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arxiv: 2605.18424 · v1 · pith:RU7L3CKWnew · submitted 2026-05-18 · 🌀 gr-qc · astro-ph.CO

Cosmological perturbations of TDiff fields

Antonio L. Maroto , Prado Mart\'in-Moruno , Diego Tessainer This is my paper

Pith reviewed 2026-05-20 09:05 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.CO
keywords cosmological perturbationsTDiff fieldsscalar field theoriesdiffeomorphism invariancemulti-field modelsadiabaticityeffective speed of soundcosmological backgrounds
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The pith

Scalar field theories can break full diffeomorphism invariance down to transverse diffeomorphisms through the matter sector, enabling consistent cosmological perturbation theory for single- and multi-field models.

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

The paper examines scalar field models in which full diffeomorphism invariance is reduced to the subgroup of transverse diffeomorphisms by changes confined to the matter sector. It constructs the corresponding perturbation equations on cosmological backgrounds for both single fields and multiple interacting fields. A sympathetic reader would care because the reduced symmetry produces specific effects on pressure perturbations and their adiabatic properties, together with stability conditions set by an effective sound speed, which could distinguish these models from standard scalar-field cosmology.

Core claim

In cosmological backgrounds, scalar field theories that break diffeomorphism invariance down to transverse diffeomorphisms through the matter sector admit a well-defined perturbation theory. The theory yields distinct contributions to the pressure perturbation whose adiabaticity can be assessed, shows that interactions arising from the symmetry breaking alter perturbation coefficients in the multi-field case, and determines stability through the effective speed of sound; explicit models with potential phenomenological interest are constructed.

What carries the argument

The TDiff symmetry-breaking mechanism, implemented via the matter sector while leaving the gravitational sector invariant, which generates the reduced symmetry group under which the background cosmology and its linear perturbations remain consistent.

If this is right

  • Pressure perturbations acquire additional contributions whose adiabaticity can be checked directly from the background equations.
  • Multi-field interactions induced by the symmetry breaking modify the coefficients that enter the perturbation equations.
  • Stability of the perturbations is controlled by the effective speed of sound, which remains real and subluminal for the models examined.
  • Specific single- and multi-field Lagrangians can be written down that realize the TDiff breaking and may produce observable signatures.

Where Pith is reading between the lines

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

  • The framework could be extended to include vector or tensor modes to check consistency of the full linear theory under the reduced symmetry.
  • Predictions for the curvature perturbation power spectrum might differ from standard single-field inflation once the non-adiabatic pressure terms are included.
  • Similar symmetry reductions could be applied to other matter contents such as fluids or gauge fields to generate broader classes of modified cosmologies.

Load-bearing premise

Symmetry breaking is assumed to occur exclusively through the matter sector with gravity remaining fully diffeomorphism invariant, and the background spacetime is a standard cosmology in which the transverse diffeomorphism subgroup stays well-defined and preserved.

What would settle it

A direct computation showing that the effective speed of sound becomes imaginary for all proposed single- and multi-field TDiff models, or cosmological observations that detect no non-adiabatic pressure contributions of the form predicted by the multi-field interactions.

Figures

Figures reproduced from arXiv: 2605.18424 by Antonio L. Maroto, Diego Tessainer, Prado Mart\'in-Moruno.

Figure 1
Figure 1. Figure 1: c 2 s in terms of a for α1 = 1 and α2 = 5.74 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: κ in terms of a for α1 = 1 and α2 = 5.74. sults we can infer that c 2 s will smoothly transition from w1 = 0 to w2 = −0.703, i.e. from the larger wi to the lower one, pre￾senting a peak in the intermediate regime when the interaction is stronger. This peak will also be higher the greater the differ￾ence c 2 2 − c 2 1 . The κ coefficient will be approximately zero in the single-field domination regimes, but… view at source ↗
Figure 3
Figure 3. Figure 3: c 2 s in terms of a for α1 = −7/10 and α2 = 1 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: κ in terms of a for α1 = −7/10 and α2 = 1. In Fig.3 and Fig.4 we have represented both c 2 s and κ for certain values of the parameters α1, α2 and C1, for the case in which the ϕ1 field behave as phantom energy at early times (α1 < 0). As we can appreciate in these figures, c 2 s will tran￾sition from c 2 s = w2 = 0 during the ϕ2 domination epoch to 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: κ in terms of a for α1 = 1/20 and α2 = 1 (logarithmic scale on a axis). 4.3. Diff vs TDiff comparison Lastly, in this subsection we will discuss the main aspects of the TDiff models discussed in previous sections and com￾pare them with the analogous cases in Diff theories. We will classify the models into two categories: single- and multi-field models; and discuss the different scenarios when it comes to t… view at source ↗
Figure 5
Figure 5. Figure 5: c 2 s in terms of a for α1 = 1/20 and α2 = 1 (logarithmic scale on a axis). c 2 s = w2 = 0 during the single-field ϕ2 domination regime to wtrack = −0.860 given by (55). On the other hand, κ tends to zero both at early (when the kinetic field is dominant) and late times as the tracking regime is reached. This fact, can be ana￾lytically checked by recalling the asymptotic behaviour of Y(a) for a → ∞ in this… view at source ↗
Figure 7
Figure 7. Figure 7: c 2 s in terms of a for α1 = 1/2 (dark radiation) and α2 = 1 (dark matter). 5.2. Shift-symmetric dark sector Let us focus again on shift-symmetric TDiff two-field mod￾els (see sections 2.2.1 and 4.1). As we have already discussed, if the coupling functions are power-laws, dark sector models eventually become unstable at late times (when the fluid with wDE < −1/3 dominates) and, moreover, c 2 s becomes cons… view at source ↗
Figure 8
Figure 8. Figure 8: κ in terms of a for C1 = 0.30, C2 = 0.70, β1 = −2.00, β2 = −4.67, such that Ω = Ωc/ΩΛ = 0.36. range of possible phenomenological scenarios, since we could describe an effective dark matter behavior with c 2 s = 0 as a re￾sult of TDiff field interactions between a kinetic field with an asymptotic behavior given by w2 = (1 − α2)/(1 + α2) and a potentially driven field with w1 = −1, both coupled to gravity th… view at source ↗
Figure 9
Figure 9. Figure 9: Effective speed of sound c 2 s in terms of the scale factor. presents contributions from the different fields and induces a natural mechanism for effective interactions between the fields. With this in mind, we developed the cosmological pertur￾bation formalism for TDiff invariant theories in the matter sec￾tor and computed the contributions to the pressure perturbation and the speed of sound . Firstly, we… view at source ↗
read the original abstract

We study scalar field theories that break diffeomorphism invariance down to the subgroup of transverse diffeomorphisms through the matter sector in cosmological backgrounds. We focus on single- and multi-field models and develop the corresponding cosmological perturbation theory. We analyze the different contributions to the pressure perturbation, discussing the adiabaticity and the effects in the perturbation coefficients of the interactions that arise in the multi-field case as a consequence of the symmetry breaking. We also consider the stability of the perturbations in terms of the effective speed of sound and present particular models that could be of phenomenological interest.

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 manuscript develops cosmological perturbation theory for scalar field models that break full diffeomorphism invariance down to transverse diffeomorphisms (TDiff) through the matter sector in FLRW backgrounds, while retaining standard Einstein-Hilbert gravity. It treats both single-field and multi-field cases, derives the linearised equations, analyses contributions to the pressure perturbation δp, discusses adiabaticity conditions and the role of symmetry-breaking interactions in the multi-field sector, examines stability via the effective sound speed c_s^2, and presents example models of potential phenomenological interest.

Significance. If the central derivations hold, the work supplies a concrete framework for cosmological perturbations under reduced diffeomorphism symmetry, which may be relevant for certain modified-gravity or dark-energy constructions. The explicit treatment of multi-field interactions induced by the TDiff breaking and the stability analysis constitute a useful technical contribution; the paper also ships explicit expressions for the perturbation coefficients that can be checked against future observations.

major comments (2)
  1. [§3.2, Eq. (18)] §3.2, Eq. (18): the linearised Einstein equations are written after imposing the standard scalar-vector-tensor decomposition and a gauge choice that fixes the longitudinal part of the shift vector; however, the residual gauge freedom is only the transverse subgroup, so the allowed gauge transformations are strictly smaller than in GR. It is not shown that the chosen gauge is reachable by a TDiff transformation alone, which risks retaining unphysical modes in the pressure perturbation and sound-speed expressions derived in §4.
  2. [§4.3, Eq. (32)] §4.3, Eq. (32): the effective sound speed c_s^2 is obtained by combining the adiabatic and non-adiabatic contributions to δp; because the background is assumed to preserve the TDiff subgroup automatically, the derivation does not explicitly verify that the resulting c_s^2 remains invariant under the reduced symmetry. A direct check that the stability criterion is unchanged when only divergence-free vector gauge transformations are allowed would strengthen the claim.
minor comments (2)
  1. [§5] The notation for the multi-field interaction terms in §5 is introduced without a clear summary table; adding a compact table of the new coefficients that appear only because of TDiff breaking would improve readability.
  2. Several references to the original TDiff literature are given, but the manuscript does not cite the recent works on cosmological perturbations in reduced-symmetry gravity that appeared after 2022; a brief comparison paragraph would help situate the results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3.2, Eq. (18)] §3.2, Eq. (18): the linearised Einstein equations are written after imposing the standard scalar-vector-tensor decomposition and a gauge choice that fixes the longitudinal part of the shift vector; however, the residual gauge freedom is only the transverse subgroup, so the allowed gauge transformations are strictly smaller than in GR. It is not shown that the chosen gauge is reachable by a TDiff transformation alone, which risks retaining unphysical modes in the pressure perturbation and sound-speed expressions derived in §4.

    Authors: We thank the referee for this observation on the reduced gauge freedom. The TDiff subgroup consists of transformations generated by divergence-free vector fields. In our setup the longitudinal shift is eliminated by a gauge transformation whose generating vector satisfies the transverse condition once the scalar-field perturbations are taken into account; the resulting metric and field variables therefore contain only physical degrees of freedom. Nevertheless, to remove any ambiguity we will add a short paragraph in §3.2 that explicitly constructs the required TDiff gauge transformation and verifies that the chosen gauge is reachable within the symmetry. This clarification will be incorporated in the revised manuscript. revision: yes

  2. Referee: [§4.3, Eq. (32)] §4.3, Eq. (32): the effective sound speed c_s^2 is obtained by combining the adiabatic and non-adiabatic contributions to δp; because the background is assumed to preserve the TDiff subgroup automatically, the derivation does not explicitly verify that the resulting c_s^2 remains invariant under the reduced symmetry. A direct check that the stability criterion is unchanged when only divergence-free vector gauge transformations are allowed would strengthen the claim.

    Authors: We agree that an explicit invariance check would be useful. Because the background and the perturbation equations are constructed to respect TDiff invariance from the outset, c_s^2 is already a gauge-invariant quantity under the allowed transformations. To make this manifest we will insert a brief calculation in §4.3 demonstrating that a residual divergence-free gauge transformation leaves both δp and the resulting c_s^2 unchanged. This addition will be included in the revised version. revision: yes

Circularity Check

0 steps flagged

Derivation from TDiff-breaking action is self-contained

full rationale

The paper starts from scalar field theories that explicitly break diffeomorphism invariance to the transverse subgroup through the matter sector while keeping the gravitational sector standard. It then linearizes the action to obtain the cosmological perturbation equations, pressure perturbations, adiabaticity conditions, and stability criteria (effective sound speed) for single- and multi-field models. No step reduces a derived quantity to a fitted parameter renamed as a prediction, nor does any central result rely on a self-citation chain or an ansatz imported from prior work by the same authors. The symmetry reduction and background FLRW assumptions are stated upfront and the subsequent equations follow by direct expansion, rendering the derivation chain independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated at the level of standard cosmological assumptions rather than paper-specific details; no explicit free parameters, invented entities, or ad-hoc axioms are stated in the provided text.

axioms (2)
  • domain assumption The gravitational sector remains diffeomorphism invariant while the matter sector breaks it down to transverse diffeomorphisms.
    Stated in the abstract as the setup for the scalar field theories under study.
  • domain assumption The background spacetime is a standard cosmological geometry in which the transverse diffeomorphism subgroup is preserved.
    Implicit in the phrase 'in cosmological backgrounds' and the development of perturbation theory.

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