--- title: GLPV Theories in Beyond-Horndeski Gravity url: https://www.emergentmind.com/topics/glpv-theories type: topic --- # GLPV Theories in Beyond-Horndeski Gravity 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 [1508.06364][1512.06497][1408.0670]. 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 [1503.06539][1912.01768]. ## 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 [1408.0670]. 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 [1512.06497][2509.05027]. 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 [1802.02728][1503.06539]. ## 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 [1508.06364][1512.06497]. - **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 \(|\alpha_H|\lesssim 10^{-3}\) [1512.06497][1712.04731]. - **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 [1802.02728][1912.01768][1712.04731]. - **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 [2004.07021]. - **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 [2509.05027]. ## 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 \(\alpha_H\) (\( \lesssim 10^{-3}\)) [1712.04731][1912.01768]. - 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 [1712.04731]. - Stability of linear perturbations enforcing positive kinetic and negative gradient terms for scalar and tensor modes is necessary for viable background and structure formation [1802.02728][1811.10170]. ## 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 [1412.6210][1506.08561]. 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 [2509.05027]. 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 [1912.01768]. 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 [1802.02728][1503.06539]. 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 [1811.10170]. 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 [1712.04731][1508.06364][1512.06497]. --- **References:** [1408.0670] Hamiltonian structure of scalar-tensor theories beyond Horndeski [1503.06539] Observational signatures of the theories beyond Horndeski [1506.08561] Cosmological disformal transformations to the Einstein frame and gravitational couplings with matter perturbations [1508.06364] Existence and disappearance of conical singularities in Gleyzes-Langlois-Piazza-Vernizzi theories [1512.06497] Conical singularities and the Vainshtein screening in full GLPV theories [1712.04731] Vainshtein Screening in Scalar-Tensor Theories before and after GW170817 [1802.02728] Dark energy scenario consistent with GW170817 in theories beyond Horndeski gravity [1811.10170] Cosmology of a higher derivative scalar theory with non-minimal Maxwell coupling [1912.01768] Cosmological evolution of viable models in the generalized scalar-tensor theory [2004.07021] Higher-Order Spectra of Weak Lensing Convergence Maps in Parameterized Theories of Modified Gravity [2509.05027] Unique gravitational wave signatures of GLPV scalar-tensor theories [1412.6210] Disformal invariance of cosmological perturbations in a generalized class of Horndeski theories Source: https://www.emergentmind.com/topics/glpv-theories