Isotropy theorem for arbitrary-spin cosmological fields
aa r X i v : . [ g r- q c ] J un Isotropy theorem for arbitrary-spin cosmological fields
J.A.R. Cembranos, ∗ A.L. Maroto, † and S.J. N´u˜nez Jare˜no ‡ Departamento de F´ısica Te´orica, Universidad Complutense de Madrid, 28040 Madrid, Spain (Dated: July 31, 2018)We show that the energy-momentum tensor of homogeneous fields of arbitrary spin in an expand-ing universe is always isotropic in average provided the fields remain bounded and evolve rapidlycompared to the rate of expansion. An analytic expression for the average equation of state isobtained for Lagrangians with generic power-law kinetic and potential terms. As an example weconsider the behavior of a spin-two field in the standard Fierz-Pauli theory of massive gravity. Theresults can be extended to general space-time geometries for locally inertial observers.
PACS numbers: 98.80.-k, 98.80.Cq
INTRODUCTION
One of the main limitations on the use of vectorsor higher-spin fields in cosmology is the high degree ofisotropy of the universe on large scales [1]. A homoge-neous field of non-zero spin generically breaks isotropyby selecting preferred directions in space.However in recent years there has been a growing in-terest in the possibility of using vectors fields (abelian ornon-abelian) as dark matter [2], dark energy [3] or infla-ton candidates [4]. In these cases the anisotropy problemis avoided thanks to the use of particular field configura-tions (temporal components, triads, etc [5–7]) that guar-antee an isotropic energy-momentum tensor. Also a moregeneral result has been proved which shows that in thecase of bounded fields which evolve rapidly as comparedto the rate of expansion, the temporal average of theenergy-momentum tensor is always isotropic for any fieldconfiguration. This means, that even anisotropic fieldconfigurations such as a linearly polarized field wouldgive rise in average to an isotropic energy-momentumtensor. This result was obtained by using a generaliza-tion of virial theorem and applies both to abelian [8] andnon-abelian [9] theories, with arbitrary potentials andwith or without gauge-fixing terms.The generality of this result for homogeneous vectorssuggests that the isotropy property could be a generalfeature of any field theory for arbitrary spin with theonly requirements of large scale homogeneity, bounded-ness and rapid evolution. In this work we prove that thisis indeed the case and present a general isotropy theoremfor arbitrary-spin cosmological fields.Unlike previous works in which explicit Lagrangiandensities were used, in the case of generic theories asthose we will consider in this work, the explicit depen-dence of the Lagrangian on the metric tensor is not fixeda priori. This means that we cannot use the Hilbert formof the energy-momentum tensor: T µν = − √ g δSδg µν (1)as our starting point. In order to avoid this difficulty, we will make use of the so called Belinfante-Rosenfeld[10, 11] energy-momentum tensor, which allows to relatethe Hilbert energy-momentum with the canonical one bymeans of the use of some extra terms. Unexpectedly, wewill show how this relation between the canonical andHilbert forms is intimately related to the anisotropy is-sue.For clarification, let us thus start by briefly re-viewing the standard Belinfante-Rosenfeld approach inMinkowski space-time [12] and consider a Lagrangiandensity depending only on the fields (labelled by A ) andtheir gradients: L ≡ L (cid:2) φ A , ∂ µ φ A (cid:3) , (2)Under an infinitesimal x -dependent translation x µ → x µ + δa µ ( x ), the field and its gradient change as [12]: δφ A = δa µ ( x ) ∂ µ φ A ( x ) , (3) δ∂ µ φ A ( x ) = δa ν ( x ) ∂ ν ∂ µ φ A ( x )+ ∂ µ [ δa ν ( x )] ∂ ν φ A ( x ) . (4)By imposing:0 = δ Z d x L = − Z d x δa ν ∂ µ Θ µν , (5)we obtain that the canonical energy-momentum tensor,defined as Θ µν = − η µν L + ∂ L ∂ ( ∂ µ φ A ) ∂ ν φ A (6)is conserved: ∂ µ Θ µν = 0 . (7)This tensor is nothing but the Noether current associatedto the symmetry under space-time translations. Noticethat although it is conserved, Θ µν is not necessarily sym-metric.However, this current is not unique, and we can add anew piece: ∂ ρ ˜Θ ρµν , (8)with ˜Θ ρµν antisymmetric in the first two indices. Thisnew piece does not modify the value of the Noethercharge because it is a total derivative, neither its timeconservation because of its antisymmetry, Q ν = Z d x (Θ ν + ∂ ρ ˜Θ ρ ν )= Z d x (Θ ν + ∂ i ˜Θ i ν ) = Z d x Θ ν ; (9) dQ ν dt = Z d x ( ∂ µ Θ µν + ∂ µ ∂ ρ ˜Θ ρµν ) = 0 . (10)We are interested in a symmetric energy-momentum ten-sor, i.e. that required to appear on the right hand side ofEinstein equations. The new piece that must be addedread [13]: T µν = Θ µν − ∂ ρ ( S ρµν + S µνρ − S νρµ ) , (11)with S µνρ = Π µA Σ νρ φ A , (12)where Σ νρ are the antisymmetric Lorentz group genera-tors in the corresponding representation andΠ µA = ∂ L ∂ ( ∂ µ φ A ) (13)is the generalized momentum associated to φ A . T µν isthe symmetric Belinfante-Rosenfeld energy-momentumtensor which agrees with the Hilbert energy-momentumtensor obtained from variations with respect to the met-ric (1) as shown in [10, 11, 14].Both, the canonical energy-momentum tensor Θ µν andthe Belifante-Rosenfeld tensor T µν can be written in acurved space-time in a straightforward way by using min-imal coupling, simply changing ordinary derivatives bycovariant ones, i.e. we will work with: T µν = Θ µν + ∇ ρ ˜Θ ρµν = Θ µν − ∇ ρ ( S ρµν + S µνρ − S νρµ ) . (14)Notice, that the form of the Lagrangian guarantees thatonly first derivatives of the fields will appear in Θ µν . HOMOGENEOUS FIELDS AND VIRIALTHEOREM
Following [8] and [9], we can use a generalization ofthe virial theorem in order to obtain interesting resultsfor the average energy-momentum tensor of homogeneousfields φ A ( t ). Before writing the most general theorem,let us consider a Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) metric for simplification: ds = dt − a ( t ) d~x . (15) With these assumptions, the ˜Θ ρµν tensor is also homo-geneous.Our aim is taking the temporal average of the energymomentum tensor during periods T ≪ H − , where H is the Hubble parameter H = ˙ a/a . Particularly, we areinterested in the average value of ∇ ρ ˜Θ ρµν as this termwill be the cause of the anisotropies. D ∇ ρ ˜Θ ρµν E = 1 T Z t + T t dt ′ (cid:16) ∇ ρ ˜Θ ρµν (cid:17) ( t ′ ) , (16)with ∇ ρ ˜Θ ρµν = ∂ ˜Θ µν + (cid:16) Γ ρδρ ˜Θ δµν + Γ µδρ ˜Θ ρδν + Γ νδρ ˜Θ ρµδ (cid:17) . (17)We can neglect the term in brackets on the right handside of the equation (17) if the temporal derivative islarger than the expansion rate, i.e. ∂ ˜Θ ≫ H ˜Θ. If thesystem oscillates with an effective period τ , ∂ Θ ∼ τ − Θ,then the condition for neglecting that term will be: τ − ≫ T − ≫ H . (18)In this limit the energy-momentum tensor expressed incomponents reads T = Π A ∂ φ A − L − ∂ (cid:0) S (cid:1) = Π A ∂ φ A − L ; (19) T j = − ∂ (cid:0) S j + S j − S j (cid:1) = 0 ; (20) T jj = − g jj L − ∂ (cid:0) S jj + S jj − S j j (cid:1) = − g jj L − ∂ (cid:16) Π jA Σ j φ A (cid:17) ; (21) T jk = − ∂ (cid:16) Π A Σ jk φ A + Π jA Σ k φ A + Π kA Σ j φ A (cid:17) , (22)with k = j . The antisymmetry of the Lorentz groupgenerators, Σ µν , has been used for simplification.On the other hand, (16) becomes D ∇ ρ ˜Θ ρµν E = 1 T Z t + T t dt ′ ∂ ˜Θ µν ( t ′ )= ˜Θ µν ( t + T ) − ˜Θ µν ( t ) T . (23)As can be seen from (23), if the field evolution is pe-riodic or bounded, the right-hand side vanishes as com-pared to h T i for sufficiently large T . In fact, the ratiocan be estimated as D ∇ ρ ˜Θ ρµν E / h T i ∼ O ( τ / T ). Thatleads us to the following average energy-momentum ten-sor: h T i = h Π A ∂ φ A − Li ; (24) h T j i = T j = 0 ; (25) h T jj i = h− g jj Li ; (26) h T jk i = 0 ; k = j , (27)which is explicitly isotropic. Notice that as commentedbefore, the anisotropies in the exact (non-averaged) ten-sor indeed come from the new terms that must be addedin the Belinfante-Rosenfeld approach in order to get thesymmetric expression.Moreover, using these results we can also express theaverage equation of state in this suggestive form: ω = h p ih ρ i = hLih Π A ∂ φ A − Li = hLihHi , (28)with H the Hamiltonian of the system.There are other ways of writing this quantity: ω = h Π A ∂ φ A ihHi − . (29)Or by using the equation ∂ φ A = ∂ H ∂ Π A : ω = h Π A ∂ H ∂ Π A ihHi − . (30)Another form is reached by using the Euler-Lagrangeequation for φ A , ∇ µ Π µA = ∂ L ∂φ A : ω = h ∂ (cid:0) Π A φ A (cid:1) − ∂ Π A φ A ihHi − h− ∂ L ∂φ A φ A ihHi − , (31)where we have also applied the extension of the virialtheorem to Π A φ A , i.e. h ∂ (cid:0) Π A φ A (cid:1) i = 0.From (30) and (31), it can be seen that the followingaverage equation is satisfied h Π A ∂ H ∂ Π A + ∂ L ∂φ A φ A i = 0 . (32)The last equation results very helpful when consider-ing theories where the kinetic and potential terms addseparately as simple power-laws in the following form H = (cid:0) λ AB g Π A Π B (cid:1) n T + (cid:0) M AB φ A φ B (cid:1) n V , (33)where λ AB and M AB are constant matrices. In such acase, Equation (32) relates T and V in the following form h T i = n V n T h V i . (34)By using (31), we can obtain an analytic expression for ω independent of initial conditions or particular polariza-tion of φ A : ω = 2 n V h V ih T + V i − n V n V n T − . (35)Notice that this result is also independent of the fieldspin. For instance, for the usual case with n T = 1, thebehaviour of the equation of state is the same as that forscalar [15] or vector [8, 9] fields: ω = n V − n V + 1 . (36) Note that fast oscillating fields can have associated anegative effective equation of state parameter. In thissense, they are potential new models of dark energy orinflation. Indeed, we have shown that this result does notdepend on the spin. Similar approaches for scalar fieldshave been already considered in the literature [16, 17].Another potential interest of these results comes from thepossibility of avoiding the anisotropy typically expectedduring the reheating period in inflationary models basedon vectors or higher-spin fields. A SPIN-2 EXAMPLE
As an example, we will apply the previous results tothe Fierz-Pauli theory of massive gravity on a curvedspace-time background given by the Lagrangian L = M P l h ∇ α h µν ∇ α h µν − ∇ α h αµ ∇ β h µβ + 2 ∇ α h αµ ∇ µ h ββ − ∇ α h µµ ∇ α h νν − m g (cid:16) h µν h µν − (cid:0) h µµ (cid:1) (cid:17)i . (37)The momentum of this field can be written asΠ µν = ∂ L ∂ ( ∇ h µν ) = M P l h ∇ h µν − δ µ ∇ α h αν ) + δ µ ∇ ν ) h αα + g µν ∇ α h α − g µν ∇ h αα i , (38)where A ( µ B ν ) = ( A µ B ν + A ν B µ ) / h µν is symmetric, the mo-menta and the Lagrangian take the formΠ µ = 0 ;Π ij = M P l ∂ h ij , i = j ;Π ii = − M P l X j = i ∂ h jj ; (39) L = M P l h ∂ h ij ∂ h ij − ∂ h ii ∂ h jj (40) − m g (cid:16) h µν h µν − (cid:0) h µµ (cid:1) (cid:17)i , where we have neglected the expansion rate with respectto the temporal variation of the field. We will also needthe explicit expression for the Hamiltonian. Under thesame assumptions, we can write H ≡ Π µν ∂ h µν − L = M P l h ∂ h ij ∂ h ij (41) − ∂ h ii ∂ h jj + m g (cid:16) h µν h µν − (cid:0) h µµ (cid:1) (cid:17)i . Note that we are assuming a minimal gravitational coupling forthe spin-2 field. There are more general options [18] but they arenot relevant for the isotropy theorem presented in this analysis.
As it can be seen, the Lagrangian and the Hamiltoniantake the classical structure L = T − V and H = T + V .If the field evolves under the conditions for applying thevirial theorem, then (32) holds. Consequently, (cid:28) Π µν ∂ H ∂ Π µν + ∂ L ∂h µν h µν (cid:29) = (cid:28) Π µν ∇ h µν + ∂ L ∂h µν h µν (cid:29) = h T − V i = 0 , (42)where one of the Hamilton equations has been used inthe first equality. We can conclude that the behaviour ofthe field will be that of non-relativistic matter by usingthe last average equation and (28): ω = hLihHi = h T − V ih T + V i = 0 . (43)Therefore, given the weak coupling to matter fields, a ho-mogeneous spin-two massive graviton can contribute tothe dark matter density. The massive graviton has beenalready studied as a dark matter candidate by assumingan isotropic stochastic background [19, 20]. However,even an anisotropic coherent evolution could be takeninto account as a viable model since, as shown before, itdoes not introduce an important amount of anisotropy inthe background geometry. GENERAL GEOMETRICAL BACKGROUNDSAND DISCUSSION
Finally, let us extend this result to a more generalspace-time geometry by considering an inertial observerlocated at x µ = 0 and write the metric around it usingRiemann normal coordinates: g µν ( x ) = η µν + 13 R µανβ x α x β + . . . (44)If the following conditions hold:1. The Lagrangian depends only on the fields andtheir gradients.2. The field evolves rapidly: | R γλµν | ≪ ( ω A ) , and | ∂ j S µνρ | ≪ | ∂ S µνρ | , for j = 1 , , ω A isthe characteristic frequency of φ A .3. S µνρ , i.e. φ A and Π A , remains bounded in the evo-lution.then, the second condition implies that if the averagingtimes satisfy | R γλµν | ≪ T − ≪ ( ω A ) , (46) we are in a normal neighborhood and we can neglectthe second term in (44) so that we can work locally ina Minkowskian space-time. In the normal neighborhoodof the observer, ˜Θ ρµν can also be considered as a ho-mogeneous field. In such a region, it is then possibleto rewrite all the above equations in Minkowski space-time ( a ( t ) = 1). Accordingly, it is possible to neglect theright-hand side in (23) and prove that the mean valueof the energy-momentum tensor is isotropic. Thus, if os-cillations are fast compared to the curvature scale, theaverage energy-momentum tensor takes the perfect fluidform for any locally inertial observer. Acknowledgements
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