ISO(4,1) Symmetry in the EFT of Inflation
Paolo Creminelli, Razieh Emami, Marko Simonović, Gabriele Trevisan
IISO(4,1) Symmetry in the EFT of Inflation
Paolo Creminelli, Razieh Emami,
2, 1
Marko Simonovi´c,
3, 4 and Gabriele Trevisan Abdus Salam International Centre for Theoretical Physics,Strada Costiera 11, 34151, Trieste, Italy School of Physics, Institute for Research in Fundamental Sciences (IPM),P. O. Box 19395-5531, Teheran, Iran SISSA, via Bonomea 265, 34136, Trieste, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34136, Trieste, Italy
In DBI inflation the cubic action is a particular linear combination of the two, otherwise indepen-dent, cubic operators ˙ π and ˙ π ( ∂ i π ) . We show that in the Effective Field Theory (EFT) of inflationthis is a consequence of an approximate 5D Poincar´e symmetry, ISO(4,1), non-linearly realized bythe Goldstone π . This symmetry uniquely fixes, at lowest order in derivatives, all correlation func-tions in terms of the speed of sound c s . In the limit c s →
1, the ISO(4,1) symmetry reduces to theGalilean symmetry acting on π . On the other hand, we point out that the non-linear realization ofSO(4,2), the isometry group of 5D AdS space, does not fix the cubic action in terms of c s . Motivations.
The study of non-linearly realized sym-metries in the context of inflation has proven to be apowerful tool to make model-independent predictions. Aspontaneously broken symmetry is manifested in rela-tions among operators with different number of fields: forexample, in the framework of the EFT of inflation [1] onefinds a relation between the kinetic term and the cubicoperators, as a consequence of the non-linear realizationof time diffeomorphisms. This implies that in any modelwith small speed of sound c s (cid:28)
1, one has parametri-cally large non-Gaussianities ∝ c − s . This regime is stillallowed by observations, although severely constrainedby the beautiful Planck data [2].In this note we study the consequences of the non-linear realization of ISO(4,1), the 5D Poincar´e symmetry,in the EFT of inflation. The motivation is twofold. Onone hand this symmetry is typical of inflationary mod-els based on brane constructions, where the position ofa brane moving in an extra dimension plays the role ofthe inflaton. Although the inflationary solution (spon-taneously) breaks ISO(4,1), the dynamics of perturba-tions is constrained by the non-linearly realized symme-tries. On the other hand, observations are only sensitiveto small perturbations around the inflating solution andtheir dynamics is encoded in the EFT of inflation. It isthen of interest to study the possible symmetries that canbe imposed in this theory. In this respect ISO(4,1) nat-urally stands out, since it contains both the 4D Poincar´egroup and the shift symmetry of the inflaton, which isusually imposed to justify slow-roll and the consequentapproximate scale-invariance of the spectrum. We willshow, for example, that the relation between the cubicoperators ˙ π and ˙ π ( ∂ i π ) which occurs in DBI inflation[3] does not require any UV input, but it is just a conse-quence of the ISO(4,1) symmetry at the level of the EFTof inflation. Nonlinear realization of ISO(4,1).
In general, thehomogeneous inflaton background φ ( t ) breaks the4D Poincar´e symmetry to translations and rotations:ISO(3,1) → ISO(3). (We here concentrate on scales much shorter than the Hubble scale H , where spacetime canbe considered flat; we will consider gravity later on.) Atleading order in slow-roll, the inflaton φ is also endowedwith an approximate shift symmetry φ → φ + c and asolution φ ( t ) = vt preserves a combination of this shiftsymmetry and time translations.Perturbations around this background can beparametrized by the Goldstone mode πφ ( (cid:126)x, t ) = φ ( t + π ( (cid:126)x, t )) = v · ( t + π ) (1)and the most general action compatible with the symme-tries reads S = (cid:90) d x (cid:0) a π + a ˙ π + a ( ∂ i π ) + f ˙ π + f ˙ π ( ∂ i π ) + g ˙ π + g ˙ π ( ∂ i π ) + g ( ∂ i π ) + · · · (cid:1) . (2)All the constants are time independent as a consequenceof the residual shift symmetry .Let us now impose the extra symmetry. We want to en-large ISO(3,1) × shift (11 generators) to a 15-dimensionalgroup, ISO(4,1). The additional four transformations actas δφ = ω µ x µ + φ ω µ ∂ µ φ . (3)These are rotations and boosts in the 5th dimension, ifwe interpret φ as a coordinate in the extra dimension,for example describing the position of a brane. The shiftsymmetry of φ is interpreted as translation in the 5th di-mension to complete the isometry group of 5D flat space.However, the geometric interpretation is not mandatory The observed deviation from exact scale-invariance [4] impliesthat the shift symmetry (and therefore the whole ISO(4,1) inthe following) is not exact, but slightly broken by corrections oforder slow-roll. We neglect these corrections in the paper. Notice that we are using a parametrization where the 4D coor-dinates do not transform and the symmetry only acts on fields. a r X i v : . [ h e p - t h ] F e b and we may remain agnostic about the origin of this sym-metry. These transformations act on the Goldstone π as δπ = 1 v δφ = ω µ x µ + v · ( t + π )( ω µ ∂ µ π + ω ) , (4)where in the last equality we have reabsorbed 1 /v intothe definition of ω µ . Demanding that the action (2) isinvariant under these additional transformations imposessome conditions on the coefficients a , a , a , . . . ( ). Ifwe focus on the variation of the action quadratic in π , weget the following relations a = − a (1 − v ) , f = a v − v , f = − a v . (5)The first equation says that the speed of propagation of π excitations, the “speed of sound” c s is related to v as c s = 1 − v . (6)From the 5D geometrical point of view, this is a con-sequence of the relativistic sum of velocities. Here it issimply a consequence of the ISO(4,1) symmetry in theEFT of inflation. The cubic action is fixed by the secondand third relation, so that up to cubic order the action(up to an overall coefficient) reads S = (cid:90) d x (cid:18) ˙ π − c s ( ∂ i π ) + 1 − c s c s (cid:0) ˙ π − c s ˙ π ( ∂ i π ) (cid:1)(cid:19) . (7)This is exactly the same result one gets in DBI inflation[3], but here we see that one does not need any UV input:this action follows from the ISO(4,1) symmetry in theEFT of inflation.As we are going to discuss later, these results will notchange when gravity is taken into account. In the nota-tion of [5] S = (cid:90) d x √− g ˙ HM (1 − c − s ) (cid:20) − a ˙ π ( ∂ i π ) + (cid:18) c c s (cid:19) ˙ π (cid:21) , (8)the coefficient ˜ c (that is in general free), is fixed byISO(4,1): ˜ c = (1 − c s ). In terms of the relative co-efficient between the two operators A ≡ − ( c s + ˜ c ), thesymmetry fixes A = −
1. The Planck limits [2] on theseparameters are shown in Fig. 1. In general φ ( t ) = c + vt , but because of the shift symmetry theconstant can be set to zero without loss of generality. Notice that the tadpole term a will in general be different fromzero, since the background solution will also be affected by termswhich are not ISO(4,1) symmetric, as a potential term and theHubble friction. Anyway a does not enter in the conditionsbelow since its variation, eq. (4), is a total derivative. FIG. 1:
Planck limits [2]: the 68%, 95% and 99.7% regionsin the parameter space ( c s , ˜ c ) (above) and ( c s , A ) (below).The red line shows the prediction if one imposes the ISO(4,1)symmetry (the same as in DBI inflation). We can go to higher order and set to zero the cubicvariation of the action (2). We get a simple system ofalgebraic equations whose solution is g = a − c s c s (cid:18) − c s (cid:19) ,g = − a − c s c s (cid:18) − c s (cid:19) , g = a
14 (1 − c s ) . (9)Again all the coefficients are completely fixed in terms ofa single parameter, the speed of sound c s . This does notcome as a surprise: the only operator with one derivativeper field, that linearly realizes the 4D Poincar´e groupand non-linearly realizes ISO(4,1) is the brane tensionoperator S = M (cid:90) d x (cid:16) − (cid:112) ∂φ ) (cid:17) , (10)so it is not surprising that everything is fixed for oper-ators with one derivative per field. One can check thatexpanding (10) around φ = vt one gets operators whichsatisfy (5) and (9). Still it is nice to see the constraintsdirectly at the level of the EFT of inflation, without as-suming to be able to extrapolate far from the inflationarysolution.One can also explore the consequences of ISO(4,1) foroperators with more derivatives. If we look at operatorswith two derivatives on one of the π ’s then the effectiveaction starts with cubic terms (quadratic terms are totalderivatives) and reads S = (cid:90) d x (cid:0) λ ˙ π ∂ i π + λ ( ∂ i π ) ∂ i π + µ ˙ π ∂ i π + µ ˙ π ( ∂ i π ) ∂ i π + · · · (cid:1) . (11)Using the transformation (4) we can easily find the rela-tions among λ , λ , µ and µ λ = − c s λ , µ = 43 1 − c s c s λ , µ = ( c s − λ . (12)As a check, one can start from the brane picture andconsider an operator with one extra derivative on π com-pared to the brane tension: there is only one, the ex-trinsic curvature of the brane. This gives the followingoperator which non-linearly realizes ISO(4,1) [6] S = M (cid:90) d x
11 + ( ∂φ ) ∂ µ ∂ ν φ∂ µ φ∂ ν φ . (13)Indeed, expanding (13) around φ = vt we find that thecubic action for the Goldstone is S = M (cid:90) d x (cid:18) − c s c s ˙ π ∂ i π + ∂ µ ∂ ν π∂ µ π∂ ν π (cid:19) , (14)which satisfy the constraints (12). The limit of Galilean symmetry and the coupling withgravity.
The ISO(4,1) transformation (4) contains a di-mensionless parameter v , which can be interpreted in a5D picture as the brane velocity in the bulk. As we dis-cussed, this parameter fixes the speed of sound of per-turbations, eq. (6). One can consistently take the limit v → . This is a group contraction and Notice that this simply corresponds to the non-relativistic limit,when the brane motion is slow compared to the speed of light.This does not imply that the 4D Poincar´e symmetry is restored. in this limit the symmetry does not act on coordinatesanymore and it thus commutes with the 4D Poincar´egroup. It reduces to an internal symmetry acting on π only δπ = ω µ x µ . (16)This is the Galilean symmetry studied in [7], whose im-plications for the EFT of inflation have been discussedin [8] (see also [9]). This symmetry requires c s = 1 andforbids all interactions with a single derivative per field.All interactions come from higher derivative terms. Forexample in eq. (14), for c s = 1 we have only the secondoperator which can be written as ( ∂π ) (cid:3) π , i.e. the cubicGalileon.So far we discussed the ISO(4,1) symmetry inMinkowski space, without including gravity. Ultimatelywe are interested in calculating correlation functions dur-ing inflation, so that the coupling with gravity cannot beneglected. Similarly to what happens in the case of theGalilean symmetry discussed above, gravity breaks theISO(4,1) symmetry . This implies that the symmetryis not a good one for the background evolution, since ingeneral the Hubble friction plays an important role. Thisis an additional motivation to formulate the symmetrydirectly in the EFT of inflation as a non-linearly real-ized symmetry for π on scales much shorter than Hubble,without reference to the background solution.Another point to address is whether the actions for π derived above can be used, once minimally coupled togravity, to calculate observables during inflation or grav-ity will completely change the picture. The breaking ofthe symmetry due to gravity will manifest in two ways.First of all, graviton radiative corrections will induce op-erators which do not respect the symmetry. This effectis arguably small, as suppressed by powers of M Pl . Sec-ond, in calculating π loops on a gravitational background,non-invariant terms will also be generated. These oper-ators will be invariant under a shift of π , as the shiftsymmetry is compatible with the coupling with gravity,but not fully ISO(4,1) invariant. As these terms ariseonly on a curved background they will contain powers ofthe Riemann tensor, schematically( R µνρσ ) n ( ∂π ) m . (17)On a quasi de Sitter background R (cid:39) H , so we expectthese terms to be suppressed with respect to the ones we Indeed the transformation of π under a 4D boost parametrizedby β i is given by δπ = β i x i + ˙ πβ i x i + ∂ i β i t . (15)This does not depend on v and is still non-linearly realized for v → On a curved background, one cannot consistently define the con-stant vector ω µ that appears in eq. (4): this shows that thesymmetry is ill-defined in the presence of gravity. considered above by powers of ( H/ Λ) (cid:28)
1, where Λ isthe UV cut-off of the theory.These corrections can become relevant if the coefficientof some operator is unnaturally large. For example, theeffect of the induced gravity term on a brane is studiedin [10, 11] and the conclusion is that the cubic actionis in general not uniquely fixed in terms of c s : a differ-ent linear combination of the operators ˙ π and ˙ π ( ∂ i π ) is possible, giving in particular an orthogonal shape ofnon-Gaussianity. This is at first surprising as the modelrespect the ISO(4,1) symmetry we are discussing. How-ever, the deviations are indeed due to cubic operatorswith more than three derivatives in the EFT of infla-tion [11]: in curved space some of these derivatives canbe traded for the curvature scale H and one is left withonly three derivatives on π . However a basic tenet of theEFT approach is that operators of higher dimension givesmall corrections: if they induce O (1) changes, it is notclear why one can neglect all the other higher dimensionalterms. ISO(4,1) or SO(4,2)?
In DBI inflation [3] a probebrane lives in an AdS throat and non-linearly realizesthe SO(4,2) group, so that one may wonder why we didnot consider this group instead of ISO(4,1). One simpleanswer is that during inflation the brane does not movemuch in units of the AdS radius L , so that the differencebetween flat and curved bulk is immaterial. It is stillinteresting to understand whether SO(4,2) would give thesame predictions.The answer is no. It is straightforward to check, for ex-ample supplementing the DBI action with other SO(4,2)-invariant operators like the AdS conformal Galileons [6],that the nice predictions of ISO(4,1) are lost. In par-ticular the speed of sound is not fixed in terms of thevelocity v in the bulk and the cubic operators ˙ π and˙ π ( ∂ i π ) can appear in a general linear combination. Thefact that c s is not fixed in terms of v may come as asurprise: after all it simply comes from the relativisticsum of velocities and this should apply locally also inAdS. This intuition however requires that higher deriva-tive operators are suppressed by a cutoff scale Λ (cid:29) L − :in this case only the tension of the brane is importantand we get back to the DBI inflation case. When, onthe other hand, Λ ∼ L − higher derivative operators areunsuppressed, the brane is a thick object in comparisonwith the AdS radius: it will not follow geodesics and wedo not expect the same predictions as for DBI inflation,though the SO(4,2) symmetry is preserved.All this can also be seen at the level of the EFT.The most general action allowed by the symmetry upto quadratic order is S EFT = (cid:90) d x (cid:0) a π + a ˙ π + a ( ∂ i π ) + m π (cid:1) , (18) where all the coefficients are now time dependent. As inthe ISO(4,1) case, a background solution with constantvelocity is not in general a solution, therefore we have tokeep a that will be cancelled by additional terms whichare not SO(4,2) symmetric. The non-linear transforma-tion of π that realizes SO(4,2) is δπ = 1˙ φ (cid:18) ω µ x µ φ + ω µ x µ x ν ∂ ν φ − x ω µ ∂ µ φ + 12 ω µ ∂ µ φ − φ ω µ ∂ µ φ (cid:19) . (19)Again, requiring the invariance of the action under thistransformation leads to a set of three constraints on thecoefficients m −
12 ˙ a + ¨ φ φ a = 0 , a + 4 ˙ a − ∂ t (cid:32) a ¨ φ φ ˙ φ (cid:33) − φ ˙ φ m = 0 , a + 4 a + 2 ∂ t (cid:32) a φ − ˙ φ φ ˙ φ (cid:33) − φ φ ˙ φ a = 0 . (20)It is straightforward to verify that, for a constant ˙ φ ,these constraints do not fix the relation among the coef-ficients of the kinetic term and therefore c s .A particular case in which these constraints actually fixthe form of the speed of sound, is the one in which thebackground preserves an SO(4,1) subgroup of SO(4,2)[12]. This happens for a background solution φ ( t ) = α t − . In this case the tadpole term is absent and π ismassless so that the constraints in (20) greatly simplify.The speed of sound is then fixed by the last constraintand the residual dilation symmetry [12]: c s = 1 − α − . Conclusions . Imposing an ISO(4,1) symmetry in theEFT of inflation leaves (at leading order in derivative) asingle free parameter, the speed of sound c s . It shouldbe straightforward to study the consequences of ISO(4,1)in the EFT of multi-field inflation [13]: this symmetry isindeed at play in multi-field DBI models [14]. Acknowledgements.
It is a pleasure to thank M. Seronefor useful discussions, N. Bartolo and M. Liguori for cor-respondence about Planck data. [1] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan,and L. Senatore, “The Effective Field Theory of Infla- tion,” JHEP , 014 (2008) [arXiv:0709.0293 [hep-th]]. [2] P. A. R. Ade et al. [Planck Collaboration], “Planck2013 Results. XXIV. Constraints on primordial non-Gaussianity,” arXiv:1303.5084 [astro-ph.CO].[3] M. Alishahiha, E. Silverstein, D. Tong and , “DBI in thesky,” Phys. Rev. D , 123505 (2004) [hep-th/0404084].[4] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013results. XXII. Constraints on inflation,” arXiv:1303.5082[astro-ph.CO].[5] L. Senatore, K. M. Smith and M. Zaldarriaga, “Non-Gaussianities in Single Field Inflation and their OptimalLimits from the WMAP 5-year Data,” JCAP , 028(2010) [arXiv:0905.3746 [astro-ph.CO]].[6] C. de Rham and A. J. Tolley, “DBI and the Galileon re-united,” JCAP , 015 (2010) [arXiv:1003.5917 [hep-th]].[7] A. Nicolis, R. Rattazzi and E. Trincherini, “The Galileonas a local modification of gravity,” Phys. Rev. D ,064036 (2009) [arXiv:0811.2197 [hep-th]].[8] P. Creminelli, G. D’Amico, M. Musso, J. Norena andE. Trincherini, “Galilean symmetry in the effective the-ory of inflation: new shapes of non-Gaussianity,” JCAP , 006 (2011) [arXiv:1011.3004 [hep-th]].[9] C. Burrage, C. de Rham, D. Seery and A. J. Tol- ley, “Galileon inflation,” JCAP , 014 (2011)[arXiv:1009.2497 [hep-th]].[10] S. Renaux-Petel, S. Mizuno and K. Koyama, “Pri-mordial fluctuations and non-Gaussianities from multi-field DBI Galileon inflation,” JCAP , 042 (2011)[arXiv:1108.0305 [astro-ph.CO]].[11] Sb. Renaux-Petel, “DBI Galileon in the Effective FieldTheory of Inflation: Orthogonal non-Gaussianities andconstraints from the Trispectrum,” arXiv:1303.2618[astro-ph.CO].[12] K. Hinterbichler, A. Joyce, J. Khoury and G. E. J. Miller,“DBI Realizations of the Pseudo-Conformal Universe andGalilean Genesis Scenarios,” JCAP , 030 (2012)[arXiv:1209.5742 [hep-th]].[13] L. Senatore and M. Zaldarriaga, “The Effective FieldTheory of Multifield Inflation,” JHEP , 024 (2012)[arXiv:1009.2093 [hep-th]].[14] 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