aa r X i v : . [ h e p - t h ] J a n UCB-PTH-09/36,IPMU09-0165
Anisotropic scale invariant cosmology
Yu Nakayama
Berkeley Center for Theoretical Physics,University of California, Berkeley, CA 94720, USAandInstitute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8582, Japan
Abstract
We study a possibility of anisotropic scale invariant cosmology. It is shown thatwithin the conventional Einstein gravity, the violation of the null energy conditionis necessary. We construct an example based on a ghost condensation model thatviolates the null energy condition. The cosmological solution necessarily containsat least one contracting spatial direction as in the Kasner solution. Our cosmologyis conjectured to be dual to, if any, a non-unitary anisotropic scale invariant Eu-clidean field theory. We investigate simple correlation functions of the dual theoryby using the holographic computation. After compactification of the contracting di-rection, our setup may yield a dual field theory description of the winding tachyoncondensation that might solve the singularity of big bang/crunch of the universe.
Introduction
Gauge/gravity correspondence [1], or more broadly holography [2][3] is a key to understandnon-perturbative features of quantum gravity. Cosmology is a natural arena where wecan apply the holographic technique to understand the time evolution of the universe,the landscape program, and the initial singularity of the big bang. While the foundationof the cosmological applications of gauge/gravity correspondence is less established thanthe standard AdS/CFT correspondence, there are many ambitious attempts including[4][5][6][7].On the other hand, anisotropic scale invariant gravity solutions have attracted a lot ofattentions these days, primarily focusing on their applications to condensed matter physics[8]. The solution has a non-relativistic dispersion relation and it has been argued that itmay give a dual description of the strongly coupled limit of the Lifshitz-like scale invariantfield theories [9]. Furthermore, an alternative proposal for ultra-violet completion ofgeneral relativity has been pushed forward based on the anisotropic non-relativistic gravityaction [10][11][12]. The breaking of the Poincar´e invariance has played a significant rolein such examples. The Poincar´e invariance, which we believe to be true in the low-energylimit of our daily lives may not be the fundamental principle of physics. General relativitydoes not require that the solution should be Poincar´e invariant, and neither does the stringtheory. It is an emergent symmetry. Besides, the cosmological time evolution explicitlybreaks the Poincar´e invariance.In this paper, we would like to combine the idea of the holographic cosmology andthe anisotropic scale invariance within the conventional general relativity. We examinea possibility of anisotropic scale invariant cosmology. The cosmological solution has aholographic interpretation of the Euclidean anisotropic scale invariant field theory thatmight be realized in a condensed matter system.The inspection of the Einstein equation tells us that the anisotropic scale invariantcosmology is only possible within the conventional Einstein gravity when the null energycondition is violated in the matter sector. While there could be many possible waysto introduce rather exotic matters to avoid the constraint, we show one particular ex-ample based on the ghost scalar action with time-like condensation [15]. The resultinganisotropic cosmology always contains at least one contracting spacial direction much likethe Kasner solution of the vacuum Einstein equation.1he dual field theory is most presumably non-unitary as indicated by the holographiccorrelation functions. However, it turns out that this non-unitarity is exactly what isneeded to tame the long range growing correlation functions of the dual field theory.Typically, a contracting spatial direction in the holographic cosmology would lead to aninstability of the dual field theory, but the pure imaginary scaling dimension makes itbehave oscillating rather than growing.In Euclidean field theories, the unitarity, or more precisely the reflection positivity, isnot the holy grail at all. There are many physical examples where the unitarity is violated.The conventional AdS/CFT may not be suitable to provide a dual gravity description ofsuch systems, and the cosmological setup can be useful for this purpose. Unfortunately,the consistency of such cosmological models beyond the gravity approximation is a delicateissue, and we may eventually need an embedding in the string theory or something ultra-violet completed. Our approach is rather bottom up, and ultra-violet completion of thesystem will be studied elsewhere in the future.
Let us study the Lifshitz-like anisotropic scale invariant cosmology in (1+3) dimension.Generalization to higher (lower) dimension will be obvious. Our scale invariant cosmo-logical ansatz for the metric is given by ds = − dt t + dx t a + dy t b + dz t c . (2.1)The metric is invariant under the anisotropic scaling t → λt , x → λ a x , y → λ b y , and z → λ c z . The ansatz is consistent with the translational invariance in ( x, y, z ) as well asthe parity invariance x i → − x i . By a coordinate transformation, one can always chooseone of the three dynamical scaling exponents (say a ) to be one. A special choice a = b = c corresponds to de-Sitter space.We first show that except for this particular de-Sitter case, which is isotropic, theanisotropic scale invariant cosmology (2.1) is only possible within the conventional Ein-stein gravity when the null energy condition is violated. To see this, we compute theEinstein tensor for the metric (2.1) as: G tt = bc + ab + act , G xx = − b + bc + c t a yy = − c + ca + a t b , G zz = − a + ab + b t c . (2.2)Now, the null energy condition demands G µν k µ k ν ≥ k µ . By taking k µ = ( √ t, t a , t b , t c ), we obtain − ( a − b ) − ( b − c ) − ( c − a ) ≥ , (2.3)which is only possible when a = b = c . Thus, except for the special case of de-Sittercosmology, the anisotropic scale invariant cosmology is inconsistent with the null energycondition. In order to realize the anisotropic scale invariant cosmology within the Einstein gravity,therefore, it is necessary to break the null energy condition. While there could be manyways to do this, here, we would like to investigate the possibly by using the ghost matter[15]: S = Z d x √− gF ( ∂ µ φ∂ µ φ ) , (2.4)where non-trivial F ( X ) with X = ∂ µ φ∂ µ φ introduces generic higher derivative interactionconsistent with the shift symmetry φ ( x ) → φ ( x ) + Λ. The ansatz for scalar field φ for thescale invariant cosmology without breaking the translational invariance is φ = p log t . (2.5)As in [16], we have to gauge the constant shift of the scalar field φ ( x ) → φ ( x ) + Λ so thatthe scaling transformation is a symmetry of the ansatz.The equation of motion for φ is solved either by F ′ ( − p ) = 0 or a + b + c = 0. Wefocus on the latter case a + b + c = 0. The field configuration (2.5) now supplies theadditional negative energy in the energy momentum tensor in addition to the cosmolog-ical constant when F ′ ( − p ) < T µν = − ˜Λ g µν + diag( F ′ ( − p ) , , , a + b + c = 0, which coincides with the condition that the scalar field ansatz(2.5) solves the ghost equation of motion. A related Bianchi I cosmology was studied in [13][14], where the violation of the null energy conditionand its (in)stability were investigated. We would like to thank I. Aref’eva for the correspondence. The former solution leads to a = b = c , and hence the de-Sitter space, which is identical to the scaleinvariant but non-conformal scalar field configuration studied in [16]. a = b = c , the config-uration with assumed unitarity in the boundary theory is inconsistent with Polchinski’stheorem that states the unitary Lorentz and scale invariant theory must be conformallyinvariant [17][18][19]. As we will see in the next section, the boundary theory for ourconfiguration is most presumably not unitary, and it is not entirely clear why the gravitytheory should be so, either. On the other hand, we might be able to come up with a bettermatter sector that is consistent with unitarity while violating the null energy condition torealize the anisotropic scale invariant cosmology. In the computation of the correlationfunctions, therefore, we only focus on the universal geometric background and will notdiscuss the ghost matter sector to be on the optimistic side.The condition a + b + c = 0 means that at least one spatial direction is contracting. Onemay compactify the contracting direction so that the visible universe is expanding whilethe internal space is contracting. In particular, when a = b = − / c , the anisotropiccosmology admits additional rotational invariance, and the symmetry algebra is Wickrotated version of the Lifshitz-like scale invariant theory whose gravity dual was firstproposed in [8]. A difference here is the “dynamical critical exponent” is now negative.In their setup, the energy condition has yielded a constraint that the dynamical criticalexponent is greater than one. Their constraint may have a physical meaning due to thefiniteness of the speed of light. Thanks to the Euclidean signature of our dual theory,such constraints cannot appear here.Our metric resembles the Kasner solution of the vacuum Einstein equation: ds = − dτ + τ a dx + τ b dy + τ c dz , (2.6)where a + b + c = 1 and a + b + c = 1. Note that one of the exponents is always negative For instance, the orientifold in string theory can violate the null energy condition, so the violationof the null energy condition itself may not be inconsistent with the consistency of quantum theories ofgravity. The continuous scale invariance will be broken by the compactification. On the other hand, we mayexpect winding tachyon condensation that might cure the big bang/crunch singularity of our cosmology[20][21]. The discussions in the following sections might give a holographic dual descriptino of such ascenario. t = e τ as ds = − dτ + e aτ dx + e bτ dy + e cτ dz . (2.7)One may regard it as an exponentially expanding/contracting version of the Kasner uni-verse.It is well-known that by further relaxing the condition of “flatness”, the alternatingfeature of the Kasner regime appears near the singularity at the beginning of the universe.It would be interesting to study a similar situation in our anisotropic scale invariantcosmology. The gauge/gravity correspondence is a non-perturbative way to understand the dual fieldtheories. Alternatively, one may understand the nature of quantum gravity from the dualfield theories. Originally, it was proposed in the negatively curved space like AdS space[22][23], while some attempts have been done to generalize it in the cosmological setup[4]. We would like to use the holographic technique to compute correlation functions tounderstand the nature of the dual field theory of our anisotropic scale invariant cosmology,if any. At the same time, the holographic computation further reveals some peculiarfeatures of the anisotropic scale invariant cosmology.We introduce a conventional scalar field ϕ with mass m that is minimally coupled tothe Einstein gravity: S = Z d x √− g (cid:0) ∂ µ ϕ∂ µ ϕ − m ϕ (cid:1) (3.1)to compute the holographic correlation functions among the operator O associated withthe scalar ϕ .The equation of motion for the scalar is given by t ∂ t ϕ + t∂ t ϕ − t a ∂ x ϕ − t b ∂ y ϕ − t c ∂ z ϕ + m ϕ = 0 . (3.2)From the translational invariance, it is convenient to go to the momentum space ϕ =˜ ϕ ( t, k ) e ik x x + ik y y + ik z z . A standard holographic recipe to compute the two-point functiongives (here ǫ is an IR cutoff) h O ( k ) O ( p ) i = δ ( k + p ) h ˜ G ( t, − k ) √− gg tt ∂ t ˜ G ( t, k ) i ∞ ǫ (3.3)5y using the bulk boundary propagator ˜ G associated with (3.2) so that˜ ϕ ( t, k ) = ˜ G ( t, k ) ˜ ϕ (0 , k ) . (3.4)We could not find an analytic expression for the most general solution, so we firstdiscuss the asymptotic form of the solution. Without loosing generality, we assume c < ≤ b ≤ a : if two of the exponents are negative, one can perform the coordinatetransformation t → t − to retain the inequality. For t ≪
1, the solution for non-zero k x is given by the Bessel functions:˜ ϕ = J ± ima (cid:18) k x t a a (cid:19) = (cid:18) k x t a a (cid:19) ± ima " ∓ ima Γ(1 ± ima ) − − ∓ ima (cid:0) k x t a a (cid:1) Γ(2 ± ima ) + · · · . (3.5)On the other hand, for t ≫
1, the solution for non-zero k z is given by˜ ϕ = J ± imc (cid:18) k z t c c (cid:19) = (cid:18) k x t c c (cid:19) ± imc " ∓ imc Γ(1 ± imc ) − − ∓ imc (cid:0) k z t c c (cid:1) Γ(2 ± imc ) + · · · . . (3.6)Note that unlike the Euclidean case, there is no compelling principle to choose a particularlinear combination of the solution of the equation of motion: the choice will be reflected inthe ambiguity to choose propagators (and vacuum) in the dual field theory. For instance,in the conventional (Euclidean) AdS/CFT setup, it is customary to choose a particularlinear combination given by the modified Bessel function:˜ ϕ = K ima (cid:18) k x t a a (cid:19) (3.7)near t ≪
1. That would correspond to choosing the Feynman propagator in the conven-tional AdS/CFT correspondence.The scaling dimension of the operator is given by∆( O ) = ± im (3.8)The imaginary scaling dimension for real m suggests the dual field theory is not unitary.Another possibility is that the theory is unitary, but the spectrum probed by the holog-raphy does not have corresponding states as in imaginary conformal dimension operatorsin Liouville theory (e.g. [25] for a review). In this case, the analytic continuation of theconformal dimension is motivated, and we will come back to these points later. Recently, more detailed studies on the anisotropic conformal boundary conditions have been presentedin [24]. h O ( k x ) O ( p x ) i = δ ( k x + p x ) 1 k ± ima x , h O ( k y ) O ( p y ) i = δ ( k y + p y ) 1 k ± imb y , h O ( k z ) O ( p z ) i = δ ( k z + p z ) 1 k ± imc z . (3.9)The sign of the scaling dimension in (3.8) corresponds to the choice of propagators.Another particular solvable case is a = 2 b . For a = 2, the explicit form of the solutionfor k z = 0 is ϕ = c e − i kxt t im U (cid:18) i k y k x + 12 + im , im, ik x t (cid:19) + c e − i kxt t im L (cid:18) − i k y k x − − im , im, ik x t (cid:19) (3.10)by using confluent hypergeometric functions (see appendix A for details). A suitablechoice of the solution corresponds to the analytic continuation of the two-point functionin the Lifshitz geometry studied in [8] (see also [26]): h O ( k x , k y ) O ( p x , p y ) i = δ ( k x + p x ) δ ( k y + p y ) k − imx Γ (cid:16) i k y k x + − im (cid:17) Γ (cid:16) i k y k x + + im (cid:17) (3.11)up to an overall normalization factor.So far, all the operators corresponding to massive scalar had the pure imaginary con-formal dimension. However, we could introduce a scalar field with the negative masssquared that would correspond to real conformal dimension. In dS/CFT [4], the corre-sponding statement would be to study “unstable” scalar modes that would correspond tothe real conformal dimension. Actually, the conventional recipe of the AdS/CFT corre-spondence to compute correlation functions can be best suited in this “unstable” rangeof mass parameters because the distinction between the normalizable modes and non-normalizable modes are clearly displayed, and the ambiguity to choose the propagator isless apparent.Yet another application of the “tachyonic mode” here is the winding tachyon conden-sation studied in [20][21]. Our prescription gives a holographic dual description of the7inding tachyon condensation by identifying the scalar mode as the winding tachyon (upon T-duality).In our case, (3.9) can be analytically continued to h O ( k x ) O ( p x ) i = δ ( k x + p x ) 1 k ˜ ma x , h O ( k y ) O ( p y ) i = δ ( k y + p y ) 1 k ˜ mb y , h O ( k z ) O ( p z ) i = δ ( k z + p z ) 1 k ˜ mc z , (3.12)where ˜ m = ± im >
0. In particular, the two-point function would be growing in large z direction, which may suggest an instability of the dual field theory. Similarly, thetwo-point function (3.13) can be analytically continued to ˜ m = ± im > h O ( k x , k y ) O ( p x , p y ) i = δ ( k x + p x ) δ ( k y + p y ) k ˜ mx Γ (cid:16) i k y k x + + ˜ m (cid:17) Γ (cid:16) i k y k x + − ˜ m (cid:17) (3.13)and can be directly compared to the one obtained in [8].We emphasize that unlike the conformal field theories, the two-point functions of theanisotropic scale invariant field theories are not uniquely determined from the symmetryalone, so our computation gives a precise prediction of the two-point functions of thedual anisotropic scale invariant field theory. At the same time, however, any non-minimalcoupling of the scalar fields to the background geometry as well as the background matterfield would change the form of the correlation functions, so a specification of the coupling isneeded to fully determine the two-point functions. Such specification is outside the scopeof the bottom up effective field theory approach taken in this paper, but the consistency ofthe effective field theory approach demand that such corrections should be small comparedwith the leading order behavior studied in this section. In this paper, we investigated the possibility of anisotropic scale invariant cosmologywithin the conventional Einstein gravity. We have shown that it is only possible when thenull energy condition is violated. Given a difficulty to break the null energy condition, it8ould be very important to find an embedding in the ultraviolet completed quantum the-ories of gravity. Note that the reference [27] suggested that even a less exotic anisotropicLifshitz geometry is difficult to realize in the string theory. From the dual field theory perspective, in particular for the Euclidean field theories,the unitarity or reflection positivity is not the central dogma. The unitarity is a crucialissue to understand the consistency of the quantum gravity, but for the time-dependentholography like dS/CFT or our anisotropic cosmology, it is yet to be investigated howthe unitarity of the bulk theory is encoded in the boundary theory. On the other hand,it is very important to understand how to construct gravity dual of non-unitary fieldtheories from the gravity viewpoint because there are plenty of non-unitary, or non-reflection positive examples of interesting condensed matter systems. Our construction isone approach in this direction.Finally, it would be interesting to study the realization of anisotropic scale invariantcosmology in the anisotropic gravity. Since the anisotropic scale invariance is alreadyencoded in the action, it may be more suitable to discuss the anisotropic cosmology therethan in the conventional Einstein gravity where the violation of the null energy conditionis needed. Acknowledgements
The author would like to thank B. Freivogel for stimulating discussions on the energyconditions. The work was supported in part by the National Science Foundation underGrant No. PHY05-55662 and the UC Berkeley Center for Theoretical Physics and WorldPremier International Research Center Initiative (WPI Initiative), MEXT, Japan. A physical reasoning behind the “no-go theorem” is obscure. Indeed, a smeared (integrated) versionof the equations motion do not have any obstruction to obtain Lifshitz type geometry [28] (see also [29]). A cosmology model with z = 5 / Confluent Hypergeometric functions
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