Spatially covariant gravity with two degrees of freedom: A perturbative analysis up to cubic order

Abstract: There has been considerable interest in constructing modified gravity theories that propagate only two degrees of freedom (DOFs), corresponding to the tensorial gravitational waves of general relativity. Within the framework of spatially covariant gravity (SCG), the conditions for obtaining 2-DOF theories can be derived from Hamiltonian constraint analysis, but it is generally difficult to translate those conditions into explicit SCG Lagrangians, especially when the Lagrangian depends nonlinearly on the extrinsic curvature. In this work, we adopt an alternative perturbative approach. We consider polynomial-type SCG Lagrangians up to $d=3$, where $d$ denotes the total number of derivatives in each monomial, and expand them around a cosmological background. By requiring the scalar mode to be eliminated up to cubic order in perturbations, we derive the corresponding conditions on the coefficient functions in the Lagrangian. We find five explicit Lagrangians that propagate only 2 DOFs up to cubic order in perturbations around a cosmological background. These theories therefore provide concrete candidate 2-DOF SCG models, at least at the perturbative level up to cubic order.

• The paper derives a perturbative framework that eliminates the scalar graviton, ensuring only the two tensor modes propagate through cubic degeneracy conditions.
• It employs the ADM formalism to construct and classify SCG Lagrangians up to third-order derivatives, resulting in five explicit models with tight coefficient constraints.
• The study connects its findings to established models like the extended cuscuton and MMG, while highlighting challenges for extending the analysis beyond cubic order.

Spatially Covariant Gravity with Two Degrees of Freedom: Perturbative Construction and Analysis up to Cubic Order

Introduction and Context

This paper rigorously addresses the formulation and explicit construction of four-dimensional spatially covariant gravity (SCG) theories propagating only two local degrees of freedom (DOF)—the transverse-traceless tensor polarizations of the metric—up to cubic order in perturbations about cosmological backgrounds. The motivation is clear: observational constraints from gravitational wave detectors (LIGO/Virgo/KAGRA) robustly select pure tensor modes and disfavor additional scalar degrees of freedom in the gravitational sector. The theoretical landscape is further shaped by Lovelock’s theorem, which singles out general relativity (GR) as the unique, ghost-free, metric-based, spacetime-diffeomorphism-invariant theory in four dimensions with second-order field equations. To circumvent the assumption of full diffeomorphism invariance, SCG restricts the invariance to spatial hypersurfaces, enabling new classes of gravity theories.

While Hamiltonian (constraint) analyses specify general conditions for theories with a definite number of propagating DOF, explicitly constructing nontrivial Lagrangians corresponding to two DOF—especially those with nonlinear extrinsic curvature dependence—is technically opaque. The authors employ a direct perturbative method to derive sufficient conditions under which the scalar mode is eliminated order by order in perturbation theory, focusing on actions containing all polynomial monomials up to third order in spatial and temporal derivatives ($d\leq 3$).

SCG Action and Perturbative Framework

The work uses the Arnowitt–Deser–Misner (ADM) formalism, with the Lagrangians constructed from the lapse $N$, shift $N^i$, the spatial metric $h_{ij}$ and its associated curvature, and the extrinsic curvature $K_{ij}$ and its traces. The general SCG action considered takes the form

$$S = \int dt\, d^3x\, N \sqrt{h}\, \mathcal{L}(t, N, h_{ij}, K_{ij}, R_{ij}, \nabla_i),$$

where monomial terms are classified according to their derivative order $d$. All possible monomials with $d \leq 3$ are included with general time- and lapse-dependent coefficients.

Key to the perturbative strategy is the expansion of the action around a flat FLRW cosmological background to quadratic and cubic order in the scalar perturbations ($A,\, B,\, \zeta$). The auxiliary shift and lapse perturbations $A$ and $N$0 are eliminated via their equations of motion, yielding an effective action for the potential scalar graviton mode $N$1. If all kinetic and propagation terms for $N$2 vanish up to the desired perturbative order, the theory propagates only the two transverse-traceless tensor modes at that order.

Quadratic-Order (Linear) Analysis and Classification

Extending the prior work (Hu et al., 2021), which derived quadratic (second-order) conditions for 2-DOF SCG up to $N$3, the present authors focus on the general monomial action up to cubic order. At quadratic level, the condition for absence of the $N$4 propagating mode is encoded in a degeneracy determinant $N$5 involving the Hessian with respect to temporal and spatial derivatives of the perturbations. Setting $N$6 yields two classes of solutions for allowed coefficient structures:

• Case I: Vanishing certain cubic acceleration terms in the Lagrangian (specific combinations of coefficients, e.g., $N$7).
• Case II: Nonvanishing but fine-tuned combinations, leading to spatial derivative operators acting nontrivially on the lapse (more general but more constrained).

These conditions are necessary but not sufficient for the absence of the scalar at higher orders.

Cubic-Order Analysis and Explicit Construction

The main technical advance is the cubic-order perturbative expansion. The effective action for $N$8 at cubic order contains a tower of terms with different numbers of time and spatial derivatives. Importantly, the authors show that the coefficients of all terms with two or more time derivatives must vanish independently to prevent reappearance of the scalar graviton at the nonlinear level. The cubic degeneracy conditions further restrict the allowed functional dependence of the coefficients.

After a detailed algebraic analysis, including the explicit computation of the scalar kinetic and gradient structure terms, the authors obtain five explicitly parametrized SCG Lagrangians in which the scalar graviton is absent at both linear and cubic order in cosmological perturbations:

• Solutions A1 and A2: Admitting a GR-compatible infrared limit, containing appropriate quadratic kinetic structures and nonlinear curvature couplings.
• Solutions B1, B2, and C: Propagate only two DOF up to cubic order but lack a proper GR low-energy limit.

No admissible 2-DOF solutions exist in Case II: inclusion of cubic monomials with nonlinear lapse derivatives inevitably regenerates the scalar upon imposing cubic-order degeneracy.

Embedding and Comparison with Known Theories

The constructed actions contain as subclasses the quadratic 2-DOF SCG analyzed previously (Gao et al., 2019, Hu et al., 2021), the covariant and extended cuscuton theories (Iyonaga et al., 2018), and various minimally modified gravity (MMG) Lagrangians, including certain limits of GLPV and k-essence models in the unitary gauge. The agreement extends both at the level of Lagrangian structure and the required degeneracy conditions. In contrast to MMG constructions that rely on auxiliary constraint fields or explicit Hamiltonian engineering, the present approach is manifestly Lagrangian and perturbative.

The explicit mapping between the constructed forms and the extended cuscuton branches is provided in detail, showing the perturbative method not only reproduces but also generalizes these models in the polynomial sector considered.

Theoretical and Phenomenological Implications

The explicit construction of 2-DOF SCG Lagrangians up to cubic order critically clarifies:

• The necessity of nonlinear derivative relations among coefficient functions for enforcing scalar degeneracy beyond quadratic order.
• The incompatibility of lapse-velocity-dependent cubic constructions (Case II) with nonlinear 2-DOF propagation, underscoring the rigidity of the scalar constraint beyond quadratic truncation.
• The embedding of previously known quadratic and cuscuton-type theories in a broader, fully Lagrangian polynomial monomial framework.

Practically, the admissible actions (A1, A2) are candidates for consistent modifications of gravity compatible with present gravitational wave and cosmological constraints, allowing for controlled departures from GR in the nonlinear and high-energy regimes.

Theoretically, the method establishes a template for perturbative (order-by-order) construction of gravity theories with controlled DOF content in the Lagrangian language, sidestepping the otherwise complicated Dirac constraint algebra.

Outlook and Future Directions

A major open question pertains to the sufficiency of the cubic-order conditions: whether higher-order (quartic, quintic, etc.) perturbative corrections or genuinely nonperturbative effects might revive the scalar DOF, as generically occurs in higher-derivative or degenerate gravity models. Extending the analysis to quartic and beyond, or finding a covariant, nonperturbative characterization in the Lagrangian language, remains a priority for mathematical gravity and effective field theory studies.

From a phenomenological perspective, the constructed actions should be subjected to detailed analysis of post-Friedmannian dynamics, gravitational wave propagation, and (when matter is included) structure formation, especially to assess possible observational signatures that distinguish them from GR and the extended cuscuton class.

Conclusion

This work provides a systematic, explicit, Lagrangian-based construction of SCG theories propagating uniquely the two tensor polarizations around FLRW backgrounds up to cubic order, by deriving and imposing algebraic and differential degeneracy constraints. The link to established 2-DOF frameworks is made explicit, while the limitations imposed by nonlinear perturbative consistency are identified and characterized. This perturbative construction constitutes a solid methodological and theoretical expansion of the SCG program and is likely to inform future developments in model building, Hamiltonian constraint analysis, and dark sector phenomenology.

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Reference:

"Spatially covariant gravity with two degrees of freedom: A perturbative analysis up to cubic order" (2604.14490)