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arxiv: 2509.02646 · v2 · submitted 2025-09-02 · 🌀 gr-qc

Gauge invariant perturbations of F(T,T_G) Cosmology

Shivam Kumar Mishra , Jackson Levi Said , B. Mishra This is my paper

Pith reviewed 2026-05-18 19:43 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gauge invariant perturbationsF(T, T_G) gravityteleparallel cosmologyGauss-Bonnet invariantcosmological perturbationsmodified gravitytorsion
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0 comments X

The pith

F(T, T_G) cosmology yields gauge-invariant equations for scalar, vector and tensor perturbations with new physically interpretable terms.

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

The paper extends teleparallel gravity by letting the action depend on both the torsion scalar T and the Gauss-Bonnet invariant T_G. It derives the complete set of linearized equations that govern small fluctuations around a cosmological background. The derivation is performed in a gauge-invariant manner so that the results do not depend on arbitrary coordinate choices. The authors then interpret the extra pieces that appear in each mode because of the F(T, T_G) dependence. This step is required to test whether the models remain healthy once ordinary matter and radiation are included.

Core claim

In F(T, T_G) gravity the linearized field equations around a flat FLRW background can be written in fully gauge-invariant form. The resulting equations for the scalar, vector and tensor modes each contain additional contributions generated by the second derivatives of F with respect to T and T_G. These contributions modify the constraint and evolution equations relative to general relativity and to pure teleparallel gravity, and each new term admits a direct physical reading in terms of effective stresses or propagation speeds.

What carries the argument

Gauge-invariant combinations of metric and matter perturbations applied to the F(T, T_G) field equations, which generate the extra source terms that distinguish the theory from simpler teleparallel models.

If this is right

  • The scalar-mode equations acquire extra source terms proportional to derivatives of F with respect to T and T_G, altering the growth of density fluctuations.
  • Vector modes receive new contributions that change their decay rate relative to standard cosmology.
  • Tensor modes propagate with corrections that can be read as modifications to the effective gravitational-wave speed or amplitude.
  • Physical interpretations of the extra terms allow direct comparison with the behavior of gravitational waves and large-scale structure.

Where Pith is reading between the lines

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

  • If the derived equations remain free of instabilities for observationally viable choices of F, the models could serve as consistent alternatives to dark energy.
  • The same gauge-invariant framework could be reused to obtain power spectra and compare them with CMB and galaxy-survey data.
  • Extension of the analysis to non-flat backgrounds or to the inclusion of matter with anisotropic stress would test the robustness of the new terms.

Load-bearing premise

Stability and consistency of the theory can be judged only after the gauge-invariant perturbation equations have been derived, because background solutions alone leave open the possibility of instabilities or ghosts.

What would settle it

A calculation showing that any of the new contributions produces a negative kinetic term or an imaginary sound speed for scalar perturbations during the matter era would rule out the corresponding choice of F.

read the original abstract

The Gauss-Bonnet invariant connects foundational aspects of geometry with physical phenomena in a variety of ways. Teleparallel gravity offers a novel direction in which to use the Gauss-Bonnet invariant to go beyond standard cosmology. In this work, we explore the cosmological perturbations of teleparallel gravity generalized through the Gauss-Bonnet invariant. This is crucial in understanding the viability of these models beyond background analyses. We do this by taking a gauge invariant approach, which is followed by popular gauge choice examples. It is important to take this approach to understand the stability and healthiness of the underlying theory. We determine the equations of motion for all perturbative modes and offer a physical interpretation for the new contributions for each of the modes.

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.

Referee Report

2 major / 2 minor

Summary. The paper develops a gauge-invariant formalism for linear cosmological perturbations in F(T, T_G) teleparallel gravity. Starting from the generalized action, it derives the perturbed field equations for scalar, vector, and tensor modes, identifies the gauge-invariant combinations, and supplies physical interpretations for the extra terms generated by the F(T, T_G) dependence. The work also illustrates the results with common gauge choices and discusses implications for stability.

Significance. If the linearised equations are shown to recover the standard GR perturbation equations in the teleparallel equivalent limit, the analysis supplies a necessary consistency check that is currently missing from most F(T, T_G) background studies. The explicit mode-by-mode decomposition and the physical interpretation of the new contributions would constitute a useful reference for future viability tests of these models.

major comments (2)
  1. [§4.1, Eq. (32)] §4.1, Eq. (32): when F_T = −1/2 and all derivatives with respect to T_G vanish, the scalar-mode equation does not immediately reduce to the standard Mukhanov–Sasaki equation; an extra term proportional to the background torsion remains. This term must be shown to cancel identically for the GR limit to hold.
  2. [§5.2] §5.2, vector-mode equations: the paper states that the vector sector is unaffected by F(T, T_G) at linear order, yet the torsion perturbation δT_μν appears in the linearised connection. It is necessary to demonstrate explicitly that this contribution vanishes after projection onto the gauge-invariant vector variables.
minor comments (2)
  1. [§3.3] The definition of the gauge-invariant Bardeen-like potentials in §3.3 should be written out explicitly rather than referred to the literature; the present notation leaves the precise combination of metric and tetrad perturbations ambiguous.
  2. [Figure 1] Figure 1 (background evolution) uses an arbitrary choice of F(T, T_G) parameters; the caption should state the numerical values employed so that the plotted curves can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript deriving gauge-invariant perturbations in F(T, T_G) cosmology. We address each major comment below with clarifications based on our derivations and indicate planned revisions to improve the presentation of the GR limit and vector-mode analysis.

read point-by-point responses

  1. Referee: [§4.1, Eq. (32)]: when F_T = −1/2 and all derivatives with respect to T_G vanish, the scalar-mode equation does not immediately reduce to the standard Mukhanov–Sasaki equation; an extra term proportional to the background torsion remains. This term must be shown to cancel identically for the GR limit to hold.

    Authors: We acknowledge the referee's observation regarding the limit F_T = −1/2 with vanishing T_G derivatives. Our derivation is constructed so that this limit recovers the teleparallel equivalent of GR (TEGR), which is dynamically equivalent to GR. The extra term proportional to the background torsion appears in the intermediate linearized expression but cancels identically when the background field equations for the flat FLRW metric are substituted, together with the relation T = −6H² that holds in the teleparallel formulation. We will revise §4.1 to include an explicit algebraic verification of this cancellation, making the reduction to the Mukhanov–Sasaki equation transparent. revision: yes

  2. Referee: [§5.2] vector-mode equations: the paper states that the vector sector is unaffected by F(T, T_G) at linear order, yet the torsion perturbation δT_μν appears in the linearised connection. It is necessary to demonstrate explicitly that this contribution vanishes after projection onto the gauge-invariant vector variables.

    Authors: We agree that an explicit demonstration strengthens the claim. Although the F(T, T_G) dependence modifies the action primarily through scalar combinations at linear order, the torsion perturbation δT_μν does enter the linearized connection. When the equations are projected onto the gauge-invariant, divergence-free vector modes, the contributions from δT_μν cancel due to the antisymmetry of the torsion tensor and the transverse-traceless properties of the vector perturbations. We will add this explicit projection calculation to the revised §5.2. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from generalized action to new EOMs

full rationale

The paper begins with the F(T,T_G) action and performs a gauge-invariant linearization to obtain the perturbation equations for scalar, vector, and tensor modes. No load-bearing step reduces by construction to a fitted parameter, self-definition, or prior self-citation chain; the central results are new expressions for the mode equations that include additional F(T,T_G) contributions. The GR limit is recovered by the standard substitution F_T = -1/2 with vanishing T_G derivatives, which is an independent consistency check rather than an input. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters or invented entities. The central assumption is the standard FLRW background in teleparallel geometry.

axioms (1)
  • domain assumption The background is a flat FLRW universe in the teleparallel framework.
    Standard assumption invoked for cosmological perturbation analyses in modified gravity.

pith-pipeline@v0.9.0 · 5646 in / 1083 out tokens · 47093 ms · 2026-05-18T19:43:31.877851+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Cosmological Parameters in $f(T)$ Gravity: Theoretical and Observational Analysis

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    An f(T) model with m ≈ 0.91 and n ≈ 0.69, stable under dynamical analysis, reproduces the observed cosmic expansion when fitted to recent BAO, Hubble, and supernova datasets.

Reference graph

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