Stability Analysis of the Possible Consistent Model of Parity Violations in the Symmetric Teleparallel Gravity: Generalized Background Solutions

Yeheng Tong

REVIEW 3 major objections 4 minor 38 references

Stability Analysis of the Possible Consistent Model of Parity Violations in the Symmetric Teleparallel Gravity: Generalized Background Solutions

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that the ghost-free condition previously derived for a parity-violating symmetric teleparallel gravity model must be supplemented by an additional coefficient relation, 2c2+c5=0, on one of the three allowed cosmological…

desk verdict A substantive extension of a parity-violating STG model with a new ghost-free condition on one background family, but the decisive vector-sector algebra is compressed and needs verification. read the letter →

arxiv 2505.15169 v1 pith:BBDQ4FT4 submitted 2025-05-21 gr-qc astro-ph.CO

Yeheng Tong This is my paper · ORCID

classification gr-qcastro-ph.CO PACS 04.50.Kd98.80.-k

keywords symmetricteleparallelgravityparityviolationghostinstabilitycosmologicalperturbationsnon-metricityvectormodesmodified

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

The paper extends an earlier study of a symmetric teleparallel gravity model that adds parity-violating couplings between a scalar field and the non-metricity tensor to the Einstein-equivalent action. It identifies three families of flat FRW backgrounds consistent with the cosmological principle, each tied to a different affine connection, and analyzes linear scalar, vector, and tensor perturbations on each. The central result is that on two of the families the previously found ghost-free condition b1 = 2c1+2c2−c4−c5 = 0 remains sufficient, while on the third family the vector perturbations become dynamical and one polarization turns into a high-energy ghost unless an additional condition, 2c2+c5 = 0, is imposed. If correct, the viable parameter space of the model is smaller than previously thought whenever the third background family is physically admissible; the tensor sector still exhibits velocity birefringence with no ghosts on all three backgrounds.

What carries the argument

The analysis rests on the three families of background non-metricity tensors derived from the cosmological-principle constraint $\mathcal{L}_\varsigma Q=0$ (Eq. 15), which force the non-metricity into the form (17) with three time-dependent functions $A,B,C$; solving the curvature-free and torsionless conditions (24–26) yields the degenerate family (27) and two non-degenerate families (28, 29). The perturbed connection is then expanded to second order as $\Gamma^\lambda_{\mu\nu}=\bar\Gamma^\lambda_{\mu\nu}+\bar\nabla_\mu\bar\nabla_\nu u^\lambda-\bar\nabla_\mu\bar\nabla_\nu u^\rho\bar\nabla_\rho u^\lambda$ (Eq. 33), and the quadratic actions for scalar, vector, and tensor perturbations are computed in the 'coincident gauge on the perturbation level' (36). On family 2, the vector action (66) acquires $E_i$-dependent terms proportional to $F$ that, after solving the constraint for $B_i$, produce the kinetic coefficient $z^2_A = \lambda_A (2c_2+c_5)\bar\varphi'^2 a^2 k^2 / (2 a^2 k + 4\lambda_A (2c_2+c_5)\bar\varphi'^2)$ (Eq. 68), whose sign determines the ghost.

What would settle it

Recompute the quadratic vector action on the non-degenerate family 2 backgrounds without imposing $b_1=0$ and explicitly integrate out $B_i$; if the resulting coefficient of $|E'|^2$ does not match Eq. (68), the additional condition $2c_2+c_5=0$ is an artifact. Alternatively, check whether family-2 backgrounds with $F\neq0$ satisfy the full connection equations of motion when $U^\rho$ is not assumed parallel to $\nabla^\rho\varphi$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the consistency of this parity-violating symmetric teleparallel gravity model depends on which cosmological background the connection chooses. The author constructs three families of flat FRW background solutions from the requirement that the non-metricity tensor respect the cosmological principle, and shows that the ghost-free combination b1=2c1+2c2−c4−c5=0 — derived previously for the simplest, degenerate background — remains sufficient on the degenerate family and on the first non-degenerate family. On the second non-degenerate family, however, the vector modes of the metric become propagating and acquire a kinetic term whose sign flips for one circular polarization at high wavenumber, so a new condition, 2c2+c5=0 (with φ′ not zero), is needed to eliminate the ghost. The paper also finds that tensor perturbations stay ghost-free on all three backgrounds with helicity-dependent dispersion relations, and that scalar perturbations coincide with those of a minimally coupled scalar in GR.

Load-bearing premise

The result's load-bearing premise is that the second-order expansion of the perturbed connection (Eq. 33) and the unshown reduction from the vector action to the kinetic coefficient (68) are both correct, together with the assumption that the affine connection respects the cosmological principle.

Editorial extensions

If this is right

If the third background family is physically allowed, the model's parameter space shrinks: parity-odd couplings must satisfy both $b_1=0$ and $2c_2+c_5=0$.
On the degenerate and first non-degenerate backgrounds the previously derived ghost-free condition is unchanged, so earlier phenomenological constraints on $c_1$ and $c_5$ remain valid there.
On all three backgrounds tensor perturbations are ghost-free and exhibit velocity birefringence, but the family-2 dispersion relation gains an extra $F$-dependent term that could distinguish the background through gravitational-wave observations.
Scalar perturbations are unaffected by the parity-violating terms at quadratic order, so the model keeps the standard GR-plus-scalar evolution for curvature perturbations.
If one judges the non-degenerate family 2 to be unphysical, the additional condition is unnecessary; otherwise it is essential to avoid ghost modes.

Reading between the lines

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

A natural step not taken in the paper is to check whether the condition $2c_2+c_5=0$ also removes ghosts for non-FRW backgrounds, which would indicate a deeper structure in the parity-odd coefficient space.
If the early universe ever passed through a family-2 phase, the dynamical vector modes could source primordial vector perturbations or leave imprints in CMB polarization spectra — a testable consequence of taking the third background seriously.
The paper leaves open the fate of the seventh parity term that vanishes at quadratic order on FRW backgrounds; testing it on anisotropic or spherical backgrounds might reveal whether the three-family classification is part of a larger pattern.
A complementary derivation of the ghost condition directly from the equations of motion, rather than from the quadratic action, would independently confirm or refute the necessity of the extra condition.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's my read on Tong's paper.

The genuinely new thing is the classification of three families of connection backgrounds and the discovery that on one of them, family 2, the previously derived ghost-free condition b1=0 is not enough; you also need 2c2+c5=0. The tensor sector on that family also gets a modified dispersion relation. That's a real extension of the author's earlier work, not a routine rerun.

What the paper does well: the background classification from the curvature-free conditions is clean and internally consistent. The scalar and tensor sectors on the two 'safe' families match the earlier results, which is a good consistency check. The paper is also honest about its limitations: it states the connection is assumed to respect the cosmological principle (so the classification may not be exhaustive), and it explicitly says the new condition is unnecessary if family 2 is not physically admissible. That caveat is not buried.

The soft spot is where it matters: the step from the vector action (66) to the kinetic coefficient (68). The paper says 'conduct the same discussion as in the previous subsections' and does not display the Fourier decomposition, the constraint equation for B_A, its solution, or the substitution that gives z_A^2. The new ghost condition hangs entirely on that reduction. A sign error or a missed factor would change the condition. The stress-test note's preliminary direct reduction suggests the qualitative ghost is plausible, but the exact coefficient in (68) is unchecked. This is a genuine gap, though not necessarily fatal.

The other qualification is that the paper does not settle whether family 2 is physically allowed. If it isn't, the new condition is moot. So the result is conditional, but the condition is stated clearly.

Who should read this: researchers working on symmetric teleparallel gravity, parity-violating gravity models, or cosmological perturbations in non-Riemannian geometries. It is a technical paper with a narrow audience.

My recommendation: send it to peer review. A referee with enough patience to redo the vector-sector algebra (or ask the author to expand it) can verify the central claim. The paper is coherent, the limitations are explicit, and the new condition is worth checking. I wouldn't cite it in my own work without an independent check, but it deserves a serious referee.

Referee Report

3 major / 4 minor

Summary. The paper studies a symmetric teleparallel gravity model with parity-violating couplings between gravity and a scalar field, extending earlier work by the same group. It classifies flat FRW background solutions into three families of affine connections, labelled degenerate, non-degenerate family 1, and non-degenerate family 2, according to the solutions of Eqs. (24)-(26). The paper then analyzes linear scalar, vector, and tensor perturbations around each family. It claims that scalar perturbations are unchanged from GR, tensor perturbations show velocity birefringence with no ghost, and vector perturbations are non-dynamical on the degenerate and non-degenerate family-1 backgrounds once the previous ghost-free condition b1=0 holds. The central new claim is that on non-degenerate family 2 the vector modes become dynamical and ghost-like unless an additional condition 2c2+c5=0 is imposed, and that the tensor dispersion relation acquires an additional F-dependent term.

Significance. If the central claim were correct, the paper would be a substantive extension of the authors' earlier ghost-free analysis: it would show that the previously derived condition b1=0 is background-dependent and is insufficient on a whole family of classically allowed cosmological backgrounds, thereby reducing the viable parameter space of the model. The background classification in Section III is a useful and internally consistent contribution, and the paper is honest about the assumption that the affine connection respects the cosmological principle. However, the main new result is undermined by a concrete algebraic issue: the F-dependent terms in the displayed quadratic actions on family 2 vanish identically by a vector identity. As written, the paper therefore does not establish that vector modes become dynamical on family 2 or that 2c2+c5=0 is needed. The manuscript also does not display the crucial Fourier-space reduction from the vector action to the claimed kinetic coefficient, so the central conclusion is not independently checkable.

major comments (3)

[Sec. IV.D.3, Eqs. (65)-(66)] The central claim is unsupported as written because the F-dependent terms that are supposed to generate the new ghost condition vanish identically. For any smooth vector field V, epsilon^{ijk} V_{j,i} V_k = V dot (curl V) = 0. Therefore the term (2c2+c5) F epsilon^{ijk} B_{j,i} B_k in Eq. (66) is identically zero, and the term (2c1+c4) F epsilon^{ijk} E_{j,i l} E_{k,l} is zero as well because, for each l, it equals (partial_l E) dot [curl(partial_l E)]. Once b1=0, the action in Eq. (66) is actually independent of F, so the claimed reduction to Eqs. (67)-(68) with z_A^2 proportional to (2c2+c5) F cannot follow. Unless these F terms are misprints with a genuinely different index structure, the paper's main new result is invalid.
[Sec. IV.D.2, Eqs. (61)-(64)] The same vector identity invalidates the claimed F correction to the tensor dispersion on family 2. The term epsilon^{ijk} h_{jl,i} h_{kl} vanishes because, for each fixed l, it is V^l dot (curl V^l) with V^l_j = h_{jl}. Consequently the extra term -8 lambda_A F (2c1+c4) phi'^2/(a^2 k) in Eq. (64) is spurious as written, and the tensor perturbation result on family 2 should coincide with the degenerate and family-1 results. This is a secondary issue, but it indicates a systematic error in the reduction of the parity-violating terms on background (29).
[Sec. IV.D.3, Eqs. (66)-(68)] Even if the F-dependent terms were not identically zero, the decisive algebra is omitted. The text says 'conduct the same discussion as in the previous subsections' and then jumps from the position-space action (66) to the Fourier-space result (67) with the specific coefficients z_A^2 and w_A^2 in (68) and (70). The revision must display the Fourier decomposition of Eq. (66), the constraint equation obtained by varying B_A, the explicit solution for B_A, the substitution back into the action, and the cancellation of the E'_A terms. Without these steps, the sign and the exact denominator in Eq. (68) cannot be verified by the reader.

minor comments (4)

[Sec. III.A, Eq. (30)] The scalar-field equation of motion is written as phi'' + 2 H phi' + a^2 phi = 0, but the potential term should be a^2 V_phi (the derivative of V with respect to phi) rather than a^2 phi.
[Sec. IV.B.3, after Eq. (46)] The text refers to 'the constraint equation (46)', but the constraint equation is actually Eq. (45); Eq. (46) is the Fourier-mode expansion. This cross-reference should be corrected.
[Sec. III.A, Eq. (19)] The statement that the hyper-potential sum vanishes automatically when U^rho is proportional to nabla^rho phi would benefit from a one-sentence explanation: the contraction with epsilon and the symmetry of the product of gradients makes the expression vanish. The current wording 'It is obvious' understates a step that is central to satisfying the connection equations.
[Secs. IV.B-IV.D] Several quadratic actions are presented after the phrase 'after tedious calculations' (e.g., Eqs. (43), (56), and (65)). Given that the main new result depends on the detailed form of these actions, at least the key intermediate expressions for the perturbed non-metricity tensor should be provided or placed in an appendix so that the displayed actions can be checked.

Circularity Check

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No circularity: the new ghost-free condition 2c2+c5=0 is derived from a fresh quadratic-action computation on background family 2, not by fitting or by re-defining the input condition b1=0.

full rationale

The derivation chain is: action (8), background families (27)-(29), perturbed connection expansion (33), quadratic actions, vector action (66), and then the kinetic coefficient z_A^2 in (68). The claimed new result, that the additional condition 2c2+c5=0 is needed on non-degenerate family 2, follows from the sign of z_A^2 at large k, which in turn comes from the surviving (2c2+c5)F term in Eq. (66) after setting b1=0. This is a genuine perturbative derivation, not a restatement of an input: the target condition is not contained in the prior ghost-free condition b1=0, and no parameter is fitted to produce it. The only notable self-citation is Ref. [22], which supplies b1=0 and is explicitly adopted as an assumption in Sec. II (Eq. 13). That is a premise, not the output being tested; the paper's contribution is precisely to test whether b1=0 is sufficient on the new background. The passage in Sec. V noting that 2c2+c5=0 would be unnecessary if family 2 is not physically permissible is an honest limitation, not an imported conclusion. The main weakness is the unshown reduction from Eq. (66) to Eqs. (67)-(68), but an omitted algebraic step is a verifiability or correctness concern, not circularity: the paper does not define z_A^2 to equal the ghost-free condition, nor does it tune coefficients to force the advertised result. No uniqueness theorem is imported, no ansatz is smuggled in via citation, and no known result is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The model's coefficients and the arbitrary function F(eta) are free inputs from the action and background classification. The paper adds no new physical entities. The main axioms are the cosmological principle applied to the connection, the operator basis from prior work, the assumed ghost-free condition b1=0, and the validity of the perturbative expansion.

free parameters (2)

coupling constants c1, c2, c4, c5, c6
Coupling constants of the parity-violating action (8), introduced in Eq. (8); their values are not fixed by data. The ghost-free conditions derived in the paper are constraints on combinations of these constants (b1=0 and 2c2+c5=0).
background function F(eta)
Arbitrary smooth function labeling members of each background family (Eqs. 27-29); the non-degenerate family 2 requires F != 0. The ghost-free condition on that family must hold for every F.

assumptions (7)

standard math Standard Scalar-Vector-Tensor decomposition of metric perturbations with transverse conditions (Sec. IV.A)
Used throughout the perturbation analysis to separate scalar, vector, and tensor modes.
standard math Algebraic identities (7) relating the parity-violating terms M_a, cited from Ref. [24]
These identities justify reducing the seven couplings to five and define the operator basis used in the action (8).
domain assumption The affine connection respects the cosmological principle through a vanishing Lie derivative of the non-metricity tensor, Eq. (15)
This restricts non-metricity to the form (16) following Ref. [26]. If the connection need not respect the cosmological principle, other background families are possible.
domain assumption Spatially flat FRW metric and a single minimally coupled scalar field as the matter content (Eqs. 14 and 8)
The whole calculation is performed on this background; other geometries or matter sectors are not considered.
domain assumption Ghost-free condition b1 = 2c1+2c2-c4-c5 = 0 from previous work is imposed throughout, Eq. (13)
The paper explicitly states 'We shall assume this condition to always hold in subsequent calculations'; the central result asks whether this condition is also sufficient on new backgrounds.
ad hoc to paper Second-order expansion of the perturbed connection, Eq. (33), and the 'coincident gauge on the perturbation level' (36) correctly capture all physical modes
The quadratic actions, especially the vector action (67), depend on this expansion and gauge choice. An error here would propagate to the ghost condition.
ad hoc to paper Scalar, vector, and tensor perturbations decouple on all three background families
Stateed in Sec. IV.A as 'it can be verified' without derivation; needed to treat the quadratic actions independently.

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Cite this review

Pith. "Pith review of Stability Analysis of the Possible Consistent Model of Parity Violations in the Symmetric Teleparallel Gravity: Generalized Background Solutions." pith.science (2026). https://pith.science/paper/BBDQ4FT4

@misc{pith2026250515169,
  author       = {Pith},
  title        = {Pith review of: Stability Analysis of the Possible Consistent Model of Parity Violations in the Symmetric Teleparallel Gravity: Generalized Background Solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBDQ4FT4}},
  note         = {Machine review of arXiv:2505.15169}
}

read the original abstract

In this paper, we consider a symmetric teleparallel gravity model that extends the general relativity equivalent model by several parity violating interactions between the gravitational field and a scalar field. We derive three different families of background solutions in flat FRW universe, with three classes of different connections. Through investigations on the linear cosmological perturbations, we show that one of the vector modes of this model will evolve into a ghost field at high energy, and the ghost instability can be cancelled only under specific combinations of the coefficients. On two of three families of backgrounds, such combination remains the same as the one we have investigated in our previous work; while on the other family of background, one additional condition should be taken into consider.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 25 canonical work pages

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Degenerate Family:A=HandC=F, so that ¯Qαµν = 2a2 (Hηµνdα +F dµdνdα) ; (27) 6 and whenB ̸= 0,

work page
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Non-degenerate Family 1:A=H+F,B=FandC=−F ′/F−F, so that ¯Qαµν =a 2 2 (H+F)η µνdα +F(η αµdν +η ανdµ)−2 F ′ F +F dµdνdα ; (28)

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Non-degenerate Family 2:A=H,B=F, andC=F ′/F−F, so that ¯Qαµν =a 2 2Hηµνdα +F(η αµdν +η ανdµ) + 2 F ′ F −F dµdνdα .(29) In these three families of solutions, the specific functional form of the smooth functionF=F(η) determines the distinct members within each family of solutions, leading to physically and mathematically differentiated cases. Specifically, ...

work page
[4]

(5) vanish up to the second order

Scalar perturbations For the scalar perturbations, it is not difficult to find that all the parity violating terms in Eq. (5) vanish up to the second order. So, they are at least third order quantities and have no contribution to the quadratic action. Therefore, the quadratic action for the scalar perturbations of the model (8) is the same as the one in G...

work page
[5]

Tensor perturbations For tensor perturbations, only the first and the fifth parity violating terms of Eq. (5) have non-zero contributions to the quadratic action: SP V1=− Z d4x2c 1 ϕ′2ϵijk hjl,ih′ kl , SP V5= Z d4x c5 ϕ′2ϵijk hjl,ih′ kl ,(38) 8 whereϵ ijk is the 3-dimensional anti-symmetric symbol andϵ 123 =−1. So the full quadratic action for tensor pert...

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However, these terms do not contribute to the quadratic action sinceb 1 = 0

Vector perturbations After some tedious calculations, we find the following four of five parity violating terms have contributions to the quadratic action for the vector perturbations: SP V1= Z d4x2c 1 ¯ϕ′2ϵijk Bj,iB′ k −F Bj,iBk −E j,ilE′ k,l , SP V2= Z d4x2c 2 ¯ϕ′2ϵijk (Bj,iB′ k −F Bj,iBk), SP V4=− Z d4x c4 ¯ϕ′2ϵijk (Bj,iB′ k −F Bj,iBk −E j,ilBk,l), SP ...

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Similarly, this quadratic action is free of effect of the perturbationsu 0 andufrom the connection

Scalar perturbations For scalar perturbations, same as the case in the last subsection, the parity violating terms have no contribution to the quadratic action, so the quadratic action for the scalar perturbations of the model (8) is also the same as the one in GR with a minimally coupled scalar field: S(2) S = Z d4x a2 ¯ϕ′2 2H2 ζ ′2 −∂ iζ∂ iζ ,(52) where...

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Tensor perturbations The calculation yields the same result with the one on the degenarate background for tensor perturbations: S(2) T = Z d4x a2 8 h′ ijh′ ij −h ij,khij,k + 1 2 b2 ¯ϕ′2 ′ ϵijk hjl,ihkl .(53) Then rewrite the effective action (53) in the Fourier space: S(2) T = X A=L,R Z dη d3⃗k a2 8 h′Ah′A∗ −ω 2 AT hAhA∗ ,(54) with ω2 AT =k 2 1 + 4λA a2k ...

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Similarly, these terms do not contribute to the quadratic action sinceb 1 = 0

Vector perturbations After some calculations, one can obtain the parity violating terms that contribute to the quadratic action for the vector perturbations: SP V1= Z d4x2c 1 ¯ϕ′2ϵijk Bj,iB′ k +CB j,iBk −E j,ilE′ k,l , SP V2= Z d4x2c 2 ¯ϕ′2ϵijk (Bj,iB′ k +CB j,iBk), SP V4=− Z ...
[10]

Similarly, this quadratic action is free of effect of the perturbationsu 0 andufrom the connection

Scalar perturbations For scalar perturbations, same as the cases of degenerate backgrounds and non-degenerate backgrounds 1, the parity violating terms have no contribution to the quadratic action, so the quadratic action for the scalar perturbations of the model (8) is also t...
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Tensor perturbations The tensor perturbations on this family of backgrounds (29) behave differently from the rest two families of back- grounds. In this case,M 4 also has contribution together withM 1 andM 5: SP V1= Z d4x2c 1 ¯ϕ′2ϵijk (hjl,ih′ kl +F hjl,ihkl), SP V4= Z d4x c4 ...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Degenerate Family:A=HandC=F, so that ¯Qαµν = 2a2 (Hηµνdα +F dµdνdα) ; (27) 6 and whenB ̸= 0,

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[2] [2]

Non-degenerate Family 1:A=H+F,B=FandC=−F ′/F−F, so that ¯Qαµν =a 2 2 (H+F)η µνdα +F(η αµdν +η ανdµ)−2 F ′ F +F dµdνdα ; (28)

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[3] [3]

Non-degenerate Family 2:A=H,B=F, andC=F ′/F−F, so that ¯Qαµν =a 2 2Hηµνdα +F(η αµdν +η ανdµ) + 2 F ′ F −F dµdνdα .(29) In these three families of solutions, the specific functional form of the smooth functionF=F(η) determines the distinct members within each family of solutions, leading to physically and mathematically differentiated cases. Specifically, ...

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[4] [4]

(5) vanish up to the second order

Scalar perturbations For the scalar perturbations, it is not difficult to find that all the parity violating terms in Eq. (5) vanish up to the second order. So, they are at least third order quantities and have no contribution to the quadratic action. Therefore, the quadratic action for the scalar perturbations of the model (8) is the same as the one in G...

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[5] [5]

Tensor perturbations For tensor perturbations, only the first and the fifth parity violating terms of Eq. (5) have non-zero contributions to the quadratic action: SP V1=− Z d4x2c 1 ϕ′2ϵijk hjl,ih′ kl , SP V5= Z d4x c5 ϕ′2ϵijk hjl,ih′ kl ,(38) 8 whereϵ ijk is the 3-dimensional anti-symmetric symbol andϵ 123 =−1. So the full quadratic action for tensor pert...

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[6] [6]

However, these terms do not contribute to the quadratic action sinceb 1 = 0

Vector perturbations After some tedious calculations, we find the following four of five parity violating terms have contributions to the quadratic action for the vector perturbations: SP V1= Z d4x2c 1 ¯ϕ′2ϵijk Bj,iB′ k −F Bj,iBk −E j,ilE′ k,l , SP V2= Z d4x2c 2 ¯ϕ′2ϵijk (Bj,iB′ k −F Bj,iBk), SP V4=− Z d4x c4 ¯ϕ′2ϵijk (Bj,iB′ k −F Bj,iBk −E j,ilBk,l), SP ...

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[7] [7]

Similarly, this quadratic action is free of effect of the perturbationsu 0 andufrom the connection

Scalar perturbations For scalar perturbations, same as the case in the last subsection, the parity violating terms have no contribution to the quadratic action, so the quadratic action for the scalar perturbations of the model (8) is also the same as the one in GR with a minimally coupled scalar field: S(2) S = Z d4x a2 ¯ϕ′2 2H2 ζ ′2 −∂ iζ∂ iζ ,(52) where...

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[8] [8]

Tensor perturbations The calculation yields the same result with the one on the degenarate background for tensor perturbations: S(2) T = Z d4x a2 8 h′ ijh′ ij −h ij,khij,k + 1 2 b2 ¯ϕ′2 ′ ϵijk hjl,ihkl .(53) Then rewrite the effective action (53) in the Fourier space: S(2) T = X A=L,R Z dη d3⃗k a2 8 h′Ah′A∗ −ω 2 AT hAhA∗ ,(54) with ω2 AT =k 2 1 + 4λA a2k ...

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