Galileons on Cosmological Backgrounds
GGalileons on Cosmological Backgrounds
Garrett Goon ∗ , Kurt Hinterbichler † and Mark Trodden ‡ Center for Particle Cosmology, Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Dated: September 4, 2018)
Abstract
We construct four-dimensional effective field theories of a generalized DBI galileon field, thedynamics of which naturally take place on a Friedmann-Robertson-Walker spacetime. The theo-ries are invariant under non-linear symmetry transformations, which can be thought of as beinginherited from five-dimensional bulk Killing symmetries via the probe brane technique throughwhich they are constructed. The resulting model provides a framework in which to explore thecosmological role that galileons may play as the universe evolves. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] S e p ontents I. Introduction
II. Review of the brane construction for DBI galileons III. DBI Galileons on a Gaussian normal foliation
IV. DBI Galileons on cosmological spaces
V. Solutions, fluctuations, and small field limits π symmetries 20C. Galileon-like limits 21 VI. Conclusion A. Explicit expression for L B. Invariance of the tadpole term under global symmetries eferences I. INTRODUCTION
Galileons are four-dimensional higher-derivative field theories originally discovered as branebending modes in decoupling limits of higher-dimensional induced-gravity models such asthe Dvali-Gabadadze-Porrati (DGP) model [1–3]. Galileons have two key features: theirequations of motion are second order (despite the appearance of higher derivatives in theaction), and they possess novel non-linear global symmetries. Generalized and abstractedaway from these origins [4], the galileons now describe a class of theories with interestingproperties including radiative stability [2, 5, 6], the successful implementation of the Vain-shtein mechanism, and the presence of self-accelerating vacuum solutions (see Sec. 4.4 of[7] for a review). As ghost free, higher derivative field theories, the galileons have havebeen applied to inflation, late time acceleration, and a variety of other cosmological appli-cations [6, 8–20]. They have been used as alternative theories to inflation [21–23], kineticbraiding theories [24, 25], and appear naturally in ghost-free theories of massive gravity([26–29], see [30] for a review). The theory has been extended to the multi-galileon case[5, 31–33], supersymmetrized [34], and generalized to p -forms [35].Recent progress has been made towards covariantizing the galileons and putting them oncurved backgrounds. Naive covariantization of the galileons leads to third order equationsof motion, a problem which is solvable by introducing appropriate non-minimal couplingsbetween the galileons and the curvature tensor [25, 35, 36]. However, this constructiondestroys the the interesting global symmetries of the flat-space theory (see however [37]).A method exists to put the galileons on a fixed curved background while preserving theglobal symmetries [38–40]. The method is based on a geometric interpretation in whichthe galileon field π ( x ) is interpreted as an embedding function describing the position ofa 3-brane living in a Minkowski bulk [41]. The galileon terms arise from the small fieldlimit of 4D Lovelock invariants [42] and the 4D boundary terms associated with the 5DLovelock invariants (the Myers terms [43, 44]). By extending this geometric constructionto allow for an arbitrary 5D bulk geometry and and an arbitrary brane embedding [38, 39],3t is possible to put the galileons on any embeddable background. The non-linear shiftsymmetries for π are then inherited from the isometries of the bulk geometry (for a shortreview on generalizing galileons, see [45]).The purpose of the present paper is to apply the brane construction to cosmological FRWspacetimes, and to identify the non-linear symmetries of the resulting theories (this possi-bility was commented on in [40]). In what follows, we construct galileons on an FRW back-ground embedded in a flat 5D bulk, so that the symmetry group will be the 15-dimensionalPoincare group of 5D flat space, of which the 6 symmetries of FRW (spatial translationsand rotations) will be linearly realized.After a short review of the general geometric construction of the galileons, we introducebulk coordinates that define a foliation of 5D Minkowski space by spatially flat FRW slices,and we present expressions for the galileon Lagrangians and their symmetries. These areghost-free higher derivative scalar theories that live on an FRW space with an arbitrarytime dependence for the scale factor, and which possess 9 non-linearly realized shift-likesymmetries. In the process, we provide expressions for a general Gaussian-normal embed-ding, of which FRW in flat space is just one example. For FRW, we display the shorterminisuperspace Lagrangians, which themselves may be useful in a number of cosmologicalsettings. Additionally, we discuss the small π limits and explore the existence and stabilityof simple solutions for π . Conventions and notation
We use the mostly plus metric signature convention. Tensors are symmetrized with unitweight, i.e T ( µν ) = ( T µν + T νµ ). Curvature tensors are defined by [ ∇ µ , ∇ ν ] V ρ = R ρσµν V σ and R µν = R ρµρν , R = R µµ . 4 I. REVIEW OF THE BRANE CONSTRUCTION FOR DBI GALILEONS
The general geometric construction of galileons living on arbitrary curved backgrounds wasderived in [38, 39] and will be briefly reviewed here. The procedure begins with a fixed 5Dmetric G AB ( X ) and a 3-brane defined by the embedding functions X A ( x ), A ∈ { , , , , } where x µ , µ ∈ { , , , } are the coordinates native to the hypersurface. The induced metricand extrinsic curvature on the brane are¯ g µν = e Aµ e Bν G AB ,K µν = e Aµ e Bν ∇ A n B , (1)where e Aµ = ∂X A ∂x µ are the tangent vectors to the brane, n A is the spacelike normal vector tothe brane, and ∇ A is the covariant derivative with respect to the 5D metric G AB .The action on the brane is an action for the embedding variables X A ( x ), and is chosen tobe a diffeomorphism scalar constructed from ¯ g µν , its covariant derivative and its curvaturetensor, as well as the extrinsic curvature tensor, S = (cid:90) d x √− ¯ g L (¯ g µν , ¯ ∇ µ , ¯ R αβµν , K µν ) , (2)so that it is invariant under under gauge symmetries which are reparameterizations of thebrane coordinates, δX A = ξ µ ( x ) ∂ µ X A . (3)Given any bulk Killing vector K A ( X ) satisfying the bulk Killing equation K C ∂ C G AB + ∂ A K C G CB + ∂ B K C G AC = 0 , (4)both the induced metric and the extrinsic curvature tensor (1), and therefore the action (2),are invariant under the action of the global symmetry δ K X A = K A ( X ) . (5)We fix the gauge symmetry by choosing X µ ( x ) = x µ , X ( x ) = π ( x ) , (6)5hereby yielding an action solely for π ( x ), S = (cid:90) d x √− ¯ g L (¯ g µν , ¯ ∇ µ , ¯ R αβµν , K µν ) (cid:12)(cid:12)(cid:12) X µ = x µ , X = π ( x ) , (7)which has no remaining gauge symmetry.However, a global symmetry transformation (5) will generally ruin the gauge choice (6) andto re-fix the gauge we must make a compensating coordinate transformation on the braneby using (3) with ξ µ = − K µ . Thus the combined transformation δπ = − K µ ( x, π ) ∂ µ π + K ( x, π ) (8)is a global symmetry of the gauge fixed action (7).Aside from their symmetries, the other defining characteristic of galileon field theories is theabsence of derivatives higher than second order in the equations of motion. Generic choicesfor the Lagrangian in (7) will not meet this requirement, but the Lovelock terms and theMyers boundary terms will [41]. In 4D there are only four such terms: L = −√− ¯ g, L = √− ¯ gK, L = −√− ¯ g ¯ R, L = 32 √− ¯ g (cid:20) − K + K µν K − K µν − (cid:18) ¯ R µν −
12 ¯ R ¯ g µν (cid:19) K µν (cid:21) , (9)where all contractions of indices are performed using the induced metric ¯ g µν and its inverse.In addition, there exists a zero derivative “tadpole” term which is not of the form (2) butwhich obeys the same symmetries. This term can be interpreted as the proper volumebetween an X = const . surface and the brane position π ( x ), S = (cid:90) d x (cid:90) π ( x ) dπ (cid:48) (cid:112) − det G AB ( π (cid:48) , x ) . (10)As we show in Appendix B, this term also respects the global symmetries (8). III. DBI GALILEONS ON A GAUSSIAN NORMAL FOLIATION
In this section we calculate the Lagrangians (9) and (10) in the general case of a backgroundmetric which is in Gaussian normal form. The FRW galileon will be a special case of this6eneral form, and we will specialize to it in later sections.The background metric in Gaussian normal form is G AB dX A dX B = f µν ( x, w ) dx µ dx ν + dw . (11)Here X = w denotes the Gaussian normal transverse coordinate, and f µν ( x, w ) is an arbi-trary metric on the leaves of the foliation defined by the constant w surfaces. Recall thatin the physical gauge (6), the transverse coordinate of the brane is set equal to the scalarfield, w ( x ) = π ( x ). This extends our earlier analysis [38, 39], by relaxing the condition thatthe extrinsic curvature of constant π slices be proportional to the induced metric. A. Induced quantities and other ingredients
The induced metric is ¯ g µν = f µν + ∂ µ π∂ ν π, (12)and its inverse is ¯ g µν = f µν − γ ∂ µ π∂ ν π , (13)where γ ≡ / (cid:112) ∂π ) , (14)and the indices on the derivatives are raised with f µν , the inverse of f µν .To calculate the extrinsic curvature we need to find the normal vector n A , which satisfies n A e Bν G AB = 0 ,n A n B G AB = 1 , (15)where e Bν = ∂X B ∂x ν are the tangent vectors to the brane. Solving these equations in the gauge(6) yields n A = γ ( − ∂ µ π, . (16)The extrinsic curvature is given by K µν = e Aµ e Bν ∇ A n B , (17)7hich can be written as K µν = e Bν ∂ µ n B − e Aµ e Bν Γ CAB n C . The ∇ A is a covariant derivative of the bulk metric and so the Christoffel Γ CAB must becalculated with X = w . The replacement w → π ( x ) is then made at the end of thecalculation. Using the bulk coordinates in the form (32), the non-zero 5D Christoffels, Γ ABC ,are Γ λµν = Γ λµν ( f ) , Γ µν = − f (cid:48) µν , Γ µ ν = 12 f µλ f (cid:48) λν , (18)where primes denote derivatives with respect to π . Note that on the right-hand side of thefirst line, the Christoffels of f µν are to be calculated with the π dependence held fixed. Theextrinsic curvature then reads K µν = − γ ∇ µ ∇ ν π + 12 γf (cid:48) µν + γ∂ λ π∂ ( µ πf (cid:48) ν ) λ , (19)where ∇ µ is the covariant derivative calculated from f µν at fixed π .The only remaining components needed to calculate the Lagrangians (9) are expressions forthe induced curvature, ¯ R ρσµν , which arise in L and L . At this point, we will specialize toa flat bulk for which the 5D curvature tensor vanishes, so that the induced curvature tensorcan be expressed solely in terms of the extrinsic curvature tensor and induced metric via theGauss-Codazzi equations, R (5) ABCD e Aµ e Bν e Cρ e Dσ = 0 = ¯ R µνρσ − K µρ K νσ + K µσ K νρ . (20)The expressions for L and L in (9) then reduce to L = −√− ¯ g (cid:2) K − K µν (cid:3) , (21) L = √− ¯ g (cid:2) K − K µν K + 2 K µν (cid:3) . (22)These are all the elements necessary for the calculation of the Lagrangians.8 . The Lagrangians We now present the explicit forms for the DBI galileon Lagrangians. In all cases, we usethe definition γ = 1 / (cid:113) ∂π ) to replace ( ∂π ) in favor of γ (recall that indices on thederivatives are raised with f µν ). In addition, we employ a shorthand notation. We defineΠ µν = ∇ µ ∇ ν π , where the covariant derivative ∇ µ is calculated from f µν at fixed π . f (cid:48) µν denotes the derivative of f µν ( x, π ) with respect to π . We use angular brackets (cid:104) . . . (cid:105) todenote traces of the enclosed product as matrices, with all contractions performed using f µν . For example, we have (cid:104) f (cid:48) (cid:105) = f µν ∂ π f µν , (cid:104) Π f (cid:48) (cid:105) = Π µν f νλ ( ∂ π f λσ ) f σµ , (cid:104) Π (cid:105) = Π µν f νλ Π λσ f σρ Π ρκ f κµ . (23)In addition, when π appears within a angled bracket, it does so only at both ends, anddenotes contraction with ∇ µ π , for example, (cid:104) πf (cid:48) π (cid:105) = ∇ µ π f µν ( ∂ π f νλ ) f λσ ∇ σ π, (cid:104) π Π f (cid:48) π (cid:105) = ∇ µ π f µν Π νλ f λσ ( ∂ π f σρ ) f ρκ ∇ κ π . (24)Employing this notation, the Lagrangians (10) and (9) are calculated to be (no integrationsby parts have been made in obtaining these expressions) L = (cid:90) π ( x ) dπ (cid:48) (cid:113) − det f µν ( x, π (cid:48) ) , L = − (cid:112) − f γ , L = (cid:112) − f (cid:104) − (cid:104) Π (cid:105) + 12 (cid:104) f (cid:48) (cid:105) + γ (cid:18) (cid:104) π Π π (cid:105) + 12 (cid:104) πf (cid:48) π (cid:105) (cid:19) (cid:105) , L = (cid:112) − f (cid:104) − (cid:104) πf (cid:48) π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) π Π π (cid:105) γ − (cid:104) π Π π (cid:105) γ + 2 (cid:104) π Π π (cid:105)(cid:104) Π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) πf (cid:48) π (cid:105) γ + (cid:104) Π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) f (cid:48) (cid:105) γ − (cid:104) Π (cid:105) γ + (cid:104) f (cid:48) (cid:105) γ − (cid:104) Π f (cid:48) (cid:105) γ + (cid:104) f (cid:48) (cid:105)(cid:104) Π (cid:105) γ + (cid:104) Π (cid:105) γ + (cid:104) πf (cid:48) π (cid:105) γ (cid:105) , = (cid:112) − f (cid:104) (cid:104) π Π π (cid:105)(cid:104) Π (cid:105) γ + 34 (cid:104) f (cid:48) (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) Π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ + 34 (cid:104) f (cid:48) (cid:105) (cid:104) π Π π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) π Π π (cid:105) γ + 3 (cid:104) Π f (cid:48) (cid:105)(cid:104) π Π π (cid:105) γ + 6 (cid:104) π Π π (cid:105) γ + 3 (cid:104) f (cid:48) (cid:105)(cid:104) π Π π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) π Π π (cid:105)(cid:104) Π (cid:105) γ − (cid:104) π Π π (cid:105)(cid:104) Π (cid:105) γ − (cid:104) π Π π (cid:105)(cid:104) Π (cid:105) γ + 38 (cid:104) f (cid:48) (cid:105) (cid:104) πf (cid:48) π (cid:105) γ + 32 (cid:104) Π (cid:105) (cid:104) πf (cid:48) π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) πf (cid:48) π (cid:105) γ + 32 (cid:104) Π f (cid:48) (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) Π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) Π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) π Π π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) πf (cid:48) π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ − (cid:104) π Π f (cid:48) Π π (cid:105) γ + 3 (cid:104) πf (cid:48) π (cid:105)(cid:104) π Π f (cid:48) π (cid:105) γ + (cid:104) f (cid:48) (cid:105) γ − (cid:104) Π (cid:105) γ + 32 (cid:104) f (cid:48) (cid:105)(cid:104) Π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) f (cid:48) (cid:105) γ + (cid:104) f (cid:48) (cid:105) γ
4+ 32 (cid:104) f (cid:48) (cid:105)(cid:104) Π f (cid:48) (cid:105) γ − (cid:104) Π f (cid:48) (cid:105) γ − (cid:104) Π f (cid:48) Π f (cid:48) π (cid:105) γ − (cid:104) f (cid:48) (cid:105) (cid:104) Π (cid:105) γ + 34 (cid:104) f (cid:48) (cid:105)(cid:104) Π (cid:105) γ − (cid:104) Π f (cid:48) (cid:105)(cid:104) Π (cid:105) γ − (cid:104) Π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) Π (cid:105) γ + 3 (cid:104) Π (cid:105)(cid:104) Π (cid:105) γ + 3 (cid:104) πf (cid:48) π (cid:105) γ − (cid:104) f (cid:48) (cid:105)(cid:104) πf (cid:48) π (cid:105) γ + 32 (cid:104) Π (cid:105)(cid:104) πf (cid:48) π (cid:105) γ + 3 (cid:104) πf (cid:48) π (cid:105) γ (cid:105) . (25)The only dynamical field present is π , and it enters the Lagrangians both explicitly, andimplicitly through the metric f µν ( x, π ) and its covariant derivatives. Despite the complicatedhigher derivative structure of these Lagrangians, the equations of motion will contain at mostsecond order time derivatives, so that they describe only the π degree of freedom. C. Global symmetries
As mentioned in the introduction, if the bulk metric possesses Killing vectors K A ( X ), thenthe induced metric and extrinsic curvature, and hence actions of the form (7), are invariantunder the transformations (8).The algebra of Killing vectors of G AB contains a subalgebra consisting of those Killingvectors for which K = 0. This is the subalgebra of Killing vectors which are parallel to thefoliation of constant w surfaces, which generates the subgroup of isometries which preservethe foliation. For such a Killing vector, the µ K µ is independent of w , and the µν components of the Killing equations tell us that10 µ ( x ) is a Killing vector of f µν ( x, w ), for any w . We choose a basis of this subalgebra withelements indexed by I , K A I ( X ) = K µ I ( x ) A = µ A = 5 . (26)We now extend this basis to a basis for the algebra of all Killing vectors by adding a suitablychosen set of linearly independent Killing vectors with non-vanishing K . We index thesewith I , so that ( K I , K I ) is a basis of the full algebra of Killing vectors. From the 55component of Killing’s equation, we see that K must be independent of w , so we may write K ( x ).A generic symmetry transformation takes the form δ K X A = a I K A I ( X ) + a I K I ( X ) , (27)where a I and a I are constant parameters. It induces the gauge preserving shift symmetry(8) ( δ K + δ g, comp ) π = − a I K µ I ( x ) ∂ µ π + a I K I ( x ) − a I K µI ( x, π ) ∂ µ π , (28)demonstrating that the K I symmetries are linearly realized, whereas the K I symmetriesare non-linearly realized, corresponding to the spontaneous breaking of the bulk symmetryalgebra down to the subalgebra which preserves the leaves of the foliation. If the bulk metric(11) has Killing vectors, the Lagrangians (25) will have the symmetries (28). IV. DBI GALILEONS ON COSMOLOGICAL SPACES
We now specialize to the case where the brane metric is FRW. We thus need a Gaussian-normal foliation of 5D Minkowski space by FRW slices.11 . Embedding 4D FRW in 5D Minkowski
We consider the case of a spatially flat FRW 3-brane embedded in 5D Minkowski space.Starting from the bulk Minkowski metric with coordinates Y A ds = − (cid:0) dY (cid:1) + (cid:0) dY (cid:1) + (cid:0) dY (cid:1) + (cid:0) dY (cid:1) + (cid:0) dY (cid:1) , (29)we make a change to coordinates to t, x i , w , where i = 1 , , , Y = S ( t, w ) (cid:18) x − H a (cid:19) − (cid:90) dt ˙ HH a ,Y i = S ( t, w ) x i ,Y = S ( t, w ) (cid:18) x − − H a (cid:19) − (cid:90) dt ˙ HH a . (30)Here, a ( t ) is an arbitrary function of t which will become the scale factor of the 4D space,and overdots denote derivatives with respect to t . We have defined x ≡ x i x j δ ij , H ≡ ˙ a/a ,and S ( t, w ) ≡ a − ˙ aw. (31)The lower limits on the integrals in (30) are arbitrary, and different choices merely shift theembedding. In the case of power law expansions a ( t ) ∼ t α , α >
0, taking the lower limit tobe zero puts the big bang at the origin of the embedding space.In these new coordinates, the Minkowski metric reads ds = − n ( t, w ) dt + S ( t, w ) δ ij dx i dx j + dw , (32)where n ( t, w ) ≡ − ¨ a ˙ a w . (33)On any w = const . slice, the induced metric is d ˜ s = − n ( t, w ) dt + S ( t, w ) δ ij dx i dx j , (34) This is the transformation used in [46], except that we have not imposed a Z symmetry. IG. 1: The embedding of an FRW brane in 5D Minksowski space for the case a ( t ) = t / . and so after a slice by slice time redefinition n ( t, w ) dt = dt (cid:48) we verify that we have indeedfoliated M with spatially flat FRW slices. Furthermore, the coordinates are Gaussiannormal with respect to this foliation. A plot of the embedding in the case a ∼ t / is shownin Fig.(1).In the FRW case, the first two galileon Lagrangians (25) read (no integrations by parts havebeen made) L = a π − a (cid:16) a + a ¨ a (cid:17) π a + a (cid:16) ˙ a + a ¨ a (cid:17) π −
14 ˙ a (cid:16) ˙ a + 3 a ¨ a (cid:17) π + 15 ¨ a ˙ a π , L = − (1 − ¨ a ˙ a π )( a − ˙ aπ ) (cid:115) − (cid:18) − ¨ a ˙ a π (cid:19) − ˙ π + ( a − ˙ aπ ) − ( (cid:126) ∇ π ) . (35)We relegate the expression for L to Appendix A, due to its complexity, and opt not to writeout explicit expressions for L and L due to their even more unmanageable length.13 . Global symmetries for FRW As reviewed in Section III C, identifying the relevant global symmetries reduces to the taskof finding the Killing vectors of the bulk Minkowski metric in the brane-adapted coordinates(32), separating the Killing vectors into those with vanishing K components, denoted by K A I , and those which have non-vanishing K ’s, denoted by K AI .Let Y A be the cartesian coordinates used in (29) with associated basis vectors ¯ ∂ A . TheKilling vectors in the Y A coordinates take the form of the ten rotations and boosts, L AB ,and the five translations P A , L AB = Y A ¯ ∂ B − Y B ¯ ∂ A , P A = − ¯ ∂ A . (36)After rewriting these Killing vectors in terms of the brane-adapted coordinates { t, x i , w } andthe associated basis vectors { ∂ t , ∂ i , ∂ w } , we find the following combinations which containno K component, L ij = x i ∂ j − x j ∂ i , −
12 [ L i + L i ] = − ∂ i . (37)These generate the three rotations and three spatial translations of the FRW leaves. Theyare the K A I .The remaining vectors form the K AI , which we take to be the following combinations, v i = 12 [ L i − L i ] = 12 x i ˙ a (cid:34)(cid:90) dt ˙ HH a (cid:35) ∂ w + x i (cid:0) a − ˙ aπ + ˙ a (cid:82) dt ˙ HH a (cid:1) a − π ¨ a ∂ t − (cid:34) x i x i ˙ a + 14 ˙ a + (cid:82) dt ˙ HH a a − π ˙ a (cid:35) ∂ i + (cid:88) j (cid:54) = i (cid:20) − x i x j ∂ j + x j x j ∂ i (cid:21) ,k i = − P i = 1 a − π ˙ a ∂ i + x i ˙ a (cid:0) ˙ aπ ¨ a − ˙ a ∂ t − ∂ w (cid:1) ,q = −
12 [ P + P ] = ˙ a (cid:0) ∂ w + ˙ a ˙ a − π ¨ a ∂ t (cid:1) ,u = −
12 [ P − P ] = x ˙ a −
14 ˙ a ∂ w + x ˙ a + 14 ˙ a − π ¨ a ∂ t − a − π ˙ a (cid:88) i x i ∂ i ,s = L = (cid:34) a − π ˙ a + ˙ a (cid:82) dt ˙ HH a π ¨ a − ˙ a (cid:35) ∂ t − ˙ a (cid:34)(cid:90) dt ˙ HH a (cid:35) ∂ w + (cid:88) i x i ∂ i , (38)14here H = ˙ a/a , x = δ ij x i x j , and the summation convention has been suspended. Thelower limits on the integrals should be the same as those in (30).The non-linear symmetries of the π field are then obtained from (8), δ v i π = 12 x i ˙ a (cid:90) dt ˙ HH a − x i (cid:0) a − ˙ aπ + ˙ a (cid:82) dt ˙ HH a (cid:1) a − π ¨ a ˙ π + (cid:34) x i x i ˙ a + 14 ˙ a + (cid:82) dt ˙ HH a a − π ˙ a (cid:35) ∂ i π − (cid:88) j (cid:54) = i (cid:20) − x i x j ∂ j π + x j x j ∂ i π (cid:21) ,δ k i π = x i ˙ a (cid:18) ˙ a ˙ π ˙ a − π ¨ a − (cid:19) − ∂ i πa − π ˙ a ,δ q π = ˙ π ˙ a π ¨ a − ˙ a + ˙ a,δ u π = x ˙ a −
14 ˙ a − x ˙ a + 14 ˙ a − π ¨ a ˙ π + 12 a − π ˙ a (cid:88) i x i ∂ i π,δ s π = − ˙ a (cid:90) dt ˙ HH a + (cid:16) a − ˙ aπ + ˙ a (cid:82) dt ˙ HH a (cid:17) ˙ π ˙ a − π ¨ a − (cid:88) x i ∂ i π, (39)where the replacement w → π ( x µ ) was performed.These non-linear transformations are the FRW analogues of the shift symmetries of the flatspace galileon. They are symmetries of the Lagrangians (35) and (A1) as well the L , L which we did not write out. Together with the spatial rotation and translation symmetriesof FRW, the commutation relations of these transformations are those of the 5D Poincaregroup. These are complicated and highly non-linear transformations, and without the braneformalism it would be nearly impossible to guess their form. C. Minisuperspace Lagrangians
For cosmological applications where we are not considering fluctuations, we may be mostinterested in the limiting case in which spatial gradients are set to zero, so that π = π ( t ). Inthis minisuperspace approximation, the Lagrangians simplify significantly, and we displaytheir full forms here. In displaying these, the numerators are ordered by increasing powersof π , and then by patterns of derivatives on the π fields. No integrations by parts have been15ade. L = a π − a (cid:16) a + a ¨ a (cid:17) π a + a (cid:16) ˙ a + a ¨ a (cid:17) π −
14 ˙ a (cid:16) ˙ a + 3 a ¨ a (cid:17) π + 15 ¨ a ˙ a π , L = − (cid:16) a − π ˙ a (cid:17) (cid:114)(cid:16) − π ¨ a ˙ a (cid:17) − ˙ π , L = (cid:104) a ˙ a + a ¨ a ˙ a + (cid:0) − a ˙ a − a ¨ a ˙ a − a ¨ a ˙ a (cid:1) π − a ˙ a ˙ π − a ˙ a ¨ π + (cid:0) a + 21 a ¨ a ˙ a + 15 a ¨ a ˙ a + a ¨ a (cid:1) π + (6 a ˙ a + 6 a ¨ a ˙ a − a ... a ˙ a + a ¨ a ˙ a ) π ˙ π + ( − a ˙ a − a ¨ a ˙ a ) ˙ π + (3 a ˙ a + a ¨ a ˙ a ) π ¨ π + ( − a ˙ a − a ¨ a ˙ a − a ¨ a ˙ a ) π + ( − a − a ¨ a ˙ a + 3 a ... a ˙ a − a ¨ a ˙ a ) π ˙ π + (6 a ˙ a + 9 a ¨ a ˙ a ) π ˙ π + ( − a ˙ a − a ¨ a ˙ a ) π ¨ π + 3 a ˙ a ˙ π + (9 a ˙ a ¨ a + 11 ˙ a ¨ a ) π + (6¨ a ˙ a − a ... a ˙ a + 9 a ¨ a ˙ a ) π ˙ π + ( − a − a ¨ a ˙ a ) π ˙ π + ( ˙ a + 3 a ¨ a ˙ a ) π ¨ π − a ˙ a π ˙ π − a ¨ a π + ( ˙ a ... a − a ¨ a ) π ˙ π + 5 ˙ a ¨ aπ ˙ π − ˙ a ¨ a ¨ ππ + 3 ˙ a ˙ π π (cid:105) / (cid:104) ˙ a (cid:0)(cid:0) ˙ π − (cid:1) ˙ a + 2 π ¨ a ˙ a − π ¨ a (cid:1) (cid:105) , L = (cid:104) − a ˙ a − a ¨ a ˙ a + (6 ˙ a + 30 a ¨ a ˙ a + 12 a ¨ a ˙ a ) π + 6 a ˙ a ˙ π + 6 a ˙ a ¨ π + ( − a ˙ a − a ¨ a ˙ a − a ¨ a ) π + ( − a − a ¨ a ˙ a + 6 a ... a ˙ a − a ¨ a ˙ a ) π ˙ π + (6 a ˙ a + 12 a ¨ a ˙ a ) ˙ π + ( − a ˙ a − a ¨ a ˙ a ) π ¨ π + (30¨ a ˙ a + 18 a ¨ a ˙ a ) π + (12¨ a ˙ a − a ... a ˙ a + 18 a ¨ a ˙ a ) π ˙ π + ( − a − a ¨ a ˙ a ) π ˙ π + (6 ˙ a + 12 a ¨ a ˙ a ) π ¨ π − a ˙ a ˙ π −
12 ˙ a ¨ a π + (6 ˙ a ... a −
12 ˙ a ¨ a ) π ˙ π + 18 ˙ a ¨ aπ ˙ π − a ¨ a ¨ ππ + 6 ˙ a π ˙ π (cid:105) / (cid:104) ˙ a ( ˙ a ( ˙ π + 1) − π ¨ a ) (cid:115)(cid:18) − π ¨ a ˙ a (cid:19) − ˙ π (cid:105) , = (cid:104) − a − a ¨ a ˙ a + (36¨ a ˙ a + 36 a ¨ a ˙ a ) π + 6 ˙ a ˙ π + 18 a ˙ a ¨ π + ( − a ˙ a − a ¨ a ˙ a ) π + ( − a ˙ a + 18 a ... a ˙ a − a ¨ a ˙ a ) π ˙ π + (6 ˙ a + 36 a ¨ a ˙ a ) ˙ π + ( −
18 ˙ a − a ¨ a ˙ a ) π ¨ π + 24 ˙ a ¨ a π + (24 ˙ a ¨ a −
18 ˙ a ... a ) π ˙ π + 18 ˙ a ¨ aπ ¨ π −
42 ˙ a ¨ aπ ˙ π − a ˙ π (cid:105) / (cid:104) ˙ a ( ˙ π + 1) − π ¨ a (cid:105) . (40)The π equations of motion derived from these are second order in time derivatives. Asbefore, the scale factor a ( t ) describes the fixed background cosmological evolution, and doesnot represent a dynamical degree of freedom.Of the symmetries (39), only δ q is free of explicit dependence on the spatial coordinates. Itis a symmetry of the Lagrangians (40), δ q π = ˙ π ˙ a π ¨ a − ˙ a + ˙ a. (41) V. SOLUTIONS, FLUCTUATIONS, AND SMALL FIELD LIMITS
In this section, we explore the existence and stability of simple solutions for π . In particular,we focus on the properties of the possible π = 0 solutions. A. Simple solutions and stability
Retaining all temporal and spatial derivatives, we expand the Lagrangians to second orderin π , and find, after much integration by parts,17 = a π − (cid:16) ¨ aa ˙ a + 3 ˙ aa (cid:17) π + O (cid:0) π (cid:1) , (42) L = (cid:18) a ˙ a + a ¨ a ˙ a (cid:19) π + 12 a ˙ π − a (cid:16) (cid:126) ∇ π (cid:17) − (cid:0) ¨ aa + ˙ a a (cid:1) π + O (cid:0) π (cid:1) , (43) L = 6 (cid:0) a ˙ a + a ¨ a (cid:1) π + 3 ˙ aa ˙ π − (cid:18) a + a ¨ a ˙ a (cid:19) (cid:16) (cid:126) ∇ π (cid:17) − (cid:0) a ¨ aa + ˙ a (cid:1) π + O (cid:0) π (cid:1) , (44) L = 6 (cid:0) ˙ a + 3 a ˙ a ¨ a (cid:1) π + 9 ˙ a a ˙ π − (cid:18) ˙ a a + 2¨ a (cid:19) (cid:16) (cid:126) ∇ π (cid:17) −
12 ˙ a ¨ aπ + O (cid:0) π (cid:1) , (45) L = 24 ˙ a ¨ a π + 12 ˙ a ˙ π −
12 ¨ a ˙ aa (cid:16) (cid:126) ∇ π (cid:17) + O (cid:0) π (cid:1) . (46)Note that at quadratic order all the higher derivative terms have cancelled out up to totalderivative, a consequence of the fact that the equations of motion are second order.Consider a theory which is an arbitrary linear combination of the five Lagrangians, L = (cid:88) n =1 c n L n , (47)where the c n are (dimensionful) constants. If π = 0 is to be a solution to the full equationsof motion, the linear terms in L must vanish, which gives the condition c a + c (cid:18) a ˙ a + a ¨ a ˙ a (cid:19) + 6 c (cid:0) a ˙ a + a ¨ a (cid:1) + 6 c (cid:0) ˙ a + 3 a ˙ a ¨ a (cid:1) + 24 c ˙ a ¨ a = 0 . (48)For generic values of the c n , this is a non-linear second order equation for a ( t ) which can besolved to yield a background for which π = 0 is a solution. If we look for standard power-lawsolutions, a ( t ) = ( t/t ) α , the condition (48) becomes (cid:2) c ( α − α + 6 c (4 α − α t + 6 c α (2 α − t + c (4 α − t + c t (cid:3) (cid:18) tt (cid:19) α = 0 . (49)Each power of t must vanish independently, so we see that the only non-trivial power-lawsolutions are for α = 1 , / , / , /
4. For these solutions, the corresponding c n must benon-zero and the others must be set to zero.18 c c c c c A B C H τ c c t c t c ( t/t ) / t c ( t/t ) / − t c ( t/t ) / / c t c ( t/t ) / t c ( t/t ) / − t c ( t/t ) / √ c c ( t/t ) / c ( t/t ) / − t c ( t/t ) / √ TABLE I: Lagrangian coefficients, stability coefficients, and time scale comparisons for fluctuationsabout π = 0 for all possible non-trivial power law solutions a ( t ) = ( t/t ) n . To test the stability around a given solution, we look at the quadratic part of the Lagrangian,which has the following form, L = 12 A ( a ( t ) , c n ) ˙ π − B ( a ( t ) , c n )( (cid:126) ∇ π ) − C ( a ( t ) , c n ) π , (50)where A ( a ( t ) , c n ) = c a + 6 c ˙ aa + 18 c ˙ a a + 24 c ˙ a ,B ( a ( t ) , c n ) = c a + 2 c (cid:18) a + a ¨ a ˙ a (cid:19) + 6 c (cid:18) ˙ a a + 2¨ a (cid:19) + 24 c ¨ a ˙ aa ¨ a,C ( a ( t ) , c n ) = c (cid:18) ¨ aa ˙ a + 3 ˙ aa (cid:19) + 6 c (cid:0) ¨ aa + ˙ a a (cid:1) + 6 c (cid:0) a ¨ aa + ˙ a (cid:1) + 24 c ˙ a ¨ a . (51)The stability of the theory against ghost and gradient instability, which is catastrophic atthe shortest length scales, requires A > B ≥
0. Freedom from tachyon-like instabilitiesrequires C ≥
0. However a tachyonic instability where
C < A ¨ π + ˙ A ˙ π − B ∇ π + Cπ = 0 . Thus,the time scale τ associated with a tachyonic mass term is given by τ = (cid:112) A/ | C | and thetachyonic instability is tolerable if Hτ (cid:38) a ( t ) ∼ t , the choice c > A > B >
0, at which point we necessarily have19 < τ H ∼ π = 0solution becomes c + 4 Hc + 12 c H + 24 c H + 24 c H = 0 , (52)and the coefficients (51) of the quadratic part are A ( a ( t ) , c i ) = a e Ht (cid:0) c + 6 c H + 18 c H + 24 c H (cid:1) ,B ( a ( t ) , c i ) = a e Ht (cid:0) c + 6 c H + 18 c H + 24 c H (cid:1) ,C ( a ( t ) , c i ) = − a e Ht H (cid:0) c + 6 c H + 18 c H + 24 c H (cid:1) . (53)All the coefficients share a common factor, so the field is either a ghost or a tachyon, inagreement with the findings in Section V.A of [38]. Comparing the tachyon time scaleagainst 1 /H gives Hτ = 1 /
2, so the tachyon time scale is approximately the Hubble time.This would be disastrous for inflation, since the instability would manifest itself after onee-fold, but it may be tolerable for late-time cosmic acceleration.
B. Small π symmetries The small π limits of the symmetries (39) expanded to lowest order in π , are δ v i π = 12 x i (cid:90) dt ˙ HH a ˙ a,δ k i π = − x i ˙ a,δ q π = ˙ a,δ u π = x ˙ a −
14 ˙ a ,δ s π = − ˙ a (cid:90) dt ˙ HH a . (54)In the case where π = 0 is a solution, these are symmetries of the quadratic action for π .Otherwise, they are symmetries of the action linear in π .20 . Galileon-like limits When we generate galileon theories by foliating a maximally symmetric bulk by maximallysymmetric branes, as in [39], there exist small field limits which greatly simplify the La-grangians (25). To take these limits, we form linear combinations ¯ L n = (cid:80) nm =1 c n,m L m ofthe original Lagrangians, with constant coefficients c n,m chosen such that a perturbative ex-pansion of L n around a constant background π → π + δπ begins at O ( δπ n ). In particular,as first shown in [41], when applied to the case of a flat brane in a flat bulk, this procedurereproduces the flat space galileons of [4].The ability to carry out such an expansion appears to be an artifact of maximal symmetry.The small π limit in the present case of a flat bulk and an FRW brane does not, for general a ( t ), admit a choice of c n,m with the above mentioned properties.One case which does work is a ( t ) ∼ e Ht , corresponding to a de Sitter brane, which hasmaximal symmetry. The induced metric on any w = const hypersurface is ds = (1 − Hw ) (cid:2) − dt + e Ht d(cid:126)x (cid:3) = (1 − Hw ) g ( dS ) µν dx µ dx ν , (55)where g ( dS ) µν is the 4D de Sitter metric in inflationary coordinates, and so we are simplyfoliating 5D minkowski by dS , returning to the setup of a maximally symmetric brane in amaximally symmetric bulk. In the gauge (6), the induced metric becomes¯ g µν = ( − Hπ ) g ( dS ) µν + ∂ µ π∂ ν π . (56)If we then make the field redefinition ˜ π = − Hπ and switch to coordinates ˆ x µ = Hx µ ,the Lagrangians calculated from the induced metric (56) and associated extrinsic curvaturetake the forms of those in Sec. IV.C of [38], from which small ˜ π limits can be constructed. VI. CONCLUSION
The probe-brane construction has facilitated the development of entirely new four-dimensional scalar effective field theories with nontrivial symmetries stemming from the21illing symmetries of the higher-dimensional bulk. The simplest example of this construc-tion [41] yields flat space galileons [4], of which the DGP cubic term represents the simplestnontrivial interaction term. In general, however, a much richer structure is possible, de-pending on the geometries of the bulk and the brane. In previous work [38, 39] we have laidout the general framework for deriving new four-dimensional field theories in this way, andhave applied the method to the examples in which bulk and brane are maximally symmetricspaces.In this paper, we have extended the construction to background geometries with Gaussiannormal foliations, of which the cosmological FRW spacetimes are a particularly useful exam-ple. We have derived the relevant operators allowed in the Lagrangians, and identified thehighly nontrivial symmetry transformations under which they are invariant. These generalexpressions are much longer for FRW spacetimes than they are for maximally symmetricones. By specializing to the minisuperspace approximation, in which the galileons dependonly on cosmic time, we are able to provide somewhat more compact versions suitable forunderstanding the effects of galileons on the background cosmology. However, more com-plicated questions, such as those involving spatially dependent galileon perturbations, willrequire the full expressions. It is possible that integrations by parts would greatly simplifythe expressions, but we have not attempted these here.We have sought interesting small-field limits of the Lagrangians and their symmetry trans-formations, as was done for galileons propagating on maximally symmetric backgrounds.Due to the fewer isometries of FRW, the analogous expressions do not seem to exist, exceptin the special cases in which the FRW space coincides with de Sitter.Finally, we have studied the stability of simple solutions, namely π = 0 with a ( t ) = ( t/t ) n ,and find that given a correct sign for coefficients in the Lagrangians, all four possible solutionsare stable, at least on the time scales of the background. One of the four cases leads to amassless field without any gradient energy and the remaining three cases lead to scalar fieldswith tachyonic masses but the associated time scales are large enough to avoid the potentialinstability. For exponential scale factor growth, the π = 0 solution also leads to a tachyonwhose time scale is again large enough to stabilize the theory for one e-fold. Acknowledgements
Appendix A: Explicit expression for L Here we present the full expression for L in the FRW case. No integrations by parts havebeen made. L = (cid:110) ˙ a ¨ aa + 3 ˙ a a − π ˙ a ¨ a a − π ˙ a ¨ aa − π ˙ a a − πa ˙ a + ( ∇ π ) a ˙ a − ¨ πa ˙ a + 18 π a ˙ a + 46 π a ¨ a ˙ a + 19 π a ¨ a ˙ a + π a ¨ a + π ˙ π ˙ a ¨ a a − π ˙ π ˙ a a (3) a + 6 π ˙ π ˙ a ¨ aa + 12 π ˙ π ˙ a a − π ˙ a ¨ aa + π ¨ π ˙ a ¨ aa − π ˙ a a + 5 π ¨ π ˙ a a + ( ∇ π ) ˙ a ¨ aa − ∇ π ) π ˙ a ¨ aa + 4( ∇ π ) ˙ a a − ∇ π ) π ˙ a a − π a ˙ a − π a ¨ a ˙ a − π a ¨ a ˙ a − π a ¨ a ˙ a − π ˙ πa ˙ a − π ˙ πa ¨ a ˙ a + 5 π ˙ πa a (3) ˙ a − π ˙ πa ¨ a ˙ a + 12 π ˙ π a ˙ a − π ¨ πa ˙ a + 3( ∇ π ) π a ˙ a − ∇ π ) πa ˙ a + 13 π ˙ π a ¨ a ˙ a − π ¨ πa ¨ a ˙ a + 9( ∇ π ) π a ¨ a ˙ a − ∇ π ) πa ¨ a ˙ a + 3( ∇ π ) π a ¨ a ˙ a − ∇ π ) πa ¨ a ˙ a + 3 ˙ π a ˙ a − ∇ π ) ˙ πa ˙ a − ( ∇ π ) ˙ π a ˙ a + 2 ∇ ˙ π · ∇ π ˙ πa ˙ a − ( ∇ π ) ¨ πa ˙ a + ( ∇ π ) ( ∇ π ) a ˙ a − δ ij δ kl ∂ i π∂ j ∂ k π∂ l πa ˙ a + 3 π ˙ a + 41 π a ¨ a ˙ a + 74 π a ¨ a ˙ a + 22 π a ¨ a ˙ a + 12 π ˙ πa ˙ a + 36 π ˙ πa ¨ a ˙ a − π ˙ πa a (3) ˙ a + 22 π ˙ πa ¨ a ˙ a − ( ∇ π ) π ˙ a + 4( ∇ π ) π ˙ a − π ˙ π a ˙ a + 10 π ¨ πa ˙ a − π ˙ π a ¨ a ˙ a + 10 π ¨ πa ¨ a ˙ a ∇ π ) π a ¨ a ˙ a + 27( ∇ π ) π a ¨ a ˙ a − ∇ π ) π a ¨ a ˙ a + 18( ∇ π ) π a ¨ a ˙ a − ( ∇ π ) π a ¨ a + ( ∇ π ) π a ¨ a − π ˙ π a ˙ a + 8( ∇ π ) π ˙ πa ˙ a + 8( ∇ π ) π ˙ πa ¨ a ˙ a − ( ∇ π ) π ˙ πa a (3) ˙ a + ( ∇ π ) π ˙ πa ¨ a ˙ a + 3( ∇ π ) π ˙ π a ˙ a − ∇ ˙ π · ∇ ππ ˙ πa ˙ a + 3( ∇ π ) π ¨ πa ˙ a − ( ∇ π ) ( ∇ π ) π ˙ a + πδ ij δ kl ∂ i π∂ j ∂ k π∂ l π ˙ a + ( ∇ π ) π ˙ π a ¨ a ˙ a − ∇ ˙ π · ∇ ππ ˙ πa ¨ a ˙ a + ( ∇ π ) π ¨ πa ¨ a ˙ a − ∇ π ) ( ∇ π ) πa ¨ a ˙ a + 3 πδ ij δ kl ∂ i π∂ j ∂ k π∂ l πa ¨ a ˙ a − π ¨ a ˙ a − π a ¨ a ˙ a − π a ¨ a ˙ a − π ˙ π ˙ a − π ˙ πa ¨ a ˙ a + 10 π ˙ πa a (3) ˙ a − π ˙ πa ¨ a ˙ a + 12 π ˙ π a ˙ a − π ¨ πa ˙ a + 3( ∇ π ) π ¨ a ˙ a − ∇ π ) π ¨ a ˙ a + 38 π ˙ π a ¨ a ˙ a − π ¨ πa ¨ a ˙ a + 9( ∇ π ) π a ¨ a ˙ a − ∇ π ) π a ¨ a ˙ a + 3( ∇ π ) π a ¨ a ˙ a − ∇ π ) π a ¨ a ˙ a + 18 π ˙ π a ˙ a − ∇ π ) π ˙ π ˙ a − ∇ π ) π ˙ πa ¨ a ˙ a + 3( ∇ π ) π ˙ πa a (3) ˙ a − ∇ π ) π ˙ πa ¨ a ˙ a − ∇ π ) π ˙ π a ˙ a + 6 ∇ ˙ π · ∇ ππ ˙ πa ˙ a − ∇ π ) π ¨ πa ˙ a + 3( ∇ π ) ( ∇ π ) π ¨ a ˙ a − ∇ π ) π ˙ π a ¨ a ˙ a + 6 ∇ ˙ π · ∇ ππ ˙ πa ¨ a ˙ a − ∇ π ) π ¨ πa ¨ a ˙ a − π δ ij δ kl ∂ i π∂ j ∂ k π∂ l π ¨ a ˙ a + 3( ∇ π ) ( ∇ π ) π a ¨ a ˙ a − π δ ij δ kl ∂ i π∂ j ∂ k π∂ l πa ¨ a ˙ a + 17 a ˙ a ¨ a π + 11 ˙ a ¨ a π + 6 π ˙ π ¨ a ˙ a − π ˙ πaa (3) ˙ a + 17 π ˙ πa ¨ a ˙ a − π ˙ π ˙ a + π ¨ π ˙ a − π ˙ π a ¨ a ˙ a + 5 π ¨ πa ¨ a ˙ a − ∇ π ) π ¨ a ˙ a + 14( ∇ π ) π ¨ a ˙ a − ∇ π ) π a ¨ a ˙ a + 11( ∇ π ) π a ¨ a ˙ a − π ˙ π a ˙ a + 8( ∇ π ) π ˙ π ¨ a ˙ a − ∇ π ) π ˙ πaa (3) ˙ a + 11( ∇ π ) π ˙ πa ¨ a ˙ a + ( ∇ π ) π ˙ π ˙ a − ∇ ˙ π · ∇ ππ ˙ π ˙ a + ( ∇ π ) π ¨ π ˙ a + 3( ∇ π ) π ˙ π a ¨ a ˙ a − ∇ ˙ π · ∇ ππ ˙ πa ¨ a ˙ a + 3( ∇ π ) π ¨ πa ¨ a ˙ a − ∇ π ) ( ∇ π ) π ¨ a ˙ a + 3 π δ ij δ kl ∂ i π∂ j ∂ k π∂ l π ¨ a ˙ a − ( ∇ π ) ( ∇ π ) π a ¨ a + π δ ij δ kl ∂ i π∂ j ∂ k π∂ l πa ¨ a − π ˙ a ¨ a + π ˙ π ˙ a a (3) − π ˙ π ˙ a ¨ a + 5 π ˙ π ¨ a ˙ a − π ¨ π ¨ a ˙ a + ( ∇ π ) π ¨ a ˙ a − ∇ π ) π ¨ a ˙ a + 3 π ˙ π ˙ a + ( ∇ π ) π ˙ πa (3) ˙ a − ∇ π ) π ˙ π ¨ a ˙ a − ( ∇ π ) π ˙ π ¨ a ˙ a + 2 ∇ ˙ π · ∇ ππ ˙ π ¨ a ˙ a − ( ∇ π ) π ¨ π ¨ a ˙ a + ( ∇ π ) ( ∇ π ) π ¨ a ˙ a − π δ ij δ kl ∂ i π∂ j ∂ k π∂ l π ¨ a ˙ a (cid:111) / (cid:110) ˙ a ( a − ˙ aπ ) ˙ π − ˙ a ( ˙ a − ¨ aπ ) (cid:16) ( a − ˙ aπ ) + ( (cid:126) ∇ π ) (cid:17)(cid:111) , (A1)where ( (cid:126) ∇ π ) = δ ij ∂ i π∂ j π and (cid:126) ∇ π = δ ij ∂ i ∂ j π .24 ppendix B: Invariance of the tadpole term under global symmetries Here we show that the tadpole term (10) has the global symmetries (8). Under the π symmetry (8), the shift of the tadpole term is δS = (cid:90) d x (cid:112) − G ( x, π ) (cid:2) K ( x, π ) − K µ ( x, π ) ∂ µ π (cid:3) , (B1)where G ( x, π ) ≡ det G AB ( x, π ). We will show that the integrand of (B1) is a total derivativeby showing that its Euler-Lagrange variation vanishes. Taking a general variation of theright hand side gives (cid:90) d x (cid:112) − G ( π, x ) (cid:110) G AB ∂ π G AB δπ (cid:2) K ( x, π ) − K µ ( x, π ) ∂ µ π (cid:3) + (cid:2) ∂ π K ( x, π ) δπ − ∂ π K µ ( x, π ) δπ∂ µ π − K µ ( x, π ) ∂ µ δπ (cid:3) (cid:111) = (cid:90) d x (cid:112) − G ( π, x ) (cid:110) G AB ∂ π G AB (cid:2) K ( x, π ) − K µ ( x, π ) ∂ µ π (cid:3) + ∂ π K ( x, π ) − ∂ π K µ ( x, π ) ∂ µ π + ∂ µ K µ ( x, π )+ ∂ π K µ ( x, π ) ∂ µ π + 12 K µ G AB ∂ µ G AB + 12 K µ G AB ∂ π G AB ∂ µ π (cid:111) δπ = (cid:90) d x (cid:112) − G ( π, x ) (cid:110) G AB ∂ π G AB K ( x, π ) + K µ G AB ∂ µ G AB + 2 ∂ π K ( x, π ) + 2 ∂ µ K µ ( x, π ) (cid:111) δπ . 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