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Gauge invariant perturbations in teleparallel Horndeski gravity

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives the gauge-invariant cosmological perturbations of the teleparallel analogue of Horndeski gravity and finds nine propagating degrees of freedom in the general BDLS theory.

desk verdict The gauge-invariant perturbation actions for BDLS teleparallel Horndeski are a genuine new reference, but the gauge-fixed catalogue disagrees with its own scalar DoF count and the missing coefficients sit in an unversioned repo. read the letter →

arxiv 2412.01349 v2 pith:6PLGP3QS submitted 2024-12-02 gr-qc

Bobomurat Ahmedov , Maria Caruana , Jackson Levi Said , Konstantinos F.Dialektopoulos , Abdurakhmon Nosirov , Zinovia Oikonomopoulou , Odil Yunusov This is my paper · ORCID
classification gr-qc PACS 04.50.Kd98.80.-k
keywords teleparallelgravityHorndeskicosmologicalperturbationsgaugeinvarianceghostinstabilitiesdegreesoffreedomBDLStheory
topics Dark Energy
open problems Dark Energy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper works out, in gauge-invariant form, the cosmological perturbations of the BDLS action, the teleparallel analogue of Horndeski gravity. It claims that the most general BDLS theory carries nine propagating degrees of freedom around a flat Friedmann-Lemaitre-Robertson-Walker background: two scalar modes, one pseudoscalar mode, two pairs of vector modes, and the two tensor polarisations. It also derives the quadratic actions for each sector and the coefficient inequalities that prevent ghost and gradient instabilities. The result matters because it converts the BDLS framework from a background-only theory into one whose perturbations can be compared with cosmological observations and implemented in numerical codes.

What carries the argument

The central object is the tetrad perturbation decomposition of Eq. (49), which splits the sixteen tetrad components into five scalars ($\Phi$, $\psi$, $B$, $\beta$, $E$), one pseudoscalar $\sigma$, three vectors ($u_i$, $v_i$, $w_i$), one pseudovector $V_i$, and the tensor $h_{ij}$, with the background tetrad fixed in the Weitzenböck gauge. Gauge-invariant combinations $X_1,\dots,X_4$ for scalars and $Y_i$, $Z_i$ for vectors remove coordinate freedom; then the auxiliary fields $X_2$, $X_4$, and $Y_i$ are integrated out, and the kinetic matrix is diagonalised at high $k$ so that the propagating modes and their sound speeds can be read off from the resulting coefficients.

What would settle it

Recompute the second-order action with a general perturbed spin connection around the same FLRW tetrad and check whether any new kinetic terms appear; if they do, the nine-degree-of-freedom count changes. A more direct observational test would be the detection of a propagating vector or pseudoscalar polarisation in the cosmological gravitational-wave background, which the standard two-tensor GR prediction cannot produce.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the BDLS action, despite its many coupling functions, has a tractable scalar-vector-tensor decomposition: after eliminating auxiliary fields in the high-k limit, the scalar sector leaves two propagating scalars, the pseudoscalar sector leaves one mode, the vector sector leaves two pairs of modes, and the tensor sector leaves the two usual polarisations. The corresponding gauge-invariant quadratic actions are Eqs. (54), (60), and (61), and the ghost and gradient stability requirements reduce to Eqs. (80)-(82) for scalars, Eqs. (89)-(91) for vectors, Eq. (64) for tensors, and $B_1 > 0$ for the pseudoscalar. The paper also confirms that subclasses reduce the count: GR and $f(T)$ give two degrees of freedom, $f(\varphi,X,T)$ and Horndeski give three, generalized teleparallel dark energy and generalized scalar-tensor theory give three, and New General Relativity gives eight.

Load-bearing premise

The whole calculation keeps the spin connection fixed in the Weitzenböck gauge and perturbs only the tetrad; if the spin connection carries its own physical perturbations, the mode count and stability conditions could be incomplete.

Editorial extensions

If this is right

  • The full BDLS theory has nine propagating degrees of freedom: two scalars, one pseudoscalar, two vector pairs, and two tensor polarisations; known subclasses reduce this count, from two for GR and $f(T)$ up to eight for New General Relativity.
  • Ghost and gradient stability become concrete coefficient checks: the scalar sector is stable when Eqs. (80)-(82) hold, the vector sector when Eqs. (89)-(91) hold, the tensor sector when Eq. (64) holds, and the pseudoscalar requires $B_1 > 0$.
  • The tensor sound speed $c_T^2 = D_2/D_1$ is generically not unity and depends on the teleparallel couplings, so gravitational-wave observations can directly constrain the BDLS Lagrangian.
  • The gauge catalogue in Appendix B provides flat, unitary, Newtonian, and synchronous versions of the same perturbation equations, allowing the same physics to be implemented in different observer frames.
  • Observational codes can now use these actions to compute power spectra and stability conditions for specific BDLS models, which is the step needed to confront the framework with structure-formation data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: because the background tetrad is fixed in the Weitzenböck gauge, a full calculation that also perturbs the spin connection is the direct next test; new kinetic terms would change the nine-mode count.
  • Beyond the paper: the gauge catalogue is detailed enough to be coded directly into a Boltzmann solver, and comparing the resulting CMB and matter power spectra with the standard cosmological model would quantify how strongly the extra scalar and vector modes are suppressed.
  • Beyond the paper: the same elimination procedure can be applied at higher order in perturbations, opening the way to bispectra in BDLS; the present second-order action is the necessary first step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper constructs the quadratic cosmological perturbations of the BDLS action, the teleparallel analogue of Horndeski gravity, on a flat FLRW background. It performs an SVT decomposition of tetrad perturbations, defines gauge-invariant variables for the scalar, vector, and tensor sectors, and derives gauge-invariant quadratic actions. From these actions the paper extracts propagating degrees of freedom and formulates ghost and Laplacian stability conditions in the high-k limit, concluding that the full BDLS theory carries 9 propagating degrees of freedom: 2 scalars, 1 pseudoscalar, 2 vector pairs, and 2 tensors. Appendix B provides a catalogue of common gauge choices (flat, unitary, Newtonian, synchronous, and vector gauges), with final coefficients partly stored in an external repository. The central unresolved issue is that the Newtonian and synchronous gauge actions in Appendix B contain three propagating scalar modes, while the gauge-invariant scalar analysis of Section IV gives two; this discrepancy is acknowledged but deferred to future work.

Significance. If correct, the paper would provide a useful reference for cosmological perturbations in teleparallel Horndeski gravity, complementing the earlier gauge-specific analysis of Ref. [4] and giving practitioners gauge-invariant results and stability conditions for a broad class of models. The explicit comparison with subclasses (GR, f(T), Horndeski, NGR) in Table I is a valuable summary. However, the advertised catalogue is undermined by the internal inconsistency between the gauge-invariant scalar sector and the Newtonian/synchronous gauge results, whose resolution is essential before the 9-DoF claim can be considered established. The paper also has the strength of connecting to published stability conditions for tensor and vector sectors and of providing an openly available repository for the lengthy coefficients, although the dependence on a non-archival repository is a weakness.

major comments (3)
  1. [Appendix B.1.c and B.1.d, Eqs. (B7)-(B10); compare Section IV.A, Eqs. (69)-(73)] The Newtonian gauge action (B7)-(B8) and the synchronous gauge action (B9)-(B10) each yield three dynamical scalar modes after eliminating auxiliary fields, whereas the gauge-invariant analysis in Section IV.A yields exactly two propagating scalars, Ψ1 and Ψ2, after eliminating X2 and X4. The manuscript acknowledges this discrepancy but states only that it does not occur for widely used subclasses and defers resolution to future work. A gauge fixing cannot change the number of physical degrees of freedom, so the discrepancy must be resolved in the present manuscript: either the third mode is pure gauge due to residual gauge freedom (which must be demonstrated explicitly), or the gauge-invariant elimination of X2 and X4 misses a propagating mode (which would invalidate the central DoF count). Merely noting the issue and deferring it is not sufficient for a paper whose central claim is the 9-DoF count and whose Appendix B is advertised as a catalogue.
  2. [Section III and Conclusion, around Eq. (49)] The perturbation calculation fixes the spin connection to the Weitzenböck gauge and perturbs only the tetrad components according to Eq. (49). The Conclusion explicitly limits the analysis to the Weitzenböck gauge for the spin connection. If the spin connection acquires perturbations that carry physical degrees of freedom, the derived actions and the 9-DoF count would be incomplete. This is a legitimate scope limitation, but it must be stated more prominently in the abstract and in the statement of the central claim, because the phrase "full BDLS framework" in the Conclusion overstates the validity of the result under this assumption.
  3. [Appendix B, Eqs. (B3), (B6), (B8), (B10)] The final reduced actions in the flat, unitary, Newtonian, and synchronous gauges are stated with coefficients δ˜Fi, δ˜Ui, δ˜Ni, and δ˜Si all deferred to Ref. [37], an external GitHub repository. For a catalogue paper whose stated purpose is to give practitioners ready-to-use perturbed actions in different gauges, this is a major omission: the central results cannot be checked or used without downloading a repository that is not part of the archival record. The coefficients should be included in the manuscript or in an arXiv ancilliary file, or the verbal claims about the gauge catalogue must be substantially weakened.
minor comments (7)
  1. [Title and abstract] The title header contains a typo: "g ravity" should read "gravity", and the title itself should be checked for spacing.
  2. [Introduction, first paragraph] The phrase "For several deacdes" is a typo for "For several decades", and the prose would benefit from a careful proofreading pass.
  3. [Section II.B, paragraph after Eq. (28)] The word "representes" should be "represents".
  4. [Section III.B, Eq. (52)] The action is written in position space with terms such as β/a^2 ∇^2(...), but later Fourier transforms are used; it would help to state explicitly the sign conventions for ∇^2 and the Fourier normalization early in the section.
  5. [Section III.D, Eq. (58)] The vector action uses both ∇v and (∇v)^2 where v likely denotes a vector magnitude; the notation is ambiguous and should define v, w, V as vectors, e.g., v = v_i, and clarify that ∇ denotes the spatial gradient.
  6. [Appendix B.1.d, Eq. (B5h)] There is a typo in the coefficient U8: "GTeleTtvec" should likely be "GTele,Tvec" or "GTele,T Tvec" depending on the intended derivative; this should be corrected.
  7. [Appendix B, general] The final coefficients of the gauge-fixed actions are only available via an external GitHub repository; even if the repository is retained, it would be preferable to include a stability or version identifier for the exact version used.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the gauge-invariant perturbation derivation is self-contained, and the Newtonian/synchronous gauge discrepancy is a consistency gap rather than a circular reduction.

full rationale

The derivation chain is self-contained in the sense relevant to circularity. The paper starts from the externally defined BDLS action (Eq. 22) and prior field equations (Ref. [16]), perturbs the tetrad according to the SVT decomposition (Eq. 49), constructs gauge-invariant variables from the gauge transformations (Eqs. 51 and 53), expands the action to quadratic order (Eqs. 52, 54, 55, 58, 61), solves the auxiliary-field constraints (Eqs. 67-68 and 84), and then reads off propagating DoFs and stability coefficients (Eqs. 69-91). No parameter is fitted to data and renamed a prediction, and no dynamical variable or action coefficient is defined in terms of the claimed output of nine DoFs. The self-citations are not load-bearing in a circular sense: Ref. [13] is used as a consistency check, Ref. [4] is prior gauge-specific work that the present gauge-invariant calculation generalizes, and Ref. [37] supplies coefficients of gauge-fixed actions rather than the logical premise of the central DoF count. The unresolved mismatch noted in Appendix B, where Newtonian and synchronous gauges give three scalar DoFs while the gauge-invariant reduction gives two, is explicitly acknowledged by the authors as requiring further investigation. That is a consistency and completeness concern, not a definitional equivalence between input and output, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free numerical parameters are fitted. The core assumptions are the BDLS Lagrangian, the FLRW background with a vanishing spin connection, the completeness of the tetrad decomposition, and the high-k limit for stability. These are stated in the text but are not independently verified within the paper.

assumptions (4)
  • domain assumption The BDLS action, Eq. (22), is the correct teleparallel analog of Horndeski gravity with second-order field equations.
    Invoked in Sec. II and used throughout; the field equations are not rederived here, but the action is accepted from Refs. [15, 16, 18].
  • domain assumption The background is a flat FLRW spacetime with the diagonal tetrad in the Weitzenböck gauge and vanishing spin connection.
    Sec. II.B and Sec. III; used to construct the background Friedmann equations and the perturbation of Eq. (47).
  • domain assumption The perturbed tetrad of Eq. (49) provides the most general linear perturbation respecting spatial rotations.
    Sec. III; if the decomposition misses or double-counts modes, the DoF count and the actions would be incomplete.
  • domain assumption The high-k limit is sufficient for the ghost and gradient stability analysis.
    Stated in Sec. IV; the full reduced scalar action is omitted and stability is assessed at leading order in momentum k.

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Pith. "Pith review of Gauge invariant perturbations in teleparallel Horndeski gravity." pith.science (2026). https://pith.science/paper/6PLGP3QS

@misc{pith2026241201349,
  author       = {Pith},
  title        = {Pith review of: Gauge invariant perturbations in teleparallel Horndeski gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PLGP3QS}},
  note         = {Machine review of arXiv:2412.01349}
}
read the original abstract

We present in the form of a catalogue of the cosmological perturbations within the Bahamonde- Dialektopoulos-Levi Said (BDLS) theory, which serves as the teleparallel counterpart of Horndeski gravity. To understand structure formation in cosmological models, it is essential to study both the background and perturbative aspects of their cosmology. While extensive analysis of both Horndeski gravity and its teleparallel analog exists in the literature, a quantitative understanding requires a detailed examination of their cosmological perturbations. We review here all the different gauges for the scalar, vector and tensor perturbations of a cosmological background up to second order and we hope this will help people who work with observations, to incorporate it in existing codes.

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