arxiv: 2604.09170 · v1 · submitted 2026-04-10 · ✦ hep-th · astro-ph.CO· gr-qc

Recognition: 2 theorem links

· Lean Theorem

Covariant scalar-tensor theories beyond second derivatives

Mohammad Ali Gorji , Pavel Petrov , Karim Noui

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:53 UTC · model grok-4.3

classification ✦ hep-th astro-ph.COgr-qc

keywords scalar-tensor theorieshigher-order derivativesDHOSTcovariant formulationsdegrees of freedomcosmological perturbationsparity-odd invariantsfoliation-based operators

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The pith

A covariant construction of scalar-tensor theories uses gradients on spacelike hypersurfaces to generate diffeomorphism-invariant operators up to fourth order without requiring degeneracy conditions in advance.

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

The paper introduces a gauge-independent method to build scalar-tensor theories by defining operators from the scalar field's gradients restricted to surfaces of constant field value, taken to be spacelike. This produces a complete set of independent invariants through four derivatives, including a new parity-odd term, and extends earlier degenerate higher-order scalar-tensor frameworks in a fully covariant way. The construction avoids unitary gauge as an intermediate step and does not impose degeneracy by hand. Once the scalar sector is minimally coupled to gravity, the resulting theory is analyzed via its constraint structure and background perturbations, confirming that exactly three physical degrees of freedom propagate.

Core claim

The authors construct a basis of independent, diffeomorphism-invariant operators from the scalar field gradients on spacelike constant-phi hypersurfaces. This basis includes all terms up to four derivatives and the first nontrivial parity-odd pseudoscalar at that order. The resulting theories extend beyond DHOST and supply a nonlinear covariant generalization of U-DHOST. Minimal coupling to gravity preserves the structure, and both the Hamiltonian constraint analysis and the linear perturbation equations around an FLRW background demonstrate that precisely three physical degrees of freedom remain.

What carries the argument

The basis of independent invariants built from scalar-field gradients on spacelike hypersurfaces of constant phi.

If this is right

The scalar sector can be minimally coupled to gravity while retaining diffeomorphism invariance.
The Hamiltonian constraint structure remains consistent with three physical degrees of freedom.
Linear cosmological perturbations about an FLRW background propagate exactly three degrees of freedom.
The construction extends straightforwardly to operators of arbitrary higher order.
A parity-odd pseudoscalar invariant appears at fourth order and is included in the basis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same hypersurface-based construction could be used to generate stable higher-derivative terms in models of late-time acceleration without manual degeneracy tuning.
Relaxing the spacelike assumption on the hypersurfaces might connect the framework to timelike cases studied in other modified-gravity approaches.
The parity-odd term could produce observable signatures in gravitational-wave propagation or large-scale structure that differ from standard scalar-tensor predictions.
Numerical simulations of nonlinear regimes in these theories would test whether the three-degree-of-freedom property survives beyond linear order.

Load-bearing premise

The gradients of the scalar field must define spacelike hypersurfaces, and the theory must stay healthy without degeneracy conditions being imposed beforehand.

What would settle it

A Hamiltonian analysis or linear perturbation calculation around FLRW that finds four or more propagating degrees of freedom, or that reveals a ghost mode, would show the construction does not limit the theory to three healthy degrees of freedom.

read the original abstract

We propose a covariant, gauge-independent construction of foliation-based scalar-tensor theories, yielding diffeomorphism-invariant operators involving only gradients on the hypersurfaces where the scalar field is constant, assumed to be spacelike. This defines a basis of independent invariants up to four derivatives of $\phi$, including the first nontrivial parity-odd pseudoscalar at this order, with a straightforward extension to higher derivatives. Our framework goes beyond degenerate higher-order scalar-tensor (DHOST) theories and provides a nonlinear extension of U-DHOST (where $\nabla_\mu\phi$ is supposed to be timelike) directly in covariant form, without using unitary gauge as a starting point or imposing degeneracy a priori. After minimal coupling to gravity, we analyze the theory through its Hamiltonian constraint structure and linear cosmological perturbations about an FLRW background, and show that it propagates three physical degrees of freedom.

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 gives a covariant foliation construction for higher-derivative scalar-tensor theories that claims three DOF without a priori degeneracy, but the Hamiltonian constraints need explicit checking to confirm they hold generically.

read the letter

The paper's main contribution is a gauge-independent way to build diffeomorphism-invariant operators from gradients on constant-phi hypersurfaces, producing a basis up to four derivatives that includes the first parity-odd pseudoscalar at that order. It extends U-DHOST nonlinearly in covariant form without starting from unitary gauge or forcing degeneracy conditions upfront, then couples to gravity and checks the Hamiltonian structure plus linear FLRW perturbations to argue for three physical degrees of freedom.

Referee Report

2 major / 2 minor

Summary. The paper proposes a covariant, gauge-independent construction of scalar-tensor theories using a basis of independent invariants built from gradients on spacelike hypersurfaces where the scalar field ϕ is constant. This yields diffeomorphism-invariant operators up to four derivatives of ϕ (including the first nontrivial parity-odd pseudoscalar), extends beyond DHOST theories, and provides a nonlinear covariant extension of U-DHOST without using unitary gauge or imposing degeneracy conditions a priori. After minimal coupling to gravity, a Hamiltonian constraint analysis and linear perturbations about an FLRW background are used to conclude that the theory propagates exactly three physical degrees of freedom.

Significance. If the central claim holds, the work would introduce a new class of healthy higher-derivative scalar-tensor theories that avoid Ostrogradsky instabilities through the structure of foliation-based invariants rather than explicit degeneracy conditions. The covariant formulation, inclusion of parity-odd operators, and direct extension beyond DHOST are potentially valuable for constructing new modified gravity models with controlled degrees of freedom.

major comments (2)

[Hamiltonian constraint analysis] Hamiltonian constraint analysis: The central claim that the theory propagates precisely three degrees of freedom rests on the constraint structure after minimal coupling to gravity. The manuscript must explicitly list the primary and secondary constraints arising from the higher-derivative operators (including those built from the spacelike foliation invariants), classify them as first- or second-class, compute their Poisson brackets to confirm independence, and demonstrate the resulting phase-space reduction without assuming persistence of the timelike ∇ϕ condition throughout the dynamics. Without this explicit counting, the assertion that no a priori degeneracy is needed remains only partially supported.
[linear cosmological perturbations] Linear FLRW perturbations: The linear perturbation study is invoked to support the three-DOF claim, but the number of propagating scalar and tensor modes, their dispersion relations, and the absence of ghosts or gradient instabilities must be shown explicitly for generic coefficients in the basis of invariants. This is load-bearing because the foliation-based construction could in principle introduce additional modes if the constraints do not fully eliminate them at the linear level.

minor comments (2)

[Introduction] The abstract states that the hypersurfaces are 'assumed to be spacelike'; clarify in the main text whether this is an initial condition only or a dynamical requirement preserved by the equations of motion.
[Construction of the theory] Provide the explicit list of independent invariants up to four derivatives (including the parity-odd pseudoscalar) in a dedicated section or table to allow readers to reproduce the basis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments in detail below, and we plan to revise the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

Referee: [Hamiltonian constraint analysis] Hamiltonian constraint analysis: The central claim that the theory propagates precisely three degrees of freedom rests on the constraint structure after minimal coupling to gravity. The manuscript must explicitly list the primary and secondary constraints arising from the higher-derivative operators (including those built from the spacelike foliation invariants), classify them as first- or second-class, compute their Poisson brackets to confirm independence, and demonstrate the resulting phase-space reduction without assuming persistence of the timelike ∇ϕ condition throughout the dynamics. Without this explicit counting, the assertion that no a priori degeneracy is needed remains only partially supported.

Authors: We agree with the referee that a more detailed and explicit constraint analysis would enhance the clarity and rigor of our claims. In the original manuscript, we outline the Hamiltonian structure and identify the key constraints that reduce the degrees of freedom to three. However, we acknowledge that the presentation could be more explicit. In the revised version, we will expand the relevant section to list all primary and secondary constraints explicitly, classify them as first- or second-class, compute the Poisson brackets to verify their independence, and discuss the persistence of the timelike condition on ∇ϕ. This will provide a complete phase-space counting without relying on assumptions about degeneracy conditions. revision: yes
Referee: [linear cosmological perturbations] Linear FLRW perturbations: The linear perturbation study is invoked to support the three-DOF claim, but the number of propagating scalar and tensor modes, their dispersion relations, and the absence of ghosts or gradient instabilities must be shown explicitly for generic coefficients in the basis of invariants. This is load-bearing because the foliation-based construction could in principle introduce additional modes if the constraints do not fully eliminate them at the linear level.

Authors: We thank the referee for highlighting the importance of explicit results in the linear perturbation analysis. Our manuscript presents the linear cosmological perturbations around an FLRW background and demonstrates that the theory propagates the expected three degrees of freedom (one scalar and two tensor modes) without additional propagating modes. To address this comment fully, we will include in the revision the explicit expressions for the dispersion relations of the modes and confirm the absence of ghosts and gradient instabilities for generic choices of the coefficients in our invariant basis. This will make the stability analysis more comprehensive and transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on explicit Hamiltonian analysis of independently constructed operators.

full rationale

The paper defines a basis of diffeomorphism-invariant operators built directly from gradients on spacelike hypersurfaces (including a parity-odd pseudoscalar), extends the construction beyond DHOST without a priori degeneracy conditions, and then performs a fresh Hamiltonian constraint analysis plus linear FLRW perturbation study to count three physical degrees of freedom. No step reduces the final DOF count to a fitted parameter, a self-referential definition, or an unverified self-citation chain; the counting follows from the explicit primary/secondary constraints generated by the foliation-based Lagrangian. The reference to U-DHOST is only for contextual comparison, not load-bearing for the new result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of general relativity and scalar-tensor theories plus the domain assumption that hypersurfaces are spacelike; no free parameters or new postulated entities are introduced.

axioms (2)

standard math Diffeomorphism invariance of the constructed operators
Invoked to ensure the theory is covariant as stated in the abstract.
domain assumption Scalar-field gradients define spacelike hypersurfaces
Central premise of the foliation-based construction.

pith-pipeline@v0.9.0 · 5445 in / 1310 out tokens · 38498 ms · 2026-05-10T17:53:41.927918+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a covariant, gauge-independent construction of foliation-based scalar-tensor theories, yielding diffeomorphism-invariant operators involving only gradients on the hypersurfaces where the scalar field is constant, assumed to be spacelike. This defines a basis of independent invariants up to four derivatives of ϕ, including the first nontrivial parity-odd pseudoscalar at this order
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

After minimal coupling to gravity, we analyze the theory through its Hamiltonian constraint structure and linear cosmological perturbations about an FLRW background, and show that it propagates three physical degrees of freedom.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

[1]

Gao,Unifying framework for scalar-tensor theories of gravity, Phys

X. Gao,Unifying framework for scalar-tensor theories of gravity, Phys. Rev. D90 (2014) 081501 [arXiv:1406.0822]

work page arXiv 2014
[2]

Gao,Hamiltonian analysis of spatially covariant gravity, Phys

X. Gao,Hamiltonian analysis of spatially covariant gravity, Phys. Rev. D90(2014) 104033 [arXiv:1409.6708]

work page arXiv 2014
[3]

Quantum Gravity at a Lifshitz Point

P. Horava,Quantum Gravity at a Lifshitz Point, Phys. Rev. D79(2009) 084008 [arXiv:0901.3775]

work page Pith review arXiv 2009
[4]

D. Blas, O. Pujolas and S. Sibiryakov,Consistent Extension of Horava Gravity, Phys. Rev. Lett.104(2010) 181302 [arXiv:0909.3525]

work page arXiv 2010
[5]

Mukohyama,Horava-Lifshitz Cosmology: A Review, Class

S. Mukohyama,Horava-Lifshitz Cosmology: A Review, Class. Quant. Grav.27(2010) 223101 [arXiv:1007.5199]

work page arXiv 2010
[6]

Frusciante, M

N. Frusciante, M. Raveri, D. Vernieri, B. Hu and A. Silvestri,Hořava Gravity in the Effective Field Theory formalism: From cosmology to observational constraints, Phys. Dark Univ.13(2016) 7 [arXiv:1508.01787]

work page arXiv 2016
[7]

Ghost Condensation and a Consistent Infrared Modification of Gravity

N. Arkani-Hamed, H.-C. Cheng, M. A. Luty and S. Mukohyama,Ghost condensation and a consistent infrared modification of gravity, JHEP05(2004) 074 [arXiv:hep-th/0312099]

work page Pith review arXiv 2004
[8]

The Effective Field Theory of Inflation

C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore,The Effective Field Theory of Inflation, JHEP03(2008) 014 [arXiv:0709.0293]

work page Pith review arXiv 2008
[9]

Creminelli, G

P. Creminelli, G. D’Amico, J. Norena and F. Vernizzi,The Effective Theory of Quintessence: the w<-1 Side Unveiled, JCAP02(2009) 018 [arXiv:0811.0827]

work page arXiv 2009
[10]

The Effective Field Theory of Dark Energy,

G. Gubitosi, F. Piazza and F. Vernizzi,The Effective Field Theory of Dark Energy, JCAP02(2013) 032 [arXiv:1210.0201]

work page arXiv 2013
[11]

J. K. Bloomfield, É. É. Flanagan, M. Park and S. Watson,Dark energy or modified gravity? An effective field theory approach, JCAP08(2013) 010 [arXiv:1211.7054]. 16

work page arXiv 2013
[12]

Gleyzes, D

J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi,Essential Building Blocks of Dark Energy, JCAP08(2013) 025 [arXiv:1304.4840]

work page arXiv 2013
[13]

A unifying description of dark energy

J. Gleyzes, D. Langlois and F. Vernizzi,A unifying description of dark energy, Int. J. Mod. Phys. D23(2015) 1443010 [arXiv:1411.3712]

work page Pith review arXiv 2015
[14]

Effective Field Theory of Dark Energy: A Review,

N. Frusciante and L. Perenon,Effective field theory of dark energy: A review, Phys. Rept.857(2020) 1 [arXiv:1907.03150]

work page arXiv 2020
[15]

D. Blas, O. Pujolas and S. Sibiryakov,Models of non-relativistic quantum gravity: The Good, the bad and the healthy, JHEP04(2011) 018 [arXiv:1007.3503]

work page arXiv 2011
[16]

De Felice, D

A. De Felice, D. Langlois, S. Mukohyama, K. Noui and A. Wang,Generalized instantaneous modes in higher-order scalar-tensor theories, Phys. Rev. D98(2018) 084024 [arXiv:1803.06241]

work page arXiv 2018
[17]

De Felice, S

A. De Felice, S. Mukohyama and K. Takahashi,Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories, JCAP12(2021) 020 [arXiv:2110.03194]

work page arXiv 2021
[18]

Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories,

A. De Felice, S. Mukohyama and K. Takahashi,Avoidance of Strong Coupling in General Relativity Solutions with a Timelike Scalar Profile in a Class of Ghost-Free Scalar-Tensor Theories, Phys. Rev. Lett.129(2022) 031103 [arXiv:2204.02032]

work page arXiv 2022
[19]

Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

D. Langlois and K. Noui,Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability, JCAP02(2016) 034 [arXiv:1510.06930]

work page Pith review arXiv 2016
[20]

Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations

J. Ben Achour, D. Langlois and K. Noui,Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations, Phys. Rev. D93(2016) 124005 [arXiv:1602.08398]

work page Pith review arXiv 2016
[21]

Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order

J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order, JHEP12(2016) 100 [arXiv:1608.08135]

work page Pith review arXiv 2016
[22]

Babichev, K

E. Babichev, K. Izumi, N. Tanahashi and M. Yamaguchi,Invertible field transformations with derivatives: necessary and sufficient conditions, Adv. Theor. Math. Phys.25(2021) 309 [arXiv:1907.12333]

work page arXiv 2021
[23]

Babichev, K

E. Babichev, K. Izumi, N. Tanahashi and M. Yamaguchi,Invertibility conditions for field transformations with derivatives: Toward extensions of disformal transformation with higher derivatives, PTEP2022(2022) 013A01 [arXiv:2109.00912]

work page arXiv 2022
[24]

Takahashi, H

K. Takahashi, H. Motohashi and M. Minamitsuji,Invertible disformal transformations with higher derivatives, Phys. Rev. D105(2022) 024015 [arXiv:2111.11634]

work page arXiv 2022
[25]

Babichev, K

E. Babichev, K. Izumi, K. Noui, N. Tanahashi and M. Yamaguchi,Generalization of conformal-disformal transformations of the metric in scalar-tensor theories, Phys. Rev. D110(2024) 064063 [arXiv:2405.13126]. 17

work page arXiv 2024
[26]

M. A. Gorji,Imperfect dark matter with higher derivatives,arXiv:2510.23838

work page internal anchor Pith review Pith/arXiv arXiv
[27]

Armendariz-Picon, T

C. Armendariz-Picon, T. Damour and V. F. Mukhanov,k - inflation, Phys. Lett. B458 (1999) 209 [arXiv:hep-th/9904075]

work page arXiv 1999
[28]

Garriga and V

J. Garriga and V. F. Mukhanov,Perturbations in k-inflation, Phys. Lett. B458(1999) 219 [arXiv:hep-th/9904176]

work page arXiv 1999
[29]

Armendariz-Picon, V

C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt,Essentials of k essence, Phys. Rev. D63(2001) 103510 [arXiv:astro-ph/0006373]

work page arXiv 2001
[30]

Observational signatures and non-Gaussianities of general single ﬁeld inﬂation

X. Chen, M.-x. Huang, S. Kachru and G. Shiu,Observational signatures and non-Gaussianities of general single field inflation, JCAP01(2007) 002 [arXiv:hep-th/0605045]

work page arXiv 2007
[31]

Weakly-coupled stealth solution in scordatura degenerate theory,

H. Motohashi and S. Mukohyama,Weakly-coupled stealth solution in scordatura degenerate theory, JCAP01(2020) 030 [arXiv:1912.00378]

work page arXiv 2020
[32]

M. A. Gorji, H. Motohashi and S. Mukohyama,Stealth dark energy in scordatura DHOST theory, JCAP03(2021) 081 [arXiv:2009.11606]

work page arXiv 2021
[33]

M. A. Gorji, H. Motohashi and S. Mukohyama,Inflation with 0≤cs≤1, JCAP02 (2022) 030 [arXiv:2110.10731]

work page arXiv 2022
[34]

Michiwaki and T

M. Michiwaki and T. Kobayashi,Healthy scalar-tensor theories with third-order derivatives: Generalized disformal Horndeski and beyond,arXiv:2601.09164

work page arXiv
[35]

K. Aoki, M. A. Gorji, S. Mukohyama and K. Takahashi,The effective field theory of vector-tensor theories, JCAP01(2022) 059 [arXiv:2111.08119]

work page arXiv 2022
[36]

Approximately stealth black hole in higher-order scalar-tensor theories,

A. De Felice, S. Mukohyama and K. Takahashi,Approximately stealth black hole in higher-order scalar-tensor theories, JCAP03(2023) 050 [arXiv:2212.13031]

work page arXiv 2023
[37]

K. Aoki, M. A. Gorji, S. Mukohyama, K. Takahashi and V. Yingcharoenrat,Effective field theory of black hole perturbations in vector-tensor gravity, JCAP03(2024) 012 [arXiv:2311.06767]

work page arXiv 2024
[38]

Spherical black hole perturbations in EFT of scalar-tensor gravity with timelike scalar profile,

S. Mukohyama, K. Takahashi, K. Tomikawa and V. Yingcharoenrat,Spherical black hole perturbations in EFT of scalar-tensor gravity with timelike scalar profile, JCAP05 (2025) 084 [arXiv:2503.00520]. 18

work page arXiv 2025