Galileons from Lovelock actions
Open accessAbstract
We demonstrate how, for an arbitrary number of dimensions, the Galileon actions and their covariant generalizations can be obtained through a standard Kaluza-Klein compactification of higher-dimensional Lovelock gravity. In this setup, the dilaton takes on the role of the Galileon. In addition, such compactifications uncover other more general Galilean actions, producing purely second-order equations in the weak-field limit, now both for the Galileon and the metric perturbations.
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Written from the paper’s full text.
TL;DR
The paper shows that covariant Galileon actions and generalizations arise from standard Kaluza-Klein compactification of higher-dimensional Lovelock gravity, with the dilaton playing the role of the Galileon. It reproduces the Deffayet-Deser-Esposito-Farese covariant Galileons in two equivalent forms and uncovers additional Galilean scalar-tensor couplings, including a new \(D=4\) term.
Key findings
- Compactifying the \(d\)-th Lovelock invariant in \(D+N\) dimensions yields an action whose terms independently lead to at most second-order equations; the dilaton controls the extra-dimensional size and acts as the Galileon.
- For arbitrary dimension \(D\), the construction reproduces the fully covariant Galileons of Deffayet, Deser, and Esposito-Farese in two equivalent versions, the \(L\)-form and a new \(K\)-form; for \(D=4\), the explicit \(K_0\)–\(K_3\) expressions (16)–(19) match the original covariant Galileons after partial integration.
- The classification gives \(D/2\) or \((D-1)/2\) Lovelock invariants for even or odd \(D\), \(D\) Galilean terms nontrivial on flat space, and \(D(D-2)/4\) or \((D-1)^2/4\) flat-space-trivial Galilean terms for even or odd \(D\).
- For \(D=4\), besides the known Ricci and Gauss-Bonnet couplings (21)–(22), the Lovelock construction produces two extra Galilean scalar-tensor terms (23)–(24); the second, (24), is stated to be new to the literature.
Methods
- Analytic derivation: start from \(D+N\)-dimensional Lovelock gravity, use a standard Kaluza-Klein metric ansatz factorizing the dimensions, and decompose the \(d\)-th Lovelock invariant into lower-dimensional curvature and scalar terms.
- Classify terms by weak-field counting of fields and derivatives, defining Galilean terms as those with purely second-order equations in the weak-field limit; derive a total-derivative relation between \(K_n\) and \(L_{n+1}\) terms.
- Specialize to \(D=4\) and write down all possible Galilean terms generated by Lovelock compactifications; no observational data or sample is used.
Limitations
- Stated: generic compactifications of Lovelock invariants give order-one coefficients for all terms, making standard Lovelock compactifications unnatural as a physical mechanism for Galilean IR modified gravity; recovering ordinary gravity at short distances would require heavy suppression of terms like (22)–(24), possible only with fine-tuning of Lovelock combinations and extra-dimensional curvature.
- Stated: the construction is not immediately generalizable to general \(p\)-form Galileons with \(p>1\); extensions to multifield Galileons and vector fields are expected but not worked out.
- Evident from the text: the paper is a formal/classification study and does not perform cosmological phenomenology, stability analysis, or observational tests of the new term (24); it explicitly says exploring (24) would be worthwhile.
Model-written summary. Check it against the paper before relying on it.
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Cites
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Claims the paper makes
0 of 18 cited sources checked · 0 supported · 0 flagged
- Findingreasoning
In this setup, the dilaton takes on the role of the Galileon.
“In this setup, the dilaton takes on the role of the Galileon.”
- Also stated in: § VII. CONCLUSIONS · plain
“With the dilaton field playing the role of the Galileon, such compactifications reproduce exactly all the covariant Galileon actions of DDE.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- The authors state this as an assumption: The dilaton phi controls the size of the extra dimensions.
- Also stated in: § VII. CONCLUSIONS · plain
- Findingown data
The decomposition of the action for the standard Kaluza-Klein ansatz for the metric, with phi identified as the dilaton controlling the size of extra dimensions, shows that different terms in the action independently lead to second-order equations for phi and the D-metric g.
“We find that different terms in the action independently lead to second-order equations for and the D-metric g .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- The authors state this as an assumption: The dilaton phi controls the size of the extra dimensions.
- Findingreasoning
For an arbitrary number of dimensions, the Galileon actions and their covariant generalizations can be obtained through a standard Kaluza-Klein compactification of higher-dimensional Lovelock gravity.
“We demonstrate how, for an arbitrary number of dimensions, the Galileon actions and their covariant generalizations can be obtained through a standard Kaluza-Klein compactification of higher-dimensional Lovelock gravity.”
- Also stated in: § I. INTRODUCTION · hedged
“In this paper, we provide a different connection with Lovelock gravity, now for arbitrary dimension D.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Also stated in: § I. INTRODUCTION · hedged
- Findingown data
For D=4, all possible Galilean terms that can be generated by Lovelock compactifications can be written down explicitly.
“In Sec.VI, we specialize to the case D ¼ 4 and write down explicitly all possible Galilean terms that can be generated by Lovelock compactifications.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Findingreasoning
For a given n, the other terms L_n^p in the p-summation come with more powers of curvature, and are therefore of higher order in the weak-field expansion.
“Notice that for a given n, the other terms L n p in the p-summation come with more powers of curvature, and are therefore of higher order in the weak-field expansion.”
Quote in the paperSubstantial claim
- Findingreasoning
Only the terms L_n^p with p = n - d lead to purely second-order equations in the weak-field limit.
“This means that only the terms L n p with p ¼ n À d lead to purely second-order equations in the weak-field limit.”
Quote in the paperSubstantial claim - Findingreasoning
The second-order nature of the action does not change by adding (or canceling) any prefactor f(phi).
“Secondly, we prove that the second-order nature of the action does not change by adding (or canceling) any prefactor fðÞ.”
Quote in the paperSubstantial claim - Findingreasoning
Up to surface terms, the full action that arises from the compactification of the d-th Lovelock invariant becomes equation (13).
“Up to surface terms, the full action that arises from the compactification of the d-th Lovelock invariant then finally becomes:”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Findingreasoning
The different K_n and L_{n+1} terms are identical up to a total derivative.
“A final subtlety in distinguishing the different independent types of terms at the level of the equations, is that the different K n and L nþ1 terms that we discussed above, are actually identical up to a total derivative.”
Quote in the paperSubstantial claim - Findingreasoning
The equations that arise from each K_n and L_n term in (8) separately are by themselves second-order.
“But for our purposes, it is interesting that the equations that arise from each K n and L n term in (8) separately, are actually by themselves also second-order.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Findingcitation
On flat space L_d ~ L_d^0, which is precisely the form of the general flat-space Galileon put forward by DDE.
“On flat space we have L d $ L d 0 , which is precisely the form of the general flat-space Galileon that was put forward by DDE.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Findingreasoning
The different terms in the action give different equations as they all have a different power of phi.
“Furthermore, it is clear now that the different terms do give different equations as they all have a different power of .”
Quote in the paperSubstantial claim - Findingreasoning
The Galilean terms are those with n >= d.
“The more interesting Galilean terms are those with n !d.”
Quote in the paperSubstantial claim - Findingreasoning
The only terms belonging to the flat-space nontrivial category are the L_n's with n = d.
“Since the degree in h of L n nÀd is n À d, the only terms belonging to the first category are the L n 's with n ¼ d.”
Quote in the paperSubstantial claim - Findingreasoning
From the bound 1 <= n <= D for n=d terms, there are a total of D different Galilean terms in D dimensions that are non-trivial on flat space.
“From (10b) we find the bound 1 n D for such terms, leading to a total of D different Galilean terms in D dimensions, that are not-trivial on flat space.”
Quote in the paperSubstantial claim - Findingreasoning
There exists an infinite number of different non-Galilean terms of the type 1 <= n < d.
“From the bound (10b) on n, one finds that there exists an infinite number of different terms of this type.”
Quote in the paperNot a substantial claim - Findingreasoning
For the Gauss-Bonnet invariant only the case multiplied by phi is nontrivial since the invariant by itself is a total derivative.
“For the Gauss-Bonnet invariant only the latter case is nontrivial since the invariant by itself is a total derivative.”
Quote in the paperSubstantial claim - Findingreasoning
The Lovelock setup generates a new class of scalar-tensor couplings that lead to second-order equations.
“Furthermore, our Lovelock setup does not only reproduce the covariant Galileons, it also generates a new class of scalar-tensor couplings that lead to second-order equations.”
Quote in the paperSubstantial claim - Findingreasoning
In this setup these terms follow automatically from the Lovelock construction with, up to an overall factor (d-1)!, the same coefficients C_d^p that were found by DDE.
“In our setup these terms follow automatically from the Lovelock construction with, up to an overall factor (d À 1)!, the same coefficients C d p that were found by DDE.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Findingnone given
In the weak-field limit, which is of nonzero order in metric perturbations, the equations for these new couplings are purely second-order.
“And in the weak-field limit, which in this case is of nonzero order in the metric perturbations, the equations are purely second-order.”
Quote in the paperSubstantial claim - Findingreasoning
There exist D(D-2)/4 or (D-1)^2/4 different terms of the n>d type, for D even or odd.
“From the bound (10b) on n, one finds that there exist DðD À 2Þ=4 or ðD À 1Þ 2 =4 different terms of this type, for D even or odd.”
Quote in the paperNot a substantial claim - Findingreasoning
Upon one partial integration of the last term in (19), the K-form expressions are exactly the original D=4 covariant Galileons obtained in [9].
“Upon one partial integration of the last term in ( 19), these are exactly the original D ¼ 4 covariant Galileons that were obtained in [9].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Findingreasoning
The Lovelock construction produces another equally elegant form for which the equations are identical to those generated by the original Galileon actions.
“In addition, our Lovelock construction produces another equally elegant form, for which the equations are identical to those generated by the original Galileon actions.”
Quote in the paperSubstantial claim - Findingreasoning
For D=4, the original scalar field covariant Galileons of [9,10] appear in two equivalent forms: the L-form found by DDE and a new K-form.
“As we discussed above, we find them appearing in two equivalent forms: the L-form that was found by DDE and a new K-form.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Findingreasoning
The Lovelock construction uncovers an equally elegant, equivalent form for the fully covariant Galileons since L_d ~ K_{d-1} at the level of the equations.
“Furthermore, the Lovelock construction uncovers an equally elegant, equivalent form for the fully covariant Galileons since by (11) we have that L d ffi K dÀ1 at the level of the equations.”
Quote in the paperSubstantial claim - Findingnone given
These n>d terms have n-d > 0 powers of curvature at the lowest order in the weak-field expansion L_n ~ L_n^{n-d}.
“These have n À d > 0 powers of curvature at the lowest order in the weak-field expansion L n $ L n nÀd .”
Quote in the paperSubstantial claim - Findingreasoning
For D=4 there are only two such terms (n>d).
“For D ¼ 4 there are only two such terms.”
Quote in the paperSubstantial claim - Findingnone given
For D=4, the two nonzero Lovelock terms are the Ricci scalar and the Gauss-Bonnet invariant, appearing multiplied by some prefactor f(phi).
“For D ¼ 4, the two nonzero terms are the Ricciscalar and the Gauss-Bonnet invariant, that appear multiplied by some prefactor fðÞ.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Findingnone given
For D=4 there exist only two of these scalar-tensor couplings (23) and (24).
“Interestingly, for D ¼ 4 there exist only two of these scalar-tensor couplings ( 23) and (24).”
Quote in the paperNot a substantial claim - Findingreasoning
The Lovelock setup reveals flat-space trivial Galilean terms for n > d.
“The Lovelock setup also reveals flat-space trivial Galilean terms (for n > d).”
Quote in the paperSubstantial claim - Findingnone given
For generic compactifications of general combinations of different Lovelock invariants, the coefficients for all different terms in the action will be order one.
“For generic compactifications of general combinations of different Lovelock invariants, the coefficients for all different terms in the action will be order one.”
Quote in the paperNot a substantial claim - Premisereasoning
For a given Lovelock invariant, at the lowest nontrivial order in the weak-field expansion in the metric perturbations h, the equations are purely second-order.
“And, for a given Lovelock invariant, at the lowest nontrivial order in the weak-field expansion in the metric perturbations h (¼ g À ), the equations are purely second-order.”
Quote in the paperSubstantial claim - Premisecitation
Lovelock invariants are the unique scalars, constructed solely out of the metric, that generate Lagrangian equations which are at most of second order.
“To see how the connection could emerge, let us recall that the Lovelock invariants are the unique scalars, constructed solely out of the metric, that generate Lagrangian equations which are at most of second order [12].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
The Lovelock construction uncovers other Galilean actions that in the weak-field limit are of nonzero order in h, and that can be thought of as interpolations between the original scalar field Galileon actions and the pure gravity Lovelock actions.
“But, we also uncover other Galilean actions that in the weakfield limit are of nonzero order in h, and that can be thought of as the interpolations between the original scalar field Galileon actions and the pure gravity Lovelock actions.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
- Interpretationnone given
To the authors' knowledge, the second extra Galilean term (24) is new to the literature.
“To our knowledge, the second extra Galilean term ( 24) is new to the literature.”
Quote in the paperNot a substantial claim - Interpretationreasoning
The fact that all different constructions lead to identical expressions speaks in favor of their purported uniqueness.
“The fact that all different constructions lead to identical expressions, clearly speaks in favor of their purported uniqueness.”
Quote in the paperSubstantial claim - Interpretationreasoning
This makes standard Lovelock compactifications rather unnatural as a physical mechanism for Galilean IR modified gravity, which only requires the original Galilean terms (16)-(19), together with the Ricciscalar (20) and Ricciscalar coupling (21).
“This makes standard Lovelock compactifications rather unnatural as a physical mechanism for Galilean IR modified gravity, which only requires the original Galilean terms ( 16)-( 19), together with the Ricciscalar (20) and Ricciscalar coupling (21).”
Quote in the paperSubstantial claim - Interpretationreasoning
To recover ordinary gravity at short distances, the coefficients of the other terms like (22)-(24) would have to be heavily suppressed.
“To recover ordinary gravity at short distances, the coefficients of the other terms like ( 22)-( 24) would have to be heavily suppressed.”
Quote in the paperSubstantial claim - Interpretationreasoning
This suppression is possible in principle, but only at the cost of some fine-tuning of both the Lovelock combinations and the extra-dimensional curvature.
“This is possible in principle, but only at the cost of some fine-tuning of both the Lovelock combinations and the extra-dimensional curvature.”
Quote in the paperSubstantial claim
- May combine two claims. 13
- Classified as probably not a substantial claim. 5
May combine two claims.
FindingreasoningSuch compactifications uncover other more general Galilean actions, producing purely second-order equations in the weak-field limit, now both for the Galileon and the metric perturbations.
“In addition, such compactifications uncover other more general Galilean actions, producing purely second-order equations in the weak-field limit, now both for the Galileon and the metric perturbations.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
May combine two claims.
FindingcitationThese n=d terms are the terms studied by DDE in [10], and the paper recovers the very same expressions.
“These are the terms that were studied by DDE in [10], and we indeed recover the very same expressions.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
Findingown dataSome of the obtained terms are Galilean, and among these are precisely the fully covariant Galileon actions of [10], in two equivalent versions.
“Some of the obtained terms are indeed Galilean, and amongst these we find precisely the fully covariant Galileon actions of [10], in two equivalent versions.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- The authors state this as an assumption: The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
May combine two claims.
FindingreasoningThere can not be any cancellation of higher derivatives between different terms in the r-summation; these terms independently lead to second-order equations.
“This implies that there can not be any cancellation of higher derivatives between different terms in the r-summation, or, in other words, that these terms independently lead to second-order equations.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningThe L_n terms with 1 <= n < d are non-Galilean, with first-order derivatives in the equations already in the weak-field limit.
“Inspection of the summation bound (10b) for p leads us then to the conclusion that the L n terms with 1 n < d are non-Galilean, with first-order derivatives in the equations already in the weak-field limit.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningGalileons have now been produced for general dimension D, either through explicit construction by DDE, or through Lovelock compactifications, and this in two forms (L_d and K_{d-1}).
“Taken together, Galileons have now been produced for general dimension D, either through explicit construction by DDE, or through Lovelock compactifications, and this in two forms (our L d and K dÀ1 ).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
FindingcitationThe specific compactification mentioned also generates the original Galileon (17) and the nonminimal Gauss-Bonnet coupling (22).
“Note that this specific compactification also generates the original Galileon ( 17) and of course the nonminimal Gauss-Bonnet coupling (22).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone givenThey add to the well known and well explored Ricciscalar (21) and Gauss-Bonnet (22) couplings.
“They add to the well known and well explored Ricciscalar (21) and Gauss-Bonnet (22) couplings.”
Quote in the paperNot a substantial claimMay combine two claims.
FindingcitationThe other terms L_d^p in L_d are trivial on flat space, but are required to maintain second-order equations for both the metric and phi on a general background.
“The other terms L d p in L d are trivial on flat space, but as DDE found, they are required to maintain second-order equations for both the metric and equations on a general background.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
Findingnone givenAny Lovelock term leads to purely second-order equations in the weak-field limit, both by itself and multiplied by a single power of phi.
“Indeed, one can easily verify that any Lovelock-term leads to purely second-order equations in the weak-field limit, both by itself and multiplied by a single power of .”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationFully covariant Galileon actions have equations that are at most second-order, with purely second-order equations at lowest order in the weak-field expansion.
“This is also precisely the case for the fully covariant Galileon actions, for which the equations are, at most, of second-order, with purely second-order equations at lowest order (in this case zero) in the weak-field expansion.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [10]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
InterpretationreasoningThese new terms are Galilean and can be interpreted as interpolations between scalar field Galileons and pure gravity Lovelock invariants.
“In that sense these terms are Galilean, and we can interpret them as interpolations between the scalar field Galileons and the pure gravity Lovelock invariants.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationFor D=4, Galileons were obtained through a similar, but not identical, explicit construction in [9] and through a brane setup in [11].
“Moreover, for D ¼ 4, they were obtained through a similar, but not identical, explicit construction in [9] and through a brane setup in [11].”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [9]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [11]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationMore recently, term (23) was constructed directly by demanding second-order equations [16].
“More recently, the term (23) was constructed directly, by demanding second-order equations [16].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [16]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe Galilean character of the Gauss-Bonnet coupling (22) was already exposed in [6], in the context of cosmological backreaction for Gauss-Bonnet gravity.
“The Galilean character of the Gauss-Bonnet coupling ( 22) was already exposed in [6], in the context of the cosmological backreaction for Gauss-Bonnet gravity.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [6]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe phenomenology of term (23) is already explored to some extent [15-19].
“The phenomenology of the term ( 23) is already explored to some extent [15][16][17][18][19].”
Quote in the paperNot a substantial claim0 of 3 sources checked- cites [15]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [16]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [17]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
InterpretationreasoningThis is a new class of scalar-tensor couplings, and can be thought of as interpolations between the original scalar field Galilean actions and the pure spin 2 Lovelock actions.
“This is a new class of scalar-tensor couplings, and we can think of them as interpolations between the original scalar field Galilean actions and the pure spin 2 Lovelock actions.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).
Interpretationnone givenThere is no immediate objection against these extra scalar-tensor couplings in the context of Galilean inflation.
“On the other hand, there is no immediate objection against these extra scalar-tensor couplings in the context of Galilean inflation.”
Quote in the paperNot a substantial claim
The authors state this as an assumption.
The metric takes a standard Kaluza-Klein factorized form with a D-dimensional metric g, a dilaton phi, and an N-dimensional internal metric gamma_ab.
“For the metric, we take a standard Kaluza-Klein anzats that factorizes the dimensions: g AB ¼ g ðx Þ 0 0 e ðx Þ ab ðx i Þ ;(6)”
The authors state this as an assumption.
The internal metric gamma_ab is an arbitrary background metric for the auxiliary space.
“We take an arbitrary background-metric ab for the auxiliary space and we will consider the equations for and g that are generated by the d-th order Lovelock invariant:”
The authors state this as an assumption.
The dilaton phi controls the size of the extra dimensions.
“with identified as the dilaton that controls the size of the extra dimensions.”
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Kamada, K., Kobayashi, T., Yamaguchi, M., & Yokoyama, J. (2010) · arXiv · doi:10.48550/arxiv.1012.4238
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Germani, C., & Kehagias, A. (2010) · arXiv · doi:10.48550/arxiv.1012.0853
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Bianchi type I anisotropic power-law solutions for the Galileon models
Do, T. Q., & Kao, W. F. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.96.023529 · Cited by 25 · see citing works
Stetsko, M. M. (2019) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.99.044028 · Cited by 25 · see citing works
On dark energy models of single scalar field
Li, M., Qiu, T., Cai, Y.-F., & Zhang, X. (2012) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2012/04/003 · Cited by 25 · see citing works
Chatzistavrakidis, A., Khoo, F. S., Roest, D., & Schupp, P. (2017) · Journal of High Energy Physics · doi:10.1007/jhep03(2017)070 · Cited by 21 · see citing works
Black brane in asymptotically Lifshitz spacetime and viscosity/entropy ratios in Horndeski gravity
Brito, F. A., & Santos, F. F. (2020) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/129/50003 · Cited by 21 · see citing works
Wang, H., Qiu, T., & Yun-song Piao, A. (2011) · Physics Letters B · doi:10.1016/j.physletb.2011.12.016 · Cited by 21 · see citing works
Conformal couplings of a scalar field to higher curvature termsPreprint
Oliva, J., & Ray, S. (2012) · Cited by 21 · see citing works
Vainshtein mechanism in Gauss-Bonnet gravity and Galileon aether
Gannouji, R., & Sami, M. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.85.024019 · Cited by 20 · see citing works
Causality violations in Lovelock theories
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Covariant Galileon and a consistent self-accelerating universe
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One-loop corrections to vector Galileon theory
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Self-anisotropizing inflationary universe in Horndeski theory and beyond
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Rotating black hole with a probe string in Horndeski gravity
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HYM-flation: Yang–Mills cosmology with Horndeski coupling
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Spherical cows in the sky with fab four
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- Authors
- Van Acoleyen, K., & Van Doorsselaere, J.
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2011-04-15
- Volume
- 83
- Issue
- 8
- ISSN
- 0556-2821 (ISSN-L)
- 1550-7998
- 1550-2368
- OpenAlex
- W2149642517
- Semantic Scholar
- d4309376b03725ee547051a31f199c31e5bb685a
- Cited by
- 136
- References
- 46
- Citations by year
- 2026: 1 · 2025: 4 · 2024: 7 · 2023: 3 · 2022: 7 · 2021: 6 · 2020: 9 · 2019: 6 · 2018: 4 · 2017: 9 · 2016: 12 · 2015: 11 · 2014: 12 · 2013: 14 · 2012: 17
- FWCI
- 9.69
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Ghent University Academic Bibliography (Ghent University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Galileon, Lovelock gravity, Kaluza-Klein compactification, higher-dimensional gravity, dilaton, second-order equations of motion, weak-field limit, metric perturbations, modified gravity
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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