Evading Cosmological Strong Coupling in Non-minimally Coupled Vector Gravity

Antonio De Felice; Atsushi Naruko; Nagisa Hiroshima; Seishi Enomoto

REVIEW 3 major objections 5 minor 19 references

A divergence-squared vector interaction lifts the zero-speed scalar mode and restores healthy high-frequency dynamics in non-minimally coupled Proca cosmology.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-14 06:52 UTC pith:RSCAUYWH

load-bearing objection The quadratic rescue of the c_s^{2}=0 mode is real and cleanly shown; the title still outruns the undetermined EFT cutoff the authors themselves flag. the 3 major comments →

arxiv 2607.11125 v1 pith:RSCAUYWH submitted 2026-07-13 hep-th astro-ph.COgr-qchep-ph

Evading Cosmological Strong Coupling in Non-minimally Coupled Vector Gravity

Antonio De Felice , Seishi Enomoto , Nagisa Hiroshima , Atsushi Naruko This is my paper

classification hep-th astro-ph.COgr-qchep-ph

keywords Proca theoriesnon-minimal vector-tensor gravityscalar perturbationsstrong couplingde Sitter fixed pointghost-free cosmologyvector condensate

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Non-minimally coupled Proca theories on an FLRW background were known to produce an extra scalar mode with identically vanishing sound speed, a signature of scale-dependent strong coupling that threatens the effective-field-theory description of vector dark energy or inflation. This Letter shows that adding the derivative interaction controlled by ξ₃—the square of the vector divergence—lifts that degeneracy. In an open region of the four-parameter space the scalar kinetic matrix becomes positive definite and the two high-frequency propagation speeds are real and positive, so the specific quadratic-order strong-coupling signature is removed. The same region contains an explicit, stable de Sitter fixed point with non-vanishing temporal condensate, surrounded by a finite basin of attraction in which the no-ghost and high-frequency stability conditions hold. The authors also map a subtle non-uniform limit: any fixed non-zero condensate eventually develops a UV ghost, yet that ghost scale is pushed to ever higher momenta as the condensate approaches zero, so the exactly vanishing-condensate branch remains regular for ξ₃ < 0. Whether the ghost lies outside the theory’s cutoff is left for a future determination of that cutoff.

Core claim

The operator ξ₃(∇_μ X^μ)² changes the scalar kinetic structure of non-minimally coupled Proca gravity so that, in an open region of (ξ₁,ξ₂,ξ₃,ξ₄) with non-vanishing temporal condensate A₀, the kinetic matrix is positive definite and the high-frequency squared speeds c_{s±}² are real and positive. At quadratic order this eliminates the c_s² = 0 strong-coupling signature. An explicit stable de Sitter fixed point with A₀ ≠ 0 lies inside that healthy region.

What carries the argument

The four-parameter action containing the new term −(ξ₃/2)(∇_μ X^μ)²; after integrating out the non-dynamical metric constraints, the resulting 2 × 2 scalar kinetic matrix K_{ij} and the high-k WKB dispersion relation that together diagnose ghosts and gradient instabilities.

Load-bearing premise

That quadratic positivity of the kinetic matrix and real high-frequency speeds are enough to claim the strong-coupling problem is evaded, even though the paper never computes the EFT cutoff and shows that any fixed non-zero condensate develops a UV ghost whose scale may still lie inside that cutoff.

What would settle it

An explicit calculation of the EFT cutoff around a healthy A₀ ≠ 0 background that places the UV ghost scale below the cutoff, or a numerical evolution of scalar perturbations that reintroduces vanishing or imaginary high-k speeds inside the claimed open parameter region.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The quadratic rescue of the c_s^{2}=0 mode is real and cleanly shown; the title still outruns the undetermined EFT cutoff the authors themselves flag.

read the letter

The new piece is concrete: adding the ξ₃(∇_μ X^μ)² term lifts the identically vanishing scalar sound speed that the authors (and others) found in non-minimal Proca. At quadratic order they get a non-degenerate kinetic matrix that can be positive definite, with real positive high-k speeds, and they exhibit an explicit attractive de Sitter point with A₀ ≠ 0 inside that region.

What they do well is the calculation itself. Background equations, constraint elimination, high-k Sylvester conditions (13)–(14), the WKB quartic, tensor/vector checks, and the linear homogeneous stability analysis around de Sitter are consistent. The numerical example (ξ₁=ξ₃=1, ξ₂=0, Ā₀/M_Pl=√0.2, H̄/M_Pl=0.1) is not a singular tuning; they also show a finite basin. The non-uniform A₀→0 vs k→∞ limit is honestly stated rather than papered over. Citation pattern is normal for this group’s line of work; the problem setup is self-cited but the enlarged action is recomputed from scratch.

The soft spot is exactly the one they flag after (17) and in the Conclusions: for any fixed nonzero A₀ the formal UV of det K is ghostly, the ghost scale only recedes as A₀→0, and whether it sits above the EFT cutoff is left open. So the quadratic signature of the old strong-coupling problem is removed, but the title-level claim that cosmological strong coupling is “evaded” still rests on an untested assumption about the cutoff. That is a real gap, not a fatal one for a Letter; it is the natural next calculation.

This is for people already working on generalized Proca / vector dark energy who need to know whether the c_s^{2}=0 obstruction is structural. It is not a broad cosmology paper. I would send it to referees: the quadratic result is solid enough to deserve that scrutiny, and the cutoff caveat is already written into the text. Worth engaging if you care about the viability of non-minimal vector cosmologies.

Referee Report

3 major / 5 minor

Summary. The Letter enlarges non-minimally coupled Proca gravity by a divergence-squared operator ξ₃(∇_μ X^μ)² and recomputes the FLRW background and quadratic scalar, vector, and tensor perturbations. It shows that this interaction lifts the identically vanishing scalar sound speed reported in prior non-minimal Proca analyses: in an open region of (ξ₁,ξ₂,ξ₃,ξ₄) with A₀≠0 the 2×2 scalar kinetic matrix is positive definite (high-k conditions (13)–(14)) and the WKB high-frequency speeds c_{s±}² are real and positive. An explicit de Sitter fixed point (example (23)) is linearly attractive in the homogeneous sector and lies inside a finite basin where the no-ghost and high-k stability diagnostics remain positive. The authors also document a non-uniform A₀→0 limit in which any fixed nonzero condensate develops a formal UV ghost whose onset scale is pushed to higher momenta as A₀→0, while the exact A₀=0 branch has a different kinetic structure that is ghost-free only for ξ₃<0.

Significance. If the quadratic analysis holds, the work shows that the c_s²=0 strong-coupling signature of non-minimal Proca cosmology is not inevitable once the derivative sector is enlarged, and it supplies an explicit healthy de Sitter realization with A₀≠0. That is a concrete, falsifiable advance relative to the obstruction identified in the authors’ earlier work. Strengths include a consistent constraint elimination, Sylvester no-ghost conditions, WKB dispersion relation, tensor/vector checks, homogeneous linearization, and a numerical basin around the fixed point. The main limitation is that the title-level claim of “evading” cosmological strong coupling still depends on an undetermined EFT cutoff relative to the UV ghost scale the paper itself uncovers; the result is therefore best read as a controlled quadratic rescue with a clear remaining viability condition.

major comments (3)

[Title; after Eq. (17); Conclusions] Title, abstract, and the paragraph after Eq. (17) / Conclusions: the title asserts that cosmological strong coupling is “evaded,” yet for any fixed A₀≠0 the high-k determinant (14) (and the small-A₀ limit (17)) is ghostly, with the ghost onset only deferred as A₀→0. The manuscript correctly states that whether this scale lies above the EFT cutoff “depends on an independent determination of the EFT cutoff,” but no estimate of either the ghost momentum or a plausible cutoff is given, even at the explicit point (23). Without that comparison, the title-level claim remains conditional. Please either (i) compute, for the working example, the comoving/physical scale where a kinetic eigenvalue or det K changes sign as a function of A₀ and discuss it against a stated cutoff ansatz, or (ii) rephrase the title and framing to match the quadratic-order scope already used in the abstract (“removes the
[Theory and background, Eq. (1)] Theory, action (1): the operator −ξ₃/2 (∇_μ X^μ)² is introduced as the ingredient that solves the c_s²=0 problem of Ref. [12]. On FLRW the reduced scalar kinetic matrix is 2×2 and the analysis is internally consistent, but the Letter does not address whether this single operator preserves the Proca secondary-constraint structure (no Ostrogradsky ghost / correct DOF count) on a general background, or whether it should be viewed as a truncation of a larger generalized-Proca Lagrangian. Because the interpretation of the model as a controlled EFT enlargement is load-bearing for the claim that the extra scalar is “genuinely dynamical and healthy,” a short DOF/constraint comment (or an explicit embedding into known healthy vector self-interactions) is needed.
[Scalar perturbations, Eqs. (11)–(14); Fig. 1] Scalar sector, schematic kinetic matrix (11) and high-k limits (13)–(14): the no-ghost and speed conditions used to certify the healthy region are high-momentum asymptotics. For the claim that the c_s²=0 strong-coupling signature is removed along cosmological trajectories, it should be checked that, at the example point (23) and nearby basin points of Fig. 1, the kinetic eigenvalues and the WKB speeds remain healthy over the intermediate-k window relevant to the background evolution (not only in the formal k→∞ limit). A brief numerical or analytic intermediate-k diagnostic would close a gap between the UV test and the cosmological statement.

minor comments (5)

[Scalar perturbations] The explicit coefficients of K_ij, L_12, and M_ij are omitted (“We will not present the explicit form”). For reproducibility, either deposit them in the Supplemental Material or give the high-k expansions of all entries used to obtain (13)–(14) and the speeds (24).
[Figure 1] Fig. 1 (bottom): the vertical axis mixes det(K)/a⁶ and c_{s±}²; clarify units/normalization and mark the fixed-point values so that positivity is visually unambiguous across the basin.
[Homogeneous stability; Numerical basin] Notation: A_0, Ā_0, and the dimensionless A_0 ≡ A_0/M_Pl are used in close succession (e.g. around Eqs. (23) and (32)); a single consistent convention would reduce ambiguity.
[Propagation speeds, Eq. (24)] The remark that c_{s+}^2 ≃ 1.14 is not a ghost/tachyonic instability is correct; a one-sentence pointer to how this is distinguished from gradient instability in the dispersion relation (18) would help non-specialist readers.
[Supplemental Material] Supplemental Material: the Hamiltonian/momentum constraints are only schematic (h_i, m_i). Even without full expressions, stating which combinations fix α and β would make the reduction to (10) easier to follow.

Circularity Check

1 steps flagged

Problem statement rests on authors' prior c_s^{2}=0 result; the ξ₃ rescue, kinetic matrix, speeds and de Sitter attractor are recomputed independently from the enlarged action.

specific steps

self citation load bearing [Introduction, paragraph on non-minimally coupled Proca (citation [12])]
"In non-minimally coupled Proca cosmology, an additional scalar mode appears on a smooth Friedmann-Lemaître-Robertson-Walker (FLRW) background with identically vanishing propagation speed, c^{2}_s = 0 or disappears at the level of linear perturbation around the FLRW universe, signaling the strong coupling [12]."

The load-bearing premise that a c_s^{2}=0 mode exists and constitutes a strong-coupling problem is justified solely by citation to prior work whose author list overlaps (De Felice). The present paper never re-derives that mode for the ξ₃=0 truncation; it only shows that adding ξ₃ lifts it. The circularity is confined to problem setup and does not force the healthy-region claims.

full rationale

The paper's central positive claims (positive-definite K_ij for A₀≠0, real positive high-k c_{s±}^{2}, attractive de Sitter fixed point inside an open healthy region) are obtained by direct expansion of the four-parameter action (1), reduction of the quadratic scalar action (10), high-k limits (13)–(14), WKB quartic (18), and linearization of the background equations around the branch (6). These steps do not assume the conclusion and contain no fitted parameters renamed as predictions. The sole mild circularity is motivational: the existence of the zero-speed mode that is being “evaded” is imported from the authors’ own prior analysis [12] rather than re-derived. That citation is not load-bearing for the new stability window, so the overall circularity remains low (score 2). No uniqueness theorems, ansatz smuggling, or self-definitional reductions appear.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 1 invented entities

The central claim rests on a standard FLRW plus homogeneous temporal vector ansatz, classical quadratic perturbation theory as the diagnostic of strong coupling, and a four-parameter non-minimal Proca-like action with one added divergence-squared operator. No new particles or forces are postulated beyond that operator. The healthy de Sitter example depends on hand-chosen numerical values of the couplings and background condensate; the viability claim for small A₀ further depends on an undetermined EFT cutoff.

free parameters (3)

ξ₁, ξ₂, ξ₃, ξ₄ (non-minimal and mass couplings)
Four dimensionless (or mass-scaled) couplings that define the theory; the healthy region is an open set in this space, and the explicit example fixes ξ₁=ξ₃=1, ξ₂=0 by hand.
Ā₀/M_Pl and H̄/M_Pl at the de Sitter example
Background condensate and Hubble values chosen as √0.2 and 0.1 to illustrate simultaneous no-ghost, positive speeds, and attractor behavior; not predicted from a deeper principle.
Λ and ξ₄ m_X² on the de Sitter branch
Fixed by the branch conditions (27)–(28) once ξ_i, Ā₀, and H̄ are chosen; for the example, μ_X/M_Pl²=-0.12 and Λ/M_Pl²=0.033.

axioms (5)

domain assumption Classical quadratic cosmological perturbation theory on FLRW diagnoses the strong-coupling problem via vanishing sound speed / degenerate kinetic matrix.
Used throughout the Introduction and Scalar perturbations sections; follows the diagnostic of Refs. [12–15] rather than a non-perturbative or loop-level analysis.
domain assumption Spatially flat FLRW metric with homogeneous temporal vector profile X_μ dx^μ = -N A₀(t) dt is the relevant cosmological background.
Stated in Theory and background; standard for isotropic vector cosmologies but excludes anisotropic or inhomogeneous condensates.
domain assumption Absence of ghosts and gradient instabilities at quadratic order, plus homogeneous attractor behavior, is necessary for a viable cosmology.
Explicit conservative stance in the Introduction; the paper does not adopt the unitary-ghost loopholes of Refs. [6–8].
standard math High-momentum WKB analysis of the reduced 2×2 scalar system correctly captures UV propagation speeds.
Propagation speeds section and Supplemental derivation of Eq. (18).
ad hoc to paper The operator ξ₃(∇_μ X^μ)² is an allowed derivative self-interaction that can be added while remaining within the intended Proca-like EFT.
Introduced as “the new term, which solves the problems found in [12]”; its UV completion and cutoff impact are not derived here.

invented entities (1)

Four-parameter non-minimally coupled vector-tensor model with explicit (∇_μ X^μ)² term no independent evidence
purpose: Provide a concrete action in which the zero-speed scalar mode can be lifted and a healthy A₀≠0 de Sitter cosmology exhibited.
The individual operators are standard in generalized Proca / vector-tensor literature; the specific combination and the claim that ξ₃ alone cures the c_s²=0 pathology are the paper’s working model, not a new particle species.

pith-pipeline@v1.1.0-grok45 · 14199 in / 3938 out tokens · 45087 ms · 2026-07-14T06:52:49.715092+00:00 · methodology

0 comments

read the original abstract

Recent analyses of Proca theories with non-minimal curvature couplings have uncovered an additional scalar degree of freedom with an identically vanishing propagation speed, $c_s^2=0$, signaling a scale-dependent strong-coupling problem. In this {\it Letter}, we show that an additional derivative interaction can lift this zero-speed degeneracy and restore a non-degenerate quadratic dynamics for the scalar perturbations. In an open region of parameter space, the scalar kinetic matrix is positive definite and the high-frequency propagation speeds are real and positive. At quadratic order, this removes the specific signature of strong coupling associated with the $c_s^2=0$ mode. We also uncover a non-uniform limit as the temporal vector condensate approaches zero: at fixed nonzero $A_0$, the formal extreme-ultraviolet regime develops a ghost, while the kinetic structure of the exactly vanishing-condensate branch is different. The momentum scale at which this ghost appears is pushed toward increasingly high values as $A_0\to0$. Whether the ghost scale ultimately lies above the EFT cutoff depends on an independent determination of the EFT cutoff. Finally, we identify a stable de Sitter fixed point with $A_0\neq0$, surrounded by a finite region in which the scalar no-ghost and high-frequency stability conditions remain satisfied.

Figures

Figures reproduced from arXiv: 2607.11125 by Antonio De Felice, Atsushi Naruko, Nagisa Hiroshima, Seishi Enomoto.

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 14 linked inside Pith

[1]

= 0,(32) whereµ X ≡ξ 4m2 X,Ξis defined in Eq. (15), and d3 =ξ 3 µX A2 0(4ξ1 + 2ξ2 +ξ 3)−2Λ(2ξ 1 +ξ 2) ,(33) d2 = 3µX A3 0 ξ3(8ξ1 + 5ξ2 + 3ξ3) + 3(2ξ1 +ξ 2)2 + 2A0 µX ξ3 −Λ ξ3(ξ2 −2ξ 1) + 3(2ξ1 +ξ 2)2 , (34) d1 = 3 n µX A4 0 [6ξ2(2ξ1 +ξ 2) +ξ 3(22ξ1 + 16ξ2 + 9ξ3)] + 2ΛA2 0 [6ξ1(2ξ1 +ξ 2) +ξ 3(2ξ1 −ξ 2)] + 2µX A2 0(6ξ1 + 3ξ2 + 2ξ3) + 4Λξ3 o ,(35) d0 = 3A0 A...
[2]

L. H. Ford, Inflation Driven by a Vector Field, Phys. Rev. D40, 967 (1989)

1989
[3]

Armendariz-Picon, Could dark energy be vector-like?, JCAP07, 007, arXiv:astro-ph/0405267

C. Armendariz-Picon, Could dark energy be vector-like?, JCAP07, 007, arXiv:astro-ph/0405267

Pith/arXiv arXiv
[4]

J. D. Barrow and S. Hervik, Anisotropically inflating universes, Phys. Rev. D73, 023007 (2006), arXiv:gr- qc/0511127

arXiv 2006
[5]

Maleknejad, M

A. Maleknejad, M. M. Sheikh-Jabbari, and J. Soda, Gauge Fields and Inflation, Phys. Rept.528, 161 (2013), arXiv:1212.2921 [hep-th]

Pith/arXiv arXiv 2013
[6]

Beltran Jimenez, R

J. Beltran Jimenez, R. Durrer, L. Heisenberg, and M. Thorsrud, Stability of Horndeski vector-tensor inter- actions, JCAP10, 064, arXiv:1308.1867 [hep-th]

Pith/arXiv arXiv
[7]

Ewasiuk and S

C. Ewasiuk and S. Profumo, Ghost Degrees of Freedom Without Quantum Runaway: Exact Moment Bounds from an Operator Conservation Law, arXiv preprint arXiv:2604.21348 (2026), arXiv:2604.21348 [hep-th]

Pith/arXiv arXiv 2026
[8]

Deffayet, A

C. Deffayet, A. F. Jalali, A. Held, S. Mukohyama, and A. Vikman, Unitary Time Evolution and Vacuum for a Quantum Stable Ghost, arXiv preprint arXiv:2604.21823 (2026), arXiv:2604.21823 [hep-th]

Pith/arXiv arXiv 2026
[9]

Deffayet, A

C. Deffayet, A. F. Jalali, A. Held, S. Mukohyama, and A. Vikman, Quantum mechanics with a ghost: Counterexamples to spectral denseness, arXiv preprint arXiv:2604.21826 (2026), arXiv:2604.21826 [hep-th]

Pith/arXiv arXiv 2026
[10]

Heisenberg, Generalization of the Proca Action, JCAP 05, 015, arXiv:1402.7026 [hep-th]

L. Heisenberg, Generalization of the Proca Action, JCAP 05, 015, arXiv:1402.7026 [hep-th]

Pith/arXiv arXiv
[11]

De Felice, L

A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, and S. Tsujikawa, Cosmology in generalized Proca theories, JCAP06, 048, arXiv:1603.05806 [gr-qc]

Pith/arXiv arXiv
[13]

De Felice and A

A. De Felice and A. Hell, On the cosmological degrees of freedom of Proca field with non-minimal coupling to gravity, JHEP07, 228, arXiv:2503.07454 [gr-qc]

Pith/arXiv arXiv
[14]

De Felice and A

A. De Felice and A. Hell, The non-minimal 3-form cos- mology and the rise of the cuscuton, JHEP11, 132, arXiv:2509.02323 [gr-qc]

arXiv
[15]

Arkani-Hamed, H.-C

N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, Ghost condensation and a consistent in- frared modification of gravity, JHEP05, 074, arXiv:hep- th/0312099

arXiv
[16]

Cheung, P

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

Pith/arXiv arXiv
[17]

C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel.7, 5 (2004), arXiv:gr-qc/0311082

Pith/arXiv arXiv 2004
[18]

Adams, N

A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nico- lis, and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP2006(10), 014, arXiv:hep-th/0602178

Pith/arXiv arXiv
[19]

De Felice, L

A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, S. Tsujikawa, and Y.-l. Zhang, Effective gravitational couplings for cosmological perturbations in general- ized Proca theories, Phys. Rev. D94, 044024 (2016), arXiv:1605.05066 [gr-qc]

Pith/arXiv arXiv 2016
[20]

De Felice, L

A. De Felice, L. Heisenberg, and S. Tsujikawa, Ob- servational constraints on generalized Proca theories, Phys. Rev. D95, 123540 (2017), arXiv:1703.09573 [astro- ph.CO]. 6 SUPPLEMENT AL MA TERIAL Expansion of the action In the flat gauge, R= 0 &E= 0,(39) the scalar metric perturbations are given by ds2 =−N(t) 2(1 + 2α)dt2 + 2N(t)∂iβdtdx i +a 2(t)δijdxidxj...

Pith/arXiv arXiv 2017

[1] [1]

= 0,(32) whereµ X ≡ξ 4m2 X,Ξis defined in Eq. (15), and d3 =ξ 3 µX A2 0(4ξ1 + 2ξ2 +ξ 3)−2Λ(2ξ 1 +ξ 2) ,(33) d2 = 3µX A3 0 ξ3(8ξ1 + 5ξ2 + 3ξ3) + 3(2ξ1 +ξ 2)2 + 2A0 µX ξ3 −Λ ξ3(ξ2 −2ξ 1) + 3(2ξ1 +ξ 2)2 , (34) d1 = 3 n µX A4 0 [6ξ2(2ξ1 +ξ 2) +ξ 3(22ξ1 + 16ξ2 + 9ξ3)] + 2ΛA2 0 [6ξ1(2ξ1 +ξ 2) +ξ 3(2ξ1 −ξ 2)] + 2µX A2 0(6ξ1 + 3ξ2 + 2ξ3) + 4Λξ3 o ,(35) d0 = 3A0 A...

[2] [2]

L. H. Ford, Inflation Driven by a Vector Field, Phys. Rev. D40, 967 (1989)

1989

[3] [3]

Armendariz-Picon, Could dark energy be vector-like?, JCAP07, 007, arXiv:astro-ph/0405267

C. Armendariz-Picon, Could dark energy be vector-like?, JCAP07, 007, arXiv:astro-ph/0405267

Pith/arXiv arXiv

[4] [4]

J. D. Barrow and S. Hervik, Anisotropically inflating universes, Phys. Rev. D73, 023007 (2006), arXiv:gr- qc/0511127

arXiv 2006

[5] [5]

Maleknejad, M

A. Maleknejad, M. M. Sheikh-Jabbari, and J. Soda, Gauge Fields and Inflation, Phys. Rept.528, 161 (2013), arXiv:1212.2921 [hep-th]

Pith/arXiv arXiv 2013

[6] [6]

Beltran Jimenez, R

J. Beltran Jimenez, R. Durrer, L. Heisenberg, and M. Thorsrud, Stability of Horndeski vector-tensor inter- actions, JCAP10, 064, arXiv:1308.1867 [hep-th]

Pith/arXiv arXiv

[7] [7]

Ewasiuk and S

C. Ewasiuk and S. Profumo, Ghost Degrees of Freedom Without Quantum Runaway: Exact Moment Bounds from an Operator Conservation Law, arXiv preprint arXiv:2604.21348 (2026), arXiv:2604.21348 [hep-th]

Pith/arXiv arXiv 2026

[8] [8]

Deffayet, A

C. Deffayet, A. F. Jalali, A. Held, S. Mukohyama, and A. Vikman, Unitary Time Evolution and Vacuum for a Quantum Stable Ghost, arXiv preprint arXiv:2604.21823 (2026), arXiv:2604.21823 [hep-th]

Pith/arXiv arXiv 2026

[9] [9]

Deffayet, A

C. Deffayet, A. F. Jalali, A. Held, S. Mukohyama, and A. Vikman, Quantum mechanics with a ghost: Counterexamples to spectral denseness, arXiv preprint arXiv:2604.21826 (2026), arXiv:2604.21826 [hep-th]

Pith/arXiv arXiv 2026

[10] [10]

Heisenberg, Generalization of the Proca Action, JCAP 05, 015, arXiv:1402.7026 [hep-th]

L. Heisenberg, Generalization of the Proca Action, JCAP 05, 015, arXiv:1402.7026 [hep-th]

Pith/arXiv arXiv

[11] [11]

De Felice, L

A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, and S. Tsujikawa, Cosmology in generalized Proca theories, JCAP06, 048, arXiv:1603.05806 [gr-qc]

Pith/arXiv arXiv

[12] [13]

De Felice and A

A. De Felice and A. Hell, On the cosmological degrees of freedom of Proca field with non-minimal coupling to gravity, JHEP07, 228, arXiv:2503.07454 [gr-qc]

Pith/arXiv arXiv

[13] [14]

De Felice and A

A. De Felice and A. Hell, The non-minimal 3-form cos- mology and the rise of the cuscuton, JHEP11, 132, arXiv:2509.02323 [gr-qc]

arXiv

[14] [15]

Arkani-Hamed, H.-C

N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, Ghost condensation and a consistent in- frared modification of gravity, JHEP05, 074, arXiv:hep- th/0312099

arXiv

[15] [16]

Cheung, P

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

Pith/arXiv arXiv

[16] [17]

C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel.7, 5 (2004), arXiv:gr-qc/0311082

Pith/arXiv arXiv 2004

[17] [18]

Adams, N

A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nico- lis, and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP2006(10), 014, arXiv:hep-th/0602178

Pith/arXiv arXiv

[18] [19]

De Felice, L

A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, S. Tsujikawa, and Y.-l. Zhang, Effective gravitational couplings for cosmological perturbations in general- ized Proca theories, Phys. Rev. D94, 044024 (2016), arXiv:1605.05066 [gr-qc]

Pith/arXiv arXiv 2016

[19] [20]

De Felice, L

A. De Felice, L. Heisenberg, and S. Tsujikawa, Ob- servational constraints on generalized Proca theories, Phys. Rev. D95, 123540 (2017), arXiv:1703.09573 [astro- ph.CO]. 6 SUPPLEMENT AL MA TERIAL Expansion of the action In the flat gauge, R= 0 &E= 0,(39) the scalar metric perturbations are given by ds2 =−N(t) 2(1 + 2α)dt2 + 2N(t)∂iβdtdx i +a 2(t)δijdxidxj...

Pith/arXiv arXiv 2017