Horndeski Gravity

Updated 1 September 2025

Horndeski gravity is the most general scalar-tensor theory in four dimensions producing second-order field equations while avoiding Ostrogradsky ghosts.
It unifies diverse models—such as f(R), kinetic gravity braiding, and galileons—under a common framework that includes General Relativity as a special case.
The theory offers practical insights into cosmological perturbations, compact object structure, and screening mechanisms, with predictions constrained by observations.

Horndeski gravity is the most general scalar–tensor theory in four dimensions producing second-order Euler–Lagrange equations for the metric and a single scalar field. Its action encompasses broad classes of dark energy and modified gravity models, including General Relativity (GR) with a cosmological constant, $f(R)$ theories, kinetic gravity braiding, and galileons, while excluding Ostrogradsky ghosts by ensuring no extra propagating degrees of freedom. Horndeski gravity offers a unified theoretical framework to explore deviations from GR and the cosmological and astrophysical signatures of these modifications, including nontrivial structure in the growth of large-scale structure, the properties and dynamics of compact objects, and self-tuning mechanisms for the cosmological constant.

1. Horndeski Action and Theoretical Structure

The generalized Horndeski action is

$S = \int d^4x\, \sqrt{-g} \Big[\sum_{i=2}^5 \mathcal{L}_i + \mathcal{L}_m(g_{\mu\nu})\Big]$

where the Lagrangian densities $\mathcal{L}_i$ are constructed from arbitrary functions $K(\phi,X)$ , $G_3(\phi,X)$ , $G_4(\phi,X)$ , $G_5(\phi,X)$ with the kinetic term $X = -\frac{1}{2} (\nabla\phi)^2$ . The structure of the Lagrangians is chosen so that all field equations are at most second order in derivatives: \begin{align*} \mathcal{L}2 &= K(\phi, X) \ \mathcal{L}_3 &= -G_3(\phi, X)\Box\phi \ \mathcal{L}_4 &= G_4(\phi, X) R + G{4X} [(\Box\phi)² - (\nabla_\mu\nabla_\nu\phi)^2] \ \mathcal{L}5 &= G_5(\phi, X) G{\mu\nu} \nabla^{\mu\nabla^\nu\phi} - \frac{1}{6} G_{5X} [(\Box\phi)³ -3\Box\phi(\nabla_\mu\nabla_\nu\phi)² +2(\nabla_\mu\nabla_\nu\phi)^3] \end{align*} where $G_{iX}\equiv\partial G_i/\partial X$ . As a result, Horndeski gravity propagates three degrees of freedom: two tensor modes and one scalar.

The general action admits GR with a cosmological constant (i.e., only $K=-2\Lambda$ , $G_4=1/(16\pi G)$ nonzero) as a special case, as well as minimally coupled quintessence, $f(R)$ , kinetic gravity braiding, and galileon models.

2. Degrees of Freedom, Ostrogradsky Instability, and Beyond Horndeski

The Horndeski construction avoids higher-derivative ghosts, specifically the Ostrogradsky instability, by restricting to second-order field equations. Beyond Horndeski extensions introduce higher-derivative terms but evade extra propagating degrees of freedom via the existence of a hidden primary constraint—demonstrated through geometric variables intrinsic to constant- $\phi$ hypersurfaces and ADM decomposition (Crisostomi et al., 2016). However, combining beyond Horndeski and Horndeski terms of different order generically destroys this constraint, admitting a pathology. Only when mixed at the same order (quartic with quartic, etc.) can the theory be recast into standard Horndeski via generalized disformal transformations; otherwise, beyond Horndeski remains a healthy but isolated sector.

3. Parametrization of Perturbations and Cosmological Structure Formation

The linear- and second-order evolution of cosmological perturbations in four-dimensional Horndeski models is captured by the formalism of Bellini and Sawicki [2014], utilizing background-dependent time functions (parameters):

$\Omega_{m,0}$ : present matter density fraction,
$H(t)$ : Hubble parameter,
$\alpha_K(t)$ : kineticity,
$\alpha_B(t)$ : kinetic gravity braiding,
$\alpha_M(t)$ : Planck mass run rate,
$\alpha_T(t)$ : tensor speed excess.

At second order, new parameters $\alpha_4(t)$ and $\alpha_5(t)$ enter, defined by combinations of derivatives of $G_4$ and $G_5$ . Under a general parametrization, these are typically assumed proportional to $1-\tilde\Omega_m(t)$ , such that $c_i$ coefficients control deviations from the $\Lambda$ CDM limit.

Matter perturbations evolve as

$\frac{k^2}{a^2} \Psi = -\frac{3}{2} H^2 \tilde\Omega_m G_\Psi \delta_m\quad,\quad \ddot\delta_m + 2H \dot\delta_m = \frac{1}{2} G_\Psi \tilde\rho_m \delta_m$

where $G_\Psi$ is an effective Newton's constant for perturbations, determined by the $\alpha$ -functions.

For second-order matter clustering, the dark matter bispectrum at tree-level is

$B(t, k_1, k_2, k_3) = F_2(t, k_1, k_2) P(t, k_1) P(t, k_2) + \text{cyc.}$

The kernel $F_2$ contains a key parameter $C(t)$ , which in GR is $34/21$, but in Horndeski gravity is $C(t) = A_0(t) + A_5(t) c_5 + A_4(t) c_4$ (functions $A_0$ , $A_5$ , $A_4$ set by the background and lower-order $\alpha_i$ parameters).

4. Phenomenology: Dark Matter Bispectrum and Observational Constraints

The leading order bispectrum kernel in Horndeski models can only deviate appreciably from the GR value if either $c_5$ or $c_4$ is large. However, the structure of the evolution equations strongly constrains these deviations:

Naturalness condition: In "natural" models (i.e., $|c_5|,|c_4| \lesssim |c_M|,|c_T| \sim \mathcal{O}(1)$ ), a $\geq 10\%$ deviation in the bispectrum kernel implies a $\sim30\%$ discrepancy in the linear growth rate, already ruled out by observations.
Observational limits: Current constraints on the linear growth rate ( $f$ ) are at the $6$- $10\%$ level, with future surveys improving precision. Since the bispectrum is measured at lower signal-to-noise than the power spectrum or growth rate, any large deviation in the bispectrum would contradict the more precise linear measurements unless due to fine-tuned or "exotic" mechanisms such as a violation of the weak equivalence principle.
Practical conclusion: For Horndeski models that reproduce the expansion history and linear growth predicted by GR, the dark matter bispectrum can be modelled using the standard GR kernel $F_2$ . Only a detection of a large, unexplained bispectrum deviation would indicate significant new gravitational physics or extreme fine-tuning (Bellini et al., 2015).

5. Compact Objects and Strong-Field Regime

Horndeski gravity modifies the structure and dynamics of neutron stars (NSs) and black holes (BHs):

No-hair theorems: For shift-symmetric cases, static and asymptotically flat BHs admit only constant scalar field profiles (no scalar hair), unless symmetry-breaking couplings or time-dependent scalars are introduced. Models such as Einstein–dilaton–Gauss–Bonnet (EdGB) or kinetic couplings ("John" class in Fab Four) can circumvent these restrictions.
Neutron stars: Horndeski models (e.g., EdGB, nonminimally coupled kinetic gravity braiding) alter mass–radius relations, moments of inertia, and may break universality in relations like $I$ –compactness. The "John" term produces "stealth" stars with nontrivial interior scalar profiles and metrics indistinguishable from GR outside. "Paul" sector couplings generally produce pathological stellar behavior, with divergences in physical quantities at the origin, indicating that certain self-tuning mechanisms may not survive inhomogeneous spacetimes (Maselli et al., 2016).
Rotation: For slowly rotating BHs/NSs, frame-dragging functions often match GR at leading order but can deviate at higher orders, suggesting potential for future tests via gravitational wave and electromagnetic observations (Silva et al., 2016).

6. Screening Mechanisms and Einstein Gravity Limit

To reconcile cosmological modifications with local gravity constraints, Horndeski gravity generically invokes nonlinear screening:

Derivative screening (Vainshtein mechanism): Nonlinear derivative interactions become dominant in high-density regions (e.g., around massive bodies), suppressing fifth forces and effectively restoring Einstein gravity locally.
Potential screening (chameleon-like): The scalar potential's effective mass becomes large in dense regions, hiding scalar effects.
Scaling method: A scaling analysis on the scalar field $\phi = \phi_0 (1 + \alpha^q\psi)$ identifies the conditions under which the nonlinear terms screen the modifications. The Einstein gravity limit occurs when the extra terms become subdominant, corresponding to appropriate asymptotic behavior of the Horndeski functions such that, for example, $G_4 - X G_{5X} \to$ constant, and all other contributions vanish. In this regime, post-Newtonian expansions can be constructed within the screened region (McManus et al., 2016).

7. Thermodynamic and Fluid Properties, Further Constraints

Horndeski gravity admits reinterpretation as an effective dissipative fluid:

Dissipative effective fluid: The modified field equations can be rewritten as GR plus an effective stress–energy tensor with nonzero heat flux, shear and bulk viscosity. The out–of–equilibrium structure is generic except for two linear subclasses of the theory: those with either $G_3(\phi,X)=G_{4,\phi}(\phi)\ln X$ or $G_3=0$ , for which the fluid is Newtonian and gravitational waves propagate at light speed. All other cases correspond to exotic non-Newtonian rheologies and are tightly constrained by multimessenger constraints on $c_T$ (Miranda et al., 2022, Giusti et al., 2021).
Stability: Positive energy conditions, the Dolgov–Kawasaki stability (from absence of tachyonic instabilities in $R$ ), and gravitational "attractiveness" (i.e., $G_4>0$ ) place strong restrictions on the Horndeski functions. The Witten positive energy theorem further demands $G_3$ be independent of $X$ ( $G_3(\phi)$ only) for the theory to avoid ghosts (Gomes et al., 2020).
Noether symmetry selection: Symmetry criteria can further narrow the class of viable theories by enforcing factorization of the $\phi$ and $X$ dependencies and relating advanced models, such as with Gauss–Bonnet coupling or kinetic gravity braiding, to very specific functional forms (Miranda et al., 16 Aug 2024).

Summary Table: Core Features of Horndeski Gravity

Aspect	GR Limit	Horndeski Modification
Action	$\mathcal{L}=R-2\Lambda$	$\mathcal{L}=\sum_{i=2}^5 \mathcal{L}_i$
Degrees of Freedom	2 (tensor)	2 (tensor) + 1 (scalar)
Field Equation Order	Second	Second
Bispectrum Deviations	None	$\lesssim 1\%$ in viable models
Screening	Not present	Vainshtein, chameleon, kmouflage
GW Speed $c_T$	Light speed ( $c$ )	$c$ , except in disfavored subclasses

Horndeski gravity offers a robust structure for unifying theoretical modifications to gravity and dark energy, with rich phenomenology across cosmic structure, compact objects, and screening physics. However, alignment with linear structure formation, gravitational wave propagation, and local gravity tests imposes strong naturalness constraints on allowed deviations, with detailed observational and theoretical diagnostics tightly delimiting viable parameter spaces and model classes. Any confirmed detection of large deviations, such as in the matter bispectrum or GW polarizations, would necessitate reexamination of the fundamental structure of gravitational physics, potentially pointing to finely tuned or fundamentally new physics beyond the natural expectations of the Horndeski construct.