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An effective field theory ofholographic dark energy
Chunshan Lin ,
Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University,30-348 Krakow, Poland
Abstract.
A general covariant local field theory of the holographic dark energy model ispresented. It turns out the low energy effective theory of the holographic dark energy isthe massive gravity theory whose graviton has 3 polarisations, including one scalar modeand two tensor modes. The Compton wavelength is the size of the future event horizon ofthe universe. The physical interpretation for the UV-IR correspondence Λ ∼ p M p / L in theholographic dark energy model is provided in the framework of our effective field theory,where L is interpreted as the graviton’s Compton wavelength, and Λ is interpreted as theenergy scale where the scalar graviton strongly couples to itself. ontents The holographic dark energy model [1][2] is based on the simple idea that the vacuumenergy density arising from the quantum fluctuation of the UV-cut-off quantum field theoryshould relate to the boundary surface of a system in the way ρ Λ ∼ M p L − , (1.1)where M p = π G is the reduced Planck mass, L is the size of a system (thus L is essentiallythe area of the boundary surface). This relation between the energy density in a bulk andthe area of its space-time boundary surface is rooted in the holographic principle [3]. Theargument that leads to the above relation is the following. The zero-point energy divergesquartically and thus the energy density scales as ρ Λ ∼ Λ if we simply cut-off the divergenceat the UV scale Λ . However, this simple scaling is violated when the total energy in thesystem with size L , i.e. L Λ , approaches to the mass of the black hole in the same size LM p .In fact, the total energy must be bounded by the mass of the black hole from above, as theeffective quantum field theory breaks down at the schwarzschild radius scale. This energybound is stronger than the Bekenstein entropy bound [4] which inspired the proposal of theholographic principle [3]. Nowadays it is widely believed that the holographic principle isone of the most important cornerstones of quantum gravity.Assuming that the energy bound is saturated on the cosmological background, oneobtains the important UV-IR correspondence Λ ∼ q M p / L (1.2)and the vacuum energy ρ Λ comparable to our current critical energy density. The relatedidea was discussed in Ref. [5][6] with the Hubble radius as the IR cut-off. However, as– 1 –ointed out in Ref. [7], the resultant equation of state is greater than − L = H as the IR cut-off. Later in the same year, a finishing kick was made in Ref. [2]by adopting the future event horizon as the IR cut-off, which eventually gave us an accel-erated expanding solution in the Friedman equation. Interestingly, the cosmic coincidenceproblem can also be resolved by inflation in this scenario, provided the minimal number ofinflationary e-foldings. Since then, the holographic dark energy has drawn a lot of attentionand has been widely studied, see Ref. [8] for a comprehensive review on the topic. In pass-ing, I shall mention that it was pointed out very recently that the holographic dark energymodel may alleviate the Hubble tension problem [9].Given the phenomenological success, however, one of key pieces for our holographicjigsaw puzzle is still missing. Namely we do not know how to write down the generalcovariant action for the holographic dark energy model. An attempt was made in the un-published work [10] in which a mini superspace action was given. Nevertheless, due to theabsence of the general covariant action, one may easily spot several conceptual problems.For instance, the evolution of our universe in the past is dependent on the one in the future.This apparent causality violation may not imply the pathology of the model, but rather thatthe low energy effective field theory is missing. This causality violation may be partially ad-dressed at the cosmological background evolution level, given the mini superspace action[10]. However, it is still not quite clear whether the local physics violates the causality, asthe perturbation theory is also missing .In the current work, I aim at finding this important missing piece for our holographicjigsaw puzzle. As we will see in the reminder of this paper, the low energy effective fieldtheory (EFT) for the holographic dark energy model is actually a massive gravity theory.A graviton has three polarisations, namely one scalar mode, and two tensor modes. TheUV-IR correspondence eq. (1.2) stems from the strong coupling of the scalar mode abovethe energy scale where our effective field theory breaks down. One may get puzzled atthis point as the Poincare symmetry in 4 dimensional space-time implies a massive spin-2particle has 5 polarisations (including helicity 0, ± ± A preliminary analysis on the perturbative stability was conducted in Ref. [11] – 2 –
From mini superspace to general covariance
We take the flat FLRW ansatz ds = − N dt + a d x and the following mini superspaceaction [10] as our starting point, S = M p Z dt (cid:20)p − g (cid:18) R − ca L (cid:19) − λ (cid:18) ˙ L + Na (cid:19)(cid:21) + S m , (2.1)where M p ≡ π G is the reduced Planck mass, R is the 4 dimensional Ricci scalar, and L is the length scale subject to the constraint equation enforced by the Lagrangian multiplier λ , i.e. ˙ L = − N / a . We may integrate this constraint equation from the infinite past − ∞ tonowadays, namely L = Z t − ∞ − Ndt ′ a ( t ′ ) + L ( − ∞ ) , (2.2)where L ( − ∞ ) is the initial condition in the infinite past, which requires the input from newphysics (such as quantum gravity) to fully determine its value, in light of the Hawking-Penrose’s singularity theorem [18]. Therefore L ( − ∞ ) remains unknown to me due to myignorance and obtuseness. On the other hand, one may try to take the integration from theother side, namely L = Z ∞ t Ndt ′ a ( t ′ ) + L (+ ∞ ) , (2.3)given the asymptotic solution in the infinite future t = + ∞ . It turns out L (+ ∞ ) = R h ≡ aL is exactly the size of the future event horizon. It followsthe energy density of the holographic dark energy ρ Λ = M p (cid:18) ca L + λ a (cid:19) , (2.4)where the first term is the holographic term, and the second term is the dark radiation as itscales as a − .The general covariance can be recovered by stuckelberging the mini superspace action[20]. We may follow the following dictionary,1 a → g ij δ ij → g µν ∂ µ φ a ∂ ν φ b δ ab , where h φ a i = x i δ ia √ L → ϕ ( t , x ) , − ˙ L N → g µν ∂ µ ϕ∂ ν ϕ , where h ϕ i = ϕ ( t ) , (2.5)where δ ab is the metric of the Stuckelberg scalar field space, δ ia is the pullback mappingbetween the physical space-time and the scalar field space, and i , j , k ... are adopted as 3dimensional spatial coordinate indices, while a , b , c ... are adopted as the field space indices.These 4 Stuckelberg fields are not yet canonically normalized and thus they are of the lengthdimension. The field space is flat, and respects SO ( ) rotational invariance. In passing,I shall mention that the global Lorentz invariance is broken in this scalar configuration.– 3 –evertheless, one should not be worried about this Lorentz-violating vacuum expectationvalues (VEVs) of scalars as the Lorentz invariance is broken anyway on the cosmologicalbackground. For instance, CMB is not invariant under a Lorentz boost.Before writing down the general covariant action, which is pretty easy and straight-forward at this point, I shall remind readers the hierarchical structure of the theory. Wehave the Planck scale as the genetic fundamental scale in the first place, and a secondaryfundamental scale for our EFT of the dark sector, which is expected to be generated via thenon-trivial VEV of the time-like stuckelberg field ϕ , namely Λ ∼ p M p / ϕ . The low energyeffective field theory that I begin with is the following, S = Z d x p − g (cid:26) M p R − M p ϕ − h ( c + λ ) · ∂ µ φ a ∂ µ φ b δ ab + λ∂ µ ϕ∂ µ ϕ i(cid:27) + S m , (2.6)where c is a constant, λ is a Lagrangian multiplier, and S m is the action of the matter sectorwhich minimally couples to gravity. The hierarchical structure of the theory is manifest atthe action level. One would expect that the hierarchy disappears if we trace the cosmicevolution all the way backward in time to the infinite past, and it was generated later as theuniverse expands. The physical significance of the energy scale Λ ∼ p M p / ϕ will be shownin the next section.I adopt the FLRW ansatz, and the four Stuckelberg scalar fields take the space-timeVEVs in the eq. (2.5). The background equations of motion read,3 M p H = ca ϕ + λ a + ρ m , − M p ˙ H = c a ϕ + λ a + ( ρ m + p m ) , (2.7)and ˙ ϕ = − a , ˙ λ = − ca ϕ , (2.8)where ρ m and p m are the energy density and pressure of the matter sector, the lapse hasbeen absorbed into the redefinition of time, namely Ndt → dt , and I have rescaled theLagrangian multiplier λ → λ · ϕ a M p for convenience. All equations of motion presentedin Ref. [10] have been reproduced . Some detailed analyses about this set of equations,including the analytical one and the numerical one, have already been conducted in Ref.[10] and Ref. [19], and I shall not repeat it in my current work.It is quite remarkable to see that the holographic dark energy, which is seemingly nonlocal and causality-violating, actually originates from a simple and well defined local fieldtheory. The trick of the game is the spontaneous symmetry breaking, namely that I startfrom the general covariant theory, then 4 scalar fields take the non-trivial space-time VEVsand break the spatial and temporal diffeomorphism invariance around this vacuum solu-tion. According the Goldstone theorem, there is a massless boson for each generator of thesymmetry that is broken. I shall start to introduce the Goldstone excitations around thisasymmetric state, φ a = √ (cid:16) x i δ ai + π a (cid:17) , ϕ = ϕ ( t ) + π , (2.9) The constraint equation associated with the Lagrangian multiplier λ has two solutions ˙ ϕ = ± a , we adoptthe solution ˙ ϕ = − a which gives us the holographic dark energy. – 4 –here π a is the space-like Nambu-Goldstone boson, and π is the time-like one. Due tothe SO ( ) rotational symmetry, we can decompose the helicity π a = δ ia ∂ i π + ˆ π a , where π isthe longitudinal mode, and ˆ π a are two transverse modes satisfying the transverse condition ∂ i ˆ π i = M p → ∞ while keeping Λ = p M p / ϕ fixed (I neglect the timedependence of ϕ for the time being for the schematic analysis). It turns out this decouplinglimit is a very illuminative perspective allowing us to take a quick peek at the microscopicdynamics of the dark sector. Perturbatively expanding the action eq. (2.6) up to quadraticorder in the Goldstone excitations, I get the following Goldstone action in the decouplinglimit, S π ≃ Λ Z λ (cid:0) ˙ π ˙ π − ∂ i π ∂ i π (cid:1) + c + λ (cid:0) ∂ i ˙ π∂ i ˙ π − ∂ π∂ π (cid:1) + δλ (cid:0) π − ∂ π (cid:1) ,(2.10)where ∂ ≡ ∂ i ∂ j δ ij . Seemingly so far so good, but unfortunately there is a pitfall lies in theGoldstone action. Taking variation w.r.t δλ we get the constraint ∂ π = π . Inserting itback into the eq. (2.10), we get a higher order temporal derivative term S π ⊃ c + λ Z ¨ π ∂ ¨ π + ... (2.11)This term yields to a fourth order equation of motion, which requires 4 initial conditions tofully determine its evolution. It implies there are 2 scalar degrees of freedom in the system,instead of one. One of them is actually the infamous Ostragorski ghost [21][22]. The ghostinstability spoils the validity of the low energy effective field theory, and renders the theoryinconsistent. I will devote the next section to kill the ghost. I define the linear perturbation of our FLRW metric as follows, g = − N ( t ) ( + α ) , g i = N ( t ) a ( t ) ( ∂ i β + S i ) , g ij = a ( t ) (cid:20) δ ij + ψδ ij + ∂ i ∂ j E + (cid:0) ∂ i F j + ∂ j F i (cid:1) + γ ij (cid:21) . (3.1)where α , β , ψ and E are the scalar perturbations, S i and F i are the vector perturbations satis-fying the transverse condition ∂ i S i = ∂ i F i =
0, and γ ij is the tensor perturbation satisfyingthe transverse and traceless condition γ ii = ∂ i γ ij = x µ → x µ + ξ µ ( t , x ) , (3.2)these four Stuckelberg fields transform accordingly, π → π + ξ , π a → π a + ξ i δ ai . (3.3)– 5 –n the other hand, the vector Z µ defined by [23] Z ≡ − aN β + a N ˙ E , Z i ≡ δ ij (cid:0) ∂ j E + F j (cid:1) , (3.4)transforms in the same manner Z µ → Z µ + ξ µ . (3.5)Therefore, the combination Z µ + π µ is gauge invariant. It is very convenient to transform tothe unitary gauge by simply muting all Goldstone bosons, while keeping all perturbationvariables in the eq. (3.1). It can be achieved by adopting a proper diffeomorphism ξ µ ( t , x ) .In this gauge, the Goldstone bosons are eaten by the graviton and the graviton develops amass gap at the low energy spectrum. The graviton turns into a massive spin-2 particle withat most 5 polarizations (it can be less), if the Ostragorski ghost had been eliminated (whichwill be done soon!).On the other hand, we can also transform back to the Goldstone bosons’ gauge bymuting the vector Z µ , which can also be done by adopting a proper diffeomorphism ξ µ .Noted that the way back to the Goldstone bosons’ gauge is not unique, we may chooseto mute some other perturbation variables in the eq. (3.1), based on their transformationsunder the diffeomprhism. The Hamiltonian analysis was introduced by P. Dirac in the 50s and 60s [24], as a way ofcounting dynamical degrees, and quantizing mechanical systems such as gauge theories(for busy/lazy readers see the appendix A in the Ref. [25] for a digested version of themethod).It is more convenient to perform the Hamiltonian analysis in the unitary gauge where π = π i =
0, as all terms introduced by Stuckelberg fields appear only in the potentialsector in the action and thus the Legendre transformation can be easily done. In the unitarygauge, the action eq. (2.6) reduces to (I set M p = S = Z d x p − g (cid:20) R − c Λ ( t ) g ij δ ij + λ (cid:18) Λ ( t ) N − g ij δ ij (cid:19)(cid:21) . (3.6)where Λ = ϕ ( t ) − and Λ ( t ) = ˙ ϕ ( t ) , namely ϕ and ˙ ϕ should not be treated as a canon-ical variable and its velocity, and instead, they should be treated as two time-dependentfunctions subject to some constraint equations which will be derived later.The Hamiltonian can be obtained by performing the Legendre transformation. To thisend, I need to adopt the ADM decomposition, ds = − N dt + h ij (cid:16) dx i + N i dt (cid:17) (cid:16) dx j + N j dt (cid:17) , (3.7)the inverse of the metric reads, g = − N , g i = N i N , g ij = h ij − N i N j N . (3.8)– 6 –he conjugate momenta are defined in the following, Π ij = ∂ L ∂ ˙ h ij = √ h (cid:0) K ij − Kh ij (cid:1) , Π N = ∂ L ∂ ˙ N = Π i = ∂ L ∂ ˙ N i = Π λ = ∂ L ∂ ˙ λ =
0. (3.9)The Hamiltonian is obtained by performing the Legendre transformation, H = Z d x (cid:16) Π ij ˙ h ij − L + ̺ N π N + ̺ i Π i + ̺ λ Π λ (cid:17) = Z d x √ h (cid:16) H + ̺ N π N + ̺ i Π i + ̺ λ Π λ (cid:17) (3.10)where ̺ N , ̺ i , ̺ λ are Lagrangian multipliers, and Π N ≈ Π i ≈ Π λ ≈ H = Nh (cid:18) Π ij Π ij − Π (cid:19) − NR + N (cid:18) c Λ ( t ) + λ (cid:19) (cid:18) h ab − N a N b N (cid:19) δ ab − λ Λ ( t ) N − ∇ j Π ji √ h ! N i , (3.12)where R is the 3-dimensional Ricci scalar, ∇ i is the covariant derivative compatible with 3-dimensional induced metric h ij , and ≈ denotes the “weak equivalence”, namely equalitieshold on the constraint surface. The consistency conditions of these 5 primary constraintsgive rise to the following 5 secondary constraints, d Π N dt = { Π N , H } = −H ≈ d Π i dt = { Π i , H } = −H i ≈ d Π λ dt = { Π λ , H } = −C λ ≈
0, (3.13)where H is the Hamiltonian constraint, H i is the momentum constraint, and C λ is the con-straint introduced by hand at the action level, H ≡ h (cid:18) Π ij Π ij − Π (cid:19) − R + (cid:18) c Λ ( t ) + λ (cid:19) (cid:18) h ab + N a N b N (cid:19) δ ab + λ Λ ( t ) N , H i ≡ − ∇ j Π ji √ h ! − N b δ ab δ ai N (cid:18) c Λ ( t ) + λ (cid:19) , C λ ≡ N (cid:18) h ab − N a N b N (cid:19) δ ab − Λ ( t ) N . (3.14)Caution should be paid to the terms with different types of indices. For instance, in the 3dimensional hyperspace h ab and N a are scalars , δ ai is a vector, while h ij and N i are tensor– 7 –nd vector respectively. These 5 secondary constraints must be conserved in time, whichyields to the following 5 consistency conditions, d H dt = ∂ H ∂ t + {H , H } ≈ d H i dt = ∂ t H i + {H i , H } ≈ d C λ dt = ∂ t C λ + {C λ , H } ≈
0. (3.15)Whether these 5 consistency conditions generate some tertiary constraints, or they only fixthe Lagrangian multipliers in the Hamiltonian, crucially depends on the rank of the matrix M AB ≡ { φ A , φ B } , (3.16)where φ A is the whole set of constraints that we have so far, i.e. φ A = ( Π N , Π i , Π λ , H , H i , C λ ) .The commutation relations among these constraints are showed in the following, { Π N , Π N } ≈ { Π N , Π i } ≈ { Π N , Π λ } ≈ { Π N , H } 6 = { Π N , H i } 6 = { Π N , C} 6 = { Π i , Π j } ≈ { Π i , Π λ } ≈ { Π i , H } 6 = { Π i , H j } 6 = { Π i , C λ } 6 = { Π λ , Π λ } ≈ { Π λ , H } 6 = { Π λ , H i } ≈ { Π λ , C λ } ≈ {H , H } 6 = {H , H i } 6 = {H , C λ } 6 = {H i , H j } 6 = {H i , C λ } 6 = {C λ , C λ } ≈
0. (3.17)The rank of the matrix M AB is 10. Therefore the consistency conditions eq. (3.15) only fixthe Lagrangian multipliers ̺ ′ s, instead of generating new tertiary constraints. Let’s collectall of primary and secondary constraints in the total Hamiltonian and treat them on thesame footing, H tot = Z d x √ h (cid:16) H + ̺ N π N + ̺ i Π i + ̺ λ Π λ + ̺ H + ˜ ̺ i H i + ̺ λ C λ (cid:17) (3.18)where (cid:0) ̺ N , ̺ i , ̺ λ , ̺ , ˜ ̺ i , ̺ λ (cid:1) are Lagrangian multipliers, and we have absorbed the shift vec-tor into the ˜ ̺ i . The algebra closes here.The rank of the matrix also implies all of these 10 constraints are second class. Thisis the consequence of that the gauge symmetries in GR, namely the space-time diffeomor-phisms are all broken in the unitary gauge. Now let’s count the number of degrees. Inthe phase space we have got 22 degrees in the first place, where 20 degrees are from the10 independent components of g µν and their conjugate momenta, and 2 degrees are fromLagrangian multiplier λ and its conjugate momentum. On the other hand, those 10 secondclass constraints φ A remove 10 degrees in the phase space, and the leftover is22 − =
12 degrees in the phase space, (3.19)which corresponds to 6 degrees of freedom in the physical space-time. The sixth mode isthe Ostragraski ghost. – 8 – .3 Ghost elimination in the Goldstone action in the decoupling limit
The Goldstone action in the decoupling limit is a very convenient perspective for us tohunt and eventually execute the ghost. Taking a closer look at the Goldstone action eq.(2.10), I notice that the ghost arises from the kinetic term of the Goldsteon boson π , and theconstraint equating the temporal derivative of π and the gradient of the π , namely Z ∂ i ˙ π∂ i ˙ π → Z ¨ π ∂ ¨ π as ∂ π = π . (3.20)Therefore, a direct way to eliminate the higher order temporal derivative term is to eliminatethe kinetic term ∂ i ˙ π∂ i ˙ π , which can be achieved by introducing the symmetry [14] π i ( t , x ) → π i ( t , x ) + ξ i ( t ) , (3.21)where ξ i is an arbitrary function of time. This symmetry prohibits all temporal derivativeterms of the Goldstone pion π i , including the ones of the longitudinal mode π and the onesof the transverse mode ˆ π i . At the lowest dimensional operator level, the building blockwhich respects this symmetry is the following, Z ab ≡ ∂ µ φ a ∂ µ φ b − (cid:0) ∂ µ ϕ∂ µ φ a (cid:1) (cid:0) ∂ ν ϕ∂ ν φ b (cid:1) ∂ µ ϕ∂ µ ϕ . (3.22)In the unitary gauge where all Goldstone bosons are muted, we have Z ab = h ij δ ai δ bj , namelythe residual symmetry eq. (3.21) strips the shift off the g ij , and leaves us with only 3 dimen-sional induced metric h ij . Now let’s look at the leftover action, which reads S ≃ Λ Z − c ˙ π ˙ π − ( λ + c ) ∂ i π ∂ i π , (3.23)where c and λ are assumed to be positive, to ensure that the energy density of dark energyand dark radiation are positive. Therefore, the kinetic term of the leftover scalar mode π has a wrong sign. Unfortunately after removing the problematic higher order temporalderivative term, the theory still contains a ghost. This ghost arises from the gradient termof the Goldstone boson π , namely Z − c + λ ∂ π∂ π → Z − (cid:18) c + λ (cid:19) ˙ π ˙ π as ∂ π = π . (3.24)I have to flip the sign of the term ∂ π∂ π to cure this ghost pathology. The remedy is theoperator introduced in Ref. [26], ¯ δ Z ab ≡ Z ab − Z ac Z db δ cd Z cd δ cd , (3.25)which is traceless up to the linear perturbation level, and thus it does not contribute to thebackground evolution. Additionally I shall introduce the following global scaling symmetryto tighten up the structure and reduce the arbitrariness of the theory, φ a → ℓ · φ a , ϕ → ℓ · ϕ , (3.26)– 9 –here ℓ is a constant. At the lowest dimensional operator level, I write down the generalcovariant and ghost free action which respects the residual symmetry eq. (3.21), the globalsymmetry eq. (3.26) and the SO ( ) rotational symmetry in the field space as follows, S = Z d x p − g ( M p R − M p ϕ − (cid:20) ( c + λ ) · Z + λ∂ µ ϕ∂ µ ϕ + d Z · ¯ δ Z ab ¯ δ Z cd δ ac δ bd (cid:21)) + S m , (3.27)where Z ≡ Z ab δ ab , the coefficient 3/8 is chosen for the later convenience and S m is theaction of the matter sector which minimally couples to gravity. This is the main result of mycurrent work. Compared with the original action eq. (2.6), all modifications are operated atthe perturbation level, and do not alter the background evolution which is subject to the setof equations in the eq. (2.7) (2.8). The Goldstone action for the scalar graviton reads S ⊃ Z − ( c + d ) ˙ π ˙ π − ( λ + c ) ∂ i π ∂ i π , (3.28)with the ghost free condition, c + d <
0. (3.29)This ghost free condition will be reproduced later in the full perturbation analysis in theunitary gauge, where the Goldstone bosons are muted.
We adopt the metric perturbation decomposition in the eq. (3.1). Due to the background SO ( ) rotational invariance, the scalar perturbations, the vector perturbations and the ten-sor perturbations completely decouple from each other at the linear perturbation level. Wealso adopt the unitary gauge and mute all Goldstone bosons.The massive gravity is an analog of the Higgs mechanism in the particle physics, wherethe Higgs field resides in an asymmetric state. In the unitary gauge, the Goldstone pionsin this symmetry broken phase are eaten by the gauge bosons and consequently the gaugebosons become massive. We expect the same phenomena should also occur in the massivegravity, namely once we mute the Goldstone bosons, the scalar graviton and gravitationalwaves are massive in the unitary gauge. The quadratic action of the scalar perturbation is obtained after a straightforward computa-tion, S ( ) scalar = Z d x ( L EH + L mass ) , (3.30)where L EH is the contribution from the Einstein-Hilbert action, L EH M p a = − ψ + k a (cid:20) ψ − a ˙ E ( H ψ − ψ ) − a HE ( ˙ ψ + H ψ ) (cid:21) + α (cid:20) k β Ha + k a (cid:0) ψ − a H ˙ E (cid:1) + H ˙ ψ − H α (cid:21) − k β ˙ ψ a , (3.31)– 10 –nd the L mass is the contribution from the dark sector, or in other word the graviton massterm, L mass = M p ϕ − (cid:20) − c + d k aE − ck aE ( α + ψ ) + ca ψ ( α + ψ ) (cid:21) + λ " − k E a + k E ( ψ − α ) a + ( α + ψ ) a + δλ (cid:18) ψ − α a − k E a (cid:19) . (3.32)I have absorbed the lapse into the redefinition of time, and rescaled the Lagrangian mul-tiplier λ → λ · ϕ a M p for convenience once again. Taking the variation of the action withrespect to the non-dynamical variables α , δλ , and β , I get the following three constraintequations, H (cid:0) ψ − k ˙ E (cid:1) + c ψ a ϕ + k a (cid:18) ψ − cE a ϕ (cid:19) + λ a M p (cid:0) ψ − k E (cid:1) − δλ a M p + α λ a M p − H ! + k H β a = k E + φ − ψ = H α − ˙ ψ =
0. (3.33)Substituting the solution of the above equations into the action, the quadratic action of thescalar perturbation takes the following form, S ( ) scalar = Z − ( c + d ) aM p H ϕ ˙ ψ + ... (3.34)where the eclipse denotes the potential term and gradient term. The ghost free conditionfor the scalar perturbation reads c + d <
0. (3.35)We have reproduced the ghost free condition eq. (3.29), in the different gauge. Let meredefine a new constant b via d ≡ − c − b and b > ψ c ≡ √ bM p aH ϕ ψ . (3.36)The canonically normalised scalar action reads S ( ) scalar = Z dtd ka (cid:18) ˙ ψ c ˙ ψ c − c s k a ψ c ψ c − M s ψ c ψ c (cid:19) , (3.37)where the sound speed reads c s = c b (cid:18) + ρ rad ρ hde (cid:19) , (3.38)– 11 –here ρ rad = λ /2 a is the energy density of dark radiation and ρ hde = cM p / a ϕ is theenergy density of the holographic dark energy, and the mass of the scalar mode reads M s = H " cb + c − bbHR h − − c + b + cb bH R h − c H R h + Ω rad + cb − HR h − c H R h ! + Ω (cid:18) cb Ω hde − (cid:19)(cid:21) , (3.39)where R h ≡ a ϕ is the future event horizon of the universe, Ω rad is the fraction of the darkradiation defined by Ω rad ≡ ρ rad /3 M p H , and Ω hde is the fraction of the holographic darkenergy defined by Ω hde ≡ ρ hde /3 M p H . The Compton wavelength of the scalar mode isabout Hubble radius size, up to a coefficient whose value is dependent on the parameters b and c .One may wonder whether the mass of scalar graviton is always positive along thecosmic evolution to ensure the stability in the scalar sector. This problem may require athorough numerical analysis, and thus it will not be carried out in the current work. Instead,I would like to show the tachyon freeness condition in the asymptotic de-Sitter phase where c = Ω rad → Ω hde →
1, and R h → H − . The scalar mass in the asymptotic phasereads M s → H (cid:20) b − (cid:21) . (3.40)The tachyon freeness condition translates to0 < b <
12. (3.41)
The quadratic action for the tensor perturbation reads S ( ) tensor = M p Z dtd ka (cid:20) ˙ γ ij ˙ γ ij − (cid:18) k a + M GW (cid:19) γ ij γ ij (cid:21) , (3.42)where M GW = c − b + H R h Ω rad R h . (3.43)The Compton wavelength of the tensor graviton is about the size of the future event horizon.The following tachyon freeness condition is required6 c − b + H R h Ω rad > < b <
36, which is weaker than the stability condition eq. (3.41)derived in the preceding subsection of the scalar mode analysis.The non-vanishing mass modifies the propagating speed of the gravitational waves,which is constrained with high precision − · − < c gw − < · at low redshiftregime by the multi-messenger observation GW170817 [27]. It gives us an upper bound– 12 –n the graviton mass around m g < − eV. Our graviton mass is much lower than thisupper bound by around 10 orders of magnitude. It remains challenging to directly probethe non-vanishing graviton mass effects at the late time epoch. However, the size of thefuture event horizon is much shorter during the early universe, and thus it leads to a muchlarger graviton mass, which may give rise to some interesting observational effects on thestochastic gravitational waves background. I expect there is no dynamical degrees in the vector sector, as the residual symmetry eq.(3.21) has projected out all temporal derivative of the Goldstone vector bosons. I will try toconfirm it in this subsection. The quadratic action of the vector perturbation reads S ( ) V = M p Z dtd kk a " ˙ F i ˙ F i − S i ˙ F i a + S i S i a + b − c a ϕ − λ M p a ! F i F i . (3.45)Taking the variation with respect to the Lagrangian multiplier S i , I get a ˙ F i − S i =
0. (3.46)Substituting the solution of this constraint equation back to the action, I get S ( ) V = M p Z dtd kk a b − c a ϕ − λ M p a ! F i F i . (3.47)The kinetic term of the vector mode has vanished, and the action is subject to a new con-straint F i =
0. After substituting this solution back to the action, the whole action vanishesand I conclude that there is no dynamical degree in the vector sector.
I have performed the Hamiltonian analysis in the subsection (3.2), and I found that theoriginal theory contains 6 modes. Compared to the Hamiltonian of the original theory (2.6),the modified one (3.27) strips off the term that is non-linear in the shift vector. Consequentlythe primary constraint Π i ≈ T i ≡ {H i , H } ≈
0. (3.48)Now let’s add these 3 new tertiary constraints into the constraint set φ A , and remove the Π i ≈ φ A = ( Π N , Π λ , H , H i , C λ , T i ) . (3.49)We can show that the rank of the following 10 ×
10 matrix M AB ≡ { φ A , φ B } (3.50)is 10, no new constraints are generated and the algebra closes here. Therefore, all constraintsin the eq. (3.49) are second class, and they remove 10 degrees in the phase space. On the– 13 –ther hand, the constraints Π i ≈ − × − =
6, (3.51)which corresponds to 3 dynamical degrees of freedom in the physical space-time. As wehave shown in our perturbation analysis, 2 of them are tensor modes, and the rest one is ascalar graviton.
The UV-IR correspondence is based on the assumption that the vacuum energy arising fromthe quartic divergence should not exceed the energy density of a black hole in the samesize. In some sense, the universe is a black hole as the Hubble radius coincides with itsSchwarzschild radius, and thus the bound should be saturated on the cosmological back-ground and we have the UV-IR relation Λ ∼ p M p / L .One may ask how this UV-IR correspondence appears naturally in our effective fieldtheory framework? The answer is that it should be interpreted as the scale where the scalargraviton strongly couples to itself. Let’s look at the 4-votex interaction in the decouplinglimit, Z M p ϕ − ∂ π∂ π∂ π∂ π ∼ Z M p ϕ − ˙ π ˙ π ˙ π ˙ π ∼ Z ω M p ϕ − π c π c π c π c , (3.52)where π c is the canonical normalised Goldstone boson, and ω is the frequency of π c . Thestrength of the coupling exceeds unity if ω > p M p / ϕ , and our effective field theory breaksdown. As shown in our perturbation analysis, we have learnt that ϕ is the graviton’s Comp-ton’s wavelength (remember ϕ has the length dimension, and we set the scale factor a = In the current work, a general covariant local field theory is proposed for the holographicdark energy model. I started from the mini superspace action proposed a few years ago [10],showed that the direct, and perhaps the simplest way of covariantizing the action gives riseto additional 4 degrees of freedom (6 in total including 2 degrees in the gravitational wavesector), which leads to the Ostragraski ghost instability. To remedy the ghost pathology, I in-troduced a new symmetry to prohibit the problematic terms in the theory. It has turned outthat the low energy effective field theory of the holographic dark energy model is actuallythe Lorentz-violating massive gravity theory, whose graviton has 3 polarisations including1 helicity 0 mode and 2 helicity 2 modes. The Compton wavelength of the graviton is aboutthe size of the future event horizon of our universe, as shown in the linear perturbationanalysis in the unitary gauge. To confirm the total number of dynamical degrees of ourtheory at the fully non-linear level, I have performed the Hamiltonian analysis and foundthat the momentum constraints yield to another 3 tertiary constraints. These 6 constraints,which are all second class, eliminate 3 out of 4 additional degrees and therefore there areonly 3 dynamical degrees in total in the gravity sector, including a scalar mode and two– 14 –ensor modes. Our effective field theory breaks down at the scale Λ ∼ p M p / L , where L isthe graviton’s Compton wavelength, due to the strong coupling of the scalar graviton abovethis scale, which offers a natural and physical interpretation for the UV-IR correspondencein the holographic dark energy model.Our effective field theory provides a general framework in which the holographic darkenergy model can be tested. For instance, the non-vanishing mass of the gravitationalwaves, which is small at late time epoch but sizeable during the early universe, leads toa modified stochastic gravitational waves background. On the other hand, the existence ofthe scalar gravton whose Compton wavelength is about the size of the Hubble radius (upto an order O ( ) ∼ O ( ) coefficient) during inflation may be tested by the cosmologicalcollider physics [28]. It is also very intriguing to ask what would happen if the Stuckelbergfield, say the time like one ϕ , couples to the standard model fields. Our EFT frameworkalso provides a general setup where the perturbation theory of the holographic dark energycan be developed, which allows us to investigate its impacts on the cosmological structureformation in detail. All these possibilities warrant further scrutiny.I shall end by commenting that the global Lorentz invariance is broken in our scalarfield configuration. As an analog to the Higgs mechanism in the particle physics, one mayexpect a similar symmetry restoration to occur at high energy scale, where the Lorentz in-variance is recovered and the graviton becomes massless again. However, this Higgs-likemechanism is till missing. In fact, to my best knowledge it is still an open question for allmassive gravity theories. Acknowledgement
This work is supported by the grant No. UMO-2018/30/Q/ST9/00795 from the NationalScience Centre, Poland. The author would like to thank Yi Wang for the useful discussionsand suggestions.
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