Shift-Symmetric Horndeski Action

• Shift-Symmetric Horndeski Action is a theory defined by derivative-only couplings (G_i(X)) that yields second-order field equations and a conserved Noether current.
• It underpins models of stationary and dynamic black holes, self-tuning cosmologies, and moving dark energy, with key sectors like KGB and the Gauss–Bonnet coupling.
• The framework enables effective-field-theory approaches for perturbations and stability analyses, and supports extensions to beyond-Horndeski/DHOST systems.

The shift-symmetric Horndeski action is the four-dimensional scalar–tensor action with second-order field equations in which the scalar field enjoys the internal symmetry $\phi \to \phi + c$, so that the covariant couplings depend on the scalar only through the kinetic invariant $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$. In this sector, the theory is used to analyze stationary and dynamical black-hole hair, self-tuning and braided cosmologies, moving dark energy, effective-field-theory descriptions of perturbations, and extensions to beyond-Horndeski/DHOST systems. A central structural feature is that shift symmetry organizes the scalar equation into a conserved current, while specific nonanalytic or topological couplings, most notably the linear coupling to the Gauss–Bonnet invariant, can evade standard no-hair assumptions without spoiling the second-order character of the field equations (Delgado et al., 2020).

1. Covariant form and defining structure

In one standard four-dimensional convention, the shift-symmetric Horndeski action is

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$

with

$\mathcal{L}_2 = G_2(X),$

$\mathcal{L}_3 = G_3(X)\,\Box\phi,$

$$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$

$$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$

Here $G_{iX}\equiv dG_i/dX$, $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$, and $G_{\mu\nu}$ is the Einstein tensor. This is the compact four-dimensional Horndeski form with shift symmetry implemented by $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$0 (Delgado et al., 2020). Closely related papers use the equally standard sign convention $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$1; the difference is conventional and does not alter the defining statement that the theory is fixed by the four functions $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$2 (Khoury et al., 2020).

A frequently studied restriction adds reflection symmetry $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$3. In that subclass the $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$4 and $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$5 sectors are removed, leaving only two arbitrary functions of $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$6, namely $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$7 and $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$8. The action then takes the form

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$9

which underlies several exact black-hole constructions and perturbative stability analyses (Kobayashi et al., 2014).

For late-time cosmology subject to the gravitational-wave speed condition $$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$0, the viable shift-symmetric sector is typically reduced further to Kinetic Gravity Braiding (KGB), with

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$1

This truncation is used in cosmological EFT studies, in theoretical-prior constructions, and in moving-dark-energy scenarios (Orjuela-Quintana et al., 2024).

2. Shift symmetry, conserved current, and the Gauss–Bonnet realization

Shift symmetry implies that the scalar equation of motion can be written as a Noether-current conservation law,

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$2

In the full shift-symmetric Horndeski theory, the current decomposes sector by sector,

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$3

with each contribution extracted from the total-derivative form of the scalar Euler–Lagrange equation (Khoury et al., 2020). In homogeneous cosmology this immediately yields a first integral; in axisymmetric Bianchi I backgrounds, for example,

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$4

with $$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$5 the conserved shift charge (Orjuela-Quintana et al., 2024).

A distinguished realization of shift symmetry is the Einstein–scalar–Gauss–Bonnet model with linear coupling

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$6

where

$$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$7

Although $$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$8 appears explicitly in the Lagrangian, the coupling is shift-symmetric at the level of the field equations because in four dimensions $$S=\int d^4x\,\sqrt{-g}\,\sum_{i=2}^5 \mathcal{L}_i,$$9 is a total divergence,

$\mathcal{L}_2 = G_2(X),$0

so that $\mathcal{L}_2 = G_2(X),$1 shifts the action by a topological term proportional to $\mathcal{L}_2 = G_2(X),$2 (Delgado et al., 2020). In this case the scalar equation becomes

$\mathcal{L}_2 = G_2(X),$3

or equivalently

$\mathcal{L}_2 = G_2(X),$4

in the conventions of the spinning-black-hole construction (Delgado et al., 2020).

This linear Gauss–Bonnet coupling is structurally special in the classification of shift-symmetric Horndeski theories. Theories in which all $\mathcal{L}_2 = G_2(X),$5 limits vanish admit $\mathcal{L}_2 = G_2(X),$6 on any background metric and therefore admit all GR solutions; theories with a linear $\mathcal{L}_2 = G_2(X),$7 term are exactly the class in which Minkowski with constant scalar remains admissible but generic curved GR solutions do not, because curvature sources the scalar. In that classification, the linear Gauss–Bonnet coupling is the unique obstruction to the statement that a locally Lorentz-invariant shift-symmetric Horndeski theory admits all GR solutions (Bernardo et al., 2019).

3. Black-hole ansätze and scalar hair

Shift symmetry allows scalar profiles that depend linearly on a symmetry direction without forcing the metric to share that explicit dependence. In the static, spherically symmetric sector one therefore encounters the now standard ansatz

$\mathcal{L}_2 = G_2(X),$8

or equivalent notations with $\mathcal{L}_2 = G_2(X),$9 and $\mathcal{L}_3 = G_3(X)\,\Box\phi,$0. The key mechanism is that the field equations depend only on derivatives of $\mathcal{L}_3 = G_3(X)\,\Box\phi,$1, so a metric may remain static even when the scalar contains a linear time dependence (Babichev et al., 2016). In John-type models this structure reduces the system to an algebraic master equation once the radial component of the shift current is set to zero, and it yields self-tuned Schwarzschild–(anti-)de Sitter, stealth Schwarzschild, Lifshitz, and Einstein-static branches (Babichev et al., 2016).

Exact reflection-symmetric shift-symmetric black holes can be obtained without choosing explicit forms of $\mathcal{L}_3 = G_3(X)\,\Box\phi,$2 and $\mathcal{L}_3 = G_3(X)\,\Box\phi,$3. For the ansatz

$\mathcal{L}_3 = G_3(X)\,\Box\phi,$4

the mixed metric equation enforces $\mathcal{L}_3 = G_3(X)\,\Box\phi,$5, and the remaining field equations reduce to an algebraic integrability condition for $\mathcal{L}_3 = G_3(X)\,\Box\phi,$6. This framework contains Schwarzschild–(A)dS-like black holes, Nariai limits, planar AdS black holes, Lifshitz black holes, and stealth Schwarzschild solutions as branches characterized by the functions $\mathcal{L}_3 = G_3(X)\,\Box\phi,$7, $\mathcal{L}_3 = G_3(X)\,\Box\phi,$8, and $\mathcal{L}_3 = G_3(X)\,\Box\phi,$9 built from $$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$0 and $$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$1 (Kobayashi et al., 2014).

The spinning black-hole construction in the shift-symmetric Einstein–scalar–Gauss–Bonnet model uses the stationary, axisymmetric ansatz

$$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$2

with $$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$3 and $$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$4 depending on $$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$5. These solutions are stationary, axially symmetric and asymptotically flat, with a nontrivial scalar field outside a regular event horizon; the scalar hair is secondary because the scalar charge satisfies

$$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$6

The same family displays a minimal black-hole size, a maximal dimensionless spin slightly above Kerr,

$$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$7

and phenomenological deviations from Kerr and Einstein–dilaton–Gauss–Bonnet that are of order a few percent for observables such as ISCO and light-ring frequencies (Delgado et al., 2020).

Shift symmetry also permits nonstandard compact objects outside the spherical black-hole setting. In four-dimensional black strings with translation invariance, the scalar profile

$$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$8

makes $$\mathcal{L}_4 = G_4(X) R + G_{4X}(X)\left[(\Box\phi)^2 - (\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi)\right],$$9 constant and removes the on-shell dependence on the $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$0 and $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$1 sectors. The remaining field equations become Einstein-like on the transverse three-dimensional base, while consistency fixes the scalar charge $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$2 algebraically in terms of the model functions $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$3 and $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$4 when an Einstein limit is imposed (Guajardo, 2023).

4. No-hair theorems, stability, and effective causal cones

The shift-symmetric Horndeski action is not equivalent to a general guarantee of stable scalar hair. In the reflection-symmetric subclass with $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$5, odd-parity perturbations around linearly time-dependent hairy black holes are governed by background functions $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$6, $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$7, $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$8, and $$\mathcal{L}_5 = G_5(X)\,G_{\mu\nu}\,\nabla^\mu\nabla^\nu \phi - \tfrac{1}{6} G_{5X}(X)\left[(\Box\phi)^3 - 3\,\Box\phi\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\mu\nabla^\nu \phi) + 2\,(\nabla_\mu\nabla_\nu \phi)(\nabla^\nu\nabla^\rho \phi)(\nabla_\rho\nabla^\mu \phi)\right].$$9, and stability requires $G_{iX}\equiv dG_i/dX$0, $G_{iX}\equiv dG_i/dX$1, and $G_{iX}\equiv dG_i/dX$2. For the known $G_{iX}\equiv dG_i/dX$3 branches, the near-horizon relation

$G_{iX}\equiv dG_i/dX$4

shows that at least one of the kinetic or gradient coefficients changes sign near the horizon, while the $G_{iX}\equiv dG_i/dX$5 branch has a vanishing odd-parity quadratic action and is therefore strongly coupled (Ogawa et al., 2015).

A broader instability result was obtained for Schwarzschild backgrounds with time-dependent scalar hair in general shift-symmetric Horndeski theories. In Lemaître coordinates, the exact background $G_{iX}\equiv dG_i/dX$6 yields constant $G_{iX}\equiv dG_i/dX$7 and solves the field equations under ghost-condensate conditions, but the parity-even sector obeys the universal relation

$G_{iX}\equiv dG_i/dX$8

As a result, making the radial gradient healthy forces an angular gradient instability, and vice versa. This rules out black holes with time-dependent scalar hair within the shift-symmetric Horndeski class considered there (Khoury et al., 2020).

The EFT reformulation of the same problem reaches the instability question from the bottom up. In unitary gauge around hairy Schwarzschild–de Sitter backgrounds, the quadratic EFT is built from operators involving $G_{iX}\equiv dG_i/dX$9, $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$0, and intrinsic curvatures on the scalar foliation. In the decoupling limit, operators such as $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$1, $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$2, $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$3, and higher-curvature terms impose explicit positivity conditions on the kinetic and gradient matrices, while the odd-parity sector reduces to two EFT coefficients $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$4 and $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$5 and yields a Regge–Wheeler-type master system with stability conditions $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$6 and $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$7 (Khoury et al., 2022).

Time-independent scalar hair is more constrained than linearly time-dependent hair, but not uniformly excluded. For static, spherically symmetric black holes with $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$8, reflection-symmetric theories with derivative couplings in $\Box\phi\equiv\nabla_\mu\nabla^\mu\phi$9 are generically unstable near the horizon whenever $G_{\mu\nu}$0. By contrast, cubic Galileons with the Einstein–Hilbert term admit nonasymptotically flat hairy black holes free of ghosts and Laplacian instabilities, and the linearly Gauss–Bonnet-coupled model admits asymptotically flat hairy black holes that are free of ghosts and Laplacian instabilities in the perturbative small-coupling regime (Minamitsuji et al., 2022). A separate construction with

$G_{\mu\nu}$1

and a tuning such that $G_{\mu\nu}$2 on the background provides an explicit example of a linearly time-dependent hairy solution that avoids the near-horizon odd-parity instability and obeys $G_{\mu\nu}$3 (Tretyakova et al., 2017).

The causal structure seen by scalar perturbations is likewise controlled by the action. In the decoupling limit, linear scalar perturbations propagate in an effective metric $G_{\mu\nu}$4 determined by the principal symbol of the scalar equation. If both the background metric and the scalar are stationary and the horizon has constant surface gravity, then the Killing horizon of the background metric is also a Killing horizon of the effective metric. Scalar perturbations therefore cannot escape the black-hole region seen by minimally coupled matter. When scalar stationarity is relaxed, decoupling-limit examples with mismatched effective and background horizons can be constructed (Benkel et al., 2018).

5. Cosmological realizations and phenomenological parametrizations

In cosmology, shift-symmetric Horndeski theories have been used both as self-tuning models and as late-time dark-energy parametrizations. A nonlinear minisuperspace family on flat FLRW is defined by

$G_{\mu\nu}$5

with the de Sitter self-tuning conditions

$G_{\mu\nu}$6

In terms of $G_{\mu\nu}$7 and $G_{\mu\nu}$8, the de Sitter critical point occurs at $G_{\mu\nu}$9 and $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$00, with Jacobian eigenvalues

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$01

Hence the de Sitter point is an attractor for fluids satisfying the null energy condition (Martin-Moruno et al., 2015).

After the gravitational-wave speed constraint, much of the late-time literature restricts to KGB,

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$02

In this framework, one concrete shift-symmetric ansatz is

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$03

with $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$04 fixed by field normalization. The resulting scalar equation is a conserved current equation,

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$05

so the background approaches a tracker with $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$06 at late times (Traykova et al., 2021).

The same KGB sector also supports anisotropic but homogeneous cosmologies with an inhomogeneous scalar profile,

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$07

which preserves homogeneity through a diagonal realization of translations and internal shifts. In axisymmetric Bianchi I, the momentum density obeys the universal identity

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$08

or equivalently, at leading order in conformal time,

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$09

This gives a field-theoretic realization of moving dark energy and directly connects the conserved shift charge to large-scale dark flows and the CMB dipole (Orjuela-Quintana et al., 2024).

A complementary use of the action is reconstruction. In cubic shift-symmetric Horndeski gravity with

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$10

and vanishing scalar current $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$11, the field equations imply

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$12

This provides a direct map from a chosen $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$13 to a unique model function $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$14 whenever $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$15 is invertible. The same formalism shows that a truly nondynamical dark-energy equation of state $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$16 cannot be generated in this sector, because it forces $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$17 to be constant and destroys the inversion $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$18 (Saravani et al., 2019).

The shift-symmetric action has also been mapped to phenomenological EFT-of-dark-energy variables. In the KGB sector with constant $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$19 and $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$20, one may parametrize

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$21

Sampling the underlying Lagrangian coefficients and enforcing stability yields nontrivial theoretical priors in the space $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$22; when combined with CMB, BAO, RSD, and SN data, these priors improve constraints by up to an order of magnitude, and pure-scalar shift-symmetric models remain observationally viable (Traykova et al., 2021). By contrast, an asymptotic-safety and EFT analysis of a GW170817-compatible KGB truncation finds that the Horndeski couplings are driven too small to account for dynamical dark energy under the stated assumptions, with the braiding coupling becoming irrelevant at the shifted Gaussian fixed point (Eichhorn et al., 2022).

6. Beyond-Horndeski extensions, thermodynamics, and current directions

The shift-symmetric action is often embedded into beyond-Horndeski or GLPV/DHOST systems by adding parity-preserving and parity-violating terms,

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$23

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$24

subject to the degeneracy relation

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$25

to avoid Ostrogradski ghosts (Antoniou et al., 23 Jul 2025). In the shift- and parity-preserving beyond-Horndeski sector, static spherical solutions are efficiently organized by the combination

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$26

A time-dependent scalar ansatz $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$27 then supports primary scalar charge carried by the Noether current, and homogeneous branches with $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$28 admit analytic black holes and solitons with tunable regularity. In that sector the weak energy condition can be satisfied for suitable choices of the couplings, and disformal transformations preserve shift symmetry while generating additional solution families (Bakopoulos et al., 2023).

A related development formulates theory-internal filters for Minkowski and de Sitter vacua with nontrivial scalar profiles directly from the reduced field equations of shift-symmetric Horndeski gravity. Under the static spherical ansatz with $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$29, these filters identify subclasses that admit stealth vacua, homogeneous geometries, and, after linear disformal transformations of the regularized Einstein–Gauss–Bonnet seed, solitons and black holes with primary scalar hair. In that construction the admissible parameter region is fixed by invertibility and regularity conditions on the disformal map (Bakopoulos et al., 10 Oct 2025).

Thermodynamics is equally sensitive to the detailed structure of the action. For spinning black holes in the shift-symmetric Einstein–scalar–Gauss–Bonnet model, the temperature, area, angular velocity, and entropy satisfy

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$30

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$31

and obey the modified Smarr relation

$$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$32

in the conventions of that work (Delgado et al., 2020). For homogeneous spacetimes in shift-symmetric Horndeski and beyond-Horndeski theories, Euclidean methods yield a general entropy-variation formula involving $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$33, $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$34, $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$35, $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$36, and $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$37. In the shift- and parity-symmetric beyond-Horndeski sector, the result collapses universally to Bekenstein’s area law, whereas particular parity-violating constraints can make the entropy identically vanish (Bakopoulos et al., 11 Feb 2025).

Current phenomenology continues to use the action as a platform for black-hole spectroscopy. In a recent full-theory axial analysis of scalarized black holes in a shift-symmetric Horndeski model with couplings $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$38, the master potential develops multiple extrema as the hair-sourcing parameter increases, producing modified greybody factors and altered quasinormal frequencies. In the full axial theory, increasing $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$39 raises $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$40 and makes $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$41 less negative, whereas the test-field approximation shows a different trend, demonstrating the importance of backreaction (Antoniou et al., 23 Jul 2025).

Taken together, these results define the shift-symmetric Horndeski action less as a single model than as a tightly structured theory space. Its core ingredients are the derivative-only dependence encoded by $$X \equiv -\tfrac{1}{2} g^{\mu\nu}\nabla_\mu\phi\,\nabla_\nu\phi$$42, the associated conserved shift current, and a set of special couplings—most prominently the linear Gauss–Bonnet term, the cubic braiding sector, and beyond-Horndeski degeneracy-preserving extensions—that determine whether the theory admits GR vacua, supports primary or secondary hair, preserves stability, or reduces to an effective area law in black-hole thermodynamics.