Einstein-singleton theory and its power spectra in de Sitter inflation
aa r X i v : . [ g r- q c ] J un Einstein-singleton theory and its power spectrain de Sitter inflation
Yun Soo Myung a ∗ , Taeyoon Moon a † , and Young-Jai Park b ‡ a Institute of Basic Sciences and Department of Computer Simulation, Inje UniversityGimhae 621-749, Korea b Department of Physics, Sogang University, Seoul 121-742, Korea
Abstract
We study the Einstein-singleton theory during de Sitter inflation since it providesa way of degenerate fourth-order scalar theory. We obtain an exact solution expressedin terms of the exponential-integral function by solving the degenerate fourth-orderscalar equation in de Sitter spacetime. Furthermore, we find that its power spectrumblows negatively up in the superhorizon limit, while it is negatively scale-invariant inthe subhorizon limit. This suggests that the Einstein-singleton theory contains theghost-instability and thus, it is not suitable for developing a slow-roll inflation model.
PACS numbers: 04.50.Kd,98.80.Cq,98.80.JkKeywords: singleton, power spectrum, de Sitter inflation
Typeset Using L A TEX ∗ e-mail address: [email protected] † e-mail address: [email protected] ‡ e-mail address: [email protected] Introduction
The single-field inflation is still known to be a promising model for describing the slow-roll(quasi-de Sitter) inflation [1] when one chooses an appropriate potential like the Starobinskypotential which originates from f ( R ) = R + R gravity [2]. This Einstein-scalar theorycorresponds to a second-order tensor-scalar theory.Our next question is to consider an Einstein-(higher-order) scalar theory even thoughone may worry about a ghost state. For this purpose, it was interesting to compute thepower spectrum of a massive singleton (other than inflaton) generated during de Sitter(dS) inflation because its equation belongs to a fourth-order equation. In order to computethe power spectrum, one has to choose the Bunch-Davies vacuum in the subhorizon limitof z → ∞ . In addition, one needs to quantize the singleton canonically as the inflatondid. However, it is hard to obtain a fully exact solution to the fourth-order equationin dS spacetime. Instead, the authors in [3] have investigated the massive singleton toshow the dS/LCFT correspondence in the superhorizon limit of z → × k in the superhorizonlimit [4]. This might show that the dS/LCFT correspondence works for obtaining thepower spectra in the superhorizon limit. Nevertheless, the limitation of these works is thattheir computations are valid only in the superhorizon limit because of difficulty in solvinga fourth-order differential equation in whole range z .In this work we obtain an exact solution and compute a complete power spectrum ofsingleton by solving the degenerate fourth-order scalar equation, which describes a prop-agation of a massless singleton during dS inflation and by requiring the Pais-Uhlenbeckquantization scheme for a degenerate fourth-order oscillator [5, 6, 7]. It turns out that thesingleton power spectrum blows negatively up in the superhorizon limit, while it is neg-atively scale-invariant in the subhorizon limit. This suggests that the Einstein-singletontheory is not a candidate for a slow-roll inflation because its power spectrum might showghost-instability. 2 Einstein-singleton theory
We introduce the Einstein-singleton theory where a dipole ghost pair φ and φ are mini-mally coupled to Einstein gravity. The starting action is a second-order scalar-tensor theorygiven by S ES = S E + S S = Z d x √− g h(cid:16) R κ − (cid:17) − (cid:16) ∂ µ φ ∂ µ φ + µ φ (cid:17)i , (1)where S E is introduced to feed the dS inflation with Λ > S S represents the singletontheory composed of two scalars φ and φ [8, 9, 10, 11, 12]. Here we have κ = 8 πG = 1 /M with the reduced Planck mass M P and µ is a coupling parameter.After the metric variation, the Einstein equation is given by G µν + κ Λ g µν = κT µν (2)with the energy-momentum tensor T µν = 2 ∂ µ φ ∂ ν φ − g µν (cid:16) ∂ µ φ ∂ µ φ + µ φ (cid:17) . (3)Importantly, two scalar fields are coupled to be ∇ φ = 0 , ∇ φ = µφ , (4)which lead to a degenerate fourth-order equation ∇ φ = 0 . (5)It can describe a fourth-order scalar theory because S S reduces to the fourth-order scalartheory when eliminating an auxiliary field φ as [13] S = 12 µ Z d x √− g ∇ φ ∇ φ , (6)which provides (5) directly. Choosing the vanishing scalars, the solution of dS spacetimecomes out as ¯ R = 4 κ Λ , ¯ φ = ¯ φ = 0 . (7)Explicitly, dS-curvature quantities are given by¯ R µνρσ = H (¯ g µρ ¯ g νσ − ¯ g µσ ¯ g νρ ) , ¯ R µν = 3 H ¯ g µν (8)3ith a Hubble parameter H = p κ Λ /
3. We select the dS background explicitly by choosinga conformal time η ds = ¯ g µν dx µ dx ν = a ( η ) [ − dη + δ ij dx i dx j ] , (9)where the conformal and cosmic scale factors are given by a ( η ) = − Hη , a ( t ) = e Ht . (10)During the dS inflation, a ( η ) goes from small to a very large value like a f /a i ≃ , whichcorresponds to the fact that the conformal time η = − /a ( η ) H runs from −∞ (subhorizon)to − R at η = − ǫ is connected to the isometry group SO(1,4) of dS space. In this case, thedS isometry group acts as conformal group when fluctuations are superhorizon [3]. Hence,correlators are expected to be constrained by conformal invariance. Actually, a slice ( R )at η = − ǫ is employed to calculate the power spectrum in the superhorizon limit. On theother hand, one introduces the Bunch-Davies vacuum to compute the power spectrum inthe subhorizon limit of η → −∞ .We wish to choose the Newtonian gauge of B = E = 0 and ¯ E i = 0 for cosmologicalperturbation around the dS background (9). In this case, the cosmologically perturbedmetric can be simplified to be ds = a ( η ) h − (1 + 2Ψ) dη + 2Ψ i dηdx i + n (1 + 2Φ) δ ij + h ij o dx i dx j i (11)with transverse-traceless tensor ∂ i h ij = h = 0. Furtherore, two scalar perturbations aredefined by φ = 0 + ϕ , φ = 0 + ϕ . (12)In order to obtain the perturbed Einstein equations, one can linearize the Einstein equation(2) directly around the dS spacetime as δR µν ( h ) − H h µν = 0 → ¯ ∇ h ij = 0 , (13)which describes a massless gravitational wave propagation. Concerning two-metric scalarsΨ and Φ, their linearized Einstein equations imply that they are not physically propagatingmodes. In addition, we note that there is no coupling between { Ψ , Φ } and { ϕ , ϕ } because4igure 1: Penrose diagram of dS spacetime with the UV/IR boundaries ( ∂ dS ∞ / ) locatedat η = −∞ and η = −
0. A slice ( R ) near η = −∞ is introduced to compute the powerspectrum in the subhorizon limit, while a slice ( R ) at η = − ǫ is employed to calculate thepower spectrum in the superhorizon limit.of ¯ φ = ¯ φ = 0 in dS inflation. The vector Ψ i is also a non-propagating mode since it hasno kinetic term. The relevant linearized equations are those for two scalars¯ ∇ ϕ = 0 , (14)¯ ∇ ϕ = µϕ , (15)which are combined to provide a degenerate fourth-order scalar equation¯ ∇ ϕ = 0 . (16)This is our main equation to be solved to obtain the power spectrum of a massless singletonduring dS-inflation.It seems appropriate to comment that Eqs.(14)-(16) are different from those of a massivesingleton in [4]: ( ¯ ∇ − m ) ϕ = 0 , ( ¯ ∇ − m ) ϕ = µϕ , ( ¯ ∇ − m ) ϕ = 0. We could notsolve the massive singleton equation in the whole range of η ∈ [ −∞ , − Propagation of massless singleton
In order to compute the complete power spectrum, we have to know the solution to singletonequations (15) and (16) in the whole range of η ∈ [ −∞ , − ϕ i can be expanded in Fourier modes φ i k ( η ) ϕ i ( η, x ) = 1(2 π ) Z d k φ i k ( η ) e i k · x . (17)Eq.(15) leads to " d dη − η ddη + k φ k ( η ) = 0 . (18)Introducing a new variable z = − kη , Eq.(18) can be rewritten as h d dz − z ddz + 1 i φ k ( z ) = 0 (19)whose positive-frequency solution with the normalization 1 / √ k is given by φ k ( z ) = H √ k ( i + z ) e iz . (20)This is the typical solution of a massless scalar propagating on dS spacetime.On the other hand, plugging (17) into (16) leads to the fourth-order scalar equation h η d dη − η ddη + k η i φ k ( η ) = 0 . (21)This equation can be expressed in terms of z as " d dz + 2 (cid:16) − z (cid:17) d dz + 4 z ddz + (cid:16) − z (cid:17) φ k = 0 (22)whose full solution is found to be φ k ( z ) = h ˜ c ( i + z ) + ˜ c n i + ( z − i ) e − iz Ei(2 iz ) oi e iz (23)with two complex coefficients ˜ c and ˜ c . This is one of our main results which states thatthe solution (23) is an exact solution to the fourth-order equation (16). The c.c. of φ k is also a solution to (22). Here, Ei(2 iz ) is the exponential-integral function of a purelyimaginary number defined by [14]Ei(2 iz ) = Ci(2 z ) + i Si(2 z ) − i π , (24)6 - - - - - H L H L Figure 2: Cosine-integral and Sine-integral functions as functions of z . In the super horizonlimit of z →
0, one finds that Ci[2 z ] → γ + ln[2 z ] and Si[2 z ] →
0. On the other hand, onefinds that Ci[2 z ] → sin[2 z ]2 z and Si[2 z ] → π − cos[2 z ]2 z in the subhorizon limit of z → ∞ .where the cosine-integral and sine-integral functions are given byCi(2 z ) = − Z ∞ z cos tt dt −→ ( z → γ + ln[2 z ] + Σ ∞ k =1 ( − k (2 z ) k k (2 k )! z → ∞ : sin(2 z )2 z + O z , (25)Si(2 z ) = Z z sin tt dt −→ ( z → ∞ k =1 ( − k − (2 z ) k − (2 k − k − z → ∞ : − cos(2 z )2 z + π + O z (26)with the Euler’s constant γ = 0 .
577 . Their behaviors are depicted in Fig. 2. We note thatEi(2 iz ) satisfies the fourth-order equation( z − i ) z d Ei(2 iz ) dz − iz d Ei(2 iz ) dz + 2 z ( i − z − iz − z ) d Ei(2 iz ) dz − i − z − iz + 2 z ) d Ei(2 iz ) dz = 8 e iz (27)and its asymptotic behaviors are given byEi(2 iz ) −→ z → γ + ln[2 z ] − iπ z → ∞ : − h i z + z ) i e iz (28)obtained from (24) together with (25) and (26).It is worth to point out that the solution (23) is suitable for choosing the Bunch-Daviesvacuum to give quantum fluctuations because it shows φ k ( z ) → z →∞ h(cid:16) ˜ c + 32 ˜ c (cid:17) i + ˜ c z i e iz . (29)7hen, Eq.(22) in the subhorizon limit of z → ∞ reduces to a degenerate fourth-orderequation which appeared in conformal gravity [15] h d dz + 1 i φ k , ∞ ( z ) = 0 (30)whose solution is given by φ k , ∞ ( z ) = ( c ′ + c ′ z ) e iz . (31)We note that after redefining ˜ c and ˜ c , Eq.(29) leads to Eq.(31). The undeterminedconstants c ′ and c ′ shows a feature of solution to the fourth-order equation (30) when onecompares these with the fixed solution (20) to the second order equation.On the other hand, in the superhorizon limit of z →
0, Eq.(22) reduces to " d dz − z d dz + 4 z ddz φ k , = 0 , (32)whose solution is given by φ k , = ¯ c + ¯ c ln[2 z ] (33)with arbitrary constants ¯ c and ¯ c . Especially, the presence of ln[2 z ] dictates that (33) isthe solution to the fourth-order equation (32). In deriving Eq.(32) from Eq.(22), we neglectthe last term of − z because it is subdominant in the limit of z →
0. We note that the fullsolution (23) reduces to Eq.(33) in the limit of z → φ k , = i h ˜ c + (2 − γ + i π c i − i ˜ c ln[2 z ] , (34)when choosing ¯ c = − i ˜ c , ¯ c = i h ˜ c + (cid:16) − γ + iπ (cid:17) ˜ c i . (35)Finally, we may determine one coefficient ˜ c by making use of Eq.(15) together withEqs.(20) and (23): ˜ c = − µ H √ k . (36)However, ˜ c remains undetermined, but it will be determined by the Wronskian conditionin the next section. 8 Power spectra
The power spectrum is the variance of singleton fluctuations due to quantum zero-pointfluctuations. It is easily defined by the zero-point correlation function which could becomputed when one chooses the Bunch-Davies vacuum state | i in the subhorizon limit.The defining relation is given by h | ˆ ϕ a ( η,
0) ˆ ϕ b ( η, | i = Z dkk P ab , (37)where k = √ k · k is the comoving wave number. Quantum fluctuations were created onall length scales with wave number k . Cosmologically relevant fluctuations start their livesinside the Hubble radius which defines the subhorizon: k ≫ aH . On later, the comovingHubble radius 1 / ( aH ) shrinks during inflation while keeping the wavenumber k constant.Eventually, all fluctuations exit the comoving Hubble radius, they reside on the superhorizonregion of k ≪ aH after horizon crossing.For fluctuations of a massless scalar ( ¯ ∇ δφ = 0) and tensor ( ¯ ∇ h ij = 0) with differ-ent normalization originate on subhorizon scales and they propagate for a long time onsuperhorizon scales. This can be checked by computing their power spectra P δφ = H (2 π ) [1 + z ] , (38) P h = 2 × (cid:16) M P (cid:17) × P φ = 2 H π M [1 + z ] . (39)To compute the singleton power spectrum, we have to know the commutation relationsand the Wronskian condition. The canonical conjugate momenta are given by π = a ϕ ′ , π = a ϕ ′ . (40)The canonical quantization is accomplished by imposing equal-time commutation relations:[ ˆ ϕ ( η, x ) , ˆ π ( η, y )] = iδ ( x − y ) , [ ˆ ϕ ( η, x ) , ˆ π ( η, y )] = iδ ( x − y ) . (41)The two operators ˆ ϕ and ˆ ϕ are expanded in terms of Fourier modes as [6, 13, 15]ˆ ϕ ( z, x ) = 1(2 π ) Z d k N "(cid:16) i ˆ a ( k ) φ k ( z ) e i k · x (cid:17) + h . c . , (42)ˆ ϕ ( z, x ) = 1(2 π ) Z d k ˜ N "(cid:16) ˆ a ( k ) φ ( z ) + ˆ a ( k ) φ ( z ) (cid:17) e i k · x + h . c . (43)9ith N and ˜ N the normalization constants. Plugging (42) and (43) into (41) determinesthe relation of normalization constants as N ˜ N = 1 / k and commutation relations betweenˆ a a ( k ) and ˆ a † b ( k ′ ) as [ˆ a a ( k ) , ˆ a † b ( k ′ )] = 2 k − ii ! δ ( k − k ′ ) , (44)where we observe a Jordan cell structure. This is the typical commutation relations ap-peared when one quantizes a degenerate Pais-Uhlenbeck fourth-order oscillator [6]. Herethe commutation relation of [ˆ a ( k ) , ˆ a † ( k ′ )] is implemented by the Wronskian condition. TheWronskian condition for φ k ( z ) and φ k ( z ) leads to a (cid:16) φ k dφ ∗ k dz − φ ∗ k dφ k dz + φ ∗ k dφ k dz − φ k dφ ∗ k dz (cid:17) = r k H h i (˜ c − ˜ c ∗ ) − (˜ c + ˜ c ∗ ) (cid:16) z + 3 z (cid:17)i = 1 k . (45)To satisfy the above relation, let us impose˜ c = − ˜ c ∗ , ˜ c = − iH √ k . (46)At this stage, it is worth to note that the Wronskian normalization condition wasoriginally designed for the second-order theory. In the subhorizon limit of z → ∞ , thefourth-order contribution is nothing, while it blows up unless ˜ c is purely imaginary in thesuperhorizon limit of z →
0. Hence, we may neglect the fourth-order contribution to theWronskian condition by choosing ˜ c to be purely imaginary. Considering (36), one maydetermine ˜ c = − i H √ k (47)by choosing µ = 2 iH . We note here that choosing ˜ c = i H √ k leads to the positive powerspectrum ( P >
0) in the whole range z , which contradicts to the negative power spectrumof a fourth-order scalar theory.Then, we could easily find that P = 0 , P ( z ) = P ( z ) = k π | φ k | = H (2 π ) [1 + z ] . (48)10 P H z L - - - - - P H z L Figure 3: Power spectra P and P as functions of z for H = (2 π ) . In the superhorizonlimit of z →
0, one finds that P → P → −∞ . On the other hand, P → ∞ and P → − z → ∞ .However, the power spectrum P takes a complicated form P ( z ) ≡ P (1)22 ( z ) + P (2)22 ( z )= k π h | φ k | + i ( φ k φ ∗ k − φ k φ ∗ k ) i = (cid:18) H π (cid:19) h z − n z + 4 i ˜ c √ k H + 2 i ˜ c √ k H Re [ f ( z )] oi = − (cid:18) H π (cid:19) h Re [ f ( z )] i , (49)where f ( z ) is given by f ( z ) = e iz ( i + z ) Ei( − iz ) . (50)This is another of our main results: power spectrum of massless singleton is explicitlyexpressed in terms of the exponential-integral function. Fig. 3 indicates the behaviors of P ( z ) and P ( z ) generated during dS inflation. We note that the former shows a typicalpower spectrum for a massless scalar ( δφ, ϕ ) or graviton ( h ), while the latter indicates apower spectrum of the singleton ( ϕ ). It is reasonable to assist that the power spectrum of P is negative because it corresponds to that of a purely fourth-order scalar theory. Thatis, one could not avoid to find ghost-instability when computing the power spectrum of afourth-order derivative scalar theory during dS inflation.In the subhorizon limit of z → ∞ , one finds a negatively scale-invariant spectrum P z →∞ = − (cid:18) H π (cid:19) (51)11ecause Re [ f ( z )] → − in this limit. We note that the power spectrum of the scale-invariant scalar tensor theory is given by [16] P SIST = 12(2 π ) . (52)Taking f ( z ) in the superhorizon limit of z → f ( z ) → z → h − γ − ln[2 z ] + (1 − γ ) z i + i h − π z − πz i , (53)the power spectrum (49) of massless singleton leads to P z → ( z ) = − (cid:18) H π (cid:19) (cid:16) − γ − ln[2 z ] (cid:17) (54)= 43 (cid:18) H π (cid:19) (cid:16) ln[ z ] − . (cid:17) , which explains why P ( z ) blows up negatively as z → P z → = (cid:18) H π (cid:19) . (55) We have obtained the exact solution and computed the complete power spectrum (49) ofa singleton expressed in term of the exponential-integral function by solving the degener-ate fourth-order equation and by requiring the Pais-Uhlenbeck quantization scheme for adegenerate fourth-order oscillator.Its two asymptotic behaviors are quite different from those [(52) and (55)] of a masslessscalar. In the subhorizon limit z → ∞ , the power spectrum (51) of a singleton is a negativelyscale-invariant one which is opposite to (52) of scale-invariant scalar-tensor theory [16], whileit blows up (negatively divergent) in the superhorizon limit of z → • Ghost-instability of the modelSince S S in (1) reduces to the fourth-order derivative scalar theory (6), we worry about theghost-instability problem. Using the Pais-Uhlenbeck quantization scheme for a degeneratefourth-order oscillator in dS spacetime, we have found the negative power spectrum P ( z )in (49), depicted in Fig 3. In the subhorizon limit of z → ∞ , we have obtained a negativelyscale-invariant power spectrum (51) which indicates the ghost instability clearly. On theother hand, P ( z ) blows negatively up in the superhorizon limit of z →
0. This indicatesthat the singleton theory is a fourth-order derivative scalar theory which must contain aghost state. • Problem of exit mechanismThe dS inflation is driven by the cosmological constant Λ which is a non-dynamical quantity.Hence, this corresponds to an eternal inflation and thus, there is no natural way to exit theinflationary phase. This is a handicap of dS inflation. In the slow-roll inflation (quasi-dSinflation), however, the inflaton plays an essential role in exiting the inflationary phase. • Is µ = 2 iH a mass square of φ ?In order to obtain Eq. (47), we specified µ = 2 iH . Recalling the definition of µ in (1),it seems that µ plays the role of the mass square of φ . However, this is not true. µ isjust a parameter of connecting φ with φ to get the fourth-order derivative equation for φ from a mixed kinetic term. If one wishes to have a massive singleton, one has to includea potential term of m φ φ [4]: ( ¯ ∇ − m ) ϕ = 0 , ( ¯ ∇ − m ) ϕ = µϕ , ( ¯ ∇ − m ) ϕ = 0. • Slow-roll inflation in the Einstein-singleton theoryIf one wishes to consider the slow-roll inflation in the Einstein-singleton theory S ES includingthe potential of m φ φ , the Einstein equation takes the form of G µν = T mµν /M whichprovides the energy density ρ = ˙ φ ˙ φ + ( m φ φ + µφ /
2) and the pressure p = ˙ φ ˙ φ − ( m φ φ + µφ / H = ρ M and ˙ H = − ρ + p M . Even though this model is similar to two-field inflation model with thechaotic potentials, this is not the case because their full scalar equations are given by¨ φ + 3 H ( t ) ˙ φ + m φ = 0 and ¨ φ + 3 H ( t ) ˙ φ + m φ = − µφ which are combined to givea fourth-order equation of ( d dt + 3 H ( t ) ddt + m ) φ = 0. It conjectures that their slow-roll equations are quite different from those of two-field inflation. Furthermore, it requiresa non-trivial task to perform the cosmological perturbations around the slow-roll inflation13nstead of the dS inflation. Especially, it is important to define the curvature perturbation R in the Einstein-singleton theory. It was given by R = − Hδφ/ ˙ φ for the single-field inflationin spatially flat gauge, while it takes the form of R S = − H [ ϕ / ˙ φ + ϕ / ˙ φ ] for the singletoninflation. For example, the power spectrum appeared in dS spacetime with ˙ φ = ˙ φ = 0 [3, 4]was given by P mϕ ϕ ∼ z w (1 + 2 ln[ z ]) with w = 3 / − p / − m /H in the superhorizonlimit. However, we remain “cosmological perturbations of the Einstein-singleton theoryaround the slow-roll inflation” as a future work, worrying about the appearance of theghost states. This is so because the strange asymptotic behavior of power spectrum of P ϕ ϕ indicates a negatively divergent behavior in the superhorizon limit of z →
0, whichreflects that the Einstein-singleton theory includes a fourth-order derivative scalar theory.Furthermore, there is no way to avoid a ghost-instability in the whole range of z . Thus, ourresult during dS inflation suggests that the Einstein-singleton theory is not considered as amodel for developing a slow-roll inflation because a negative power spectrum of curvatureperturbation ( P R S R S <
0) persists in the slow-roll inflation. This is because ϕ satisfies afourth-order differential equation during the slow-roll inflation.14 eferences [1] P. A. R. Ade et al. [Planck Collaboration], arXiv:1502.02114 [astro-ph.CO].[2] A. A. Starobinsky, Phys. Lett. B , 99 (1980). doi:10.1016/0370-2693(80)90670-X[3] A. Kehagias and A. Riotto, Nucl. Phys. B , 492 (2012)doi:10.1016/j.nuclphysb.2012.07.004 [arXiv:1205.1523 [hep-th]].[4] Y. S. Myung and T. Moon, JHEP , 137 (2014) doi:10.1007/JHEP10(2014)137[arXiv:1407.7742 [gr-qc]].[5] A. Pais and G. E. Uhlenbeck, Phys. Rev. , 145 (1950).[6] P. D. Mannheim and A. Davidson, Phys. Rev. A , 042110 (2005) [hep-th/0408104].[7] Y. W. Kim, Y. S. Myung and Y. J. Park, Phys. Rev. D , 085032 (2013)doi:10.1103/PhysRevD.88.085032 [arXiv:1307.6932].[8] M. Flato and C. Fronsdal, Commun. Math. Phys. , 469 (1987).[9] A. M. Ghezelbash, M. Khorrami and A. Aghamohammadi, Int. J. Mod. Phys. A ,2581 (1999) [hep-th/9807034].[10] I. I. Kogan, Phys. Lett. B , 66 (1999) [hep-th/9903162].[11] Y. S. Myung and H. W. Lee, JHEP , 009 (1999) [hep-th/9904056].[12] D. Grumiller, W. Riedler, J. Rosseel and T. Zojer, J. Phys. A , 494002 (2013)[arXiv:1302.0280 [hep-th]].[13] V. O. Rivelles, Phys. Lett. B , 137 (2003) doi:10.1016/j.physletb.2003.10.039[hep-th/0304073].[14] M. Abramowitz and A. Stegun, Handbook of Mathematical functions, (Dover publi-cations, New York, 1970).[15] Y. S. Myung and T. Moon, Mod. Phys. Lett. A , no. 32, 1550172 (2015)doi:10.1142/S0217732315501722 [arXiv:1501.01749 [gr-qc]].1516] Y. S. Myung and Y. J. Park, Eur. Phys. J. C76