GLPV Theories in Beyond-Horndeski Gravity

• GLPV theories are scalar-tensor models that generalize Horndeski gravity by incorporating beyond-Horndeski operators while maintaining two tensor and one scalar degree of freedom.
• They achieve stability by enforcing degeneracy conditions that avoid Ostrogradsky ghosts, and they allow independent functional freedom in the Lagrangian structure.
• Observational predictions include unique screening via the Vainshtein mechanism, modified gravitational wave propagation, and distinct cosmological dynamics.

Gleyzes–Langlois–Piazza–Vernizzi (GLPV) Theories

Gleyzes–Langlois–Piazza–Vernizzi (GLPV) theories define a class of scalar–tensor models that generalize Horndeski gravity by introducing additional “beyond-Horndeski” operators while preserving propagation of only the usual two tensor (gravitational wave) modes plus a single scalar degree of freedom. These theories admit higher-derivative Lagrangian terms while evading Ostrogradsky instabilities via specific degeneracy conditions. GLPV theories are embedded within the broader DHOST (Degenerate Higher-Order Scalar-Tensor) landscape and have attracted sustained interest for their cosmological flexibility and distinctive screening and gravitational wave signatures.

1. Lagrangian Structure and Extension Beyond Horndeski

GLPV theories extend the covariant Horndeski action (the most general scalar–tensor theory yielding second-order field equations) by lifting two key algebraic constraints linking the Lagrangian coefficients. In ADM/unitary gauge, the general action takes the schematic form

$$S = \int d^4x \sqrt{-g}\left[ A_2(\phi,X) + A_3(\phi,X)\Box\phi + A_4(\phi,X)R + B_4(\phi,X)\,L_{\rm bH,4} + \cdots \right] + S_{\rm matter},$$

where $X = g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$, and $L_{\rm bH,4}$ denotes quartic “beyond-Horndeski” operators constructed from the Levi-Civita tensor (e.g., $$F_4(\phi,X)\epsilon^{\cdots}\epsilon_{\cdots}\cdots$$). Quintic-order terms ($A_5,B_5,F_5$) and their disformal extensions are present in the most general form but are often neglected for cosmological analyses (Felice et al., 2015, Kase et al., 2015, Lin et al., 2014).

The specific relaxation of the Horndeski constraints allows independent functional freedom in $A_4$ and $B_4$ (and also in $A_5,B_5$ for full GLPV). In terms of the effective field theory (EFT) framework, GLPV theories are associated with the “beyond-Horndeski” parameter $\alpha_H$, non-vanishing only when these relations are violated (Felice et al., 2015, Arai et al., 2019).

2. Degrees of Freedom and Hamiltonian Structure

Despite the inclusion of higher-order (in derivatives) operators, GLPV theories remain free of Ostrogradsky ghosts under suitable degeneracy conditions. Hamiltonian analysis confirms that, for the most commonly analyzed subclass with $A_5=0$, the constraint structure yields six first-class (spatial diffeomorphism) constraints and two second-class constraints, precisely removing all but two tensor and one scalar propagating degrees of freedom (Lin et al., 2014). The degeneracy ensures that no extra ghostlike excitation arises.

Comparison with Horndeski shows that the latter correspond to further restrictions that remove any “beyond-Horndeski” ($\alpha_H=0$) operators, but the constraint counting and number of degrees of freedom remain unchanged in GLPV.

3. Covariant Action and Cosmological Dynamics

The GLPV action up to quartic order is usually written as

$$\begin{aligned} S_{\rm GLPV} = \int d^4x \sqrt{-g} \bigg\{ & G_2(\phi,X) + G_3(\phi,X)\Box\phi + G_4(\phi,X)R \ & -2G_{4,X}\left[(\Box\phi)^2 - \nabla^\mu\nabla^\nu\phi \nabla_\mu\nabla_\nu\phi\right] \ & + F_4(\phi,X)\,\epsilon^{\mu\nu\rho\sigma}\epsilon_{\mu'\nu'\rho'}{}_{\!\sigma}\,\nabla_\mu\phi \nabla^{\mu'}\phi \nabla_\nu\nabla^{\nu'}\phi \nabla_\rho\nabla^{\rho'}\phi \bigg\} \ & + S_{\rm matter}, \end{aligned}$$

with two genuinely new functions $F_4(\phi,X),F_5(\phi,X)$ parametrizing the extension beyond Horndeski (Kase et al., 2015, Domènech et al., 5 Sep 2025). When these vanish, the action reduces to the standard Horndeski form. The presence of $F_4,F_5$ generates the nonzero $\alpha_H$ parameter, directly controlling the novel “beyond-Horndeski” signatures in perturbation theory and screening phenomena.

Background cosmological dynamics (Friedmann equations) and perturbations (tensor and scalar) can be expressed in terms of EFT functions $Q_t, c_t^2, Q_s, c_s^2$, governed by derivatives of the Lagrangian coefficients. GLPV theories admit cosmologically viable solutions provided ghost and gradient (Laplacian) instabilities are avoided; this is achieved by imposing positivity of appropriate kinetic terms and squared sound speeds (Kase et al., 2018, Felice et al., 2015).

4. Observational Consequences and Astrophysical Constraints

GLPV models exhibit a suite of distinctive signatures, both in high-density environments and at cosmological scales. Key features include:

• Conical Singularities: For spherically symmetric vacuum/interior solutions with constant nonzero $\alpha_H$, curvature invariants (e.g., Ricci scalar) diverge as $R(r) \sim -2\alpha_H/r^2$ at $r \to 0$, indicating a conical singularity. This pathology is eliminated in models built such that $\alpha_H(r\to 0)\to0$, e.g., via specific kinetic structure in the Lagrangian (Felice et al., 2015, Kase et al., 2015).
• Vainshtein Mechanism: Screening of the scalar “fifth force” via the Vainshtein mechanism is operative in GLPV. The scalar profile is flattened, and both $\alpha_H(r)$ and the effect of the fifth force on metric potentials are suppressed inside the Vainshtein radius. Modified gravity signatures are compatible with solar system constraints provided $|\alpha_H|\lesssim 10^{-3}$ (Kase et al., 2015, Dima et al., 2017).
• Gravitational Wave Propagation: The tensor propagation speed $c_t^2$ in GLPV can, in principle, differ from unity. However, the confirmation that GWs propagate at luminal speed ($c_t = 1$) from GW170817/GRB 170817A enforces $F_4 = 0$ (or strict relations among the Lagrangian functions), thereby restricting viable parameter space to models with negligible “beyond-Horndeski” kinetic mixing at late times (Kase et al., 2018, Arai et al., 2019, Dima et al., 2017).
• Weak Lensing and Higher-Order Statistics: GLPV modifies the growth of structure and lensing potentials, leading to departures in the bispectrum, trispectrum, and higher-order convergence moments accessible in next-generation surveys. These deviations manifest as modifications in the weak-lensing skew- and kurt-spectra, parameterizable via altered kernel coefficients in the projected statistics (Munshi et al., 2020).
• Induced Gravitational Waves: GLPV theories disformally disconnected from Horndeski propagate a new cubic scalar–scalar–tensor operator, producing an $f^5$ scaling in the frequency spectrum of induced gravitational waves—a signature not reproducible in Horndeski or disformally related models (Domènech et al., 5 Sep 2025).

5. Screening, Perturbations, and Effective Theory Parameters

The Vainshtein mechanism in GLPV is sensitive to the detailed nonlinear structure of the action:

• Inside screened regions, modifications to the Poisson equation and the metric slip parameter (ratio of the two gravitational potentials) depend on the “beyond-Horndeski” EFT function $\alpha_H$ and, in DHOST theories, on additional parameters $\beta_1, \beta_3$. GLPV is distinguished by $\beta_1 = 0$ and small $\alpha_H$ ($\lesssim 10^{-3}$) (Dima et al., 2017, Arai et al., 2019).
• In the cosmological EFT framework, GLPV theories reside at ($\alpha_H \neq 0, \beta_1=0$), nearly lying on the “Horndeski line” $\alpha_B = \alpha_M/2$. Nonzero $\beta_1$ signals full DHOST behavior.
• Observational viability now requires $\alpha_H$ to be tightly bounded, with e.g., the Hulse–Taylor binary pulsar and main-sequence stellar structure placing $|\alpha_H|\lesssim 10^{-3}$ for quartic GLPV (Dima et al., 2017).
• Stability of linear perturbations enforcing positive kinetic and negative gradient terms for scalar and tensor modes is necessary for viable background and structure formation (Kase et al., 2018, Shahidi, 2018).

6. Disformal Transformations and Classification

A defining property of GLPV is their closure under general disformal transformations

$$g_{\mu\nu} \to \Omega^2(\phi)g_{\mu\nu} + \Gamma(\phi,X)\nabla_\mu\phi\nabla_\nu\phi,$$

which generally map Horndeski into a subclass of GLPV models (Tsujikawa, 2014, Tsujikawa, 2015). The invariance of cosmological power spectra under these transformations (in unitary gauge) is maintained, up to next-to-leading order in slow-roll, provided both models have the same disformal parameters. The GLPV class naturally contains all theories reachable by such disformal maps from Horndeski, as well as a sector “disformally disconnected” from Horndeski that possesses genuinely novel phenomenology, particularly in gravitational wave physics (Domènech et al., 5 Sep 2025).

The effective field theory parameters $(\alpha_M,\, \alpha_B-\alpha_M/2,\, \beta_1)$ provide a minimal discriminatory basis for separating Horndeski, GLPV, and general DHOST subclasses (Arai et al., 2019). In GLPV, $\beta_1=0$ and deviations from $\alpha_B=\alpha_M/2$ are suppressed to $\mathcal{O}(X^2)$.

7. Cosmological Models, Stability, and Dark Energy

Quartic-order GLPV models have been constructed that yield late-time cosmic acceleration, with or without tracking behavior of the equation of state $w_{\rm DE}$. Phenomenologically, the dark energy sector is realized by combining quadratic, cubic, and quartic Galileon-like Lagrangians with properly tuned coefficients to ensure a viable expansion history, no ghosts or gradient instabilities in perturbations, and suppressed deviations from general relativity within the solar system (Kase et al., 2018, Felice et al., 2015). Models with nonminimal scalar-Maxwell couplings admit two exact de Sitter branch solutions: a healthy, Λ-driven branch and one driven directly by the GLPV scalar sector, with only the former being free of gradient instabilities (Shahidi, 2018).

Empirical limits from GW and astrophysical tests effectively require negligible "beyond-Horndeski" couplings in all viable GLPV dark energy models, but GLPV still provides a robust framework for describing marginal departures from Horndeski or for investigating screening and nonlinear phenomena (Dima et al., 2017, Felice et al., 2015, Kase et al., 2015).

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References:

(Lin et al., 2014) Hamiltonian structure of scalar-tensor theories beyond Horndeski (Felice et al., 2015) Observational signatures of the theories beyond Horndeski (Tsujikawa, 2015) Cosmological disformal transformations to the Einstein frame and gravitational couplings with matter perturbations (Felice et al., 2015) Existence and disappearance of conical singularities in Gleyzes-Langlois-Piazza-Vernizzi theories (Kase et al., 2015) Conical singularities and the Vainshtein screening in full GLPV theories (Dima et al., 2017) Vainshtein Screening in Scalar-Tensor Theories before and after GW170817 (Kase et al., 2018) Dark energy scenario consistent with GW170817 in theories beyond Horndeski gravity (Shahidi, 2018) Cosmology of a higher derivative scalar theory with non-minimal Maxwell coupling (Arai et al., 2019) Cosmological evolution of viable models in the generalized scalar-tensor theory (Munshi et al., 2020) Higher-Order Spectra of Weak Lensing Convergence Maps in Parameterized Theories of Modified Gravity (Domènech et al., 5 Sep 2025) Unique gravitational wave signatures of GLPV scalar-tensor theories (Tsujikawa, 2014) Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories