Pith. sign in

REVIEW 2 major objections 5 minor 73 references

Multi-Galileons in Curved Space

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read A multi-galileon theory on de Sitter space has a vacuum whose Goldstone modes carry no kinetic term.

desk verdict Solid construction with a genuinely interesting vacuum-dependent Boulware-Deser-ghost lesson, but the arbitrary-N claim outruns the even-N classification it rests on. read the letter →

arxiv 2505.08865 v1 pith:I7XM4J42 submitted 2025-05-13 hep-th gr-qc

Alice Garoffolo , Kurt Hinterbichler , Mark Trodden This is my paper · ORCID
classification hep-thgr-qc
keywords multi-galileonsprobebraneconstructiondeSitterspacespontaneoussymmetrybreakingGoldstonebosonsBoulware-DeserghostLovelockinvariantseffectivefieldtheory
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 builds $N$-field scalar effective field theories on four-dimensional de Sitter space by embedding a dS$_4$ brane in a higher-dimensional dS bulk with an $\mathrm{SO}(N)$ symmetry in the extra dimensions. It shows that for a window of parameters the scalar potential develops a second, $\mathrm{SO}(N)$-breaking de Sitter vacuum. Around that vacuum the $N-1$ Goldstone modes have vanishing quadratic action, so they appear only in cubic and higher interactions and are infinitely strongly coupled, while the radial mode has a kinetic term with the wrong sign. This is the paper's central claim: an explicit scalar effective field theory with two dS-invariant vacua that propagate different numbers of degrees of freedom, one with Boulware-Deser-like ghosts and one without.

What carries the argument

The central machinery is the probe brane construction, in which the fields $\pi^I$ are the bending modes of a brane embedded in a higher-dimensional bulk, combined with the restriction to the two Lovelock terms $S=\int d^4x\sqrt{-\bar g}(-a_2+a_4\bar R)$ that survive for $d=4$, at least for even co-dimension. The argument is carried by the resulting $\mathrm{SO}(N)$-invariant potential, equation (3.23), whose dimensionless parameter $C$ selects the vacua. Around the breaking vacuum, the decisive identity is the transformation law of the Goldstones: the broken internal generator acts as a constant shift $\delta\phi^I=-\rho_0\,\delta^I_J$ (for $I,J\neq N$), while the non-linearly realized dS symmetries act as galileon shifts requiring a nonzero mass; the incompatibility forces the Goldstone quadratic action to vanish.

What would settle it

Expand the $N=3$ theory, including any extra boundary terms from the full Lovelock classification, around the $\rho_0$ vacuum; a nonzero quadratic term for the Goldstone modes would falsify the central claim. A Hamiltonian analysis showing $N$ propagating degrees of freedom at linear order around the breaking vacuum would also contradict the paper's reading.

Watch

Extended reading notes

Core claim

Starting from the probe brane action $S=\int d^4x\,\sqrt{-\bar g}\,(-a_2+a_4\bar R)$, with $\bar g$ the induced metric, the authors derive a multi-field DBI-galileon theory on dS$_4$. The potential is a function of $\pi^2=\delta_{IJ}\pi^I\pi^J$ and, for $0<C<2$ with $C=12a_4/(a_2L_D^2)$, it has an $\mathrm{SO}(N)$-preserving maximum at $\pi=0$ and an $\mathrm{SO}(N)$-breaking minimum at $\pi=\rho_0$. Expanding around the $\pi=\rho_0$ vacuum, the quadratic Lagrangian contains only the radial mode $\phi^N$, with the dS galileon form $-\nabla_\mu\phi^N\nabla^\mu\phi^N+(4/L_4^2)\phi_N^2$ but a wrong-sign kinetic term; the $N-1$ Goldstones are absent at quadratic order and appear first at cubic order. The mechanism is a clash of symmetries: the broken $\mathrm{SO}(N)$ shift requires a massless Goldstone, while the dS galileon shift fixes the mass at $4/L_4^2$, so no quadratic term can satisfy both. The paper interprets this as a scalar-only example of a Boulware-Deser-like mismatch between the number of linear and non-linear degrees of freedom around one vacuum, coexisting with a healthy $\mathrm{SO}(N)$-preserving vacuum.

Load-bearing premise

The two-vacuum phenomenon rests on the completeness of the two-term action (2.18), which follows from a Lovelock classification stated for $d=4$ and at least even co-dimension; if additional independent Lovelock or boundary terms exist for odd $N$, they could restore kinetic terms for the Goldstones and remove the effect.

Editorial extensions

If this is right

  • For $0<C<2$ the theory has two de Sitter vacua: the $\pi=0$ vacuum propagates $N$ fields with mass squared $4/L_4^2$, while the $\pi=\rho_0$ vacuum propagates a single radial mode with a wrong-sign kinetic term.
  • The $N-1$ Goldstone modes are infinitely strongly coupled around the breaking vacuum because they have no kinetic term yet appear in cubic and higher interactions.
  • This gives an explicit scalar-only realisation of a background-dependent Boulware-Deser phenomenon: what looks like missing, ghostly degrees of freedom around one vacuum are healthy propagating fields around the other.
  • At the boundary $C=2$ the $\pi=0$ vacuum becomes strongly coupled and the leading term is a quartic multi-field dS galileon, the multi-field generalisation of the single-field dS galileon.

Reading between the lines

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

  • Editorial inference: if the two-term Lovelock action is not complete for odd co-dimension, an extra boundary term could supply kinetic terms for the Goldstones; checking $N=3$ directly would settle whether the two-vacuum phenomenon survives beyond even $N$, a restriction the paper only flags.
  • Editorial inference: the same symmetry clash, one shift symmetry demanding a mass and another demanding zero mass, could be a general mechanism for producing infinitely strongly coupled Goldstones in other probe brane or multi-field constructions; the authors do not claim this generality.
  • Editorial inference: a Hamiltonian or scattering-amplitude analysis around the $\rho_0$ vacuum could test whether the strong coupling hides a finite number of propagating degrees of freedom once quantum effects are included; the paper does not perform this analysis.
  • Editorial inference: if used for multi-field inflation, the strongly coupled Goldstones would change non-Gaussianities in a way distinct from standard multi-field DBI models; this application is beyond what the paper computes.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper extends the probe-brane construction of galileon and DBI effective field theories to higher co-dimension and curved backgrounds. After deriving the general derivative expansion of the brane action with a cosmological constant and an induced Einstein-Hilbert term (Eq. (2.18)), the authors specialize to a dS_d brane embedded in a dS_D bulk with an SO(N) rotational symmetry in the normal directions. For d=4 they compute the scalar potential, identify a non-trivial SO(N)-breaking vacuum for 0<C<2, and expand the action around it. The central claim is that around this vacuum the N-1 Goldstone modes have vanishing quadratic action and are infinitely strongly coupled, while the radial mode has a dS galileon kinetic term with a wrong sign. The paper interprets this as an explicit scalar EFT with two dS-invariant vacua that propagate different numbers of degrees of freedom, one of which has Boulware-Deser-like ghosts and one of which does not.

Significance. If the completeness premise holds, this is a valuable and explicit example of a scalar EFT on de Sitter space whose vacuum structure can change the propagating degrees of freedom in a symmetry-preserving way. The construction is self-contained: the Killing vectors, the derivative expansions in Appendix A, and the quadratic actions (3.29) and (3.39) are mutually consistent, and the symmetry argument for the vanishing Goldstone kinetic terms is coherent. The free parameters are model parameters, not fitted constants, and the two-vacuum phenomenon is a sharp falsifiable prediction within the model. The main weakness is that the two-term action (2.18) is justified in Section 2.2 only for d=4 and at least even co-dimension, while the central claims are stated for arbitrary N; this load-bearing premise needs to be either proven or explicitly restricted.

major comments (2)
  1. [Section 2.2 and Sections 3.4-3.6, Eqs. (3.23), (3.39)] The action (2.18) is introduced as the independent brane action on the strength of a classification that the paper itself states holds only 'in d=4, and at least in the case of even N' (Section 2.2). All subsequent computations, including the potential (3.23), the non-trivial minimum (3.25), and the vanishing Goldstone quadratic action (3.39), are presented for arbitrary N, and the abstract and conclusions make unrestricted claims. If odd N admits additional independent Lovelock or boundary terms, those terms can modify the scalar potential, shift or remove the minimum at ρ0, and contribute quadratic kinetic terms for the Goldstones, which would invalidate the central two-vacuum phenomenon. Footnote 1 addresses only the Myers boundary term in maximally symmetric bulks and does not establish completeness for odd N. The authors should either prove or explicitly cite a completeness theorem valid for all N, or restrict all claims, including the abstract and conclusions, to even N.
  2. [Section 3.6, Eqs. (3.40)-(3.42)] The symmetry argument that the Goldstone quadratic action must vanish is sound for the two-derivative sector, but the presentation could be sharpened. The text states that a would-be Goldstone kinetic term would be incompatible with the simultaneous constant-shift symmetry (3.42) and the galileon shift symmetry (3.41), and concludes that the kinetic term must vanish. This conclusion is convincing, but it is stated only at the level of the leading-order transformations; a brief explicit variation of the would-be quadratic action under (3.41) and (3.42) would make the no-go argument more transparent and would also make clear that no higher-derivative quadratic Goldstone terms are generated by the Lovelock action (2.18) at any order in the expansion of Appendix A.
minor comments (5)
  1. [Eq. (3.25)] The displayed formula for ρ0 is ambiguous: it should be written as ρ0 = sqrt((2 - sqrt(C))/(2 - C)) so that the numerator and denominator are clear.
  2. [Eqs. (3.40) and (3.41)] The index structure in these transformation laws is confusing: the left-hand sides have only a J index while the right-hand sides contain δI_J. The intended meaning is presumably δN_J for the radial mode and δA_J for the Goldstone modes, with A≠N; please rewrite with consistent indices.
  3. [Section 3.5] The phrase 'the mass term is tachyonic' in the C<2 case may confuse readers because the quadratic action (3.29) has a positive coefficient for π2; the mass squared is m2 = -4/L2 in the standard convention used in the text. Adding one sentence connecting the sign of the mass term to the conventional m2 would help.
  4. [Section 2.2, footnote 1] The phrase 'In some case there is a boundary term' should read 'In some cases'; this is a minor typo.
  5. [Section 3.2, Eq. (3.14)] The sentence 'we recognized, p_i, k_i, j_ij' appears to be missing an equation reference or a punctuation adjustment; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two-vacuum dS phenomenon is derived, not assumed; self-citations are background references backed by external work.

full rationale

The paper's derivation chain is self-contained once the two-term brane action (2.18) is adopted. The action is not fitted to the target result: a2 and a4 are free coefficients, and the parameter C = 12a4/(a2 L_D^2) is a reparametrization of their ratio, not a quantity determined by the Goldstone/potential data. The scalar potential (3.23), the vacua at pi=0 and pi=rho0, and the quadratic action (3.39) in which the Goldstones have vanishing kinetic terms are all obtained by explicit substitution and expansion of (2.18) with the dS bulk metric (3.8); the Goldstone shift transformations (3.19)-(3.42) are derived Killing symmetries, not imposed to force the result. The only self-citations, notably [15] for the higher-codimension classification, are not load-bearing in a circular sense: the two-term restriction is attributed to the external classification [42,43], and [15] is cited only for discussion. The paper itself flags the restriction with 'at least in the case of even N'; whether additional odd-N terms would alter the vacuum phenomenon is a completeness/correctness concern, not a circular one. No fitted-input-called-prediction, self-definitional, or renaming pattern is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The construction depends on the specific two-term action and on the parameter range 0 < C < 2. These are inputs from the model, not fitted quantities. No new fields, particles, or forces are posited.

free parameters (2)
  • a_2
    Coefficient of the brane cosmological constant term in the action (2.18). The paper assumes a_2 > 0; its sign controls which vacuum is healthy or ghostly, but the qualitative two-vacuum structure does not depend on a fit.
  • C = 12 a_4/(a_2 L_D^2)
    Dimensionless ratio of the two brane action coefficients. The so(N)-breaking vacuum ρ_0 exists only for 0 < C < 2, so this parameter range is a hand-chosen input that controls the central two-vacua claim.
assumptions (4)
  • domain assumption The brane action is S = ∫ d^d x √-ḡ (-a_2 + a_4 R̄), i.e. the two Lovelock terms are the complete set of independent terms for d=4, even co-dimension.
    Invoked in Section 2.2 (Eq. 2.18); based on the classification in refs [15,42,43]. The central Goldstone result is derived within this specific action, so if additional terms exist the result may change.
  • domain assumption The brane is a probe: it does not back-react on the bulk dS_D metric.
    The probe brane construction throughout Section 2 treats the bulk as fixed; standard for deriving the effective brane action.
  • domain assumption a_2 > 0 (with sign flips reversing stability statements).
    Stated in Section 3.4 before the vacuum analysis; a convention that lets the authors call one vacuum healthy and one ghostly.
  • domain assumption The foliation of dS_D is chosen so the normal directions are conformally flat, fixing the warp factor F_d as in (3.6).
    In Section 3.1 the radial coordinate is fixed by Eq. (3.4); the positive-at-small-ρ solution is selected and the remaining free parameter absorbed by rescaling ρ.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multi-Galileons in Curved Space." pith.science (2026). https://pith.science/paper/I7XM4J42

@misc{pith2026250508865,
  author       = {Pith},
  title        = {Pith review of: Multi-Galileons in Curved Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7XM4J42}},
  note         = {Machine review of arXiv:2505.08865}
}
read the original abstract

Using the probe brane construction of higher derivative effective field theories, extended to higher co-dimensions and curved spaces, we construct galileon and DBI theories on de Sitter space with N fields and an so(N) internal symmetry, non-linearly realizing the symmetries of a higher dimensional de Sitter space. In some cases, the theory admits a non-trivial vacuum that spontaneously breaks the so(N) symmetry, and around this vacuum the Goldstone modes have vanishing kinetic terms and become infinitely strongly coupled. This gives an example of a scalar effective field theory with two de Sitter vacua, one of which appears to have Boulware-Deser-like ghosts, and one of which does not.

Figures

Figures reproduced from arXiv: 2505.08865 by the authors.

Figure 1
Figure 1. The scalar potential for representative choices of [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 22 canonical work pages

  1. [15]

    Multi-field galileons and higher co-dimension branes,

    K. Hinterbichler, M. Trodden, and D. Wesley, “Multi-field galileons and higher co-dimension branes,” Phys. Rev. D82(2010) 124018,arXiv:1008.1305 [hep-th]

    arXiv 2010

  2. [1]

    The Galileon as a local modification of gravity,

    A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification of gravity,” Phys. Rev. D79(2009) 064036,arXiv:0811.2197 [hep-th]

    arXiv 2009

  3. [2]

    DBI and the Galileon reunited,

    C. de Rham and A. J. Tolley, “DBI and the Galileon reunited,” JCAP05(2010) 015, arXiv:1003.5917 [hep-th]

    arXiv 2010

  4. [3]

    Geometry of special Galileons,

    J. Novotny, “Geometry of special Galileons,” Phys. Rev. D95no. 6, (2017) 065019, arXiv:1612.01738 [hep-th]

    arXiv 2017

  5. [4]

    Effective Field Theories from Soft Limits of Scattering Amplitudes,

    C. Cheung, K. Kampf, J. Novotny, and J. Trnka, “Effective Field Theories from Soft Limits of Scattering Amplitudes,” Phys. Rev. Lett.114no. 22, (2015) 221602, arXiv:1412.4095 [hep-th]

    arXiv 2015

  6. [5]

    Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,

    F. Cachazo, S. He, and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,” JHEP07(2015) 149,arXiv:1412.3479 [hep-th]

    arXiv 2015

  7. [6]

    Hidden symmetry of the Galileon,

    K. Hinterbichler and A. Joyce, “Hidden symmetry of the Galileon,” Phys. Rev. D92 no. 2, (2015) 023503,arXiv:1501.07600 [hep-th]

    arXiv 2015

  8. [7]

    Gauged Galileons From Branes,

    G. Goon, K. Hinterbichler, A. Joyce, and M. Trodden, “Gauged Galileons From Branes,” Phys. Lett. B714(2012) 115–119,arXiv:1201.0015 [hep-th]

    arXiv 2012

Show all 73 references
  1. [8]

    Galileons with Gauge Symmetries,

    S.-Y. Zhou and E. J. Copeland, “Galileons with Gauge Symmetries,” Phys. Rev. D85 (2012) 065002,arXiv:1112.0968 [hep-th]

    2012 arXiv

  2. [9]

    Symmetries for Galileons and DBI scalars on curved space,

    G. Goon, K. Hinterbichler, and M. Trodden, “Symmetries for Galileons and DBI scalars on curved space,” JCAP07(2011) 017,arXiv:1103.5745 [hep-th]

    2011 arXiv

  3. [10]

    Galileons on Cosmological Backgrounds,

    G. Goon, K. Hinterbichler, and M. Trodden, “Galileons on Cosmological Backgrounds,” JCAP12(2011) 004,arXiv:1109.3450 [hep-th]

    2011 arXiv

  4. [11]

    A New Class of Effective Field Theories from Embedded Branes,

    G. Goon, K. Hinterbichler, and M. Trodden, “A New Class of Effective Field Theories from Embedded Branes,” Phys. Rev. Lett.106(2011) 231102,arXiv:1103.6029 [hep-th]

    2011 arXiv

  5. [12]

    de Sitter Galileon,

    C. Burrage, C. de Rham, and L. Heisenberg, “de Sitter Galileon,” JCAP05(2011) 025,arXiv:1104.0155 [hep-th]

    2011 arXiv

  6. [13]

    Shift Symmetries in (Anti) 27 de Sitter Space,

    J. Bonifacio, K. Hinterbichler, A. Joyce, and R. A. Rosen, “Shift Symmetries in (Anti) 27 de Sitter Space,” JHEP02(2019) 178,arXiv:1812.08167 [hep-th]

    2019 arXiv

  7. [14]

    Exceptional scalar theories in de Sitter space,

    J. Bonifacio, K. Hinterbichler, A. Joyce, and D. Roest, “Exceptional scalar theories in de Sitter space,” JHEP04(2022) 128,arXiv:2112.12151 [hep-th]

    2022 arXiv

  8. [16]

    Interacting hypersurfaces and multiple scalar-tensor theories,

    Y. Yu, Z. Chen, Y.-M. Hu, and X. Gao, “Interacting hypersurfaces and multiple scalar-tensor theories,” Phys. Rev. D111no. 2, (2025) 024052,arXiv:2410.12680 [gr-qc]

    2025 arXiv

  9. [17]

    Arbitraryp-form Galileons,

    C. Deffayet, S. Deser, and G. Esposito-Farese, “Arbitraryp-form Galileons,” Phys. Rev. D82(2010) 061501,arXiv:1007.5278 [gr-qc]

    2010 arXiv

  10. [18]

    Bi-galileon theory I: Motivation and formulation,

    A. Padilla, P. M. Saffin, and S.-Y. Zhou, “Bi-galileon theory I: Motivation and formulation,” JHEP12(2010) 031,arXiv:1007.5424 [hep-th]

    2010 arXiv

  11. [19]

    Bi-galileon theory II: Phenomenology,

    A. Padilla, P. M. Saffin, and S.-Y. Zhou, “Bi-galileon theory II: Phenomenology,” JHEP01(2011) 099,arXiv:1008.3312 [hep-th]

    2011 arXiv

  12. [20]

    Multi-galileons, solitons and Derrick’s theorem,

    A. Padilla, P. M. Saffin, and S.-Y. Zhou, “Multi-galileons, solitons and Derrick’s theorem,” Phys. Rev. D83(2011) 045009,arXiv:1008.0745 [hep-th]

    2011 arXiv

  13. [21]

    Goldstone’s Theorem and Hamiltonian of Multi-galileon Modified Gravity,

    S.-Y. Zhou, “Goldstone’s Theorem and Hamiltonian of Multi-galileon Modified Gravity,” Phys. Rev. D83(2011) 064005,arXiv:1011.0863 [hep-th]

    2011 arXiv

  14. [22]

    Galileons as Wess-Zumino Terms,

    G. Goon, K. Hinterbichler, A. Joyce, and M. Trodden, “Galileons as Wess-Zumino Terms,” JHEP06(2012) 004,arXiv:1203.3191 [hep-th]

    2012 arXiv

  15. [23]

    New terms for scalar multi-Galileon models and application to SO(N) and SU(N) group representations,

    E. Allys, “New terms for scalar multi-Galileon models and application to SO(N) and SU(N) group representations,” Phys. Rev. D95no. 6, (2017) 064051, arXiv:1612.01972 [hep-th]

    2017 arXiv

  16. [24]

    Spontaneously Broken Spacetime Symmetries and the Role of Inessential Goldstones,

    R. Klein, D. Roest, and D. Stefanyszyn, “Spontaneously Broken Spacetime Symmetries and the Role of Inessential Goldstones,” JHEP10(2017) 051, arXiv:1709.03525 [hep-th]

    2017 arXiv

  17. [25]

    Geometry of Multiflavor Galileon-Like Theories,

    M. P. Bogers and T. Brauner, “Geometry of Multiflavor Galileon-Like Theories,” Phys. Rev. Lett.121no. 17, (2018) 171602,arXiv:1802.08107 [hep-th]

    2018 arXiv

  18. [26]

    An Algebraic Classification of 28 Exceptional EFTs,

    D. Roest, D. Stefanyszyn, and P. Werkman, “An Algebraic Classification of 28 Exceptional EFTs,” JHEP08(2019) 081,arXiv:1903.08222 [hep-th]

    2019 arXiv

  19. [27]

    Scattering Amplitudes and Soft Theorems in Multi-Flavor Galileon Theories,

    K. Kampf and J. Novotn´ y, “Scattering Amplitudes and Soft Theorems in Multi-Flavor Galileon Theories,” JHEP12(2020) 056,arXiv:2009.07940 [hep-th]

    2020 arXiv

  20. [28]

    Shift-symmetricSO(N) multi-Galileon,

    K. Aoki, Y. Manita, and S. Mukohyama, “Shift-symmetricSO(N) multi-Galileon,” JCAP12no. 12, (2021) 045,arXiv:2110.05510 [gr-qc]

    2021 arXiv

  21. [29]

    Generalizing Galileons,

    M. Trodden and K. Hinterbichler, “Generalizing Galileons,” Class. Quant. Grav.28 (2011) 204003,arXiv:1104.2088 [hep-th]

    2011 arXiv

  22. [30]

    A formal introduction to Horndeski and Galileon theories and their generalizations,

    C. Deffayet and D. A. Steer, “A formal introduction to Horndeski and Galileon theories and their generalizations,” Class. Quant. Grav.30(2013) 214006, arXiv:1307.2450 [hep-th]

    2013 arXiv

  23. [31]

    Can gravitation have a finite range?,

    D. G. Boulware and S. Deser, “Can gravitation have a finite range?,” Phys. Rev. D6 (1972) 3368–3382

    1972

  24. [32]

    Resummation of Massive Gravity,

    C. de Rham, G. Gabadadze, and A. J. Tolley, “Resummation of Massive Gravity,” Phys. Rev. Lett.106(2011) 231101,arXiv:1011.1232 [hep-th]

    2011 arXiv

  25. [33]

    Resolving the Ghost Problem in non-Linear Massive Gravity,

    S. F. Hassan and R. A. Rosen, “Resolving the Ghost Problem in non-Linear Massive Gravity,” Phys. Rev. Lett.108(2012) 041101,arXiv:1106.3344 [hep-th]

    2012 arXiv

  26. [34]

    Theoretical Aspects of Massive Gravity,

    K. Hinterbichler, “Theoretical Aspects of Massive Gravity,” Rev. Mod. Phys.84 (2012) 671–710,arXiv:1105.3735 [hep-th]

    2012 arXiv

  27. [35]

    Massive Gravity,

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

    2014 arXiv

  28. [36]

    S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 7, 2019

    2019

  29. [37]

    The Einstein tensor and its generalizations,

    D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys.12(1971) 498–501

    1971

  30. [38]

    Higher Derivative Gravity, Surface Terms and String Theory,

    R. C. Myers, “Higher Derivative Gravity, Surface Terms and String Theory,” Phys. Rev. D36(1987) 392

    1987

  31. [39]

    Counterterms in Dimensionally Continued AdS Gravity,

    O. Miskovic and R. Olea, “Counterterms in Dimensionally Continued AdS Gravity,” JHEP10(2007) 028,arXiv:0706.4460 [hep-th]

    2007 arXiv

  32. [40]

    Action Integrals and Partition Functions in 29 Quantum Gravity,

    G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in 29 Quantum Gravity,” Phys. Rev. D15(1977) 2752–2756

    1977

  33. [41]

    Role of conformal three geometry in the dynamics of gravitation,

    J. W. York, Jr., “Role of conformal three geometry in the dynamics of gravitation,” Phys. Rev. Lett.28(1972) 1082–1085

    1972

  34. [42]

    Matching conditions for a brane of arbitrary codimension,

    C. Charmousis and R. Zegers, “Matching conditions for a brane of arbitrary codimension,” JHEP08(2005) 075,arXiv:hep-th/0502170

    2005 arXiv

  35. [43]

    Einstein gravity on an even codimension brane,

    C. Charmousis and R. Zegers, “Einstein gravity on an even codimension brane,” Phys. Rev. D72(2005) 064005,arXiv:hep-th/0502171

    2005 arXiv

  36. [44]

    Multifield DBI Inflation and Non-Gaussianities,

    M.-x. Huang, G. Shiu, and B. Underwood, “Multifield DBI Inflation and Non-Gaussianities,” Phys. Rev. D77(2008) 023511,arXiv:0709.3299 [hep-th]

    2008 arXiv

  37. [45]

    Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation,

    D. Langlois, S. Renaux-Petel, D. A. Steer, and T. Tanaka, “Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation,” Phys. Rev. D78 (2008) 063523,arXiv:0806.0336 [hep-th]

    2008 arXiv

  38. [46]

    Primordial fluctuations and non-Gaussianities in multi-field DBI inflation,

    D. Langlois, S. Renaux-Petel, D. A. Steer, and T. Tanaka, “Primordial fluctuations and non-Gaussianities in multi-field DBI inflation,” Phys. Rev. Lett.101(2008) 061301,arXiv:0804.3139 [hep-th]

    2008 arXiv

  39. [47]

    Multi-field DBI inflation: Introducing bulk forms and revisiting the gravitational wave constraints,

    D. Langlois, S. Renaux-Petel, and D. A. Steer, “Multi-field DBI inflation: Introducing bulk forms and revisiting the gravitational wave constraints,” JCAP04(2009) 021, arXiv:0902.2941 [hep-th]

    2009 arXiv

  40. [48]

    On the full trispectrum in multi-field DBI inflation,

    S. Mizuno, F. Arroja, and K. Koyama, “On the full trispectrum in multi-field DBI inflation,” Phys. Rev. D80(2009) 083517,arXiv:0907.2439 [hep-th]

    2009 arXiv

  41. [49]

    Primordial fluctuations and non-Gaussianities from multifield DBI Galileon inflation,

    S. Renaux-Petel, S. Mizuno, and K. Koyama, “Primordial fluctuations and non-Gaussianities from multifield DBI Galileon inflation,” JCAP11(2011) 042, arXiv:1108.0305 [astro-ph.CO]

    2011 arXiv

  42. [50]

    The trace formulas yield the inverse metric formula,

    R. R. Silva, “The trace formulas yield the inverse metric formula,” 2000. https://arxiv.org/abs/math-ph/9805006

    2000 arXiv

  43. [51]

    Orthogonal non-Gaussianities from Dirac-Born-Infeld Galileon inflation,

    S. Renaux-Petel, “Orthogonal non-Gaussianities from Dirac-Born-Infeld Galileon inflation,” Class. Quant. Grav.28(2011) 182001,arXiv:1105.6366 [astro-ph.CO]. [Erratum: Class.Quant.Grav. 28, 249601 (2011)]

    2011 arXiv

  44. [52]

    Trispectrum from Co-dimension 2(n) Galileons,

    M. Fasiello, “Trispectrum from Co-dimension 2(n) Galileons,” JCAP12(2013) 033, 30 arXiv:1303.5015 [hep-th]

    2013 arXiv

  45. [53]

    Group theory and de Sitter QFT: The concept of mass,

    M. Boers, “Group theory and de Sitter QFT: The concept of mass,” Master’s thesis, Groningen U., 2013

    2013

  46. [54]

    Mixed-symmetry fields in de Sitter space: a group theoretical glance,

    T. Basile, X. Bekaert, and N. Boulanger, “Mixed-symmetry fields in de Sitter space: a group theoretical glance,” JHEP05(2017) 081,arXiv:1612.08166 [hep-th]

    2017 arXiv

  47. [55]

    A note on the representations of SO(1,d + 1),

    Z. Sun, “A note on the representations of SO(1,d + 1),” Rev. Math. Phys.37no. 01, (2025) 2430007,arXiv:2111.04591 [hep-th]

    2025 arXiv

  48. [56]

    The de Sitter group and its presence at the late-time boundary,

    G. Seng¨ or, “The de Sitter group and its presence at the late-time boundary,” PoS CORFU2021(2022) 356,arXiv:2206.04719 [hep-th]

    2022 arXiv

  49. [57]

    Particles of a de Sitter Universe,

    G. S ¸eng¨ or, “Particles of a de Sitter Universe,” Universe9no. 2, (2023) 59, arXiv:2212.10626 [hep-th]

    2023 arXiv

  50. [58]

    Enayati, J.-P

    M. Enayati, J.-P. Gazeau, H. Pejhan, and A. Wang, The de Sitter (dS) Group and its Representations. An Introduction to Elementary Systems and Modeling the Dark Energy Universe. Synthesis Lectures on Mathematics & Statistics. Springer, 2023. arXiv:2201.11457 [math-ph]

    2023 arXiv

  51. [59]

    Notes on gauge fields and discrete series representations in de Sitter spacetimes,

    A. Rios Fukelman, M. Semp´ e, and G. A. Silva, “Notes on gauge fields and discrete series representations in de Sitter spacetimes,” JHEP01(2024) 011, arXiv:2310.14955 [hep-th]

    2024 arXiv

  52. [60]

    A Walk ThroughSpin(1,d+ 1),

    V. Schaub, “A Walk ThroughSpin(1,d+ 1),”arXiv:2405.01659 [hep-th]

    arXiv

  53. [61]

    Ghosts in massive gravity,

    P. Creminelli, A. Nicolis, M. Papucci, and E. Trincherini, “Ghosts in massive gravity,” JHEP09(2005) 003,arXiv:hep-th/0505147

    2005 arXiv

  54. [62]

    Ghosts, strong coupling and accidental symmetries in massive gravity,

    C. Deffayet and J.-W. Rombouts, “Ghosts, strong coupling and accidental symmetries in massive gravity,” Phys. Rev. D72(2005) 044003,arXiv:gr-qc/0505134

    2005 arXiv

  55. [63]

    Probing Scalar Effective Field Theories with the Soft Limits of Scattering Amplitudes,

    A. Padilla, D. Stefanyszyn, and T. Wilson, “Probing Scalar Effective Field Theories with the Soft Limits of Scattering Amplitudes,” JHEP04(2017) 015, arXiv:1612.04283 [hep-th]

    2017 arXiv

  56. [64]

    Extended DBI and its generalizations from graded soft theorems,

    K. Kampf, J. Novotny, and P. Vasko, “Extended DBI and its generalizations from graded soft theorems,” JHEP10(2021) 101,arXiv:2107.04587 [hep-th]

    2021 arXiv

  57. [65]

    Geometric soft theorems,

    C. Cheung, A. Helset, and J. Parra-Martinez, “Geometric soft theorems,” JHEP04 31 (2022) 011,arXiv:2111.03045 [hep-th]

    2022 arXiv

  58. [66]

    Enhanced soft limits in de Sitter space,

    C. Armstrong, A. Lipstein, and J. Mei, “Enhanced soft limits in de Sitter space,” JHEP12(2022) 064,arXiv:2210.02285 [hep-th]

    2022 arXiv

  59. [67]

    One-loop divergences in the Galileon model,

    T. de Paula Netto and I. L. Shapiro, “One-loop divergences in the Galileon model,” Phys. Lett. B716(2012) 454–460,arXiv:1207.0534 [hep-th]

    2012 arXiv

  60. [68]

    Quantum corrections in Galileon theories,

    N. Brouzakis, A. Codello, N. Tetradis, and O. Zanusso, “Quantum corrections in Galileon theories,” Phys. Rev. D89no. 12, (2014) 125017,arXiv:1310.0187 [hep-th]

    2014 arXiv

  61. [69]

    Quantum corrections to generic branes: DBI, NLSM, and more,

    G. Goon, S. Melville, and J. Noller, “Quantum corrections to generic branes: DBI, NLSM, and more,” JHEP01(2021) 159,arXiv:2010.05913 [hep-th]

    2021 arXiv

  62. [70]

    Instabilities of Spherical Solutions with Multiple Galileons and SO(N) Symmetry,

    M. Andrews, K. Hinterbichler, J. Khoury, and M. Trodden, “Instabilities of Spherical Solutions with Multiple Galileons and SO(N) Symmetry,” Phys. Rev. D83(2011) 044042,arXiv:1008.4128 [hep-th]

    2011 arXiv

  63. [71]

    Stability and superluminality of spherical DBI galileon solutions,

    G. L. Goon, K. Hinterbichler, and M. Trodden, “Stability and superluminality of spherical DBI galileon solutions,” Phys. Rev. D83(2011) 085015,arXiv:1008.4580 [hep-th]

    2011 arXiv

  64. [72]

    Behavior of perturbations on spherically symmetric backgrounds in multi-Galileon theory,

    S. Garcia-Saenz, “Behavior of perturbations on spherically symmetric backgrounds in multi-Galileon theory,” Phys. Rev. D87no. 10, (2013) 104012,arXiv:1303.2905 [hep-th]

    2013 arXiv

  65. [73]

    Flavour-kinematics duality for Goldstone modes,

    D. de Neeling, D. Roest, and S. Veldmeijer, “Flavour-kinematics duality for Goldstone modes,” JHEP10(2022) 066,arXiv:2204.11629 [hep-th]. 32

    2022 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.