arxiv: 2604.14490 · v1 · submitted 2026-04-16 · 🌀 gr-qc · hep-th

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

Yang Yu , Yu-Min Hu , Xian Gao

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Pith reviewed 2026-05-10 11:19 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords spatially covariant gravitytwo degrees of freedomperturbative analysiscubic ordercosmological backgroundmodified gravityLagrangians

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The pith

Five spatially covariant gravity Lagrangians can be constructed to propagate only two degrees of freedom up to cubic order around a cosmological background.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks explicit examples of modified gravity theories limited to the two tensor degrees of freedom found in general relativity. Within the spatially covariant gravity framework, the authors expand polynomial Lagrangians (up to three derivatives total) around a cosmological background and impose conditions that remove any scalar mode through linear, quadratic, and cubic orders in perturbations. This procedure produces five concrete Lagrangians whose coefficient functions satisfy the necessary restrictions. A sympathetic reader would care because the resulting models supply workable perturbative realizations that can be studied without introducing extra propagating degrees of freedom at low orders.

Core claim

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.

What carries the argument

Perturbative expansion of spatially covariant gravity Lagrangians around a cosmological background together with coefficient conditions that eliminate the scalar perturbation mode up to cubic order.

Load-bearing premise

Eliminating the scalar mode up to cubic order in perturbations around a cosmological background is sufficient to guarantee that the full theory propagates only two degrees of freedom.

What would settle it

Performing the full nonlinear Hamiltonian analysis on one of the five Lagrangians and finding an extra propagating scalar mode, or repeating the count on a non-cosmological background, would show that the 2-DOF property does not hold.

Figures

Figures reproduced from arXiv: 2604.14490 by Xian Gao, Yang Yu, Yu-Min Hu.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives five explicit polynomial SCG Lagrangians that remove the scalar mode up to cubic order in perturbations around a cosmological background.

read the letter

The main result is five concrete Lagrangians in the spatially covariant gravity class that satisfy the two-degree-of-freedom condition perturbatively through cubic order on an FLRW background. They reach this by taking the general SCG form, restricting to polynomial terms with total derivative count at most three, expanding the action, and solving the coefficient conditions that make the scalar perturbation drop out of the equations at that order. This turns abstract Hamiltonian constraints into usable examples, which is the practical advance over earlier SCG work that often stopped at general conditions hard to invert for explicit nonlinear dependence on extrinsic curvature. The method is direct constraint solving on the expanded equations rather than a first-principles derivation, and it stays within the stated perturbative scope without claiming full nonlinear health or background independence. The soft spots are exactly those limits: the check is only up to cubic order on one background, the polynomial truncation is an assumption that shapes the output, and there is no verification that extra modes stay absent at higher orders or on other spacetimes. No instabilities appear in the given calculation, but that leaves the full nonlinear status open. This is for people building or testing modified gravity models that want exactly two tensor modes, especially in cosmological perturbation theory. Specialists following SCG or related constructions will find the explicit forms and coefficient conditions useful as starting points. The reasoning holds together within its narrow claim, with no internal contradictions or circular fitting. It deserves peer review; a referee can ask for clearer statements on the perturbative nature and perhaps suggest extensions, but the work supplies solid, usable candidates at the level it claims.

Referee Report

0 major / 3 minor

Summary. The paper develops a perturbative method within spatially covariant gravity (SCG) to construct theories with two degrees of freedom. It considers polynomial Lagrangians up to total derivative order d=3, expands them around a cosmological background, and derives conditions on coefficient functions by requiring elimination of the scalar perturbation mode up to cubic order. This yields five explicit Lagrangians that satisfy the 2-DOF condition at this perturbative level.

Significance. If the derived conditions are correct, the work supplies concrete, explicit candidate Lagrangians for 2-DOF SCG models, which is valuable given the difficulty of performing full nonlinear Hamiltonian constraint analysis on theories with nonlinear dependence on extrinsic curvature. The perturbative approach offers a practical route to generating testable models, and the explicit forms enable immediate follow-up studies on stability, cosmological dynamics, and observational signatures.

minor comments (3)

[Abstract] The abstract and introduction would benefit from a concise statement of the precise monomials included in the d=3 polynomial ansatz (e.g., listing the independent terms involving the extrinsic curvature and spatial curvature).
[Results] The five explicit Lagrangians identified in the results section should be collected in a single table or appendix, with their coefficient conditions written out explicitly, to improve readability and allow direct comparison.
[Discussion] A short discussion of whether the derived conditions preserve the absence of ghosts or gradient instabilities at the cubic order (beyond scalar elimination) would strengthen the presentation, even if outside the main scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recognizing the value of the perturbative approach in identifying concrete candidate 2-DOF SCG Lagrangians. The recommendation for minor revision is noted, and we will incorporate any editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper imposes the condition that the scalar mode must vanish in the cubic-order perturbative equations around a cosmological background and solves for the coefficient functions in the assumed polynomial SCG Lagrangian (up to total derivative order d=3). This yields five explicit Lagrangians as direct outputs of the constraint-solving process. No step reduces by construction to a prior fit, self-definition, or load-bearing self-citation; the derivation remains self-contained within the stated perturbative framework and does not rely on external uniqueness theorems or renamed empirical patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the spatially covariant gravity framework and the validity of perturbative expansion around cosmology; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Lagrangians are restricted to polynomial form with total derivative order d ≤ 3
Chosen to make the perturbative expansion and coefficient solving tractable.
domain assumption Eliminating the scalar mode in the perturbative expansion around a cosmological background implies only two degrees of freedom
Standard assumption in cosmological perturbation theory for counting propagating modes.

pith-pipeline@v0.9.0 · 5493 in / 1446 out tokens · 37599 ms · 2026-05-10T11:19:09.906036+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

91 extracted references · 75 canonical work pages · 2 internal anchors

[1]

To this end, a higher-order perturbation analysis is necessary

by constraining the coefficients of thed= 3terms. To this end, a higher-order perturbation analysis is necessary. In the next section, we will expand the perturbed action to cubic order and require the complete disappearance of any dynamical scalar mode, thereby deriving further conditions that restrict the functional form of the Lagrangian. III. DEGENERA...
[2]

solution A1

Case I.11 We first consider Case I.11, which corresponds to ω2 = C1N C5 −2C 2N andC 4 ̸= 0.(3.32) The equation∆ I11 ˙ζ∂ 2ζ∂ 2ζ,1 = 0, with ∆I11 ˙ζ∂ 2ζ∂ 2ζ,1 = 1 (C5 −2C 2 ¯N)2 4C1C4H ¯N(1 +C 2 ¯N)3 h C1C2 ¯N2 −C 5c(1;1,0)′ 1 + 2C2(2D3 +c (1;1,0) 1 +c (1;1,0)′ 1 ) ¯N + 3C4 2C2D1 ¯N3 −C 2(C5 −4)C 5 ¯N(ω3 −ω ′ 3) −4C 2 2(C5 −1) ¯N2ω′ 3 +C 2 5(ω′ 3 −2ω 3) + 4...
[3]

Solution B1

Case I.12 We next turn to Case I.12, which corresponds toC4 = 0. For this branch, the coefficient∆I.12 ˙ζ∂ 2ζ∂ 2ζ,1 vanishes identically. We therefore begin with∆I.12 ˙ζ∂ 2ζ∂ 2ζ,2, which implies ω2 = C1C2 2(2D3 +c (1;1,0) 1 )2N(1 +C 2N) −8D2 4 +C 1C2 2 C5(1 +C 2N) (3.55) andc (1;1,0) 1 ̸=−2D 3. Substituting this expression forω2 into∆ I12 ˙ζ∂ 2ζ∂ 2ζ,3, we...
[4]

Solution C

Case I.2 Finally, we return to Case I.2, withC4 = 0. In this branch, the equations (3.30) are drastically simplified, which leads toD4 = 0. We therefore obtain another set of solutions for the coefficients, which we call “Solution C”. The corresponding action is SC = Z dtd3xN √ h h ω2 ˆK ij ˆKij + 1 3 C1N 1 +C 2N K2 + C3 N 3R+ B1 N +B 2 +c (0;3,0) 1 ˆKij ...
[5]

+ 24C4˜c(1;1,0)′ 1 ω3 + 2˜c(1;1,0) 1 (˜c(1;1,0)′ 1 ω2 + 18C4ω3 −12C 4ω′ 3) +C 2 2 ¯N C1˜c(1;1,0) 1 (3˜c(1;1,0) 1 ω3 + 4˜c(1;1,0)′ 1 ω3 −2˜c(1;1,0) 1 ω′ 3) +D 1 ¯N ˜c(1;1,0)2 1 (ω2 −ω ′
[6]

+ 12C4˜c(1;1,0)′ 1 ω3 + 2˜c(1;1,0) 1 (˜c(1;1,0)′ 1 ω2 + 6C4ω3 −6C 4ω′ 3) ,(A2) ∆I ˙ζ∂ 2ζ∂ 2ζ,2 =−H 2 ¯N 72C5 2 C2 4 ¯N4ω2 3 + 36C2 4 D1 ¯N(2ω3 −ω ′ 3) +C 2 2 ¯N C1 ¯N ˜c(1;1,0)2 1 (ω2 −ω ′
[7]

+ 36C4˜c(1;1,0)′ 1 ω3 + 2˜c(1;1,0) 1 (˜c(1;1,0)′ 1 ω2 + 30C4ω3 −18C 4ω′ 3) + 12C4 24C4ω2 3 +D 1 ¯N2(2˜c(1;1,0) 1 ω2 + 2˜c(1;1,0)′ 1 ω2 −2˜c(1;1,0) 1 ω′ 2 + 12C4ω3 −9C 4ω′ 3) +C 4 2 ¯N3 288C2 4 ω2 3 +C 1 ¯N ˜c(1;1,0)2 1 (ω2 −ω ′ 2) + 12C4˜c(1;1,0)′ 1 ω3 + 2˜c(1;1,0) 1 (˜c(1;1,0)′ 1 ω2 + 6C4ω3 −6C 4ω′ 3) + 2C3 2 ¯N2 C1 ¯N ˜c(1;1,0)2 1 (ω2 −ω ′
[8]

+ 18C4˜c(1;1,0)′ 1 ω3 + 2˜c(1;1,0) 1 (˜c(1;1,0)′ 1 ω2 + 12C4ω3 −9C 4ω′ 3) + 6C4 36C4ω2 3 +D 1 ¯N2(˜c(1;1,0) 1 ω2 + ˜c(1;1,0)′ 1 ω2 −˜c(1;1,0) 1 ω′ 2 + 3C4ω3 −3C 4ω′ 3) + 4C2 −2D 2 4 ¯N2ω2 2 + 9C2 4 2ω2 3 +D 1 ¯N2(5ω3 −3ω ′ 3) + 3C4 ¯N D1 ¯N(˜c(1;1,0) 1 ω2 + ˜c(1;1,0)′ 1 ω2 −˜c(1;1,0) 1 ω′
[9]

+C 1(2˜c(1;1,0) 1 ω3 + ˜c(1;1,0)′ 1 ω3 −˜c(1;1,0) 1 ω′ 3) , (A3) ∆I ˙ζ∂ 2ζ∂ 2ζ,1 =−4C 4H ¯N(1 +C 2 ¯N)3 C1C2 ¯N ˜c(1;1,0)′ 1 ω2 + ˜c(1;1,0) 1 (ω2 −ω ′ 2) + 3C4 D1 ¯N(ω2 −ω ′
[10]

Appendix B: Degeneracy analysis in Case II In this appendix, we present the procedure used to analyze the degeneracy conditions in Case II

+C 1(2 +C 2 ¯N)ω3 + 4C2(1 +C 2 ¯N)ω2ω3 −C 1(1 +C 2 ¯N)ω ′ 3 ,(A4) and ∆I ˙ζ∂ 2ζ∂ 2ζ,0 =−4C 2 4 ¯N(1 +C 2 ¯N)4 2C2ω2 2 +C 1(ω2 −ω ′ 2) (A5) with˜c(1;1,0) 1 :=c (1;1,0) 1 + 2D3. Appendix B: Degeneracy analysis in Case II In this appendix, we present the procedure used to analyze the degeneracy conditions in Case II. From (3.67), if ˜f3 ̸= 0, then∆1 contains...
[11]

proportional

˜f3 ̸= 0 We first consider the subcase with˜f3 ̸= 0. Since the calculations in this part rely heavily on the operator∆1 in (3.67), it is useful to recast the relevant expressions into a form that makes the dependence on∆−1 1 explicit. In particular, the solutions forAandB can be rewritten equivalently as A= 1 ∆1 2(f3 −3 ˜f3) ¯N(1 +C 2 ¯N)(C1 +C 1C2 ¯N+ 3D...
[12]

˜f3 = 0 We next consider the subcase with˜f3 = 0. Under this assumption, the solutions forAandBobtained from (3.65) and (3.66) become local, namely, A= 1 +C 2 ¯N C2H ¯N ˙ζ− (d8 + 2J3)(1 +C 2 ¯N)3 3C2H ¯N2(C1 +C 1C2 ¯N+ 3D 1H ¯N)a2 ∂2ζ,(B22) and B= f3(1 +C 2 ¯N)3 2C2H ¯N2(C1 +C 1C2 ¯N+ 3D 1H ¯N)a ˙ζ − f3(d8 + 2J3)(1 +C 2 ¯N)5 3C2H ¯N3(C1 +C 1C2 ¯N+ 3D 1H ¯...

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