and clustering properties (Gleyzes et al., 2014).

The same EFT variables also control tensor propagation. For tensor perturbations,

$$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$8

so the tensor speed is $$\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4,\mathcal{L}_5$$9 (Diez-Tejedor et al., 2018). This became especially important after multimessenger gravitational-wave observations sharply restricted $G_i(\phi,X)$0 at low redshift.

4. Non-singular cosmologies and the evasion of Horndeski no-go theorems

One of the clearest arenas in which beyond Horndeski differs qualitatively from Horndeski is non-singular cosmology. In pure Horndeski, under mild assumptions, a no-go theorem forbids geodesically complete, fully stable, non-singular cosmological solutions such as bounces or Genesis if one requires that tensor and scalar kinetic and gradient coefficients remain positive and bounded away from zero (Mironov et al., 2022). The key quantity in Horndeski is

$G_i(\phi,X)$1

whose time derivative is forced to remain positive in a stable non-singular background (Mironov et al., 2022).

Beyond Horndeski changes this structure by introducing an additional coefficient $G_i(\phi,X)$2, so that the relevant combination becomes

$G_i(\phi,X)$3

Because $G_i(\phi,X)$4 is not constrained to stay positive, $G_i(\phi,X)$5 can cross zero without forcing a singularity or instability. This invalidates the Horndeski no-go argument and permits stable NEC-violating evolutions (Mironov et al., 2022). Explicit examples were constructed in “Cosmological bounce and Genesis beyond Horndeski,” where a spatially flat bouncing solution with

$G_i(\phi,X)$6

and a Genesis solution with

$G_i(\phi,X)$7

were shown to be non-singular and stable throughout the whole evolution (Kolevatov et al., 2017). In those models, the scalar is beyond Horndeski at early times but approaches a massless scalar minimally coupled to gravity at late times (Kolevatov et al., 2017).

The disformal perspective clarifies why this does not contradict the relation to Horndeski. For a fully stable non-singular cosmology in beyond Horndeski, the same combination $G_i(\phi,X)$8 that allows $G_i(\phi,X)$9 to cross zero also appears in the denominator of the disformal mapping to Horndeski. The transformation becomes singular precisely when the non-singular solution exploits the beyond-Horndeski structure to evade the Horndeski no-go theorem (Mironov et al., 2022). This suggests that the Horndeski and beyond-Horndeski descriptions are only locally equivalent in field space, not globally equivalent along such backgrounds.

5. Matter coupling, structure formation, and dark-matter-like behavior

Beyond Horndeski modifies large-scale structure both through linear growth and through nonlinear mode coupling. At the level of linear cosmological phenomenology, the class can be characterized by one extra function of time, $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$0, in addition to the usual Horndeski functions. A phenomenological analysis found that $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$1 is directly related to damping of the matter power spectrum on both large and small scales, enhancement of the low-$X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$2 CMB temperature power spectrum, and a reduction of the lensing potential (Traykova et al., 2019). In the quasi-static approximation, the Poisson equations acquire new terms proportional to the matter velocity potential when $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$3, and these terms are absent in Horndeski (Traykova et al., 2019). The consequence is a modified growth equation with an effective gravitational coupling and an additional friction-like contribution, leading generically to suppressed structure growth for positive $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$4 in the simple parameterizations studied there (Traykova et al., 2019).

At the nonlinear level, the matter bispectrum provides a sharper discriminator. In the GLPV subclass with $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$5, the second-order density kernel takes the form

$X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$6

In GR and in Horndeski one has $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$7, whereas beyond Horndeski generates a new time-dependent coefficient $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$8 through terms proportional to $X\equiv -\frac12 \nabla_\mu\phi\nabla^\mu\phi$9 (Hirano et al., 2018). This deforms the matter bispectrum in folded triangle configurations $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$0, a deformation that is not possible within Horndeski (Hirano et al., 2018). This suggests that folded-shape bispectrum measurements offer a probe of beyond-Horndeski operators distinct from standard Horndeski signatures.

The question of whether the scalar can mimic cold dark matter was examined in a particularly symmetric GLPV sector with shift symmetry and a $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$1 mirror symmetry. In vacuum and at linear order, the scalar mode clusters exactly like standard nonrelativistic cold dark matter, and this remains true even in the subsector where the speed of gravitational waves equals that of light (Diez-Tejedor et al., 2018). However, once ordinary matter is included, the beyond-Horndeski structure induces nontrivial couplings between the scalar and baryons. In the Bellini–Sawicki variables, no parameter choice was found that makes the scalar plus baryon system indistinguishable from standard CDM plus baryons. The unavoidable scalar–baryon coupling alters baryon clustering and lensing, preventing exact CDM mimicry (Diez-Tejedor et al., 2018). A plausible implication is that beyond Horndeski can reproduce some dark-matter-like features in restricted regimes, but not a complete phenomenological replacement of CDM in a realistic matter-filled universe.

6. Compact objects, wormholes, and strong-field configurations

Beyond Horndeski also modifies the space of static and stationary solutions. In shift-symmetric Horndeski and beyond-Horndeski theories, asymptotically flat black-hole solutions with nontrivial scalar hair have been constructed (Babichev et al., 2017). The quartic beyond-Horndeski term $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$2 can support asymptotically flat hairy black holes with finite current norm, and pure quartic sectors can admit Schwarzschild or even Kerr metrics with a nontrivial scalar field as stealth solutions (Babichev et al., 2017). The same analysis showed that in suitable pure quartic sectors, any vacuum GR solution can remain a solution of the quartic theory provided certain conditions on $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$3 and $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$4 hold at a constant $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$5 (Babichev et al., 2017). This indicates that agreement with a GR background metric does not necessarily exclude a beyond-Horndeski scalar sector.

Wormholes provide a more dramatic contrast with Horndeski. In Horndeski theory, static, spherically symmetric wormholes generically suffer from ghost or gradient instabilities in parity-even perturbations (Mironov et al., 2018). Beyond Horndeski modifies the relevant kinetic coefficients by introducing $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$6-dependent terms. In the no-ghost condition, the key quantity

$${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$7

can cross zero because $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$8 changes sign while $${}^{\rm BH}\mathcal{L}_4[F_4], \qquad {}^{\rm BH}\mathcal{L}_5[F_5],$$9 stays positive (Mironov et al., 2018). This breaks the logic of the Horndeski no-go theorem and allows static wormhole solutions satisfying the analyzed ghost and radial-gradient stability conditions (Mironov et al., 2018). An explicit model with

$L^{\rm bH}_4$0

and a suitable scalar profile was constructed as an existence proof (Mironov et al., 2018).

At the same time, the wormhole literature highlights a persistent caveat. A different construction found that ghost instabilities can indeed be avoided beyond Horndeski, but only with strong fine tuning, because the function $L^{\rm bH}_4$1 entering the kinetic determinant tends to become singular at the throat unless a precise condition is imposed there (Mironov et al., 2018). That work emphasized that even if the Horndeski no-go theorem is circumvented, the resulting wormholes are likely unstable or suffer from other pathologies not fully analyzed, including possible angular or radial gradient instabilities and slow tachyonic modes (Mironov et al., 2018). This suggests that beyond Horndeski enlarges the solution space, but not every formally allowed exotic geometry is likely to be dynamically robust.

7. Quantum generation and EFT robustness

Beyond-Horndeski interactions need not arise only as classical inputs. A quantum-field-theoretic analysis showed that a specific beyond-Horndeski matter coupling is generated by quantum gravity effects even when one starts from the minimal action

$L^{\rm bH}_4$2

(Latosh, 2021). The induced local interaction is

$L^{\rm bH}_4$3

which corresponds to the general beyond-Horndeski matter coupling

$L^{\rm bH}_4$4

with specific coefficients (Latosh, 2021). The paper stressed that the amplitude generating this term is free of ultraviolet divergences and independent of UV details, so it is a universal low-energy prediction of the effective field theory of gravity (Latosh, 2021). This suggests that beyond-Horndeski couplings are radiatively natural in the EFT sense, even if absent at tree level.

A complementary EFT analysis asked which higher-derivative scalar–tensor interactions can be simultaneously ghost-free, parametrically important, and robust under quantum corrections. Using a weakly broken galileon power counting with scales $L^{\rm bH}_4$5 and $L^{\rm bH}_4$6, it found that the structurally robust class up to quadratic order in $L^{\rm bH}_4$7 is precisely a subset of quartic Horndeski plus quartic beyond Horndeski (Santoni et al., 2018). In particular, the general robust quartic action

$L^{\rm bH}_4$8

was identified with Horndeski $L^{\rm bH}_4$9 plus beyond-Horndeski $L^{\rm bH}_5$0 subject to the weakly broken galileon consistency conditions (Santoni et al., 2018). This suggests that beyond Horndeski is not just algebraically allowed by degeneracy; certain subsets are also selected by quantum consistency as especially natural EFTs.

8. Gravitational-wave constraints and the post-GW170817 landscape

The observation of GW170817 and its electromagnetic counterpart profoundly reshaped the viable space of beyond-Horndeski theories. In GLPV and related EFT descriptions, tensor propagation is controlled by

$L^{\rm bH}_5$1

and the multimessenger constraint requires $L^{\rm bH}_5$2 at low redshift (Diez-Tejedor et al., 2018). In the symmetric sector studied as a dark-matter analogue, this implies

$L^{\rm bH}_5$3

unless the functions satisfy the luminal relation

$L^{\rm bH}_5$4

which yields

$L^{\rm bH}_5$5

(Diez-Tejedor et al., 2018). A phenomenological study of beyond Horndeski likewise imposed $L^{\rm bH}_5$6 in response to GW170817 and found that this reduces the number of free parameters but does not significantly change cosmological constraints on the remaining ones (Traykova et al., 2019).

The review literature summarized the broader consequence: after GW170817, many quartic and quintic Horndeski and beyond-Horndeski interactions are either ruled out or confined to narrow subclasses with luminal tensor speed (Kobayashi, 2019). In quadratic DHOST language, requiring $L^{\rm bH}_5$7 removes large parts of the theory space, and avoiding graviton decay into scalar fluctuations constrains it further (Kobayashi, 2019). Still, a nontrivial beyond-Horndeski sector survives, particularly within the luminal subclasses relevant to late-time cosmology.

A persistent point of clarification is that luminal tensor speed does not force $L^{\rm bH}_5$8 to vanish identically. Depending on the subclass, one can retain beyond-Horndeski effects in scalar–matter mixing or nonlinear structure formation while satisfying the tensor-speed bound (Traykova et al., 2019). This suggests that the observational status of beyond Horndeski is not determined by gravitational-wave speed alone; large-scale structure, lensing, bispectrum measurements, compact-object tests, and matter couplings remain essential diagnostics.

9. Conceptual synthesis

Beyond Horndeski is best understood as a shift from a “second-order equations” criterion to a “degenerate dynamics” criterion for healthy scalar–tensor gravity. The theory permits the quartic and quintic $L^{\rm bH}_5$9 and $F_4$0 operators that lie outside the original Horndeski construction, yet through degeneracy or primary constraints it still propagates only $F_4$1 degrees of freedom (Gleyzes et al., 2014). In cosmology, this extra structure appears in the EFT parameter $F_4$2, modifies scalar–matter coupling even for minimally coupled matter, and can change the matter sound speed, the growth of structure, and the matter bispectrum in ways unavailable to Horndeski (Gleyzes et al., 2014). In the early universe, it provides a route to fully stable non-singular bounces and Genesis scenarios by evading Horndeski no-go theorems, though the disformal relation to Horndeski becomes singular precisely where that evasion occurs (Mironov et al., 2022). In strong gravity, it enlarges the family of hairy black holes and can support wormholes that evade Horndeski no-go arguments, albeit often with fine tuning or incomplete stability control (Babichev et al., 2017).

This suggests a balanced interpretation. Beyond Horndeski is neither a trivial field-redefinition artifact nor an unrestricted enlargement of scalar–tensor gravity. It is a sharply constrained extension whose healthy sectors are selected by degeneracy, by disformal equivalence classes, by gravitational-wave observations, and, in some EFT constructions, by quantum robustness (Santoni et al., 2018). Its continued relevance lies in the fact that it opens physical possibilities—stable NEC violation, distinctive matter coupling, nonlinear large-scale-structure signatures, and stealth strong-field configurations—that pure Horndeski either forbids or realizes only in restricted form (Kolevatov et al., 2017).