General second order scalar-tensor theory, self tuning, and the Fab Four
Christos Charmousis, Edmund J. Copeland, Antonio Padilla, Paul M. Saffin
aa r X i v : . [ h e p - t h ] J un General second order scalar-tensor theory, self tuning, and the Fab Four
Christos Charmousis,
1, 2
Edmund J. Copeland, Antonio Padilla, and Paul M. Saffin LPT, CNRS UMR 8627, Universit´e Paris Sud-11, 91405 Orsay Cedex, France. LMPT, CNRS UMR 6083, Universit´e Fran¸cois Rabelais-Tours, 37200, France School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK (Dated: June 17, 2011)Starting from the most general scalar-tensor theory with second order field equations in fourdimensions, we establish the unique action that will allow for the existence of a consistent self-tuning mechanism on FLRW backgrounds, and show how it can be understood as a combinationof just four base Lagrangians with an intriguing geometric structure dependent on the Ricci scalar,the Einstein tensor, the double dual of the Riemann tensor and the Gauss-Bonnet combination.Spacetime curvature can be screened from the net cosmological constant at any given momentbecause we allow the scalar field to break Poincar´e invariance on the self-tuning vacua, therebyevading the Weinberg no-go theorem. We show how the four arbitrary functions of the scalar fieldcombine in an elegant way opening up the possibility of obtaining non-trivial cosmological solutions.
In a little known paper published in 1974, G.W. Horn-deski presented the most general scalar-tensor theorywith second order field equations in four dimensions [1].Given the amount of research into modified gravity overthe last ten years or so (see [2] for a review), it seems ap-propriate to revisit Horndeski’s work. Scalar tensor mod-els of modified gravity range from Brans-Dicke gravity[3] to the recent models [4, 5] inspired by galileon theory[6], the latter being examples of higher order scalar ten-sor Lagrangians with second order field equations. Eachof these models represent a special case of Horndeski’spanoptic theory.In this letter, we study Horndeski’s theory on FLRWbackgrounds. In particular we ask whether or not thereare subclasses of [1] giving a viable self-tuning mecha-nism for solving the (old) cosmological constant prob-lem. In other words, we ask if one can completely screenthe spacetime curvature from the net cosmological con-stant. Naively one might expect this to be impossibleon account of Weinberg’s no-go theorem [7]. However,Weinberg not only assumes Poincar´e invariance at thelevel of the spacetime curvature but also at the level ofthe self-adjusting fields. Here we follow a route similarto [8] and allow our scalar field to break Poincar´e invari-ance on the self-tuning vacua, whilst maintaining a flatspacetime geometry.By demanding that the self-tuning mechanism contin-ues to work through phase transitions that cause the vac-uum energy to jump, we are able to impose some powerfulrestrictions on Horndeski’s theory. Based on equivalenceprinciple (EP) considerations, we assume that matteris only minimally coupled to the metric and then passthe model through our self-tuning filter. This reducesit to four base Lagrangians each depending on an ar-bitrary function of the scalar only. We call these baseLagrangians the Fab Four , L john = √− gV john ( φ ) G µν ∇ µ φ ∇ ν φ (1) L paul = √− gV paul ( φ ) P µναβ ∇ µ φ ∇ α φ ∇ ν ∇ β φ (2) L george = √− gV george ( φ ) R (3) L ringo = √− gV ringo ( φ ) ˆ G (4) where ˆ G = R µναβ R µναβ − R µν R µν + R is the Gauss-Bonnet combination, ε µναβ is the Levi-Civita tensor and P µναβ = − ε µνλσ R λσγδ ε αβγδ is the double dual of theRiemann tensor [9].Our results prove that any self tuning scalar-tensor the-ory (satisfying EP) must be built from the Fab Four. Theweakest of the four is Ringo since this cannot give riseto self-tuning without “a little help from [its] friends”,John and Paul. When this is the case, Ringo does have anon-trivial effect on the cosmological dynamics but doesnot spoil self-tuning. George also has difficulties in go-ing solo: when V george = const., we just have GR andno self-tuning, whereas when V george = const., we haveBrans-Dicke gravity with w = 0, which does self-tunebut is immediately ruled out by solar system constraints.Thus it is best to consider the Fab Four as combining togive a single theory, as opposed to four different theo-ries in their own right. In particular we expect that oneshould always include John and/or Paul for the reasonsgiven above, and because their non-trivial derivative in-teractions might give rise to Vainshtein effects [11] thatwould help in passing solar system tests. Chameleon ef-fects [12] may also play an important role in this regard. HORNDESKI’S THEORY
The most general second-order scalar tensor theory is S = S H [ g µν , φ ] + S m [ g µν ; Ψ n ] (5)where the Horndeski action, S H = R d x √− g L H , is ob-tained from equation (4.21) of [1], L H = δ ijkµνσ (cid:20) κ ∇ µ ∇ i φR νσjk − κ ,ρ ∇ µ ∇ i φ ∇ ν ∇ j φ ∇ σ ∇ k φ + κ ∇ i φ ∇ µ φR νσjk − κ ,ρ ∇ i φ ∇ µ φ ∇ ν ∇ j φ ∇ σ ∇ k φ (cid:21) (6)+ δ ijµν (cid:2) ( F + 2 W ) R µνij − F ,ρ ∇ µ ∇ i φ ∇ ν ∇ j φ + 2 κ ∇ i φ ∇ µ φ ∇ ν ∇ j φ (cid:3) − F + 2 W ) ,φ + ρκ ] ∇ µ ∇ µ φ + κ ( φ, ρ ) , with ρ = ∇ µ φ ∇ µ φ and δ i i ...i n µ µ ...µ n = n ! δ [ i µ δ i µ ...δ i n ] µ n . Wehave four arbitrary functions of φ and ρ , κ i = κ i ( φ, ρ )as well as F = F ( φ, ρ ), which is constrained so that F ,ρ = κ ,φ − κ − ρκ ,ρ . Note that W = W ( φ ), whichmeans that it can be absorbed into a redefinition of F ( φ, ρ ). The matter part of the action is given by S m [ g µν ; Ψ n ], where we require that the matter fields, Ψ n ,are all minimally coupled to the metric g µν . This fol-lows (without further loss of generality) from assumingthat there is no violation of equivalence principle . Thisreasoning is consistent with the original construction ofBrans-Dicke gravity [3].Here we are interested in Horndeski’s theory on FLRWbackgrounds, for which we have a homogeneous scalar, φ = φ ( t ), and a homogeneous and isotropic metric, g µν dx µ dx ν = − dt + a ( t ) (cid:20) dr − κr + r d Ω (2) (cid:21) (7)with κ being a (positive or negative) constant, specifyingthe spatial curvature. Plugging this into (6), we obtainan effective Horndeski Lagrangian in the minisuperspaceapproximation L eff H ( a, ˙ a, φ, ˙ φ ) = a X n =0 (cid:16) X n − Y n κa (cid:17) H n (8)where H = ˙ a/a is the Hubble parameter, and we have, X = − ˜ Q ,φ ˙ φ + κ , X = 3(2 κ ˙ φ − F ,φ ˙ φ + Q ˙ φ − ˜ Q ) ,X = − F + F ,ρ ˙ φ ) , X = 8 κ ,ρ ˙ φ ,Y = ˜ Q ,φ ˙ φ + 12 κ ˙ φ − F, Y = ˜ Q − Q ˙ φ where Q = ∂ ˜ Q /∂ ˙ φ = − κ and Q = ∂ ˜ Q /∂ ˙ φ =6 F ,φ − φ κ . In a cosmological setting, the matter ac-tion contributes a homogeneous and isotropic fluid with For EP to hold all matter must be minimally coupled to the same metric, ˜ g µν , and this should only be a function of g µν and φ . Dependence on derivatives is not allowed since it wouldresult in the gravitational coupling to matter being momentumdependent, leading to violations of EP. Given ˜ g µν = ˜ g µν ( g αβ , φ ),we simply compute g αβ = g αβ (˜ g µν , φ ), and substitute back intothe action (5), before dropping the tildes. Since this procedurewill not generate any additional derivatives in the equations ofmotion, it simply serves to redefine the Horndeski potentials, κ i ( φ, ρ ). energy density ρ m and pressure p m , satisfying the usualconservation law ˙ ρ m + 3 H ( ρ m + p m ) = 0.The generalized Friedmann equation follows in thestandard manner by computing the Hamiltonian densityfor the Horndeski Lagrangian, and identifying it with theenergy density, ρ m , as follows H ( a, ˙ a, φ, ˙ φ ) = 1 a (cid:20) ˙ a ∂L eff H ∂ ˙ a + ˙ φ ∂L eff H ∂ ˙ φ − L eff H (cid:21) = − ρ m (9)Since matter only couples directly to the metric, and notthe scalar, the scalar equation of motion is given by E ( a, ˙ a, ¨ a, φ, ˙ φ, ¨ φ ) = ddt (cid:20) ∂L eff H ∂ ˙ φ (cid:21) − ∂L eff H ∂φ = 0 (10)Note that this equation is always linear in both ¨ a and ¨ φ . SELF-TUNING
Since our ultimate goal is to identify those corners ofHorndeski’s theory that exhibit self-tuning, we first askwhat it means to self-tune, in a relatively model indepen-dent way. Consider our cosmological background in vac-uum. The matter sector is expected to contribute a con-stant vacuum energy density, which we identify with thecosmological constant, h ρ m i vac = ρ Λ . In a self tuning sce-nario, this should not impact on the curvature, so what-ever the value of ρ Λ , we have a Minkowski spacetime ,with H + κ/a = 0. This should remain true even whenthe matter sector goes through a phase-transition, chang-ing the overall value of ρ Λ by a constant amount. Thisextra requirement will place the strongest constraints onour theory.In order to proceed we shall take these transitions tobe instantaneous, thereby assuming that ρ Λ evolves in apiecewise constant fashion. Now consider a self-tuningsolution, H + κ/a = 0, φ = φ Λ ( t ), satisfying the “on-shell-in-a” equations of motion for the metric¯ H ( φ Λ , ˙ φ Λ , a ) = − ρ Λ (11) Different values of κ represent different slicings of Minkowksispace: κ = 0 corresponds to Minkowski coordinates; κ < κ > By ”on-shell-in- a ” we mean that we have set H = − κ/a .Weshall see later that this is indeed a consistent solution to oursystem. and the scalar,¯ E ( φ Λ , ˙ φ Λ , ¨ φ Λ , a ) = ¨ φ Λ f ( φ Λ , ˙ φ Λ , a ) + g ( φ Λ , ˙ φ Λ , a ) = 0(12)Suppose that a phase transition occurs at some arbi-trary time t = t ∗ , so that ρ Λ ( t −∗ ) = ρ Λ ( t + ∗ ). We re-quire that the scalar field is continuous at the transi-tion, φ Λ ( t −∗ ) = φ Λ ( t + ∗ ), but allow its derivative to jump,˙ φ Λ ( t −∗ ) = ˙ φ Λ ( t + ∗ ). We first consider equation (11). Thisis discontinuous on the right hand side, so it must also bediscontinuous on the left, which means that ¯ H must havesome non-trivial ˙ φ Λ dependence. Next consider equation(12). As ˙ φ Λ is discontinuous, ¨ φ Λ must run into a deltafunction at t = t ∗ . This is not supported on the righthand side of equation (12), and since t ∗ can be chosenarbitrarily, we deduce that f must vanish independentlyof g , so that (12) actually splits into two equations f ( φ Λ , ˙ φ Λ , a ) = 0 , g ( φ Λ , ˙ φ Λ , a ) = 0 (13)Focusing on the former it is clear that if f has nontrivialdependence of ˙ φ Λ then it may be discontinuous at thetransition. Since it is constrained to vanish either side ofthe transition we deduce that ∂f∂ ˙ φ Λ = 0, or equivalently f = f ( φ Λ , a ). Using this simplified form for f , we nowtake derivatives, staying on-shell-in-a, so that we have dfdt ( φ Λ , ˙ φ Λ , a ) = ∂f∂φ Λ ˙ φ Λ + ∂f∂a √− κ = 0 (14)Again, applying similar logic we now conclude that ∂f∂φ Λ = 0 or equivalently f = f ( a ). An identical lineof argument implies that g = g ( a ). What this tells usis that the on-shell-in-a scalar equation of motion (12)has lost all dependence on the scalar field φ Λ and itsderivatives. φ Λ ( t ) is fixed by the gravity equation (11),and must necessarily retain some non-trivial time depen-dence even away from transitions in order to evade theclutches of Weinberg’s theorem. More generally, in orderto cope with transitions the on-shell-in-a gravity equa-tion (11) must depend on ˙ φ Λ . The scalar equation (10)should vanish identically on a flat spacetime, and musttherefore have the schematic form. E = X n ≥ (cid:20) A n + ˜ A n ddt (cid:21) ∆ n (15)where A n = ¨ φα ( φ, ˙ φ, a ) + β ( φ, ˙ φ, a ) , ˜ A n = ¨ φ ˜ α ( φ, ˙ φ, a ) +˜ β ( φ, ˙ φ, a ) and we define∆ n = H n − (cid:18) √− κa (cid:19) n (16)which vanishes on-shell-in-a for n >
0. Since the scalarequation (10) ultimately forces self-tuning, it should notbe trivial. Furthermore, for a remotely viable cosmologyit should be dynamical in the sense that we can evolve towards H + κ/a = 0 rather than having it be true atall times. This imposes the condition that at least one ofthe ˜ A n should be non-vanishing. Note that the sum doesnot include n = 0, which is absolutely crucial in order toforce self-tuningLet us now apply the self-tuning filters to Horndeski’stheory. Using equation (10) we can infer the followingform of the minisuperspace Lagrangian in a self tuningset-up, L effself-tun = a " c ( a ) + X n =1 Z n ( φ, ˙ φ, a )∆ n . (17)In order for the on-shell-in-a gravity equation(11) to retain dependence on ˙ φ Λ we demand that P n =1 nZ n, ˙ φ (cid:16) √− κa (cid:17) n = 0. By requiring (8) to take theform (17) up to a total derivative, we find that we musthave κ <
0, and that κ = 2 V ′ ringo ( φ ) (cid:20) | ρ | ) (cid:21) − V paul ( φ ) ρκ = V ′′ ringo ( φ ) ln( | ρ | ) − V ′ paul ( φ ) ρ − V john ( φ ) [1 − ln( | ρ | )] κ = 12 V ′ john ( φ ) ln( | ρ | ) , κ = − ρ bareΛ − V ′′ george ( φ ) ρF = 12 V george ( φ ) − V john ( φ ) ρ ln( | ρ | )with V ′ george ≡ S self-tun = Z d x [ L john + L paul + L george + L ringo −√− gρ bareΛ (cid:3) + S m [ g µν ; Ψ n ] (18)where the base Lagrangians are built from the Fab Four (1) to (4). Note also the presence of the bare cosmolog-ical constant term ρ bareΛ which can always be absorbedinto a renormalisation of the vacuum energy (containedwithin S m ). This serves as a good consistency check ofour derivation. Such a term had to be allowed by theself-tuning theories – if it had not been there it wouldhave amounted to fine tuning the vacuum energy. THE COSMOLOGY OF
THE FAB FOUR
We shall now briefly present the cosmological equa-tions for the general self-tuning theory (18). To this end,we note that the minisuperspace Lagrangians for the FabFour have the desired structure given by equation (17),and that the Friedmann equations describing this cos-mology are H john + H paul + H george + H ringo = − [ ρ Λ + ρ matter ] (19)where we have absorbed ρ bareΛ into the vacuum energycontribution ρ Λ , and H john = 3 V john ( φ ) ˙ φ (cid:16) H + κa (cid:17) ) H paul = − V paul ( φ ) ˙ φ H (cid:16) H + 3 κa (cid:17) H george = − V george ( φ ) (cid:20)(cid:16) H + κa (cid:17) + H ˙ φ V ′ george V george (cid:21) H ringo = − V ′ ringo ( φ ) ˙ φH (cid:16) H + κa (cid:17) The scalar equations of motion are E john + E paul + E george + E ringo = 0 where E john = 6 ddt h a V john ( φ ) ˙ φ ∆ i − a V ′ john ( φ ) ˙ φ ∆ E paul = − ddt h a V paul ( φ ) ˙ φ H ∆ i + 3 a V ′ paul ( φ ) ˙ φ H ∆ E george = − ddt (cid:2) a V ′ george ( φ )∆ (cid:3) + 6 a V ′′ george ( φ ) ˙ φ ∆ +6 a V ′ george ( φ )∆ E ringo = − V ′ ringo ( φ ) ddt (cid:20) a (cid:18) κa ∆ + 13 ∆ (cid:19)(cid:21) We see that on-shell-in-a, H = − κ/a , Ringo’s contribu-tion to the Friedmann equation loses its dependence on ˙ φ .This explains why Ringo cannot self-tune by itself. Weshould emphasize that Ringo does not spoil self-tuningwhen John and/or Paul are also present, even though it does alter the cosmological dynamics. Note also that if V ′ george = 0, and all the other potentials are vanishing,then the scalar equation of motion becomes trivial anddoes not force self-tuning.For a generic combination of the Fab Four that includesJohn and/or Paul, we have a scalar tensor model of self-tuning. The self-tuning is forced by the scalar equationof motion, while the gravity equation links phase transi-tions in vacuum energy to discontinuities in the temporalderivative of the scalar field. On self-tuning vacua, thescalar field is explicitly time dependent, as it must be inorder to evade Weinberg’s theorem [7]. A detailed studyof Fab Four cosmology will be presented elsewhere.
DISCUSSION
In this letter we have resurrected Horndeski’s theorythat describes the most general scalar-tensor theory withsecond order field equations. We have asked which cor-ners of this theory admit a consistent self-tuning mecha-nism for solving the (old) cosmological constant problem.Remarkably, this reduces the theory down to a combina-tion of four base Lagrangians, dubbed the Fab Four . Self-tuning is made possible by breaking Poincar´e invariancein the scalar sector. There are hints at some deep underlying structure inthis theory. This merits further investigation, but fornow we note that each of the Fab Four can be associatedwith a dimensionally enhanced Euler density. This isimmediately evident for George and Ringo, whereas forJohn and Paul we note that they can both be written inthe form V ( φ ) ∇ µ φ ∇ ν φ δWδg µν , with W john = R d x √− gR and W paul = − R d x √− gφ ˆ G .Have we really solved the cosmological constant prob-lem? We have certainly evaded Weinberg’s theorem, butthere is plenty more to consider. Does the Fab Four ulti-mately give rise to a gravity theory that is phenomeno-logically consistent, in particular at the level of both cos-mology and solar system tests. This is a work in progress,but there are reasons to be guardedly optimistic, espe-cially when one considers the fact that John and Paulcontain non-trivial derivative interactions that may giverise to a successful Vainshtein effect. Relevant work in-volving three of the Fab Four was carried out in [10].We should also ask whether or not the self-tuning prop-erty of the Fab Four is spoilt by radiative corrections.Chances are it probably is spoilt by matter loops, butit is interesting to note that the self-tuning is imposedby the scalar equation of motion, and the scalar doesnot couple directly to matter. The intriguing geometricproperties of the Fab Four may also play a role here, butsuch considerations are beyond the scope of this letter.In any event, the ethos behind our approach is not tomake any grandiose claims regarding a solution of thecosmological constant problem but to ask what can beachieved in this direction at the level of a scalar tensortheory. Given that our starting point was the most gen-eral scalar tensor theory, we should be in a position tomake some reasonably general statements. As we haveshown, Weinberg’s theorem alone is not enough to ruleout possible self-tuning mechanisms, so even if the FabFour are ultimately ruled out by other considerations weshould be able to say we have learnt something aboutthe obstacles towards solving the cosmological constantproblem and how one might think about extending thescope of Weinberg’s theorem.EJC and AP acknowledge financial support from theRoyal Society and CC from STR-COSMO, ANR-09-BLAN-0157. AP notes that he was born in the samehospital as John Lennon. [1] G. W. Horndeski, Int. J. Theor. Phys. (1974) 363-384.[2] T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis,[arXiv:1106.2476 [astro-ph.CO]].[3] C. Brans, R. H. Dicke, Phys. Rev. (1961) 925-935.[4] C. Deffayet, G. Esposito-Farese, A. Vikman, Phys. Rev. D79 (2009) 084003. [arXiv:0901.1314 [hep-th]].[5] C. Deffayet, O. Pujolas, I. Sawicki, A. Vikman,JCAP (2010) 026. [arXiv:1008.0048 [hep-th]].
F. P. Silva, K. Koyama, Phys. Rev.
D80 (2009) 121301.[arXiv:0909.4538 [astro-ph.CO]]. T. Kobayashi, M. Yam-aguchi, J. ’i. Yokoyama, Phys. Rev. Lett. (2010)231302. [arXiv:1008.0603 [hep-th]]. A. De Felice, S. Tsu-jikawa, [arXiv:1008.4236 [hep-th]]. C. Deffayet, S. Deserand G. Esposito-Farese, Phys. Rev. D , 064015 (2009)[arXiv:0906.1967]. C. Deffayet, X. Gao, D. A. Steer,G. Zahariade, [arXiv:1103.3260 [hep-th]]. T. Kobayashi,M. Yamaguchi, J. ’i. Yokoyama, [arXiv:1105.5723 [hep-th]].[6] A. Nicolis, R. Rattazzi, E. Trincherini, Phys. Rev. D79 (2009) 064036. [arXiv:0811.2197 [hep-th]].[7] S. Weinberg, Rev. Mod. Phys. (1989) 1-23. [8] A. Padilla, P. M. Saffin, S. -Y. Zhou, JHEP (2010)031. [arXiv:1007.5424 [hep-th]]. A. Padilla, P. M. Saffin,S. -Y. Zhou, JHEP (2011) 099. [arXiv:1008.3312[hep-th]].[9] C. W. Misner, K. S. Thorne, J. A. Wheeler, San Francisco1973, 1279p.[10] L. Amendola, C. Charmousis and S. C. Davis, Phys. Rev.D , 084009 (2008) [arXiv:0801.4339 [gr-qc]].[11] A. I. Vainshtein, Phys. Lett. B39 (1972) 393-394.N. Kaloper, A. Padilla, N. Tanahashi, in preparation [12] J. Khoury, A. Weltman, Phys. Rev. Lett.93