Horndeski and Degenerate Higher-Order Theories

Second-order field equations are sufficient to avoid the generic Ostrogradsky degree of freedom, but they are not necessary. A higher-order scalar–tensor action can propagate only two tensor modes and one scalar when its velocity Hessian is degenerate and the associated primary constraint generates a secondary constraint. This page states that result with its hypotheses, proves the mechanism first in a finite-dimensional model, and then identifies what the degeneracy condition does—and does not—establish in Horndeski and DHOST theories.

Required background. Einstein Equations supplies the metric dynamics and Bianchi identities used in the constraint analysis.

Helpful background. Gravitational Actions and Boundary Terms supplies the integrations by parts and well-posed variational data; Formulations of General Relativity supplies the ADM split, lapse, shift, and constraint-counting language; Gravity Beyond General Relativity supplies the distinction between mode removal and full dynamical consistency; Relativistic Matter and Compact Objects supplies the matter coupling whose transformation can preserve or spoil a degenerate class.

Horndeski as the second-order benchmark

Use

$$X=-\frac12\nabla_a\phi\nabla^a\phi, \qquad \phi_{ab}=\nabla_a\nabla_b\phi.$$

In four dimensions the Horndeski action can be written as a sum $\mathcal L=\sum_{n=2}^{5}\mathcal L_n$ with

$$\begin{aligned} \mathcal L_2={}&G_2(\phi,X),\\ \mathcal L_3={}&-G_3(\phi,X)\Box\phi,\\ \mathcal L_4={}&G_4(\phi,X)R +G_{4X}\left[(\Box\phi)^2-\phi_{ab}\phi^{ab}\right],\\ \mathcal L_5={}&G_5(\phi,X)G_{ab}\phi^{ab}\\ &-\frac{G_{5X}}6\left[ (\Box\phi)^3-3\Box\phi\,\phi_{ab}\phi^{ab} +2\phi_a{}^b\phi_b{}^c\phi_c{}^a \right]. \end{aligned}$$

The relative curvature and second-derivative terms make third derivatives cancel from both the metric and scalar Euler–Lagrange equations. Hence the equations are second order even though individual variations are not. The construction and convention translation are summarized in Kobayashi 2019, §2, equations (8)–(17).

Second-order equations prevent a nondegenerate higher-time-derivative kinetic term, but they are a stronger requirement than the actual degree-of-freedom condition. Beyond-Horndeski and degenerate higher-order scalar–tensor theories allow higher-order equations whose kinetic matrix possesses a null direction. Constraints then remove the would-be extra canonical pair.

The mechanical theorem in miniature

Assume constants $a\ne0$, $b\ne0$ and a regular potential $V(q,x)$. Consider

$$L=\frac a2(\ddot q)^2 +b\ddot q\,\dot x +\frac c2\dot x^2-V(q,x).$$

Introduce $Q=\dot q$. The Hessian with respect to the highest velocities $(\dot Q,\dot x)$ is

$$H= \begin{pmatrix} a&b\\ b&c \end{pmatrix}, \qquad \det H=ac-b^2.$$

If $ac-b^2\ne0$, both velocities are determined by their momenta. Together with the Ostrogradsky pair $(q,p_q)$, the three configuration variables $(q,Q,x)$ propagate three degrees of freedom, and the Hamiltonian is linear in $p_q$.

Now assume the primary degeneracy hypothesis

$$ac-b^2=0.$$

For $c=b^2/a$, the kinetic term is

$$L_{\mathrm{kin}} =\frac a2\left(\dot Q+\frac ba\dot x\right)^2.$$

The momenta obey the primary constraint

$$\Psi =p_x-\frac baP_Q\approx0.$$

On the primary surface, a canonical Hamiltonian is

$$H_C=p_qQ+\frac{P_Q^2}{2a}+V(q,x).$$

Preserving $\Psi$ in time gives

$$\dot\Psi=\{\Psi,H_C\} =-V_{,x}+\frac ba p_q\approx0,$$

so the secondary constraint is

$$\Omega=\frac ba p_q-V_{,x}\approx0.$$

With $\{q,p_q\}=\{Q,P_Q\}=\{x,p_x\}=1$, the remaining bracket is

$$\{\Psi,\Omega\}=V_{,xx}.$$

Thus the explicit regularity condition is $V_{,xx}\ne0$ on the region being counted. The constraints form a second-class pair and reduce the six-dimensional phase space by two dimensions:

$$N_{\mathrm{dof}} =\frac12(6-2)=2.$$

This removes the extra mode without guaranteeing that the two survivors are healthy. For example, take $a\gt0$ and $V=\kappa x^2/2$ with $\kappa\gt0$. The regular change $Y=Q+(b/a)x$ rewrites the degenerate model as

$$L=\frac a2\dot Y^2 -\frac{\kappa a^2}{2b^2}(Y-\dot q)^2.$$

It is second order in $(Y,q)$ but its $\dot q^2$ coefficient is negative. The mode count and the kinetic-sign test therefore answer different questions, even in this elementary example.

The result depends on both constraints. A singular Hessian alone supplies the primary relation but does not guarantee that its preservation removes the Ostrogradsky pair; special potentials or a changing Hessian rank require a new Dirac analysis. The general mechanical argument, including primary and secondary constraints, is derived in the original calculation Langlois and Noui 2016, §II.C–D, equations (2.10)–(2.29) and reviewed in Langlois 2019, §III.A–C, equations (3.1)–(3.18).

The DHOST kinetic matrix

A quadratic DHOST action takes the schematic form

$$S=\int d^4x\sqrt{-g}\left[ P(\phi,X)+Q(\phi,X)\Box\phi+F(\phi,X)R +\sum_{I=1}^{5}A_I(\phi,X)L_I \right],$$

where the $L_I$ span terms quadratic in $\phi_{ab}$, with contractions by $g^{ab}$ and $\phi^a$. Introduce a foliation with unit normal $n^a$. The highest-velocity sector can be expressed through the normal derivative $V_*$ of $A_*=-n^a\nabla_a\phi$ and the extrinsic curvature $K_{ij}$:

take $K_{ab}=h_a{}^c h_b{}^d\nabla_c n_d$ and $V_*=n^a\nabla_a A_*$ for the principal-velocity terms. Reversing the definition of $A_*$ or $K_{ij}$ reverses the appropriate mixed coefficient but not the degeneracy condition.

$$\mathcal L_{\mathrm{kin}} =\mathcal A V_*^2 +2\mathcal B^{ij}V_*K_{ij} +\mathcal C^{ij,kl}K_{ij}K_{kl}.$$

Treating the symmetric index pairs as matrix indices, the Hessian, apart from an irrelevant common factor of two, is

$$\mathbb H= \begin{pmatrix} \mathcal A&\mathcal B^{kl}\\ \mathcal B^{ij}&\mathcal C^{ij,kl} \end{pmatrix}.$$

On a branch where $\mathcal C$ is invertible, the exact degeneracy condition is the vanishing Schur complement

$$\boxed{ \mathcal A -\mathcal B^{ij}(\mathcal C^{-1})_{ij,kl}\mathcal B^{kl}=0 }.$$

It supplies a null eigenvector of $\mathbb H$ and a primary relation among the momentum conjugate to $A_*$ and the metric momentum $\pi^{ij}$. Under the regularity hypotheses below, preserving that relation generates a secondary constraint. The pair removes the extra scalar canonical pair that a generic nondegenerate higher-derivative action would carry. The ADM kinetic matrix and quadratic degeneracy classes are derived in Langlois 2019, §IV.A–C, equations (4.1)–(4.20), with the covariant degeneracy relations reviewed in Kobayashi 2019, §4.1, especially equation (110). The full primary-to-secondary Hamiltonian count for this quadratic scalar–tensor class is proved in Langlois and Noui 2016, Hamiltonian Analysis, §§ IV.D–IV.E, PDF.

If $\mathcal C$ is itself singular, the boxed formula is not the classification condition; one must find the null space of the full Hessian directly. Different constant-rank branches can contain additional constraints or strongly coupled modes. Dividing by a kinetic coefficient that vanishes on the background can therefore erase the very branch that needs separate analysis.

Precise statement and hypotheses

The conclusion can now be stated narrowly.

For a four-dimensional scalar–tensor action with a finite derivative order, a regular ADM decomposition, a kinetic Hessian of constant rank, and a nonvanishing lapse on the selected branch, a null Hessian direction that yields an independent primary constraint and whose time preservation yields an independent secondary constraint removes one canonical pair. Subject to the usual diffeomorphism constraints, the generic quadratic DHOST branch then propagates two tensor degrees of freedom and one scalar rather than an additional Ostrogradsky scalar.

The hypotheses include the following.

1. The functions $P,Q,F,A_I$ and the background are regular enough for the Legendre and constraint analyses; the Hessian rank is constant nearby.
2. The ADM foliation and field variables are nonsingular, and any conformal or disformal transformation used to classify the action is invertible.
3. The primary and secondary constraints are independent and close under time evolution rather than fixing an inconsistent multiplier or disappearing on the background.
4. Matter is coupled through the declared physical metric. A transformation of the vacuum gravity sector must also transform that matter coupling.
5. Boundary conditions do not add or remove local modes; boundary and zero modes are counted separately.

These assumptions are why a degeneracy relation is a theorem about mode count on a regular branch, not a universal health certificate.

What the proof does not establish

Removing the Ostrogradsky pair does not imply that the remaining scalar has a positive kinetic coefficient, a positive sound-speed squared, or a benign mass. It does not prove that the gauge-fixed equations are strongly hyperbolic, that the strong-coupling scale lies above the process, or that radiative corrections preserve the chosen coefficient relations. Those are independent tests.

Many class-Ia quadratic DHOST theories are related to Horndeski by an invertible disformal transformation in vacuum. If matter was minimally coupled before the map, it generally becomes disformally coupled afterward. Declaring it minimal to the new metric changes the theory and can change the constraint structure of the coupled system. The transformation and matter-coupling qualification are discussed in Langlois 2019, §IV.E, equations (4.30)–(4.32).

A discriminating counterexample is the same mechanical model with $ac-b^2\ne0$: it has nearly identical-looking equations but no primary null relation and propagates the additional Ostrogradsky mode. Another limitation occurs at a background where a healthy kinetic eigenvalue tends to zero. The mode may cease to appear in the quadratic action while becoming strongly coupled; it has not necessarily been removed by a nonlinear constraint.

Common pitfalls

Second-order equations are necessary for health. They are sufficient for avoiding the generic higher-derivative mode, but a degenerate higher-order system can remove it by constraints.

A zero determinant completes the proof. Degeneracy gives a primary constraint. Its independence, secondary partner, and constant rank must still be established.

Vacuum disformal equivalence preserves minimal matter coupling. The matter metric transforms too. Re-minimalizing matter defines a different coupled theory.

Exercises

For the mechanical Lagrangian

$$L=\frac a2(\ddot q)^2 +b\ddot q\,\dot x +\frac c2\dot x^2-V(q,x),$$

assume $a\ne0$, $b\ne0$, and $c=b^2/a$. Derive the primary and secondary constraints and count the physical degrees of freedom. What changes when $ac-b^2\ne0$?

Solution

Set $Q=\dot q$. The momenta of the highest-velocity variables are

$$P_Q=a\dot Q+b\dot x, \qquad p_x=b\dot Q+\frac{b^2}{a}\dot x =\frac baP_Q.$$

Therefore

$$\Psi=p_x-\frac baP_Q\approx0$$

is a primary constraint. Only the combination $Y=\dot Q+(b/a)\dot x=P_Q/a$ is determined. The undetermined orthogonal velocity multiplies $\Psi$, while the canonical Hamiltonian is

$$H_C=p_qQ+\frac{P_Q^2}{2a}+V(q,x).$$

Time preservation gives

$$0\approx\dot\Psi =\{\Psi,H_C\} =-V_{,x}+\frac ba p_q,$$

so

$$\Omega=\frac ba p_q-V_{,x}\approx0.$$

Here $\{\Psi,\Omega\}=V_{,xx}$, so for $V_{,xx}\ne0$ the two constraints are second class. The variables $(q,Q,x)$ give a six-dimensional phase space; the pair leaves dimension four, or two physical configuration-space degrees of freedom. If $ac-b^2\ne0$, the Hessian is invertible, there is no primary relation, and all three configuration-space degrees of freedom propagate. The Hamiltonian’s linear $p_qQ$ term is then the nondegenerate Ostrogradsky direction. A special potential that makes $\{\Psi,\Omega\}=0$ requires continuing the constraint algorithm rather than reusing the generic count.

References

• Kobayashi, Tsutomu (2019), “Horndeski Theory and Beyond: A Review,” Reports on Progress in Physics 82, 086901. DOI; Open PDF.
• Langlois, David (2019), “Dark Energy and Modified Gravity in Degenerate Higher-Order Scalar–Tensor Theories,” International Journal of Modern Physics D 28, 1942006. DOI; Open PDF.
• Langlois, David, and Karim Noui (2016), “Degenerate Higher Derivative Theories beyond Horndeski: Evading the Ostrogradski Instability,” Journal of Cosmology and Astroparticle Physics 02, 034. DOI; Open PDF.
• Langlois, David, and Karim Noui (2016), “Hamiltonian Analysis of Higher Derivative Scalar–Tensor Theories,” Journal of Cosmology and Astroparticle Physics 07, 016. DOI; Open PDF.