GLPV Theories: Beyond Horndeski Gravity

• GLPV theories are scalar–tensor models extending Horndeski, characterized by a degenerate structure that preserves two tensor modes and one scalar mode.
• Their formulation in unitary gauge via ADM variables maintains Hamiltonian consistency and avoids the Ostrogradsky ghost by constraining the lapse and curvature terms.
• These models impact cosmological perturbations and compact-object solutions, offering novel signatures in gravitational waves, screening mechanisms, and large-scale structure.

Gleyzes–Langlois–Piazza–Vernizzi (GLPV) theories are a class of scalar–tensor modified-gravity models that extend Horndeski theory beyond the requirement of manifestly second-order field equations while still propagating only three physical degrees of freedom, namely two tensor polarizations and one scalar mode (Lin et al., 2014). In unitary gauge they are naturally formulated in ADM variables, where they retain a constrained, degenerate structure that removes the would-be Ostrogradsky mode. In the later taxonomy of scalar–tensor theories, GLPV occupies a distinguished beyond-Horndeski corner of the broader DHOST framework, and it has become a central reference point for questions about Hamiltonian consistency, disformal frame transformations, screening, compact-object solutions, and cosmological observables (Minamitsuji et al., 2019).

1. Emergence from Horndeski and effective field theory

The immediate conceptual background of GLPV is the effective field theory of a single extra scalar degree of freedom in cosmology. In the EFT formulation of dark energy and modified gravity, the starting point is the most general unitary-gauge Lagrangian built from the lapse and the extrinsic and intrinsic curvature of constant-time hypersurfaces. Focusing on linear perturbations, one can identify seven operators that lead to equations of motion containing at most two derivatives, while the homogeneous background is controlled by only three time-dependent coefficients (Gleyzes et al., 2013). Within that framework, Horndeski theory occupies a six-operator subset satisfying a specific relation between the coefficients conventionally denoted $m_4^2$ and $\tilde m_4^2$, whereas allowing $m_4^2\neq \tilde m_4^2$ produces a broader class of single-field theories with second-order linear dynamics. That broader class is the EFT precursor of what is now called GLPV (Gleyzes et al., 2013).

This EFT viewpoint is significant because it reframes the criterion for consistency. Horndeski achieves ghost freedom by arranging the covariant equations to be manifestly second order. GLPV instead allows higher-order equations at the Lagrangian level but preserves the physical spectrum through degeneracy and constraints. This distinction became foundational for later DHOST developments, where the absence of extra modes is characterized by degeneracy of the kinetic structure rather than by manifest second-order equations alone (Minamitsuji et al., 2019).

In the cosmological EFT language, GLPV is also associated with the appearance of the beyond-Horndeski parameter $\alpha_{\rm H}$. Horndeski corresponds to $\alpha_{\rm H}=0$, whereas GLPV allows $\alpha_{\rm H}\neq 0$, together with related time-dependent EFT coefficients such as $\alpha_{\rm K}$, $\alpha_{\rm B}$, $\alpha_{\rm M}$, and $\alpha_{\rm T}$ (Munshi et al., 2020). This parameterization is especially useful for linear cosmological perturbations and for connecting GLPV models to large-scale-structure observables.

2. ADM formulation and relation to Horndeski

GLPV theories are most compactly defined in unitary gauge, assuming the scalar field has a timelike gradient and choosing

$\tilde m_4^2$0

With the ADM decomposition

$\tilde m_4^2$1

the GLPV action can be written as

$\tilde m_4^2$2

with

$\tilde m_4^2$3

$\tilde m_4^2$4

$\tilde m_4^2$5

where

$\tilde m_4^2$6

and $\tilde m_4^2$7, $\tilde m_4^2$8 are the intrinsic three-curvature scalar and Einstein tensor of $\tilde m_4^2$9 (Lin et al., 2014). The functions $m_4^2\neq \tilde m_4^2$0 depend on $m_4^2\neq \tilde m_4^2$1 and $m_4^2\neq \tilde m_4^2$2, or equivalently on $m_4^2\neq \tilde m_4^2$3 in covariant language.

Horndeski is recovered by imposing specific relations among these functions. In the quartic sector,

$m_4^2\neq \tilde m_4^2$4

and in the quintic sector,

$m_4^2\neq \tilde m_4^2$5

GLPV is defined by retaining the same ADM operator structure while dropping these relations (Tsujikawa, 2014). In covariant notation, the quartic beyond-Horndeski deviation is often encoded through

$m_4^2\neq \tilde m_4^2$6

so that Horndeski corresponds to $m_4^2\neq \tilde m_4^2$7, while GLPV allows $m_4^2\neq \tilde m_4^2$8 (Felice et al., 2015).

A standard dimensionless measure of this departure is

$m_4^2\neq \tilde m_4^2$9

Pure Horndeski has $\alpha_{\rm H}$0. In GLPV, $\alpha_{\rm H}$1 measures the deformation of the scalar–curvature sector away from the Horndeski structure, including the mismatch between the $\alpha_{\rm H}$2 and $\alpha_{\rm H}$3 pieces that would combine into the Einstein–Hilbert form in general relativity (Felice et al., 2015).

This ADM representation is not merely notational. It makes explicit which combinations of lapse, extrinsic curvature, and intrinsic curvature are retained, which constraints survive, and how GLPV generalizes Horndeski while staying within the same basic unitary-gauge architecture.

3. Hamiltonian structure and degree-of-freedom counting

The defining nontrivial claim about GLPV is that it propagates only three physical degrees of freedom even though its equations are generically higher order. A full Hamiltonian proof was given for a simple but nontrivial GLPV subclass with $\alpha_{\rm H}$4 by Lin, Mukohyama, Namba, and Saitou (Lin et al., 2014).

In that subclass, the canonical momenta conjugate to $\alpha_{\rm H}$5 are linear in the extrinsic curvature,

$\alpha_{\rm H}$6

which permits inversion for $\alpha_{\rm H}$7 provided $\alpha_{\rm H}$8 (Lin et al., 2014). Because the action contains no $\alpha_{\rm H}$9 or $\alpha_{\rm H}=0$0, the primary constraints are

$\alpha_{\rm H}=0$1

Preservation in time generates secondary constraints

$\alpha_{\rm H}=0$2

where $\alpha_{\rm H}=0$3 plays the role of a lapse-related Hamiltonian constraint (Lin et al., 2014).

A key point of the analysis is that the naive momentum constraint must be corrected. The generator of spatial diffeomorphisms is not $\alpha_{\rm H}=0$4 alone but

$\alpha_{\rm H}=0$5

With this correction, $\alpha_{\rm H}=0$6 and $\alpha_{\rm H}=0$7 form the first-class set associated with spatial diffeomorphisms, while $\alpha_{\rm H}=0$8 form a second-class pair, provided

$\alpha_{\rm H}=0$9

The lapse is therefore fixed by constraints rather than generating an extra gauge symmetry (Lin et al., 2014).

The degree-of-freedom count is then straightforward. The phase space contains 20 variables: $\alpha_{\rm H}\neq 0$0 namely 10 configuration variables and 10 conjugate momenta. After gauge fixing the three spatial diffeomorphisms, one has 14 second-class constraints in total, leaving a 6-dimensional physical phase space: $\alpha_{\rm H}\neq 0$1 This corresponds to

$\alpha_{\rm H}\neq 0$2

configuration-space degrees of freedom, interpreted as two tensor modes and one scalar (Lin et al., 2014).

This result is central for the subject. It establishes that “beyond Horndeski” does not mean “beyond three degrees of freedom.” In GLPV, higher derivatives in the equations of motion do not automatically imply an extra Ostrogradsky ghost; the relevant question is whether the Hamiltonian system is degenerate and constrained.

4. Disformal transformations, Einstein frame, and DHOST embedding

Disformal transformations play a structural role in GLPV: $\alpha_{\rm H}\neq 0$3 In unitary gauge, this transformation preserves the GLPV ADM form: a GLPV action is mapped into another GLPV action with transformed coefficients $\alpha_{\rm H}\neq 0$4 (Tsujikawa, 2015). This closure under disformal maps sharply distinguishes GLPV from a merely ad hoc extension of Horndeski.

For cosmological perturbations on flat FLRW backgrounds, both the curvature perturbation and the tensor perturbation are invariant under the disformal transformation in unitary gauge: $\alpha_{\rm H}\neq 0$5 The quadratic scalar and tensor actions retain the same form, with transformed coefficients such as

$\alpha_{\rm H}\neq 0$6

and the inflationary scalar and tensor power spectra are invariant up to next-to-leading order in slow-roll (Tsujikawa, 2014).

A particularly useful frame is the Einstein frame defined by choosing $\alpha_{\rm H}\neq 0$7 and $\alpha_{\rm H}\neq 0$8 so that the tensor quadratic action takes the same form as in general relativity. In that frame,

$\alpha_{\rm H}\neq 0$9

and the effective gravitational potential seen by matter perturbations separates into a GR-like contribution plus terms induced by the disformal coupling (Tsujikawa, 2015). When the transformed action belongs to a Horndeski subclass, one has

$\alpha_{\rm K}$0

so $\alpha_{\rm K}$1 implies

$\alpha_{\rm K}$2

i.e. no anisotropic stress in the Einstein frame (Tsujikawa, 2015).

In the broader scalar–tensor classification, GLPV is naturally embedded in DHOST. In the notation of shift-symmetric quadratic-plus-cubic DHOST theories,

$\alpha_{\rm K}$3

with the GLPV identifications

$\alpha_{\rm K}$4

$\alpha_{\rm K}$5

$\alpha_{\rm K}$6

(Minamitsuji et al., 2019). In this language, pure quartic GLPV belongs to quadratic DHOST, while pure quintic GLPV belongs to cubic DHOST. The black-hole analysis of shift-symmetric DHOST theories indicates that fully general quartic-plus-quintic GLPV with both $\alpha_{\rm K}$7 and $\alpha_{\rm K}$8 cannot simultaneously be degenerate and admit the exact constant-$\alpha_{\rm K}$9 Schwarzschild or Schwarzschild–(A)dS solutions considered there; combined with gravitational-wave speed constraints, this pushes viable GLPV models toward purely quartic, effectively quadratic-DHOST sectors (Minamitsuji et al., 2019).

5. Compact objects, singularities, and black-hole solutions

A major nonperturbative issue for GLPV is the behavior of static, spherically symmetric solutions. For a metric

$\alpha_{\rm B}$0

and a regular central field profile with $\alpha_{\rm B}$1, one finds that if

$\alpha_{\rm B}$2

then the Ricci scalar behaves as

$\alpha_{\rm B}$3

near the origin. The resulting geometry is a conical, or solid-angle-deficit, singularity (Felice et al., 2015). The same conclusion holds both in vacuum Schwarzschild-like solutions and inside compact bodies under the simplifying assumption $\alpha_{\rm B}$4: a nonzero constant central limit of $\alpha_{\rm B}$5 produces a curvature divergence that cannot be removed by tuning integration constants (Felice et al., 2015).

This pathology is not generic to every GLPV model. The singularity disappears if

$\alpha_{\rm B}$6

A sufficient model-building strategy is to require that the constant-in-$\alpha_{\rm B}$7 piece of $\alpha_{\rm B}$8 vanish identically, which in the notation

$\alpha_{\rm B}$9

amounts to setting

$\alpha_{\rm M}$0

and ensuring that $\alpha_{\rm M}$1 have no constant term (Felice et al., 2015). Then $\alpha_{\rm M}$2 is proportional to positive powers of $\alpha_{\rm M}$3, and since $\alpha_{\rm M}$4 at the center, the singularity is avoided.

An explicit regular example is an extension of the covariant Galileon with a dilatonic coupling,

$\alpha_{\rm M}$5

with $\alpha_{\rm M}$6. In that case $\alpha_{\rm M}$7 as $\alpha_{\rm M}$8, the Ricci scalar remains finite, and a Vainshtein mechanism operates so that general-relativistic behavior is recovered inside a Vainshtein radius (Felice et al., 2015).

The central singularity issue is also illuminating by contrast. Beyond-generalized Proca theories reduce to shift-symmetric GLPV in the scalar limit $\alpha_{\rm M}$9, yet in the full vector theory the temporal vector component regularizes the center and removes the solid-angle-deficit singularity that would appear in the scalar GLPV limit. Near $\alpha_{\rm T}$0, the vector theory enforces regular expansions

$\alpha_{\rm T}$1

and curvature invariants remain finite (Heisenberg et al., 2016). This comparison suggests that the conical singularity is tied specifically to the scalar-only realization of the beyond-Horndeski quartic structure rather than to the formal presence of an $\alpha_{\rm T}$2-type interaction alone.

Black-hole solutions form a separate line of inquiry. Shift-symmetric GLPV theories admit exact stealth Schwarzschild and Schwarzschild–(A)dS solutions with a linearly time-dependent scalar and constant kinetic term in those subclasses satisfying specific algebraic conditions on the coupling functions. However, the DHOST analysis indicates that quintic and full quartic-plus-quintic GLPV sectors are strongly restricted by degeneracy and gravitational-wave constraints, leaving pure quartic GLPV as the most plausible surviving corner for such exact GR-metric black holes (Minamitsuji et al., 2019).

6. Cosmological perturbations, non-Gaussianity, and observational probes

At linear order, GLPV cosmology differs from Horndeski not only through background functions but through the coupling between tensor and scalar sectors. In a simple GLPV dark-energy model, a slight deviation of the tensor propagation speed squared $\alpha_{\rm T}$3 from 1 can induce a large modification of the scalar propagation speed squared $\alpha_{\rm T}$4 whenever the scalar kinetic energy density is much smaller than the matter density, which is the generic early-time situation in dark-energy models (Felice et al., 2015). A scaling solution with approximately constant $\alpha_{\rm T}$5 avoids this amplification. In the same setup, if the oscillating mode of scalar perturbations is initially suppressed, the anisotropy parameter

$\alpha_{\rm T}$6

can deviate significantly from 1; with generic initial conditions, the deviation of $\alpha_{\rm T}$7 from 1 instead produces large oscillations of $\alpha_{\rm T}$8 with a frequency related to $\alpha_{\rm T}$9 (Felice et al., 2015). These behaviors feed directly into CMB and weak-lensing observables.

In large-scale structure, GLPV effects are often encoded phenomenologically in the second-order density kernel

$\tilde m_4^2$00

with

$\tilde m_4^2$01

In the parameterization used for weak-lensing higher-order spectra, GR has $\tilde m_4^2$02, Horndeski has $\tilde m_4^2$03 with $\tilde m_4^2$04, and GLPV allows both $\tilde m_4^2$05 and $\tilde m_4^2$06 (Munshi et al., 2020). This changes the projected cumulant correlators and therefore the skew-spectrum, kurt-spectra, and related higher-order weak-lensing observables. The same modified kernel enters the integrated bispectrum of convergence maps, making squeezed weak-lensing observables a direct probe of GLPV departures from GR (Munshi et al., 2019).

During inflation, GLPV fits naturally into the unified EFT of inflation. The quadratic action retains the universal form

$\tilde m_4^2$07

with $\tilde m_4^2$08 and $\tilde m_4^2$09 required for stability, while the cubic action can be written in a basis of operators controlled by time-dependent EFT coefficients (Passaglia et al., 2018). A striking result is that GLPV does not generate new scalar bispectrum shapes beyond those already present in Horndeski; rather, it changes the time dependence and amplitudes of the same operator basis. The squeezed-limit consistency relation remains valid even when slow-roll is transiently violated, provided the single-clock assumptions of the EFT remain in force (Passaglia et al., 2018).

An analogous statement holds for primordial tensor non-Gaussianity. In Gao’s unifying ADM framework, the GLPV generalization does not introduce any new tensor cubic operators relative to Horndeski; it only loosens the relations among the coefficients of the same two basic tensor interactions. Four genuinely new tensor cubic operators appear only in theories beyond GLPV (Akita et al., 2015). When the modification of the tensor dispersion relation is small, there is only a single cubic term generating squeezed tensor non-Gaussianity, and it is precisely the term already present in general relativity; the other cubic interactions peak at equilateral configurations (Akita et al., 2015).

A more recent development concerns scalar-induced gravitational waves in GLPV theories that are disformally disconnected from Horndeski. In that case, the tensor action contains a new scalar–scalar–tensor interaction controlled by the combination

$\tilde m_4^2$10

which vanishes in Horndeski and in GLPV sectors reachable from Horndeski by allowed disformal transformations. This interaction introduces third derivatives in the source for induced tensor modes and enhances the gravitational-wave signal. For a scale-invariant primordial scalar spectrum, the induced gravitational-wave spectral density scales as

$\tilde m_4^2$11

a much steeper rise than in GR or Horndeski-related theories (Domènech et al., 5 Sep 2025). This suggests that early-universe GLPV phases disformally disconnected from Horndeski may be distinguishable through a genuinely new gravitational-wave signature.

Taken together, these results define the modern status of GLPV. The class is broader than Horndeski but narrower than completely general higher-derivative scalar–tensor theories; it is best understood as a constrained, degenerate extension whose consistency is established by Hamiltonian structure, whose relation to Horndeski is organized by disformal transformations, and whose phenomenology is sharply shaped by compact-object regularity, screening, gravitational-wave constraints, and higher-order cosmological observables (Lin et al., 2014).