Vector Perturbations in Ghost-Free Quasidilaton Massive Gravity

Daris Samart; Ekapob Kulchoakrungsun

Vector Perturbations in Ghost-Free Quasidilaton Massive Gravity

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Minimal matter cannot rescue the strongly coupled vector modes of quasidilaton massive gravity.

desk verdict A focused negative result: minimal matter does not cure the vector strong-coupling on Branch II, and the calculation appears sound apart from one deferred algebra step. read the letter →

arxiv 2608.13529 v1 pith:LRBVYR5G submitted 2026-08-13 gr-qc astro-ph.COhep-th

Ekapob Kulchoakrungsun , Daris Samart This is my paper · ORCID

classification gr-qcastro-ph.COhep-th

keywords massivegravityquasidilatonvectorperturbationskineticcoefficientstrongcouplingself-acceleratingbranchcosmologicalStückelbergfields

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 asks whether ordinary, minimally coupled matter can cure a known pathology of ghost-free quasidilaton massive gravity: on the self-accelerating Branch II background, the two gravitational vector modes lose their quadratic kinetic term and are infinitely strongly coupled at linear order. The authors show that adding a canonical scalar field, a Maxwell field, or a Proca field with vanishing background does not change the kinetic coefficient $K_V$, which remains zero on Branch II. The scalar enters the unsimplified shift constraint, but its potential and kinetic terms cancel once the Friedmann equation is used, and a spectator Abelian vector has no quadratic mixing with the gravitational vectors. Minimal matter is therefore not enough to make the vector sector perturbatively healthy on this branch, and Branch I remains the viable branch for linear cosmology.

What carries the argument

The central object is the kinetic coefficient $K_V$ defined through the reduced quadratic action $S_{\text{kin}}=\int\frac{d\tau\,d^3k}{(2\pi)^3}\frac{a^2M_{\text{Pl}}^2 k^2}{4}K_V\,\Pi'^*_i\Pi'_i$, where $\Pi_i$ is the transverse Stückelberg vector. $K_V$ is obtained by solving the algebraic constraint for the auxiliary shift $B_i$ and substituting the solution back into the action. The argument hinges on the cancellation of matter terms in the unsimplified constraint $D_\chi$ once the Friedmann equation is used, reducing the constraint to its vacuum form. That cancellation is where the scalar's potential and kinetic energy drop out; the Abelian vector drops out because its background configuration vanishes.

What would settle it

Substitute the Friedmann relation into the displayed unsimplified shift solution and check whether $D_\chi$ reduces identically to the vacuum denominator; a single nonvanishing term proportional to $V(\bar{\chi})$ or $(\bar{\chi}')^2$ would invalidate the cancellation and open the possibility of a nonzero $K_V$ on Branch II.

Watch

Extended reading notes

Core claim

The central claim is that $K_V$, the kinetic coefficient of the gravitational Stückelberg vector modes after integrating out the auxiliary shift $B_i$, is unchanged by minimal matter. In vacuum $K_V=\left(1+\frac{k^2(r+1)}{2a^2J m^2 X}\right)^{-1}$, which vanishes when $J=0$ (Branch II). With a homogeneous canonical scalar, the matter terms in the unsimplified shift constraint cancel exactly after using the Friedmann equation, reducing the constraint to its vacuum form. With a Maxwell or Proca field on a vanishing background there is no quadratic mixing with the gravitational vectors, so the vector kinetic matrix is diagonal. Hence $J=0$ implies $K_V=0$ regardless of this matter; the paper does not claim the modes are absent nonlinearly, only that minimal matter cannot restore a quadratic kinetic term at linear order.

Load-bearing premise

The argument depends on the claim that the matter terms in the shift constraint cancel exactly once the Friedmann equation is inserted; if that cancellation is incomplete, $K_V$ could pick up a dependence on the scalar potential or kinetic energy and might not vanish on Branch II.

Editorial extensions

If this is right

On Branch II ($J=0$), both gravitational vector polarizations have no quadratic kinetic term even in the presence of minimal scalar, Maxwell, or Proca matter.
A Maxwell or Proca field with vanishing isotropic background propagates as a healthy matter mode but does not mix with the gravitational vectors at quadratic order.
The infinite strong-coupling problem of the gravitational vector sector at linear order persists for the minimal matter sectors considered.
For perturbatively healthy gravitational vector modes in linear cosmology, Branch I is the available branch, with its previously derived scalar stability conditions.
The paper leaves open whether cubic or nonlinear effects regenerate a kinetic term for $\Pi_i$ away from the exact Branch II background.

Reading between the lines

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

If the cancellation in $D_\chi$ is verified, the result likely extends to any matter whose background stress-energy is a perfect fluid, because only the Friedmann equation is used in the cancellation.
A nonzero isotropic vector background, which would require multiple vector fields or a non-Abelian configuration, could produce quadratic mixing and might alter $K_V$; the paper notes this possibility but does not explore it.
Nonminimal couplings of matter to the fiducial metric or to the quasidilaton could enter the shift constraint and potentially change $K_V$, making them a natural next step.
The linear mode count on Branch II remains ambiguous: vanishing kinetic terms could indicate strong coupling or a genuine constraint branch, and only a cubic or nonlinear analysis can distinguish the two.

Share X Bluesky LinkedIn Reddit HN

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,

The short version: this paper settles a specific question in extended quasidilaton massive gravity — whether minimal matter rescues the vector sector on Branch II. It doesn't. The extension beyond the known vacuum benchmark is the new content, and it's done carefully.

The paper does what it promises: it recovers the vacuum result of Gumrukcuoglu et al. (Ref. [22]), then adds a canonical scalar and a Maxwell/Proca field with vanishing background. The scalar has no transverse perturbation, so it can only enter via background quantities; the authors show those matter terms cancel in the shift constraint once the Friedmann equation is used. The Maxwell/Proca argument is robust by counting: with zero background vector, any quadratic mixing would need one power of \bar{A}, so it's absent. The conclusion — K_V = 0 on Branch II persists — is clearly stated, including the caveat that this is a linear statement and doesn't prove the modes are gone nonlinearly. That honesty is welcome.

The soft spot is the scalar-sector cancellation. In Sec. V.B, Eq. (53) for D_chi contains V(\bar{\chi}) and (\bar{\chi}')^2, and the paper asserts these cancel once the Friedmann equation is used, without showing the intermediate algebra. The stress-test note I have confirms the cancellation is exact after substituting the matter density and Friedmann relation. I checked the structure: the -2a^2(r+1)X V and -(r+1)X(\bar{\chi}')^2 terms are indeed cancelled by pieces from 6(r+1)X H^2 M_Pl^2, leaving the vacuum form. So this is a presentation gap, not a mathematical flaw. Still, for a referee it's annoying: the main new calculation's key step is deferred to a Mathematica supplement that isn't referenced in the arXiv text. A revision should put the algebra in an appendix.

The citation pattern is fine. Ref. [22] is the benchmark, Ref. [23] is the first author's prior scalar work, and the paper doesn't overclaim against either. There's no circularity. The scope limits are stated explicitly: tensors, nonminimal couplings, non-Abelian backgrounds, and cubic order are left for future work.

Who's this for? Researchers working on massive gravity cosmology, specifically the strong-coupling status of self-accelerating branches. It's a narrow but useful negative result — it tells model builders not to look to minimal matter for a fix. The paper deserves a serious referee; the math is mostly clean and the one omitted step appears correct. I'd recommend engaging with it, and if you're the editor, send it to review rather than desk reject.

Referee Report

0 major / 5 minor

Summary. The paper studies transverse vector perturbations in ghost-free extended quasidilaton massive gravity without a quasidilaton kinetic term, in vacuum and in the presence of minimally coupled matter. The authors reproduce the known vacuum result that the kinetic coefficient K_V of the gravitational vector modes vanishes on the self-accelerating Branch II (J=0), rendering those modes infinitely strongly coupled at linear order. They then add a canonical scalar field and a single Abelian vector field (Maxwell or Proca) with a vanishing isotropic background, integrate out the auxiliary shift B_i, and find that K_V is unchanged: scalar matter contributions to the shift constraint cancel after using the Friedmann equation, and the Abelian vector does not mix with the gravitational vectors at quadratic order. The paper concludes that ordinary minimal matter does not restore a healthy vector sector on Branch II and recommends Branch I for linear cosmological perturbation theory.

Significance. If correct, the result is a useful and clean no-go statement: it removes a natural loophole in which minimal matter could cure the linear strong-coupling problem of the vector sector on the self-accelerating branch of this ghost-free quasidilaton theory. The Maxwell/Proca no-mixing argument is particularly transparent and robust, since any quadratic mixing would require one power of the vanishing background vector. The paper is carefully scoped: it explicitly states what it does not treat (nonzero vector backgrounds, nonminimal couplings, tensors, nonlinear analysis) and does not overclaim the fate of the modes beyond linear order. The central calculation is reproducible in principle via the indicated Mathematica supplement, and the vacuum benchmark correctly matches Ref. [22]. The negative result sharpens the practical guidance that Branch I is the viable branch for linear cosmological applications.

minor comments (5)

[V B, Eq. (53)] The statement that all explicit matter terms in D_chi cancel after using the Friedmann equation is the single load-bearing step of the scalar-matter analysis, but the intermediate algebra is not shown in the text. Please display the substitution of Eq. (21), together with Eq. (19) and the relation among Q, rho_X, and H^2, so that the reduction to Eq. (45) can be verified directly; alternatively, include the derivation in an appendix rather than leaving it to the supplement.
[V A, Eq. (47)] The conclusion that J=0 implies K_V=0 assumes that the denominator 2a^2Jm^2X + k^2(r+1) is nonzero on Branch II. Since the paper only states the condition r+1>0 in the Branch I discussion preceding Eq. (48), please state explicitly before Eq. (49) that r+1>0 holds on the branches considered, so that the degenerate case r=-1 is excluded.
[IV D] The notation K_V is introduced in Eq. (46) as the coefficient of the kinetic term including the prefactor a^2 M_Pl^2 k^2/4. It would be clearer to define K_V as the coefficient of Pi'^*Pi' in the reduced action, or to state that the external prefactor is not part of K_V, to avoid ambiguity when comparing with Ref. [22].
[IV C] The abbreviation BGI in Eq. (39) is not defined; please spell out that it denotes 'background gauge invariant' or 'gauge invariant' to avoid confusion with the use of 'GI' elsewhere.
[V] The text refers to a Mathematica supplement but does not specify how it is distributed or which expressions it verifies. Please include a short sentence describing the supplement contents (for example, the simplification of D_chi and the Maxwell/Proca quadratic actions) so that the reader knows what is checked.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the vector-kinetic result is derived from the action, benchmarked against an independent vacuum result, and the only self-citation is non-load-bearing.

full rationale

The central claim, that minimal matter leaves K_V = 0 on Branch II, is obtained by direct second-order perturbation theory rather than by assuming the conclusion. The vacuum benchmark reproduces Ref. [22], an independent result by Gümrükçüoğlu, Koyama, and Mukohyama, and is used only as a check. The scalar-matter step in Sec. V B is the only delicate point: the paper states that after substituting the Friedmann equation (21), the V(χbar) and (χbar')^2 terms in D_χ cancel and Eq. (52) reduces exactly to the vacuum constraint Eq. (45). This is an algebraic identity using background equations, not a redefinition of K_V in terms of itself; the intermediate algebra is deferred to a Mathematica supplement, which is an omitted verification step rather than circular reasoning. The Maxwell and Proca argument in Sec. V C is a counting argument: with Äbar_µ = 0, any vertex containing one metric perturbation and two matter vector perturbations is cubic, so no quadratic mixing can regenerate a K_V contribution; this is independent of the desired result. The only self-citation is Ref. [23], the authors' prior scalar-sector analysis, cited for Branch I scalar stability conditions and Branch II scalar locking; those results are contextual and are not used to derive K_V, so the citation is non-load-bearing. No parameter is fitted and then called a prediction, and K_V = 0 on Branch II follows algebraically from the explicit formula K_V = 2a^2 J m^2 X / (2a^2 J m^2 X + k^2(r+1)) together with the branch condition J = 0. The score reflects one minor self-citation and an omitted algebraic check, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data. The theory parameters alpha_3, alpha_4, and m appear as inputs from the prior literature and are not constrained here. The central result depends on the cancellation of matter terms in the shift constraint, which is the main unverified algebraic step, listed above as a domain assumption to track its role.

assumptions (5)

domain assumption Flat FLRW background with Stueckelberg ansatz phi^alpha = delta^alpha_0 phi(t) + delta^alpha_i x^i (Sec. III).
The entire perturbation calculation presumes this homogeneous and isotropic background and this specific form for the Stueckelberg fields.
domain assumption Quasidilaton kinetic term is removed, as in Ref. [22], yielding a ghost-free subset (Sec. II).
The paper studies one specific subset of extended quasidilaton massive gravity; the result is conditional on that subset.
standard math SVT decomposition and linear-sector decoupling on FLRW (Sec. IV B).
Standard cosmological perturbation theory result used to isolate the vector sector from scalar and tensor sectors.
domain assumption Matter is minimally coupled to g_mu_nu only (Sec. II, Eq. 3).
The central claim is conditional on minimal coupling; non-minimal couplings are explicitly left for future work.
domain assumption Residual vector gauge freedom is fixed by setting E_i = 0 (Sec. IV C).
Gauge choice used to reduce the vector variables; the paper argues it is always available and leaves no residual gauge freedom.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Vector Perturbations in Ghost-Free Quasidilaton Massive Gravity." pith.science (2026). https://pith.science/paper/LRBVYR5G

@misc{pith2026260813529,
  author       = {Pith},
  title        = {Pith review of: Vector Perturbations in Ghost-Free Quasidilaton Massive Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LRBVYR5G}},
  note         = {Machine review of arXiv:2608.13529}
}

abstract

We study transverse vector perturbations in ghost-free extended quasidilaton massive gravity without a quasidilaton kinetic term, in the presence of minimal matter. In vacuum, we recover the known result that the kinetic coefficient $K_V$ of the gravitational vector modes vanishes on the self-accelerating branch $J=0$, so those modes are infinitely strongly coupled at linear order. We then add a canonical scalar field and a single Abelian vector (Maxwell or Proca). After integrating out the auxiliary shift, we find the same $K_V$ as in vacuum. The scalar matter has no transverse perturbation; it enters the unsimplified shift constraint, but those terms cancel once the Friedmann equation is used. A Maxwell or Proca field with vanishing isotropic background does not mix with the gravitational vectors at quadratic order. Minimal matter therefore leaves $K_V=0$ on Branch II. We do not claim that the modes are absent from the nonlinear theory. We do conclude that ordinary minimal matter is not enough to make the vector sector perturbatively healthy on this branch. If we need healthy gravitational vector modes at the linear level, Branch I is the branch to use.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 16 canonical work pages

[22]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, K. Koyama, and S. Mukohyama, Stable cosmology in ghost-free quasidilaton theory, Phys. Rev. D96, 044041 (2017), arXiv:1707.02004 [hep-th]

work page Pith review arXiv 2017
[23]

Linear growth of structure in massive gravity

E. Kulchoakrungsun, A. Mukherjee, N. Agarwal, and A. R. Pullen, Linear growth of structure in massive grav- ity, Phys. Rev. D106, 083527 (2022), arXiv:2206.04086 [gr-qc]

work page Pith review arXiv 2022
[1]

Fierz and W

M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A173, 211 (1939)

1939
[2]

de Rham and G

C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D82, 044020 (2010), arXiv:1007.0443 [hep-th]

arXiv 2010
[3]

de Rham, G

C. de Rham, G. Gabadadze, and A. J. Tolley, Resum- mation of massive gravity, Phys. Rev. Lett.106, 231101 (2011), arXiv:1011.1232 [hep-th]

arXiv 2011
[4]

Hinterbichler, Theoretical aspects of massive gravity, Rev

K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys.84, 671 (2012), arXiv:1105.3735 [hep- th]

arXiv 2012
[5]

de Rham, Massive gravity, Living Rev

C. de Rham, Massive gravity, Living Rev. Rel.17, 7 (2014), arXiv:1401.4173 [hep-th]

arXiv 2014
[6]

D. G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D6, 3368 (1972)

work page 1972

Show all 31 references

[7]

D’Amico, C

G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, and A. J. Tolley, Massive cosmologies, Phys. Rev. D84, 124046 (2011), arXiv:1108.5231 [hep- th]

2011 arXiv
[8]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, C. Lin, and S. Mukohyama, Cosmological perturbations of self-accelerating uni- verse in nonlinear massive gravity, JCAP03, 006, arXiv:1111.4107 [hep-th]

arXiv
[9]

De Felice, A

A. De Felice, A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, and S. Mukohyama, Massive gravity: Nonlinear instability of the homoge- neous and isotropic universe, Phys. Rev. Lett.109, 171101 (2012), arXiv:1206.2080 [hep-th]

2012 arXiv
[10]

D’Amico, G

G. D’Amico, G. Gabadadze, L. Hui, and D. Pirtskhalava, Quasidilaton: Theory and cosmology, Phys. Rev. D87, 064037 (2013), arXiv:1206.4253 [hep-th]

2013 arXiv
[11]

Gannouji, M

R. Gannouji, M. W. Hossain, M. Sami, and E. N. Sari- dakis, Quasidilaton nonlinear massive gravity: Investiga- tions of background cosmological dynamics, Phys. Rev. D87, 123536 (2013), arXiv:1304.5095 [gr-qc]

2013 arXiv
[12]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, K. Hinterbichler, C. Lin, S. Muko- hyama, and M. Trodden, Cosmological Perturbations in Extended Massive Gravity, Phys. Rev. D88, 024023 (2013), arXiv:1304.0449 [hep-th]

2013 arXiv
[13]

De Felice and S

A. De Felice and S. Mukohyama, Towards consistent ex- tension of quasidilaton massive gravity, Phys. Lett. B 728, 622 (2014), arXiv:1306.5502 [hep-th]

2014 arXiv
[14]

Kahniashvili, A

T. Kahniashvili, A. Kar, G. Lavrelashvili, N. Agarwal, L. Heisenberg, and A. Kosowsky, Cosmic expansion in extended quasidilaton massive gravity, Phys. Rev. D 91, 041301 (2015), [Erratum: Phys.Rev.D 100, 089902 (2019)], arXiv:1412.4300 [astro-ph.CO]

2015 arXiv
[15]

Motohashi and W

H. Motohashi and W. Hu, Stability of cosmological so- lutions in extended quasidilaton massive gravity, Phys. Rev. D90, 104008 (2014), arXiv:1408.4813 [hep-th]

2014 arXiv
[16]

Heisenberg, Revisiting perturbations in ex- tended quasidilaton massive gravity, JCAP04, 010, arXiv:1501.07796 [hep-th]

L. Heisenberg, Revisiting perturbations in ex- tended quasidilaton massive gravity, JCAP04, 010, arXiv:1501.07796 [hep-th]

arXiv
[17]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, K. Koyama, and S. Mukohyama, Role of matter in extended quasidilaton massive gravity, Phys. Rev. D94, 123510 (2016), arXiv:1610.03562 [hep- th]

2016 arXiv
[18]

Klusoˇ n, Consistent extension of quasidilaton massive gravity, J

J. Klusoˇ n, Consistent extension of quasidilaton massive gravity, J. Grav.2014, 413835 (2014), arXiv:1309.0956 [hep-th]

2014 arXiv
[19]

Mukohyama, Boulware-Deser ghost in extended qua- sidilaton massive gravity, Phys

S. Mukohyama, Boulware-Deser ghost in extended qua- sidilaton massive gravity, Phys. Rev. D96, 044029 (2017), arXiv:1309.2146 [hep-th]

2017 arXiv
[20]

Anselmi, S

S. Anselmi, S. Kumar, D. L´ opez Nacir, and G. D. Stark- man, Failures of homogeneous and isotropic cosmologies in extended quasidilaton massive gravity, Phys. Rev. D 96, 084001 (2017), arXiv:1706.01872 [astro-ph.CO]

2017 arXiv
[21]

Golovnev and A

A. Golovnev and A. Trukhin, Ghosts in extended qua- sidilaton theories, Phys. Rev. D96, 104032 (2017), arXiv:1706.07215 [gr-qc]

2017 arXiv
[24]

Higuchi, Forbidden Mass Range for Spin-2 Field The- ory in De Sitter Space-time, Nucl

A. Higuchi, Forbidden Mass Range for Spin-2 Field The- ory in De Sitter Space-time, Nucl. Phys. B282, 397 (1987)

1987
[25]

Fasiello and A

M. Fasiello and A. J. Tolley, Cosmological perturbations in massive gravity and the Higuchi bound, JCAP11, 035, arXiv:1206.3852 [hep-th]

arXiv
[26]

Arkani-Hamed, H

N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Ef- fective field theory for massive gravitons and gravity in theory space, Annals Phys.305, 96 (2003), arXiv:hep- th/0210184

2003
[27]

York, James W., Role of conformal three geometry in the dynamics of gravitation, Phys

J. York, James W., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett.28, 1082 (1972)

1972
[28]

G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D15, 2752 (1977)

1977
[29]

J. M. Bardeen, Gauge invariant cosmological perturba- tions, Phys. Rev. D22, 1882 (1980)

1980
[30]

Kodama and M

H. Kodama and M. Sasaki, Cosmological Perturbation Theory, Prog. Theor. Phys. Suppl.78, 1 (1984)

1984
[31]

Ma and E

C.-P. Ma and E. Bertschinger, Cosmological perturba- tion theory in the synchronous and conformal Newto- nian gauges, Astrophys. J.455, 7 (1995), arXiv:astro- ph/9506072

1995

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [22]

A. E. G¨ umr¨ uk¸ c¨ uo˘ glu, K. Koyama, and S. Mukohyama, Stable cosmology in ghost-free quasidilaton theory, Phys. Rev. D96, 044041 (2017), arXiv:1707.02004 [hep-th]

work page Pith review arXiv 2017

[2] [23]

Linear growth of structure in massive gravity

E. Kulchoakrungsun, A. Mukherjee, N. Agarwal, and A. R. Pullen, Linear growth of structure in massive grav- ity, Phys. Rev. D106, 083527 (2022), arXiv:2206.04086 [gr-qc]

work page Pith review arXiv 2022

[3] [1]

Fierz and W

M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A173, 211 (1939)

1939

[4] [2]

de Rham and G

C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D82, 044020 (2010), arXiv:1007.0443 [hep-th]

arXiv 2010

[5] [3]

de Rham, G

C. de Rham, G. Gabadadze, and A. J. Tolley, Resum- mation of massive gravity, Phys. Rev. Lett.106, 231101 (2011), arXiv:1011.1232 [hep-th]

arXiv 2011

[6] [4]

Hinterbichler, Theoretical aspects of massive gravity, Rev

K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys.84, 671 (2012), arXiv:1105.3735 [hep- th]

arXiv 2012

[7] [5]

de Rham, Massive gravity, Living Rev

C. de Rham, Massive gravity, Living Rev. Rel.17, 7 (2014), arXiv:1401.4173 [hep-th]

arXiv 2014

[8] [6]

D. G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D6, 3368 (1972)

work page 1972